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+{"text":"\\section{Introduction}\nThe mathematical scattering theory for potential perturbations of the self-adjoint Laplacian in $L^{2}(\\mathbb{R}^{3})$ is a well developed subject. In particular, the stationary approach leads to a representation formula for the scattering matrix (see, e.g., \\cite{Y-LNM}). \nIn the case of singular perturbations, in particular, in the case of self-adjoint realizations of Laplacians with boundary conditions on hypersurfaces, similar results have been obtained in recent years, either using properties of the so-called Weyl function appearing in the resolvent formula (see \\cite{BMN}) or using a Limiting Absorption Principle (see \\cite{JST} and \\cite{JMPA}). The advantage of the approach by LAP over the former is that no trace class condition is required and so the results are not restricted to the two dimensional situation. \nThe target of the present paper is to put these separate results together and so to provide a representation formula for the scattering matrix in the case where the Laplacian is perturbed both by a regular short-range potential and by a singular one describing self-adjoint boundary conditions. In particular, we extend the abstract results in \\cite{JMPA} to a wider framework which then allows applications to such a setting. In order to develop such a strategy, at first we provide an abstract Kre\\u\\i n's type resolvent formula which, when applied to the concrete case of a potential perturbation of the Laplacian, allows for not compactly supported potentials. In particular, whenever the singular part of the perturbations is absent, this extends from compactly supported potentials in one dimension to short range potentials in three dimensions the kind of results provided in \\cite[Section 5]{BN}.  \\par\nHere, in more details, the contents of the paper.  In Section 2, following the scheme proposed in \\cite{JFA}, we provide an abstract resolvent formula for a perturbations $A_{\\mathsf B}$ of the self-adjoint $A$ by a linear combination of the adjoint of two bounded trace-like maps $\\tau_{1}:\\text{\\rm dom}(A)\\to\\mathfrak h_{1}$ and $\\tau_{2}:\\text{\\rm dom}(A)\\to\\mathfrak h_{2}$; while the kernel of $\\tau_{2}$ is required to be dense, so $\\tau_{2}^{*}$ plays the role of a singular perturbations,  no further hypothesis is required for $\\tau_{1}$ and in applications that allows $\\tau_{1}^{*}$ to represent  a regular perturbations by a short-range potential.  In Subsection 2.3, by block operator matrices and the Schur complement, we re-write the obtained resolvent formula in terms of the resolvent of the operator corresponding to the non singular part of the perturbations; that plays an important  role in the subsequent part regarding LAP and the scattering theory. In Section 3,  following the scheme proposed in \\cite{JST} and further generalized in \\cite{JMPA}, at first we provide, under suitable hypothesis, a Limiting Absorption Principle for $A_{\\mathsf B}$ (see Theorem \\ref{LAP}) and then, by a combination of such a LAP with stationary scattering theory in the Birman-Yafaev scheme and the invariance principle, we obtain a representation formula for the scattering matrix of the couple $(A_{\\mathsf B},A)$ (see Theorem \\ref{S-matrix}). Whenever $A$ is the free Laplacian in $L^{2}(\\mathbb{R}^{3})$, such a formula contains, as subcases, both the usual formula for the perturbation given by a short-range potential as given, e.g., in \\cite{Y-LNM} and the formula for the case of a singular perturbation describing self-adjoint boundary conditions on a hypersurface as given in \\cite{JMPA}. Successively, in Section 4,\nin order to apply our abstract results to the case in which $A$ is the free Laplacian and the regular part represents a perturbation by a potential, we give various regularity results for the boundary layer operators associated to $\\Delta+\\mathsf v$, where $\\mathsf v$ is a potential of Kato-Rellich type. In the last Section we present various examples, where the free Laplacian is perturbed both by a singular term, describing either separating boundary conditions (as Dirichlet and Neumann ones) or semi-transparent (as $\\delta$ and $\\delta'$ type ones), and a regular one given by a short range potential $\\mathsf v$ decaying as $|x|^{-\\kappa(1+\\epsilon)}$, $\\epsilon>0$. In order to satisfy all our hypotheses, we need $\\kappa=2$. However, all our hypotheses but a single one (see Lemma \\ref{5.4}) hold with $\\kappa=1$; we conjecture that the requirement $\\kappa=2$ is merely of technical nature and that our results are true for a short range potential decaying as $|x|^{-(1+\\epsilon)}$. Finally, let us remark that whenever one is only interested in the construction of the operators and not in the scattering theory, then it is sufficient to assume that $\\mathsf v$ is a Kato-Rellich potential (see Remark \\ref{suff}). As a byproduct of our abstract construction (see Theorem \\ref{Th-alt-res}), the operator $A_{\\mathsf B}$ corresponds, whenever $A$ is the free Laplacian, to a singular perturbation of a Schr\\\"odinger operator. However, our concern here is the scattering theory with respect to the free Laplacian, thus we regard the regular and the singular parts of the perturbation as a single object; this constitutes the main novelty of our approach.  Schr\\\"odinger operators with a Kato-Rellich potential plus a $\\delta$-like perturbation with a $p$-summable strength ($p>2$)  have been already considered in \\cite{JDE18}, while for a different construction with a bounded potential  and a $\\delta$- or a $\\delta'$-like perturbation with bounded strength we refer to \\cite{BLL}. None of such references considered the scattering matrix (however, \\cite{JDE18} provided a limiting absorption principle).\n\n\\subsection{Some notation and definition.}\n{\\ }\\par\n\\vskip5pt \\noindent $\\bullet$  $\\|\\cdot\\|_{X}$ denotes the norm on the complex Banach space $X$; in case $X$ is a Hilbert   space, $\\langle\\cdot,\\cdot\\rangle_{X}$ denotes the (conjugate-linear w.r.t. the first argument) scalar product.\n\\vskip5pt\\noindent $\\bullet$ $\\langle\\cdot,\\cdot\\rangle_{X^{*},X}$ denotes the duality (assumed to be conjugate-linear w.r.t. the first argument) between the dual couple $(X^{*},X)$.\n\\vskip5pt\\noindent $\\bullet$ $L^{*}:\\text{\\rm dom}(L^{*})\\subseteq Y^{*}\\to X^{*}$ denotes the dual of the densely defined linear operator $L:\\text{\\rm dom}(L)\\subseteq X\\to Y$; in a Hilbert   spaces setting $L^{*}$ denotes the adjoint operator.\n\\vskip5pt\\noindent $\\bullet$ $\\varrho(A)$ and $\\sigma(A)$ denote the resolvent set and the spectrum of the self-adjoint operator $A$; $\\sigma_{p}(A)$, $\\sigma_{pp}(A)$, $\\sigma_{ac}(A)$,  $\\sigma_{sc}(A)$,  denote the point, pure point, absolutely continuous and singular continuous spectra.\n\\vskip5pt\\noindent $\\bullet$ $\\mathscr B(X,Y)$, $\\mathscr B(X)\\equiv \\mathscr B(X,X)$, denote the Banach space of bounded linear operator on the Banach space $X$ to the Banach space $Y$; ${\\|}\\cdot {\\|}_{X,Y}$ denotes the corresponding norm.\n\\vskip5pt\\noindent $\\bullet$ ${\\mathfrak S}_{\\infty}(X,Y)$ denotes the space of compact operators on  $X$ to  $Y$.\n\\vskip5pt\\noindent $\\bullet$ $X\\hookrightarrow Y$ means that $X$ is continuously embedded into $Y$. \n\\vskip5pt\\noindent $\\bullet$ $\\Omega\\equiv\\Omega_{\\rm in}\\subset\\mathbb{R}^{3}$ denotes an open and bounded subset with a Lipschitz boundary $\\Gamma$; $\\Omega_{\\rm ex}:=\\mathbb{R}^{3}\\backslash\\overline\\Omega$.\n\\vskip5pt\\noindent $\\bullet$ $H^{s}(\\Omega)$ and $H^{s}(\\Omega_{\\rm ex})$ denote the scales of Sobolev spaces. \n\\vskip5pt\\noindent  $\\bullet$ $H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H^{s}(\\Omega_{\\rm ex})$.\n\\vskip5pt\\noindent $\\bullet$ $|x|$ denotes the norm of $x\\in\\mathbb{R}^{n}$.\n $\\langle x\\rangle$ denotes the function $x\\mapsto (1+|x|^{2})^{1\/2}$.\n\\vskip5pt\\noindent $\\bullet$ $L_{w}^{2}(\\mathbb{R}^{3})$, $w\\in\\mathbb{R}$, denotes the set of complex-valued functions $f$ such that $\\langle x\\rangle^{w}f\\in L^{2}(\\mathbb{R}^{3})$. \n\\vskip5pt\\noindent  $\\bullet$  $H_{w}^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H_{w}^{s}(\\Omega_{\\rm ex})$, where $H_{w}^{s}(\\Omega_{\\rm ex})$ denotes the weighted Sobolev space relative to the weight $\\langle x\\rangle^{w}$.\n\\vskip5pt\\noindent $\\bullet$ $\\gamma_{0}^{\\rm in\/\\rm ex}$ and $\\gamma_{1}^{\\rm in\/\\rm ex}$ denote the interior\/exterior Dirichlet and Neumann traces on the boundary $\\Gamma$. \n\\vskip5pt\\noindent $\\bullet$  $\\gamma_{0}:=\\frac12(\\gamma_{0}^{\\rm in}+\\gamma_{0}^{\\rm ex})$,   \n $\\gamma_{1}:=\\frac12(\\gamma_{1}^{\\rm in}+\\gamma_{1}^{\\rm ex})$.\n\\vskip5pt\\noindent $\\bullet$ $[\\gamma_{0}]:=\\gamma_{0}^{\\rm in}-\\gamma_{0}^{\\rm ex}$,   \n $[\\gamma_{1}]:=\\gamma_{1}^{\\rm in}-\\gamma_{1}^{\\rm ex}$.  \n\\vskip5pt\\noindent $\\bullet$ $S\\!L_{z}$ and $D\\!L_{z}$ denote the single- and double-layer operators. \n\\vskip5pt\\noindent $\\bullet$ $S_{z}:=\\gamma_{0}S\\!L_{z}$, $D_{z}:=\\gamma_{1}D\\!L_{z}$.\n\\vskip5pt\\noindent $\\bullet$ $D\\subset\\mathbb{R}$ is said to be discrete in the open set $E\\supset D $ whenever the (possibly empty) set of its accumulations point is contained in $\\mathbb{R}\\backslash E$; $D$ is said to be discrete whenever $E=\\mathbb{R}$.\n\\vskip5pt\\noindent $\\bullet$ Given $x\\ge 0$ and $y\\ge 0$, $x\\lesssim y$ means that there exists $c\\ge 0$ such that $x\\le c\\,y$.\n\n\\section{An abstract  Kre\\u\\i n-type resolvent formula}\\label{Sec_Krein} \n\\subsection{The resolvent formula} Let $A:\\text{\\rm dom}(A)\\subseteq \\mathsf H\\to \\mathsf H$ be a self-adjoint operator in the Hilbert space $\\mathsf H$.  We denote by $R_{z}:=(-A+z)^{-1}$, $z\\in \\varrho(A)$, its resolvent; one has $R_{z}\\in\\mathscr B(\\mathsf H,\\mathsf H_{A})$, where $\\mathsf H_{A}$ is the Hilbert space given by $\\text{\\rm dom} (A)$ equipped with the scalar product $$\\langle u,u\\rangle_{\\mathsf H_{A}}:=\\langle (A^{2}+1)^{1\/2} u,(A^{2}+1)^{1\/2}v\\rangle_{\\mathsf H}\\,.\n$$  \nLet\n$$\n\\mathfrak h_{k}\\hookrightarrow\\mathfrak h_{k}^{\\circ}\\hookrightarrow\\mathfrak h_{k}^{\\ast}\\,,\\qquad k=1,2\\,,\n$$\nbe auxiliary Hilbert spaces with dense continuous embedding; we do not identify $\\mathfrak h_{k}$ with its dual $\\mathfrak h^{*}_{k}$ (however, we use $\\mathfrak h_{k}\\equiv\\mathfrak h_{k}^{**}$) and we work with the $\\mathfrak h_{k}^{\\ast}$-$\\mathfrak h_{k}$ duality \n$\\langle\\cdot,\\cdot\\rangle_{\\mathfrak h_{k}^{\\ast},\\mathfrak h_{k}}$ defined in terms of the scalar\nproduct of the intermediate Hilbert space $\\mathfrak h_{k}^{\\circ}$. The scalar product and hence the duality are supposed to be conjugate linear with respect to the first variable; notice that $\\langle\\varphi,\\phi\\rangle_{\\mathfrak h_{k},\\mathfrak h^{\\ast}_{k}}=\\langle\\phi,\\varphi\\rangle^{*}_{\\mathfrak h_{k}^{\\ast},\\mathfrak h_{k}}$.\\par\nGiven the bounded linear maps\n$$\\tau_{k}:\\mathsf H_{A}\\rightarrow\\mathfrak h_{k}\\,,\\qquad k=1,2\\,,\n$$\nsuch that \n\\begin{equation}\\label{tau2}\n\\text{$\\text{\\rm ker}(\\tau_{2})$ is dense in $\\mathsf H$ and $\\text{\\rm ran}(\\tau_{2})$ is dense in $\\mathfrak h_{2}$,} \n\\end{equation}\nwe introduce the bounded operators \n$$\n\\tau:\\mathsf H_{A}\\to \\mathfrak h_{1}\\oplus\\mathfrak h_{2}\\,,\\qquad \\tau u:=\\tau_{1}u\\oplus\\tau_{2} u\\,,\n$$\nand\n$$\nG_{z}:\\mathfrak h_{1}^{\\ast}\\oplus \\mathfrak h_{2}^{*}\\to\\mathsf H\\,,\\qquad  \nG_{z}:=(\\tau R_{\\bar{z}})^{\\ast}\\,,\\qquad z\\in\\varrho(A)\\,.\n$$\nWe further suppose that there exist reflexive Banach spaces $\\mathfrak b_{k}$, $k=1,2$, with dense continuous embeddings $\\mathfrak h_{k}\\hookrightarrow\\mathfrak b_{k}$ (hence $\\mathfrak b_{k}^{*}\\hookrightarrow\\mathfrak h_{k}^{*}$),\nsuch that\n$\\text{\\rm ran}(G_{z}|\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*})$ is contained in the domain of definition of some (supposed to exist) $(\\mathfrak b_{1}\\oplus\\mathfrak b_{2})$-valued extension of $\\tau$ (which we denote by the same symbol) in such a way that \n\\begin{equation}\n\\tau G_{z}|\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*}\\in{\\mathscr B}(\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*},\\mathfrak b_{1}\\oplus\\mathfrak b_{2})\\,. \\label{tauG}%\n\\end{equation}\nGiven these hypotheses, \nwe set $\\mathsf B=(B_{0},B_{1},B_{2})$, with \n\\begin{equation}\\label{B012}\nB_{0}\\in{\\mathscr B}(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})\\,,\\quad B_{1}\\in\\mathscr B(\\mathfrak b_{1},\\mathfrak b_{1}^{*})\\,,\\quad\nB_{2}\\in\\mathscr B(\\mathfrak b_{2},\\mathfrak b^{*}_{2,2})\\,, \\quad\\text{$\\mathfrak b_{2,2}$ a reflexive Banach space,}\n\\end{equation}\n\\begin{equation}\\label{Bse}\n\\quad B_{1}=B_{1}^{*}\\,,\\qquad B_{0}B^{*}_{2}=B_{2}B^{*}_{0}\\,,\n\\end{equation}\nand introduce the map\n\\begin{equation}\\label{Lambda}\nZ_{\\mathsf B}\\ni z\\mapsto\\Lambda_{z}^{\\!\\mathsf B}\\in{\\mathscr B}(\\mathfrak b_{1}\\oplus\\mathfrak b_{2},\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*})\n\\,,\\qquad \n\\Lambda_{z}^{\\!\\mathsf B}:=(M_{z}^{\\mathsf B})^{-1}(  B_{1}\\oplus B_{2})  \\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{ZB}\nZ_{\\mathsf B}:=\\big\\{z\\in{\\mathbb{C}}\\backslash(-\\infty,0]: (M_{w}^{\\mathsf B})^{-1}\n\\in{\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\n\\end{equation}\n$$\nM_{z}^{\\mathsf B}:=(  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2}) \\tau G_{z} \\in {\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2})\\,.\n$$\n\\begin{theorem}\\label{Th_Krein} Suppose hypotheses \\eqref{tau2}, \\eqref{tauG}, \\eqref{B012} and \\eqref{Bse} hold and that $Z_{\\mathsf B}$ defined in \\eqref{ZB} is not empty. Then, defined $\\Lambda_{z}^{\\!\\mathsf B}$ as in \\eqref{Lambda},\n\\begin{equation}\nR_{z}^{\\mathsf B}:=R_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}\\,,\\quad z\\in\nZ_{\\mathsf B}\\,, \\label{resolvent}%\n\\end{equation}\nis the resolvent of a self-adjoint operator $A_{\\mathsf B}$ and $Z_{\\mathsf B}%\n=\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$.\n\\end{theorem}\n\\begin{proof} By \\eqref{Bse}, one gets\n\\begin{align*}\n\\big(  (  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})\\tau\nG_{\\bar{z}}\\big)(  B_{1}\\oplus B^{*}_{2})  =&(  B_{1}\\oplus B_{2})\n\\big(  (  1\\oplus B_{0}^{*})  -\\tau G_{\\bar{z}}(  B_{1}\\oplus B_{2}^{*})\\big)\\\\\n=&(  B_{1}\\oplus B_{2})\n\\big(  (  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})\\tau G_{{z}}\\big)^{*}  \\,.\n\\end{align*}\nThis entails, by the definitions \\eqref{Lambda} and \\eqref{ZB},\n\\begin{equation}\\label{PS1}\n(\\Lambda^{\\!\\mathsf B}_{z})^{*}=\\Lambda^{\\!\\mathsf B}_{\\bar z}\\,.\n\\end{equation}\nBy the resolvent identity, there follows\n\\begin{align*}\n& ((  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})\n\\tau G_{z})  -((  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})  \\tau G_{w}) \\\\ \n&=(  B_{1}\\oplus B_{2})  \\tau(G_{w}-G_{z})\n=(z-w)(  B_{1}\\oplus B_{2})  \\tau R_{w}G_{z}\\\\\n&=(z-w)(B_{1}\\oplus B_{2})  G_{\\bar{w}}^{\\ast}G_{z}\\,,\n\\end{align*}\nwhich entails \n\\begin{align*}\n& ( (  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})\n\\tau G_{w})  ^{-1}-((  1\\oplus B_{0})  -\n(B_{1}\\oplus B_{2})  \\tau G_{z})  ^{-1}  \\\\\n=&(z-w)((  1\\oplus B_{0})  -(  B_{1}\\oplus B_{2})  \\tau G_{w})  ^{-1}\n(  B_{1}\\oplus B_{2})G_{\\bar{w}}^{\\ast}G_{z}((  1\\oplus B_{0})\n -(B_{1}\\oplus B_{2})  \\tau G_{z})  ^{-1}\\,,\n\\end{align*}\nand hence\n\\begin{equation}\\label{PS2}\n\\Lambda_{w}^{\\!\\mathsf B}-\\Lambda_{z}^{\\!\\mathsf B}=(z-w)\\Lambda_{w}^{\\!\\mathsf B}G_{\\bar{w}}^{\\ast\n}G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\,.\n\\end{equation}\nBy \\eqref{PS1} and $\\eqref{PS2}$, \n$$(R_{z}^{\\mathsf B})^{*}=R_{\\bar z}^{\\mathsf B}\\,,\\qquad R_{z}^{\\mathsf B}=R_{w}^{\\mathsf B}+(w-z)R_{z}^{\\mathsf B}R_{w}^{\\mathsf B}\\,.\n$$\n(see \\cite[page 113]{JFA}). Hence, $R_{z}^{\\mathsf B}$ is the resolvent of a self-adjoint operator whenever it is injective (see, e.g., \\cite[Theorems 4.10 and 4.19]{Stone}). By \\eqref{resolvent},\n\\begin{align*}\n&(B_{1}\\oplus B_{2})\\tau R_{z}^{\\mathsf B}=(B_{1}\\oplus B_{2})\\big(1+\\tau G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}=\\big((B_{1}\\oplus B_{2})+(B_{1}\\oplus B_{2})\\tau G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}\\\\\n=&\\big((B_{1}\\oplus B_{2})+\\big((1\\oplus B_{0})-\\big((1\\oplus B_{0})-(B_{1}\\oplus B_{2})\\tau G_{z}\\big)\\big)\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}=(1\\oplus B_{0})\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}\\,.\n\\end{align*}\nThus, if $R_{z}^{\\mathsf B}u=0$ then \n$$\n0\\oplus 0=(1\\oplus B_{0})\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u=\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u\\big)_{1}\\oplus B_{0}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u\\big)_{2}\n$$\nBy\n$$\nG_{z}(\\phi_{1}\\oplus\\phi_{2})=G^{1}_{z}\\phi_{1}+G^{2}_{z}\\phi_{2}\\,, \\qquad G^{k}_{z}:=(\\tau_{k}R_{\\bar{z}})^{\\ast}\\,,\n$$\nthere follows \n\\begin{equation}\n0=R_{z}^{\\mathsf B}u=R_{z}u+G^{1}_{z}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}\nu\\big)_{1}+G^{2}_{z}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}u\\big)_{2}=R_{z}u+G^{2}_{z}\n\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}u\\big)_{2}\\,.\\label{z2}%\n\\end{equation}\nSince the denseness of $\\text{\\rm ker}(\\tau_{2})$ is equivalent to $\\text{\\rm ker}(G^{2}_{z})=\\{0\\}$\nand the denseness of $\\text{\\textrm{ran}}(\\tau_{2})$ implies\n$\\text{\\textrm{ran}}(G^{2}_{z})\\cap \\text{\\rm dom}(A)=\\{0\\}$ \n(see \\cite[Remark 2.9]{JFA}), the relation \\eqref{z2} gives $\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast\n}u\\big)_{2}=0$. Thus  $R_{z}^{\\mathsf B}u=0$ compels $R_{z}u=0$ and hence $u=0$.\\par \nFinally, the equality $Z_{\\mathsf B}=\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$ is consequence of \\cite[Theorem 2.19 and\nRemark 2.20]{CFP}.\n\\end{proof}\n\\begin{remark} By \\eqref{resolvent}, if $u\\in\\text{\\rm dom}(A_{\\mathsf B})$, then $u=u_{0}+G_{z}(\\phi_{1}\\oplus\\phi_{2})$ for some $u_{0}\\in\\mathsf H_{A}$ and $\\phi_{1}\\oplus\\phi_{2}\\in \\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*}$; hence, by \\eqref{tauG}, \n$$\n\\tau:\\text{\\rm dom}(A_{\\mathsf B})\\to \\mathfrak b_{1}\\oplus\\mathfrak b_{2}\\,.\n$$  \n\\end{remark}\n\\subsection{An additive representation}\nAt first, let us introduce the Hilbert space $\\mathsf H_{A}^{*}$ defined as the completion of $\\mathsf H$ endowed with the scalar product $$\\langle u,v\\rangle_{\\mathsf H_{A}^{*}}:=\\langle(A^{2}+1)^{-1\/2}u,(A^{2}+1)^{-1\/2}v\\rangle_{\\mathsf H}\\,.\n$$\nNotice that that $R_{z}$ extends to a bounded bijective map (which we denote by the same symbol) on $\\mathsf H_{A}^{*}$ onto $\\mathsf H$. The linear operator $A$, being a densely defined bounded operator on $\\mathsf H$ to  $\\mathsf H_{A}^{*}$, extends to a bounded operator $\\overline {\\!A}:\\mathsf H\\to\\mathsf H_{A}^{*}$ given by its closure. Moreover, denoting by $\\langle\\cdot,\\cdot\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}$  the pairing obtained by extending the scalar product in $\\mathsf H$, since $A$ is self-adjoint and since $\\text{\\rm dom}(A)$ is dense in $\\mathsf H$,\n$$\n\\langle u,Av\\rangle_{\\mathsf H}=\\langle\\, \\overline{\\!A}u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\,,\\qquad u\\in\\mathsf H\\,,\\ v\\in\\mathsf H_{A}, \\,.\n$$\nFurther, we define $\\tau^{*}:\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2}\\to\\mathsf H_{A}^{*}$ by \n\\begin{equation}\\label{tau*}\n\\langle \\tau^{*}\\phi,u\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}=\\langle\\phi,\\tau u\\rangle_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2},\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\,,\\qquad u\\in\\mathsf H_{A}\\,,\\ \\phi\\in\\\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2}\\,.\n\\end{equation}\nObviously, $\\tau^{*}(\\phi_{1}\\oplus\\phi_{2})=\\tau_{1}^{*}\\phi_{1}+\\tau^{*}_{2}\\phi_{2}$, where \n$\\tau_{k}^{*}:\\mathfrak h_{k}\\to\\mathsf H_{A}^{*}$, $k=1,2$, are defined in the same way as $\\tau^{*}$.\n\\par\nLet us notice that $R_{z}:\\mathsf H_{A}^{*}\\to \\mathsf H$ is the adjoint, with respect the pairing $\\langle\\cdot,\\cdot\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}$, of $R_{\\bar z}:\\mathsf H_{A}\\to \\mathsf H$ and it is the inverse of $(-\\overline{\\!A} +z):\\mathsf H\\to\\mathsf H_{A}^{*}$; therefore \n\\begin{equation}\\label{blw}\nG_{z}=R_{z}\\tau^{*}\\,.\n\\end{equation}\n\\begin{lemma}\n\\label{le-green}Let $A_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\subseteq\\mathsf H\\to\\mathsf H$ be the self-adjoint\noperator provided in Theorem \\ref{Th_Krein} and define\n\\begin{equation}\n\\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\rightarrow\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\,,\\quad\n\\rho_{\\mathsf B}(R_{z}^{\\mathsf B}u):=(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast\n}u\\,,\\qquad u\\in \\mathsf H\\,, \\quad z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A)\\,,\\label{rho_def}%\n\\end{equation}\nwhere $\\pi_{1}$ denotes the orthogonal projection onto the subspace\n$\\overline{\\text{\\rm ran}(\\tau_{1})}$. Then, the definition of $\\rho_{\\mathsf B}$ is\nwell-posed, i.e.,\n\\[\nR_{z_{1}}^{\\mathsf B}u_{1}=R_{z_{2}}^{\\mathsf B}u_{2}\\quad\\implies\\quad (\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}}^{\\ast}u_{1}=(\\pi_{1}^{\\ast}\\oplus 1)\n\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}u_{2}%\n\\]\nand \n\\begin{equation}\\label{GF}\n\\langle u,\\Delta_{\\mathsf B}v\\rangle_{\\mathsf H}=\\langle A\nu,v\\rangle_{\\mathsf H}+\n\\langle\\tau u,\\rho_{\\mathsf B} v\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\,,\\quad u\\in \\text{\\rm dom}(A),\\,v\\in\n\\text{\\rm dom}(A_{\\mathsf B})\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $v=R_{z}^{\\mathsf B}u=v_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}$, where $v_{z}%\n:=R_{z}u$ (hence $\\tau v_{z}=G_{\\bar{z}}^{\\ast}u$). Then\n\\begin{align*}\n&  \\langle u,A_{\\mathsf B}v\\rangle_{\\mathsf H}-\\langle A\nu,v\\rangle_{\\mathsf H}\\\\\n=  &  -\\langle u,(-A_{\\mathsf B}+z)v\\rangle_{\\mathsf H}%\n+\\langle(-A+\\bar{z})u,v\\rangle_{\\mathsf H}\\\\\n=  &  -\\langle u,(-A+z)v_{z}\\rangle_{\\mathsf H}%\n+\\langle(-A+\\bar{z})u,v_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}%\n\\rangle_{\\mathsf H}\\\\\n=  &  \\langle(-A+\\bar{z})u,G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle\n_{\\mathsf H}=\\langle\\tau u,\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}%\n\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\\\\n=  &  \\langle (\\pi_{1}\\oplus 1)\\tau u,\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle\n_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}=\\langle\\tau u,(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda\n_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\,.\n\\end{align*}\nSuppose now that $R_{z_{1}}^{\\mathsf B}u_{1}=R_{z_{2}}^{\\mathsf B}u_{2}$. Then, by the above\nidentities, one gets, for any $u\\in \\text{\\rm dom}(A)$,\n\\[\n\\langle \\tau^{\\ast}(\\pi_{1}^{\\ast}\\oplus 1)(\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}%\n}^{\\ast}u_{1}-\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}u_{2}),u\\rangle\n_{\\mathsf H_{A}^{*},\\mathsf H_{A}}=0\\,.\n\\]\nHence $\\tau^{\\ast}((\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}}%\n^{\\ast}u_{1}-(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}%\nu_{2})=0$. However, $\\text{\\rm ker}(\\tau^{\\ast})\\cap\\text{\\rm ran}((\\pi_{1}^{\\ast}\\oplus 1))=\\{0\\}$ since $\\pi_{1}^{\\ast}\\oplus 1$ is the projector onto the subspace\northogonal to $\\text{\\rm ker}(\\tau^{\\ast})$.\n\\end{proof}\nThe next Lemma provides a sort of abstract boundary conditions holding for the elements in $\\text{\\rm dom}(A_{\\mathsf B})$:\n\\begin{lemma}\\label{abc} Let $A_{\\mathsf B}$ be the self-adjoint operator in Theorem \\ref{Th_Krein}. Then, for any $z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$, one has the representation\n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u_{z}:=u-G_{z}\\rho_{\\mathsf B}u\\in \\text{\\rm dom}(A)\\}\\,,\n$$\n$$\n(-A_{\\mathsf B}+z)u=(-A+z)u_{z}\\,.\n$$\nMoreover, \n$$\nu\\in \\text{\\rm dom}(A_{\\mathsf B})\\quad\\Longrightarrow\\quad (\\pi_{1}^{*}B_{1}\\oplus B_{2})\\tau u=(1\\oplus B_{0})\\rho_{\\mathsf B} u\\,.\n$$\n\\end{lemma}\n\\begin{proof} Since $G_{z}=R_{z}\\tau^{*}$ (see \\eqref{blw} below) and $\\pi_{1}^{*}\\oplus 1$ is the projection onto the orthogonal to $\\text{\\rm ker}(\\tau^{*})$, one has $G_{z}=G_{z}(\\pi_{1}^{*}\\oplus 1) $. \nHence, $u\\in\\text{\\rm dom}(A_{\\mathsf B})$ if and only if $u=R_{z}v+G_{z}(\\pi_{1}^{*}\\oplus 1) \\Lambda_{z}^{\\mathsf B}G^{*}_{\\bar z}v=R_{z}v+G_{z}\\rho_{\\mathsf B}u$. Therefore, \n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u=u_{z}+G_{z}\\rho_{\\mathsf B}u\\,, \\ u_{z}\\in\\text{\\rm dom}(A)\\}\\,.\n$$\nMoreover, given any $u\\in\\text{\\rm dom}(A)$, $u=R^{\\mathsf B}_{z}v$, one has\n$$\n(-A+z)u_{z}=(-A+z)R_{z}v=(-A_{\\mathsf B}+z)R^{\\mathsf B}_{z}v=(-A_{\\mathsf B}+z)u\\,.\n$$\nFinally, given $u=R^{\\mathsf B}_{z}v\\in\\text{\\rm dom} (A_{\\mathsf B})$,  one has\n\\begin{align*}\n&(\\pi_{1}^{*}B_{1}\\oplus B_{2})\\tau u=(\\pi_{1}^{*}\\oplus 1)(B_{1}\\oplus B_{2})\\tau R^{\\mathsf B}_{z}v\\\\\n=&(\\pi_{1}^{*}\\oplus 1)\\big((B_{1}\\oplus B_{2})G_{\\bar z}v+ \n(B_{1}\\oplus B_{2})\\tau G_{z}\\big((1\\oplus B_{0})-(B_{1}\\oplus B_{2})\\tau G_{z}\\big)^{-1}(B_{1}\\oplus B_{2})G_{\\bar z}v\\big)\\\\\n=&(\\pi_{1}^{*}\\oplus 1)(1\\oplus B_{0})\\Lambda^{\\mathsf B}_{z}G_{\\bar z}v=(1\\oplus B_{0})(\\pi_{1}^{*}\\oplus 1)\\Lambda^{\\mathsf B}_{z}G_{\\bar z}v=(1\\oplus B_{0})\\rho_{\\mathsf B} u\\,.\n\\end{align*}\n\\end{proof} \nNow, we provide an additive representation of the self-adjoint $A_{\\mathsf B}$ in Theorem \\ref{Th_Krein}. \n\\begin{theorem}\\label{Th-add} Let $A_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\subseteq \\mathsf H\\to\\mathsf H$ be the self-adjoint operator appearing in Theorem \\ref{Th_Krein}. Then\n$$\nA_{\\mathsf B}=\\overline {\\!A}+\\tau^{*}\\!\\rho_{\\mathsf B}\\,,\n$$\nwhere $\\rho_{\\mathsf B}$ is defined in \\eqref{rho_def}. In particular, if $B_{0}^{-1}\\in\\mathscr B(\\mathfrak b_{2,2}^{*},\\mathfrak b_{2}^{*})$, then\n$$\nA_{\\mathsf B}=\\overline {\\!A}+\\tau_{1}^{*}B_{1}\\tau_{1}+\\tau_{2}^{*}B_{0}^{-1}\\!B_{2}\\tau_{2}\\,.\n$$ \n\\end{theorem}  \n\\begin{proof} By \\eqref{GF}, for any $u\\in\\text{\\rm dom}(A_{\\mathsf B})$ and $v\\in\\mathsf H_{A}$,\n\\begin{align*}\n\\langle \\Delta_{\\mathsf B}u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\equiv&\\langle \\Delta_{\\mathsf B}u,v\\rangle_{\\mathsf H}=\n\\langle u,Av\\rangle_{\\mathsf H}+\\langle\\rho_{\\mathsf B} u,\\tau v\\rangle_{\\mathfrak h^{*}_{1}\\oplus\\mathfrak h^{*}_{2},\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\\\\n=&\\langle\\, \\overline{\\!A}u+\\tau^{*}\\!\\rho_{\\mathsf B} u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\,.\n\\end{align*}\nBy Lemma \\ref{abc} and by $\\tau_{1}^{*}\\pi_{1}^{*}=(\\pi_{1}\\tau_{1})^{*}=\\tau_{1}^{*}$,\n$$\n\\tau^{*}\\!\\rho_{\\mathsf B} =\\tau^{*}(\\pi^{*}_{1}B_{1}\\tau_{1}\\oplus B_{0}^{-1}B_{1}\\tau_{2})=\n\\tau_{1}^{*}B_{1}\\tau_{1}+\\tau_{2}^{*}B_{0}^{-1}\\!B_{2}\\tau_{2}\\,.\n$$\n\\end{proof}\n\n\n\\subsection{An alternative resolvent formula.} At first, let us notice that hypothesis \\eqref{tauG}, can be re-written as\n$$\n\\tau_{j}G^{k}_{z}|\\mathfrak b_{k}\\in\\mathscr B(\\mathfrak b_{k}^{*},\\mathfrak b_{j})\\,,\\quad j,k=1,2\\,,\\qquad G^{k}_{z}:=(\\tau_{k}R_{\\bar z})^{*}\\,.\n$$\nMoreover, \n$$\n{M}_{z}^{\\mathsf B}=(1\\oplus B_{0})+(B_{1}\\oplus B_{2})\\tau G_{z}=\n\\begin{bmatrix}M_{z}^{B_{1}}&B_{1}\\tau_{1}G^{2}_{z}\\\\\nB_{2}\\tau_{2}G^{1}_{z}&M_{z}^{B_{0},B_{2}}\n\\end{bmatrix}\n$$\nwhere\n$$\nM_{z}^{B_{1}}:=1-B_{1}\\tau_{1}G^{1}_{z}\\,,\\qquad \nM_{z}^{B_{0},B_{2}}:=B_{0}-B_{2}\\tau_{2}G^{2}_{z}\\,.\n$$\nThen, supposing all the inverse operators appearing in the next formula exist, by the inversion formula for block operator matrices, one gets\n\\begin{align}\\label{MB-1}\n&({M}_{z}^{\\mathsf B})^{-1}=\n\\begin{bmatrix}(M^{B_{1}}_{z})^{-1}+(M^{B_{1}}_{z})^{-1}B_{1}\\tau_{1}G^{2}_{z}({C}_{z}^{\\mathsf B})^{-1}B_{2}\\tau_{2}G^{1}_{z}(M^{B_{1}}_{z})^{-1}&\n(M^{B_{1}}_{z})^{-1}B_{1}\\tau_{1}G^{2}_{z}({C}_{z}^{\\mathsf B})^{-1}\\\\\n({C}_{z}^{\\mathsf B})^{-1}B_{2}\\tau_{2}G^{1}_{z}(M^{B_{1}}_{z})^{-1}&({C}_{z}^{\\mathsf B})^{-1}\n\\end{bmatrix}\n\\end{align}\nwhere ${C}^{\\mathsf B}_{z}$ denotes the second Schur complement, i.e.,\n\\begin{align*}\n{C}^{\\mathsf B}_{z}:=&M_{z}^{B_{0},B_{2}}-B_{2}\\tau_{2}G^{1}_{z}\n(M_{z}^{B_{1}})^{-1}B_{1}\\tau_{1}G^{2}_{z}\\\\\n=&M_{z}^{B_{0},B_{2}}\\left(1-(M_{z}^{B_{0},B_{2}})^{-1}B_{2}\\tau_{2}G^{1}_{z}\n(M_{z}^{B_{1}})^{-1}B_{1}\\tau_{1}G^{2}_{z}\\right)\\\\\n=&M_{z}^{B_{0},B_{2}}\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{z}\\right)\\,,\n\\end{align*}\n\\begin{equation}\\label{LB1}\n\\Lambda_{z}^{\\!B_{1}}:=(1-B_{1}\\tau_{1}G^{1}_{z})^{-1}B_{1}\\,,\n\\end{equation}\n\\begin{equation}\\label{LB02}\n\\Lambda_{z}^{\\!B_{0},B_{2}}:=(B_{0}-B_{2}\\tau_{2}G^{2}_{z})^{-1}B_{2}\\,.\n\\end{equation}\nRegarding the well-posedness of \\eqref{MB-1}, taking into account the definition of ${C}^{\\mathsf B}_{z}$, one has \n$$\nZ_{\\mathsf B}=\\big\\{z\\in\\varrho(A): (M^{\\mathsf B}_{z})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\\supseteq \\widehat Z_{\\mathsf B}\\,,\n$$\nwhere\n\\begin{equation}\\label{wZB}\n\\widehat Z_{\\mathsf B}:=\\big\\{z\\in Z_{B_{1}}\\cap Z_{B_{0},B_{2}}:\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{w}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{w}\\right)^{-1}\\in\\mathscr B(\\mathfrak b_{2}^{*}),\\quad w=z,\\bar z\\big\\}\n\\end{equation}\n\\begin{equation}\\label{ZB1}\nZ_{B_{1}}:=\\big\\{z\\in\\varrho(A): (1-B_{1}\\tau_{1}G^{1}_{w})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{1})\\,,\\ w=z,\\bar{z}\\big\\}\\,,\n\\end{equation}\n\\begin{equation}\\label{ZB02}\nZ_{B_{0},B_{2}}:=\\big\\{z\\in\\varrho(A): (B_{0}-B_{2}\\tau_{2}G^{2}_{w})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\\,,\n\\end{equation}\nTherefore, supposing that $\\widehat Z_{\\mathsf B}$ is not empty, for any $z\\in\\widehat Z_{\\mathsf B}$, by \\eqref{resolvent} and by\n$$\n({C}_{z}^{\\mathsf B})^{-1}B_{2}=\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\,,\\qquad \\Sigma^{\\mathsf B}_{z}:=\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{z}\\right)^{-1}\\,,\n$$\n one has\n\\begin{align*}\n\\Lambda_{z}^{\\!\\mathsf B}=(M^{\\mathsf B}_{z})^{-1}\n\\begin{bmatrix}B_{1}&0\\\\\n0&B_{2}\\end{bmatrix}\n=&\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\\\\n\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\n\\end{bmatrix}\n\\,.\n\\end{align*}\nTherefore\n\\begin{equation}\\label{res}\nR_{z}^{\\mathsf B}=R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\\\\n\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\\Sigma^{\\mathsf B}_{z}\n\\Lambda_{z}^{\\!B_{0},B_{2}}\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\\,.\n\\end{equation}\nIn particular, taking $\\mathsf B=(1,B_{1},0)$, one gets, for any $z\\in Z_{B_{1}}$,\n\\begin{align}\\label{res1}\nR_{z}^{B_{1}}:=R_{z}^{(1,B_{1},0)}\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda_{z}^{\\!B_{1}}&0\\\\\n0&0\n\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\n=R_{z}+G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}{G^{1*}_{\\bar z}}\n\\end{align}\nwhile, taking $\\mathsf B=(B_{0},0,B_{2})$, one gets, for any $z\\in Z_{B_{0},B_{2}}$, \n\\begin{align}\\label{res2}\nR_{z}^{B_{0},B_{2}}:=R_{z}^{(B_{0},0,B_{2})}\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}0&\n0\\\\\n0&\\Lambda_{z}^{\\!B_{0},B_{2}}\n\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\n=R_{z}+G^{2}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}{G^{2*}_{\\bar z}}\\,.\n\\end{align}\nTherefore, by Theorem \\ref{Th_Krein} with $\\mathsf B=(B_{1},0,0)$, one gets\n\\begin{corollary}\\label{cor1} Let $\\tau_{1}\\in\\mathscr B(\\mathsf H_{A},\\mathfrak h_{1})$ such that $\\tau_{1}G^{1}_{z}|\\mathfrak b^{*}_{1}\\in\\mathscr B(\\mathfrak b_{1}^{*},\\mathfrak b_{1})$ and let $B_{1}\\in\\mathscr B(\\mathfrak b_{1},\\mathfrak b^{*}_{1})$ self-adjoint; suppose that  \n$Z_{B_{1}}$ defined in \\eqref{ZB1} is not empty. Then \n\\begin{equation}\\label{res1.1}\nR_{z}^{B_{1}}=R_{z}+G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}{G^{1*}_{\\bar z}}\\,,\\qquad z\\in Z_{B_{1}}\\,,\n\\end{equation}\nwhere $\\Lambda_{z}^{\\!B_{1}}$ is defined in \\eqref{LB1}, is the resolvent of a self-adjoint operator $A_{B_{1}}$ and $Z_{B_{1}}=\\varrho(A_{B_{1}})\\cap\\varrho(A)$.\n\\end{corollary}\nBy Theorem \\ref{Th_Krein} with $\\mathsf B=(B_{0},0,B_{2})$, one gets\n\\begin{corollary}\\label{cor02} Let $\\tau_{2}\\in\\mathscr B(\\mathsf H_{A},\\mathfrak h_{2})$ satisfy \\eqref{tau2} be such that $\\tau_{1}G^{1}_{z}|\\mathfrak b^{*}_{2}\\in\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2})$ and let $B_{0}\\in\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})$, $B_{2}\\in \\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2,2}^{*})$ be such that $B_{0}B_{2}^{*}=B_{2}B_{0}^{*}$; suppose that\n$Z_{B_{0},B_{2}}$ defined in \\eqref{ZB02} is not empty. Then \n\\begin{equation}\\label{res2.1}\nR_{z}^{B_{0},B_{2}}=R_{z}+G^{2}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}{G^{2*}_{\\bar z}}\\,,\\qquad z\\in Z_{B_{0},B_{2}}\n\\end{equation}\nwhere $\\Lambda_{z}^{\\!B_{0},B_{2}}$ is defined in \\eqref{LB02}, is the resolvent of a self-adjoint operator $A_{B_{0},B_{2}}$ and $Z_{B_{0},B_{2}}=\\varrho(A_{B_{0},B_{2}})\\cap\\varrho(A)$.\n\\end{corollary}\nSupposing $\\widehat Z_{\\mathsf B}\\not=\\varnothing $, by \\eqref{res}, by \\eqref{res1} and by the relations\n\\begin{align}\\label{GB1}\nG^{B_{1}}_{z}:=(\\tau_{2}R_{\\bar z}^{B_{1}})^{*}=&\n(\\tau_{2}R_{\\bar z}+\\tau_{2} G^{1}_{\\bar z}\\Lambda^{\\!B_{1}}_{\\bar z}{G^{1*}_{z}})^{*}\\\\\n=&G_{z}^{2}+G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\nonumber\n\\end{align}\n\\begin{align*}\n{G^{B_{1}*}_{\\bar z}}=\\tau_{2}R_{z}^{B_{1}}=&\n\\tau_{2}R_{z}+\\tau_{2} G^{1}_{ z}\\Lambda^{\\!B_{1}}_{z}{G^{1*}_{\\bar z}}\\\\=&G_{\\bar z}^{2*}+\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}G_{\\bar z}^{1*}\n\\end{align*}\n\\begin{align*}\n\\widehat M_{z}^{\\mathsf B}=B_{0}-B_{2}\\tau_{2}G^{B_{1}}_{z}=&B_{0}-B_{2}\\tau_{2}G_{z}^{2}+\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\\\\n=&M_{z}^{B_{0},B_{2}}+B_{2}\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\\\\n=&M_{z}^{B_{0},B_{2}}\\big(1+\\Lambda^{\\!B_{0},B_{1}}_{z}\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\big)\n\\end{align*}\n\\begin{equation}\\label{wLB1}\n\\widehat \\Lambda^{\\mathsf B}_{z}:=(\\widehat M_{z}^{\\mathsf B})^{-1}B_{2}=(B_{0}-B_{2}\\tau_{2}G^{B_{1}}_{z})^{-1}B_{2}=\\Sigma_{z}^{\\mathsf B}\\Lambda^{\\!B_{0},B_{2}}_{z}\n\\end{equation}\n one gets \n\\begin{align}\n\\Lambda_{z}^{\\mathsf B}=&\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\\\\n\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}&\n\\widehat \\Lambda^{\\mathsf B}_{z}\\,;\n\\end{bmatrix}\\label{LB-new}\\\\\n=&\\left(1+\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}&0\\\\0&\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\\begin{bmatrix}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}&\n\\tau_{1}G^{2}_{z}\\\\\n\\tau_{2}G^{1}_{z}&0\n\\end{bmatrix}\\,\\right)\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}&0\\\\0&\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\\,.\\label{LB-new2}\n\\end{align}\nTherefore \n\\begin{align}\\label{res-new}\nR_{z}^{\\mathsf B}=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\\\\n\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}&\n\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\n\\begin{bmatrix}\n{G^{1*}_{\\bar z}}\\\\ \n{G^{2*}_{\\bar z}}\n\\end{bmatrix}\\\\\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\\\\n\\widehat \\Lambda^{\\!\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}G^{1*}_{\\bar z}+\n\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\n\\end{bmatrix}\\nonumber\\\\\n=&R_{z}+G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\nG^{1}_{z}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\nonumber\\\\\n&+G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}G^{1*}_{\\bar z}+\nG^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\nonumber\\\\\n=&R^{B_{1}}_{z}+G_{z}^{B_{1}}\\widehat \\Lambda^{\\mathsf B}_{z}G_{\\bar z}^{B_{1}*}\\nonumber\\,.\n\\end{align}\nThis also entails, by \\cite[Theorem 2.19 and Remark 2.20]{CFP}, that if $\\widehat Z_{\\mathsf B}\\not=\\varnothing $, then $\\widehat Z_{\\mathsf B}=Z_{\\mathsf B}=\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$. Summing up, one has the following \n\\begin{theorem}\\label{Th-alt-res} Assume that  hypotheses \\eqref{tauG}, \\eqref{B012} and \\eqref{Bse} hold and that $\\widehat Z_{\\mathsf B}$ defined in \\eqref{wZB} is not empty. Then, for any $z\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$, the resolvent $R_{z}^{\\mathsf B}$ in \\eqref{resolvent} has the representation \\eqref{res-new} and  \n\\begin{equation}\\label{alt-res}\nR^{\\mathsf B}_{z}=R_{z}^{B_{1}}+G_{z}^{B_{1}}\\widehat \\Lambda^{\\mathsf B}_{z}G_{\\bar z}^{B_{1}*}\\,,\\qquad\nz\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})\\,,\n\\end{equation}\nwhere $R_{z}^{B_{1}}$, $G_{z}^{B_{1}}$ and $\\widehat \\Lambda^{\\mathsf B}_{z}$ are defined in \\eqref{res1.1}, \\eqref{GB1} and \\eqref{LB1}.\n\\end{theorem}\n\\begin{remark}Let us notice that the resolvent formula \\eqref{alt-res} is of the same kind of the one in \\eqref{res2.1}, whenever one replaces $A$ with $A_{B_{1}}$. \n\\end{remark}\nBy using the same kind of arguments as in the proof of Lemma \\ref{abc} and defining \n$$\n\\widehat \\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\to\\mathfrak h_{2}^{*}\\,,\\qquad \\widehat \\rho_{\\mathsf B}(R^{\\mathsf B}_{z}u):=G_{z}^{B_{1}}\\widehat\\Lambda_{z}^{\\mathsf B}G^{B_{1}*}_{\\bar z}u\\,,\n$$\none gets the following\n\\begin{lemma}\\label{alt-abc} Let $A_{\\mathsf B}$ be the self-adjoint operator in Theorem \\ref{Th-alt-res}. Then, for any $z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$, one has the representation\n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u_{z}:=u-G^{B_{1}}_{z}\\widehat\\rho_{\\mathsf B}u\\in \\text{\\rm dom}(A_{B_{1}})\\}\\,,\n$$\n$$\n(-A_{\\mathsf B}+z)u=(-A_{B_{1}}+z)u_{z}\\,.\n$$\nMoreover, \n$$\nu\\in \\text{\\rm dom}(A_{\\mathsf B})\\quad\\Longrightarrow\\quad B_{2}\\tau_{2} u=B_{0}\\widehat\\rho_{\\mathsf B} u\\,.\n$$\n\\end{lemma}\n\n\\section{The Limiting Absorption Principle and the Scattering Matrix}\\label{Sec-LAP}\nNow we suppose that $\\mathsf H=L^{2}(M,{\\mathcal B},m)\\equiv L^{2}(M)$. \nGiven a measurable $\\varphi:M\\to [1,+\\infty)$, we define the weighted $L^{2}$-space \n\\begin{equation}\\label{Lphi}\nL_{\\varphi}^{2}(M,{\\mathcal B},m)\\equiv L_{\\varphi}^{2}(M):=\\{\\text{$u:M\\to\\mathbb{C}$ measurable} : \\varphi u\\in L^{2}(M)\\}\\,.\n\\end{equation}\nBy $\\varphi\\ge 1$,  \n$$\nL^{2}_{\\varphi}(M)\\hookrightarrow L^{2}(M)\\hookrightarrow L_{\\varphi^{-1}}^{2}(M)\\simeq L_{\\varphi}^{2}(M)^{*}\\,.\n$$\nFrom now on $\\langle\\cdot,\\cdot\\rangle $ and  $\\|\\cdot\\|$ denote the scalar product and the corresponding norm on  $L^{2}(M)$; $\\langle\\cdot,\\cdot\\rangle_{\\varphi} $ and  $\\|\\cdot\\|_{\\varphi}$ denote the scalar product and the corresponding norm on  $L_{\\varphi}^{2}(M)$. \\par\nThen we introduce the following hypotheses: \\vskip5pt\\par\\noindent\n(H1) $A_{B_{1}}$ is bounded from above and there exists a positive $\\lambda_{1}\\ge\\sup\\sigma(A_{B_{1}})$, such that \n$R^{B_{1}}_{z}\\in \\mathscr B(L^{2}_{\\varphi}(M))$ for any $z\\in \\varrho(A_{B_{1}})$ such that $\\text{Re}(z)> \\lambda_{1}$; \n\\vskip5pt\\par\\noindent\n(H2) $A_{B_{1}}$ satisfies a Limiting Absorption Principle (LAP for short), i.e. there exists a (eventually empty) closed set with zero Lebesgue measure ${e}(A_{B_{1}})\\subset \\mathbb{R}$ such that, for all $\\lambda\\in \\mathbb{R}\\backslash{e}(A_{B_{1}})$, the limits\n\\begin{equation}\\label{LAP1}\nR^{B_{1},\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}R^{B_{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$ and the maps $z\\mapsto R^{B_{1},\\pm}_{z}$, where $R^{B_{1},\\pm}_{z}\\equiv R^{B_{1}}_{z}$ whenever $z\\in\\varrho(A_{B_{1}})$, are continuous on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$;\n\\vskip5pt\\par\\noindent\n(H3) for any compact set $K\\subset \\mathbb{R}\\backslash{e}(A_{B_{1}})$ there exists $c_{K}>0$ such that for any $\\lambda\\in K$ and for any $u\\in L_{\\varphi^{2}}^{2}(M)\\cap\\text{\\rm ker}(R^{B_{1},+}_{\\lambda}-R^{B_{1},-}_{\\lambda})$ one has\n\\begin{equation}\\label{(H3)}\n\\|R^{B_{1},\\pm}_{\\lambda}u\\|\\le c_{K}\\|u\\|_{\\varphi^{2}}\\,;\n\\end{equation} \nWe split next hypothesis (H4) in two separate points:\\vskip5pt\\par\\noindent\n(H4.1) $A_{\\mathsf B}$ is bounded from above;\n\\vskip5pt\\par\\noindent\n(H4.2) the embedding $\\mathfrak h_{2}\\hookrightarrow\\mathfrak b_{2}$ is compact and there exists a positive $\\lambda_{2}>\\sup\\sigma(A_{B_{1}})$, such that  $G_{z}^{B_{1}}\\in \\mathscr B(\\mathfrak h^{*}_{2},L_{\\varphi^{2+\\gamma}}^{2}(M))$ for some $\\gamma>0$ and for any $z\\in\\varrho(A_{B_{1}})$ such that $\\text{Re}(z)>\\lambda_{2}$.\n\\vskip5pt\\par\\noindent\nThen, $A_{\\mathsf B}$ satisfies a Limiting Absorption Principle as well:\n\\begin{theorem}\\label{LAP} Suppose hypotheses (H1)-(H4) hold. Then the limits\n\\begin{equation}\\label{LAP2}\nR^{\\mathsf B,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}R^{\\mathsf B}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$ for all $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, where ${e}(A_{\\mathsf B}):={e}(A_{B_{1}})\\cup\\sigma_{p}(A_{\\mathsf B})$, and \n${e}(A_{\\mathsf B})\\backslash{e}(A_{B_{1}})$ is a (possibly empty) discrete set in $\\mathbb{R}\\backslash {e}(A_{B_{1}})$; the maps $z\\mapsto R^{\\mathsf B,\\pm}_{z}$, where $R^{\\mathsf B,\\pm}_{z}\\equiv R^{\\mathsf B}_{z}$ whenever $z\\in\\varrho(A_{\\mathsf B})$, are continuous on $(\\mathbb{R}\\backslash {e}(A_{\\mathsf B}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$\n\\end{theorem}\n\\begin{proof}  We use \\cite[Theorem 3.1]{JMPA} (which builds on \\cite{Reng}). By (H1), \\eqref{alt-res} and (H4.2), $R^{B_{1}}_{z}$ and $R^{\\mathsf B}_{z}$ are in $\\mathscr B(L^{2}_{\\varphi}(M))$ and  $z\\mapsto R^{B_{1}}_{z}$ and $z\\mapsto R^{\\mathsf B}_{z}$ are continuous since pseudo-resolvents in $\\mathscr B(L^{2}_{\\varphi}(M))$; $A_{\\mathsf B}$ is bounded from above by (H4.1). Therefore hypothesis (H1) in \\cite{JMPA} holds true. Our hypotheses (H2) and (H3) coincides with the same ones in \\cite{JMPA}. By (H4.2), the embedding $\\mathfrak b^{*}_{2}\\hookrightarrow\\mathfrak h^{*}_{2}$ is compact. From $\\widehat\\Lambda^{\\mathsf B}_{z}\\in\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2}^{*})$ and \\eqref{alt-res}, follows that \n$R^{\\mathsf B}_{z}-R^{B_{1}}_{z}\\in{\\mathfrak S}_{\\infty}(L^{2}(M),L_{\\varphi^{2+\\gamma}}^{2}(M))$. Therefore hypothesis (H4) in \\cite{JMPA} holds and the statement is a consequence of \\cite[Theorem 3.1]{JMPA}.\n\\end{proof}\nLet us now assume that \\vskip8pt\\noindent\n(H5) the limits\n\\begin{equation}\\label{limG}\nG_{\\lambda}^{B_{1},\\pm}:=\\lim_{\\epsilon\\searrow   0}G^{B_{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$ and the maps $z\\mapsto G_{z}^{B_{1},\\pm}$, where $G_{z}^{B_{1},\\pm}\\equiv G^{B_{1}}_{z}$ whenever $z\\in\\varrho(A_{B_{1}})$, are continuous on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$; moreover, the linear operators $G_{z}^{B_{1},\\pm}$ are injective. \n\\vskip8pt\\noindent\nThen, by \\cite[Lemma 3.6]{JMPA}, one gets the following:\n\\begin{lemma}\\label{bound}\nAssume that (H1)-(H5) hold. Then, for any open and bounded $I$ s.t. $\\overline{I}\\subset \\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, one has \n\\begin{equation}\\label{supLambda}\n\\sup_{(\\lambda,\\epsilon)\\in I\\times (0,1)}{\\|}\\widehat\\Lambda^{\\mathsf B}_{\\lambda\\pm i\\epsilon}{\\|}_{\\mathfrak h_{2} ,\\mathfrak h_{2}^{*}}<+\\infty\\,.\n\\end{equation}\nMoreover, for any $\\lambda\\in \\mathbb{R}\\backslash{e}(A_{\\mathsf B})$,  the limits\n\\begin{equation}\\label{limLambda}\n\\widehat\\Lambda^{\\mathsf B,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}\\widehat\\Lambda^{\\mathsf B}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ,\\mathfrak h_{2} ^{*})$ and\n\\begin{equation}\\label{limRes}\nR^{\\mathsf B,\\pm}_{\\lambda}=R^{B_{1},\\pm}_{\\lambda}+G^{B_{1},\\pm}_{\\lambda}\\widehat\\Lambda^{\\mathsf B,\\pm}_{\\lambda}(G^{B_{1},\\mp}_{\\lambda})^{*}\\,.\n\\end{equation}\n\\end{lemma}\nBy the same reasoning as at the end of \\cite[proof of Theorem 5.1]{JMPA}, one can improve the result regarding \\eqref{limLambda}:  \n\\begin{corollary}\\label{limbb} Suppose hypotheses (H1)-(H5) hold. Then the limits \\eqref{limLambda} exist in $\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2}^{*})$.\n\\end{corollary}\nBefore stating the next results, we recall the following: \n\\begin{definition} Given two self-adjoint operators $A_{1}$ and $A_{2}$ in the Hilbert space $\\mathsf H$, we say that completeness holds for the scattering couple $(A_{1},A_{2})$ whenever  the strong limits \n$$\nW_{\\pm}(A_ {1},A_{2})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itA_{1}}e^{-itA_{2}}P_{2}^{ac}\n\\,,\\qquad\nW_{\\pm}(A_{2},A_{1})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itA_{2}}e^{-itA_{1}}P_{1}^{ac}\\,,\n$$ \nexist everywhere in $\\mathsf H$ and \n$$\\text{\\rm ran}(W_{\\pm}(A_ {1},A_{2}))=\\mathsf H_{1}^{ac }\\,,\\qquad\\text{\\rm ran}(W_{\\pm}(A_{2},A_{1}))=\\mathsf H_{2}^{ac }\\,,\n$$\n$$ W_{\\pm}(A_ {1},A_{1})^{*}=W_{\\pm}(A_{2},A_ {1})\\,,\n$$ \nwhere $P_{k}^{ac}$ denotes the orthogonal projector onto the absolutely continuous subspace $\\mathsf H_{k}^{ac}$ of $A_{k}$. Furthermore, we say the asymptotic completeness holds for the scattering couple $(A_{1},A_{2})$ whenever, beside completeness, one has \n$$\n\\mathsf H_{1}^{ac }=(\\mathsf H_{1}^{pp})^{\\perp}\\,,\\qquad \\mathsf H_{2}^{ac }=(\\mathsf H_{2}^{pp})^{\\perp}\\,,\n$$ \nwhere $\\mathsf H_{k}^{pp}$ denotes the pure point subspace of $A_{k}$; equivalently, whenever $\n\\sigma_{sc}(A_{1})=\\sigma_{sc}(A_{2})=\\varnothing $.\n\\end{definition}  \nOur next hypothesis is \\vskip8pt\\noindent\n(H6) completeness hold for the scattering  couple $(A_{B_{1}},A)$. \n\\vskip8pt\\noindent\n\\begin{theorem}\\label{AC} Suppose that (H1)-(H6) hold. Then completeness holds for the couple $(A_{\\mathsf B},A)$. If furthermore ${e}(A_{B_{1}})$ is a discrete set and $\\sigma_{sc}(A)=\\varnothing $,  then asymptotic completeness holds for the couple $(A_{\\mathsf B},A)$.\n\\end{theorem}\n\\begin{proof} By \\eqref{alt-res} and by the same proof as in Lemma \\ref{le-green}, one gets \n\\begin{equation}\\label{alt-gf}\n\\langle u,\\Delta_{\\mathsf B}v\\rangle_{L^{2}(M)}-\\langle A_{B_{1}}\nu,v\\rangle_{L^{2}(M)}=\n\\langle\\tau_{2} u,\\widehat \\rho_{\\mathsf B} v\\rangle_{\\mathfrak h_{2},\\mathfrak h_{2}^{*}}\\,,\\quad u\\in \\text{\\rm dom}(A_{B_{1}}),\\,v\\in\n\\text{\\rm dom}(A_{\\mathsf B})\\,,\n\\end{equation}\nwhere\n\\begin{equation*}\n\\widehat\\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\rightarrow\\mathfrak h_{2}^{*}\\,,\\quad\n\\widehat\\rho_{\\mathsf B}(R_{z}^{\\mathsf B}u):=\\widehat\\Lambda_{z}^{\\mathsf B}G^{B_{1}*}_{\\bar{z}}u\\,,\\qquad u\\in \\mathsf H\\,, \\quad z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})\\,.\n\\end{equation*}\nThen, by hypotheses (H1)-(H5) and by \\cite[Theorems 2.8 and 3.8]{JMPA} (compare \\eqref{alt-gf} and Lemma \\ref{bound} here with (2.19) and Lemma 3.6 there and notice that hypothesis (H6) there is included in our hypothesis (H4)) one gets the completeness for the couple $(A_{\\mathsf B},A_{B_{1}})$. By (H6) and the chain rule for the wave operators (see \\cite[Theorem 3.4, Chapter X]{Kato}), one then gets completeness for the scattering couple $(A_{\\mathsf B},A)$. To conclude the proof it remains to show that \n$\\sigma_{sc}(A_{\\mathsf B})=\\varnothing$. Let $\\mathsf H_{\\mathsf B}^{pp}$ denote the pure point subspace of $A_{\\mathsf B}$; given $u\\in (\\mathsf H_{\\mathsf B}^{pp})^{\\perp}$, let $\\mu_{u}^{\\mathsf B}$ be the corresponding spectral measure. By Theorem \\ref{LAP} and by standard arguments (see, e.g., the proof of \\cite[Thm. 6.1]{Agmon}), the support of the singular continuous component of $\\mu_{u}^{\\mathsf B}$ is contained in $e(A_{B_{1}})$. Since $e(A_{B_{1}})$ is discrete, such a support is empty and so $u$ has a null projection onto $\\mathsf H_{\\mathsf B}^{sc}$,  the singular continuous subspace of $A_{\\mathsf B}$. This gives $(\\mathsf H_{\\mathsf B}^{pp})^{\\perp}=\\mathsf H_{\\mathsf B}^{ac}$, where $\\mathsf H_{\\mathsf B}^{ac}$ denote the absolutely continuous subspace of $A_{\\mathsf B}$.\n\\end{proof}\n\n\\subsection{A representation formula for the scattering matrix}\nAccording to Theorem \\ref{AC}, under the assumptions there stated, the scattering operator \n$$\nS_{\\mathsf B}:=W_{+}(A_{\\mathsf B},A)^{*}W_{-}(A_{\\mathsf B},A)\n$$\nis a well defined unitary map.  Let  \n\\begin{equation}\\label{F0}\nF:L^{2}(M)_{ac}\\to\\int^{\\oplus}_{\\sigma_{ac}(A)}(L^{2}(M)_{ac})_{\\lambda}\\,d\\eta(\\lambda)\n\\end{equation}\nbe  a unitary map which diagonalizes the absolutely continuous component of $A$, i.e., a direct integral representation of $L^{2}(M)_{ac}$, the absolutely continuous subspace relative to $A$, with respect to the spectral measure of the absolutely continuous component of $A$  (see e.g. \\cite[Section 4.5.1]{BW}). We define the scattering matrix  $${\\mathcal S}^{\\mathsf B}_{\\lambda}:(L^{2}(M)_{ac})_{\\lambda}\\to (L^{2}(M)_{ac})_{\\lambda}$$ by the relation (see e.g. \\cite[Section 9.6.2]{BW})\n$$\nFS_{\\mathsf B}F^{*}u_{\\lambda}={\\mathcal S}^{\\mathsf B}_{\\lambda}u_{\\lambda}\n\\,.\n$$\nNow, following the same scheme as in \\cite{JMPA}, which uses the Birman-Kato invariance principle and the Birman-Yafaev general scheme in stationary scattering theory, we provide an explicit relation between  ${\\mathcal S}^{\\mathsf B}_{\\lambda}$ and $\\Lambda^{\\!\\mathsf B,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\!\\mathsf B}_{\\lambda+i\\epsilon}$. \\par \nGiven $\\mu\\in \\varrho(A)\\cap\\varrho(A_{\\mathsf B})$, we consider the scattering couple $(R^{\\mathsf B}_{\\mu}, R_{\\mu})$ and the strong limits \n$$\nW_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itR^{\\mathsf B}_{\\mu}}e^{-itR_{\\mu}}P^{\\mu}_{ac}\\,,\n$$ \nwhere $P^{\\mu}_{ac}$ is the orthogonal projector onto the absolutely continuous subspace of $R_{\\mu}$; we prove below that such limits exist everywhere in $L^{2}(M)$. Let $S^{\\mu}_{\\Lambda}$ the corresponding scattering operator  \n$$\nS_{\\mathsf B}^{\\mu}:=W_{+}(R^{\\mathsf B}_{\\mu},R_{\\mu})^{*}W_{-}(R^{\\mathsf B}_{\\mu},R_{\\mu})\\,.\n$$\nUsing the unitary operator $F_{\\mu}$ which diagonalizes the absolutely continuous component of $R_{\\mu}$, i.e. $(F_{\\mu}u)_{\\lambda}:=\\frac1\\lambda(Fu)_{\\mu-\\frac1\\lambda}$, $\\lambda\\not=0$ such that $\\mu-\\frac1\\lambda\\in \\sigma_{ac}(A)$, one defines the scattering  matrix  $${\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}:(L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}\\to (L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}$$  corresponding to the scattering operator $S^{\\mu}_{\\mathsf B}$ by the relation\n$$\nF_{\\mu}S^{\\mu}_{\\mathsf B}F_{\\mu}^{*}u^{\\mu}_{\\lambda}={\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}u^{\\mu}_{\\lambda}\n\\,.\n$$ \nWe introduce a further hypothesis (H7), which we split in four separate points:\\vskip5pt\\par\\noindent\n(H7.1) $A$ is bounded from above and satisfies a Limiting Absorption Principle: there exists a (eventually empty) closed set ${e}(A)\\subset\\mathbb{R}$ of zero Lebesgue measure such that for all $\\lambda\\in \\mathbb{R}\\backslash{e}(A)$ the limits\n\\begin{equation}\\label{LAP0}\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}R_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$; \n\\vskip5pt\\par\\noindent\n(H7.2) $G^{1}_{z}\\in\\mathscr B(\\mathfrak h_{1}^{*},L^{2}_{\\varphi}(M))$ for any $z\\in\\varrho(A)$ and the limits\n\\begin{equation}\\label{limG1}\nG_{\\lambda}^{{1},\\pm}:=\\lim_{\\epsilon\\searrow   0}G^{{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{1} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A)$;\n\\vskip5pt\\par\\noindent\n(H7.3) the limits\n\\begin{equation}\\label{limLB1}\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}\\Lambda^{\\!B_{1},\\pm}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{1},\\mathfrak h_{1}^{*})$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$;\n\\vskip5pt\\par\\noindent\n(H7.4) the limits\n\\begin{equation}\\label{limtG1}\n\\tau_{2} G^{1,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}\\tau_{2} G^{1}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak b^{*}_{1},\\mathfrak b_{2})$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$.\n\\begin{remark}\\label{rem3.6} By $\\tau_{2}G^{1}_{z}=\\tau_{2} (\\tau_{1}R_{\\bar z})^{*}=(\\tau_{1} (\\tau_{2}R_{ z})^{*})^{*}=(\\tau_{1}G^{2}_{\\bar z})^{*}$, hypothesis (H7.4) entails the existence in $\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{1}^{*})$, for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$,  of the limits  \n\\mathfrak b\n\\tau_{1} G^{2,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow   0}\\tau_{1} G^{2}_{\\lambda\\pm i\\epsilon}\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark} Whenever one strengthens hypotheses (H7.2) as in (H5), then, by the same kind of proof that leads to the existence of the limit \\eqref{limLambda} (see \\cite[Lemma 3.6]{JMPA}), one gets the existence of the limits requested in hypotheses (H7.3).\n\\end{remark}\n\\begin{lemma}\\label{rmH7} Suppose that (H1)-(H5) and (H7) hold. Then\n\\begin{equation}\\label{rmH7-1}\nR_{\\lambda}^{B_{1},\\pm}=R^{\\pm}_{\\lambda}+G_{\\lambda}^{{1},\\pm}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}(G_{\\lambda}^{{1},\\mp})^{*}\\,;\n\\end{equation}\n\\begin{equation}\\label{Gz2}\nG^{2}_{z}\\in \\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi}(M))\\,,\\qquad z\\in\\varrho(A_{B_{1}})\\cap\\varrho(A)\\,,\n\\end{equation} \nthe limits\n\\begin{equation}\\label{limG2}\nG_{\\lambda}^{{2},\\pm}:=\\lim_{\\epsilon\\searrow   0}G^{{2}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$ and\n\\begin{equation}\\label{limGB1}\nG_{\\lambda}^{B_{1},\\pm}=G_{\\lambda}^{2,\\pm}+G_{\\lambda}^{1,\\pm}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G_{\\lambda}^{2,\\pm}\\,;\n\\end{equation}\nthe limits \n$$\n\\Lambda_{\\lambda}^{\\!\\mathsf B,\\pm}:=\n\\lim_{\\epsilon\\searrow   0}\\Lambda_{\\lambda\\pm i\\epsilon}^{\\!\\mathsf B}\n$$\nexist in $\\mathscr B(\\mathfrak h_{1}\\oplus \\mathfrak b_{2},\\mathfrak h_{1}^{*}\\oplus \\mathfrak b_{2}^{*})$ and\n\\begin{align}\n\\Lambda_{\\lambda}^{\\!\\mathsf B,\\pm}=&\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}+\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\\\\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda_{\\lambda}^{\\!B_{1},\\pm}&\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\label{LBpm}\\\\\n=&\\left(1+\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&0\\\\0&\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\begin{bmatrix}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}&\n\\tau_{1}G^{2,\\pm}_{\\lambda}\\\\\n\\tau_{2}G^{1,\\pm}_{\\lambda}&0\n\\end{bmatrix}\\,\\right)\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&0\\\\0&\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\,.\\label{LBpm2}\n\\end{align}\n\\end{lemma}\n\\begin{proof} The relation \\eqref{rmH7-1} is an immediate consequence of \\eqref{res1.1} and (H7.1)-(H7.3). By \\eqref{GB1}, \n$$\nG_{z}^{2}=G^{B_{1}}_{z}-G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\n$$\nand \\eqref{Gz2} follows from (H4.2) and (H7.2). Then, Remark \\ref{rem3.6}, (H.5) and (H7.3) entail \n\\eqref{limG2} and \\eqref{limGB1}. Finally, \\eqref{LBpm} and \\eqref{LBpm2} are consequence of  \\eqref{LB-new}, \\eqref{LB-new2}, Corollary \\ref{limbb}, (H7.3), Remark \\ref{rem3.6} and (H7.4). \n\\end{proof}\nBefore stating the next results, let us notice the relations \n\\begin{equation}\\label{RR}\n\\left(-R_{\\mu}+z\\right)^{-1}=\\frac1z\\,\\left(1+\\frac1{z}\\,R_{\\mu-\\frac1z}\\right)\\,,\n\\quad \\left(-R_{\\mu}^{\\mathsf B}+z\\right)^{-1}=\\frac1z\\,\\left(1+\\frac1{z}\\,R^{\\mathsf B}_{\\mu-\\frac1z}\\right)\\,,\n\\end{equation}\nTherefore, by (H7.1) and Theorem \\ref{LAP}, the limits \n\\begin{equation}\\label{RRlim1}\n\\left(-R_{\\mu}+(\\lambda\\pm i0)\\right)^{-1}:=\\lim_{\\epsilon\\searrow   0}\\left(-R_{\\mu}+(\\lambda\\pm i\\epsilon)\\right)^{-1}\\,,\\quad \\lambda\\not=0\\,,\\ \\mu-\\frac1\\lambda\\in\\mathbb{R}\\backslash{e}(A)\\,,\n\\end{equation} \n\\begin{equation}\\label{RRlim2}\n\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i0)\\right)^{-1}:=\n\\lim_{\\epsilon\\searrow   0}\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i\\epsilon)\\right)^{-1}\\,,\n\\quad \\lambda\\not=0\\,,\\ \\mu-\\frac1\\lambda\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})\\,,\n\\end{equation}\nexist in $\\mathscr B(L^{2}_{\\varphi}(M),L^{2}_{\\varphi^{-1}}(M))$.\n\\begin{theorem}\\label{BK}\nSuppose that hypotheses (H1)-(H7) hold. Then the strong limits \n\\begin{equation}\\label{WR}\nW_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itR^{\\mathsf B}_{\\mu}}e^{-itR_{\\mu}}P^{\\mu}_{ac}\n\\end{equation}\nexist everywhere in $L^{2}(M)$.\nMoreover, for any  $\\lambda\\not=0$ such that $\\mu-\\frac1\\lambda\\in \\sigma_{ac}(A)\\cap(\\mathbb{R}\\backslash{e}(A_{\\mathsf B}))$, one has\n\\begin{equation}\\label{S1}\n{\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}=1-2\\pi i\\,\\mathcal L^{\\mu}_{\\lambda}\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\big(1+G^{*}_{\\mu}\\big(-R_{\\mu}^{\\mathsf B}+(\\lambda+ i0)\\big)^{-1}G_{\\mu}\\Lambda^{\\!\\mathsf B}_{\\mu}\\big)\n(\\mathcal L^{\\mu}_{\\lambda})^{*}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{SR}\n\\mathcal L^{\\mu}_\\lambda: \\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\to(L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}\\,,\\quad \n\\mathcal L^{\\mu}_{\\lambda}(\\phi_{1}\\oplus\\phi_{2}):=\\frac1\\lambda(FG_{\\mu}(\\phi_{1}\\oplus\\phi_{2}))_{\\mu-\\frac1\\lambda}\\,.\n\\end{equation}\n\\end{theorem} \n\\begin{proof}\nBy \\eqref{resolvent}, one has $R_{\\mu}^{\\mathsf B}-R_{\\mu}=G_{\\mu}\\Lambda^{\\! B}_{\\mu}G^{*}_{\\mu}$ and we can use \\cite[Theorem 4', page 178]{Y} (notice that the maps there denoted by $G$ and $V$ corresponds to our $G_{\\mu}^{*}$ and $\\Lambda^{\\! B}_{\\mu}$ respectively). Let us check that the hypotheses there required are satisfied. Since $G^{*}_{\\mu}\\in \\mathscr B(L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2})$, the operator $G_{\\mu}$ is $|R_{\\mu}|^{1\/2}$-bounded. By (H7.2) and \\eqref{Gz2}, one has $G_{z}\\in\\mathscr B(\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}, L^{2}_{\\varphi}(M))$ for any $z\\in\\varrho(A_{B_{1}})\\cap\\varrho(A)\\supset[\\lambda_{1},+\\infty)\\ni\\mu$. Therefore, by \\eqref{RRlim1}, \\eqref{RRlim2}, (H7.1), Theorem \\ref{LAP} and (H4), the limits \n$$\n\\lim_{\\epsilon\\searrow   0}\\, G^{*}_{\\mu}(-R _{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}\\,,\n$$\n$$\n\\lim_{\\epsilon\\searrow   0}\\, G^{*}_{\\mu}(-R^{\\mathsf B}_{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}\\,,\n$$\n$$\n\\lim_{\\epsilon\\searrow   0}\\, G^{*}_{\\mu}(-R^{\\mathsf B}_{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}G_{\\mu}\n$$\nexist. Therefore, to get the thesis we need to check the validity of the remaining hypothesis in \n\\cite[Theorem 4', page 178]{Y}: $G^{*}_{\\mu}$ is weakly-$R _{\\mu}$ smooth, i.e., by \\cite[Lemma 2, page 154]{Y}, \n\\begin{equation}\\label{in1.1}\n\\sup_{0<\\epsilon<1}\\epsilon\\,{\\|}G^{*}_{\\mu} (-R _{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}^{2}\\le c_{\\lambda}<+\\infty\\,,\\quad\\text{a.e. $\\lambda$}\\,.\n\\end{equation}\nBy \\eqref{RR}, this is consequence of\n\\begin{equation}\\label{in2.2}\n\\sup_{0<\\epsilon<1}\\epsilon\\,{\\|}G^{*}_{\\mu}R_{\\mu-\\frac1\\lambda\\pm i\\epsilon}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}^{2}\\le C_{\\lambda}<+\\infty\\,,\\quad\\text{a.e. $\\lambda$}\\,.\n\\end{equation}\nBy \\cite[(3.16)]{JMPA},\n\\begin{align*}\n&\\epsilon\\,{\\|}G_{\\lambda\\pm i\\epsilon}{\\|}_{\\mathfrak h^{*}_{1}\\oplus\\mathfrak h^{*}_{2},L^{2}(M)}^{2}\\\\\n\\le& \n\\frac12\\,(|\\mu-\\lambda|+\\epsilon)\\, {\\|}G_{\\mu}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi}^{2}(M)} \\left(\\|G_{\\lambda- i\\epsilon}\\|_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi^{-1}}^{2}(M)} +\n\\|G_{\\lambda+ i\\epsilon}\\|_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi^{-1}}^{2}(M)} \\right)\\,.\n\\end{align*}\nThen, \\eqref{in2.2} follows from \\eqref{limG1}, \\eqref{limG2} and the equality\n\\begin{align*}\n&{\\|}G^{*}_{\\mu} R _{z}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\n={\\|}\\tau R_{\\mu}R _{z}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}= \n{\\|}\\tau R _{z}R _{\\mu}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\\\\n=&\n{\\|}R _{\\mu}(\\tau R _{z})^{*}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L^{2}(M)}\\le\n{\\|}R _{\\mu}{\\|}_{L^{2}(M),L^{2}(M)} {\\|}G_{\\bar z}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L^{2}(M)}\\,.\n\\end{align*}\nThus, by \\cite[Theorem 4', page 178]{Y}, \nthe limits \\eqref{WR} exist everywhere in $L^{2}(M)$ and the corresponding scattering matrix is given by \\eqref{S1}, where $\\mathcal L^{\\mu}_{\\lambda}\\phi:=(F^{\\mu}G_{\\mu}\\phi)_{\\lambda}=\\frac1\\lambda(FG_{\\mu}\\phi)_{\\mu-\\frac1\\lambda}$. \n\\end{proof}\n\\begin{theorem}\\label{S-matrix} Suppose that hypotheses (H1)-(H7) hold. Then the scattering matrix of the couple $(A_{\\mathsf B},A)$ has the representation\n\\begin{equation}\\label{S-M}\n{\\mathcal S}^{\\mathsf B}_{\\lambda}=1-2\\pi i\\mathcal L_{\\lambda}\\Lambda^{\\!\\mathsf B,+}_{\\lambda}\\mathcal L_{\\lambda}^{*}\\,,\\quad \\lambda\\in\\sigma_{ac}(A)\\cap(\\mathbb{R}\\backslash{e}(A_{\\mathsf B}))\\,,\n\\end{equation}\nwhere  $\\mathcal L_\\lambda: \\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\to(L^{2}(M)_{ac})_{\\lambda}$ is the $\\mu$-independent linear operator defined by  \n$$\n\\mathcal L_{\\lambda}(\\phi_{1}\\oplus\\phi_{2}):=(\\mu-\\lambda)(FG_{\\mu}(\\phi_{1}\\oplus\\phi_{2}))_{\\lambda}\n$$\nand $\\Lambda^{\\!\\mathsf B,+}_{\\lambda}$ is given in \\eqref{LBpm}.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref {AC}, Theorem \\ref{BK} and by Birman-Kato invariance principle (see e.g. \\cite[Section II.3.3]{BW}), one has\n$$\nW_{\\pm}(A_{\\mathsf B},A)=W_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n$$\nand so\n$$\nS_{\\mathsf B}=S_{\\mathsf B}^{\\mu}\\,.\n$$\nThus, since $(F^{\\mu}u)_{\\lambda}=\\frac1\\lambda(Fu)_{\\mu-\\frac1\\lambda}$,  one obtains (see also \\cite[Equation (14), Section 6, Chapter 2]{Y})\n\\begin{equation}\\label{SS}\n{\\mathcal S}^{\\mathsf B}_{\\lambda}={\\mathcal S}^{\\mathsf B,\\mu}_{(-\\lambda+\\mu)^{-1}}\\,.\n\\end{equation}\nBy \\cite[Lemma 4.2]{JMPA}, for any $z\\not=0$ such that $\\mu-\\frac1z\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A)$, there holds\n$$\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\left(1+G^{*}_{\\mu}\\left(-R_{\\mu}^{\\mathsf B}+z\\right)^{-1}G_{\\mu}\\Lambda^{\\! B}_{\\mu}\\right)=\\Lambda^{\\!\\mathsf B}_{\\mu-\\frac1z}\\,.\n$$\nHence, whenever $z=\\lambda\\pm i\\epsilon$ and $\\mu-\\frac1{\\lambda}\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, one gets, as $\\epsilon\\downarrow 0$, \n$$\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\big(1+G^{*}_{\\mu}\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i0)\\right)^{-1}G_{\\mu}\\Lambda^{\\!\\mathsf B}_{\\mu}\\big)=\\Lambda^{\\!\\mathsf B,\\pm}_{\\mu-\\frac1{\\lambda}}\\,.\n$$\nThe proof is then concluded by Theorem \\ref{BK}, by \\eqref{SS} and by setting  \n$\\mathcal L_{\\lambda}:=\\mathcal L^{\\mu}_{{(-\\lambda+\\mu)^{-1}}}$. The operator $\\mathcal L_{\\lambda}$ is $\\mu$-independent by invariance principle (see the proof in \\cite[Corollary 4.3]{JMPA} for an explicit check).\n\\end{proof}\n\\begin{remark}\nBy \\eqref{LBpm}, \n$$\n\\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}=\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{z}&0\\\\0&0\\end{bmatrix}+\\widetilde \\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}\\,,\n$$\nwhere\n$$\n\\widetilde \\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}:=\n\\begin{bmatrix}\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\\\\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda_{\\lambda}^{\\!B_{1},\\pm}&\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\,.\n$$\nTherefore, defining \n$$\n\\mathcal L^{1}_{\\lambda}\\phi_{1}:=\\mathcal L_{\\lambda}(\\phi_{1}\\oplus 0)\\,,\n$$ \none gets\n$$\n{\\mathcal S}^{\\mathsf B}_{\\lambda}={\\mathcal S}^{B_{1}}_{\\lambda}-2\\pi i\\mathcal L_{\\lambda}\\widetilde\\Lambda^{\\!\\mathsf B,+}_{\\lambda}\\mathcal L_{\\lambda}^{*}\\,,\n$$\nwhere\n\\begin{equation}\\label{SB1}\n{\\mathcal S}^{B_{1}}_{\\lambda}=1-2\\pi i\\mathcal L^{1}_{\\lambda}\\widetilde\\Lambda^{\\!B_{1},+}_{\\lambda}(\\mathcal L^{1}_{\\lambda})^{*}\n\\end{equation}\nis the scattering matrix relative to the couple $(A_{B_{1}},A)$.  Moreover, in the case $B_{1}=0$, \ndefining \n$$\n\\mathcal L^{2}_{\\lambda}\\phi_{2}:=\\mathcal L_{\\lambda}(0\\oplus\\phi_{2})\\,,\n$$ \none gets the following representation formula for the scattering couple $(A_{B_{0},B_{2}},A)$ (compare with \\cite[Corollary 4.3]{JMPA}):\n$$\n{\\mathcal S}^{B_{0},B_{2}}_{\\lambda}=1-2\\pi i\\mathcal L^{2}_{\\lambda}\\Lambda^{\\!B_{0},B_{2},+}_{\\lambda}(\\mathcal L^{2}_{\\lambda})^{*}\\,.\n$$\nLet us further notice that, whenever $A$ is the free Laplacian in $L^{2}(\\mathbb{R}^{3})$ and $B_{1}$ corresponds to a perturbation by a regular potential as in Section 5 below, then \\eqref{SB1} gives the usual formula for the scattering matrix for a short-range potential (see, e.g., \\cite[Section 8]{Y-LNM}).\n\\end{remark}\n\n\\section{Kato-Rellich perturbations and their layers potentials}\n\n\\subsection{\\label{Sec_V}Potential perturbations}\n\nIn this section we suppose that the real-valued potential $\\mathsf v$ is of Kato-Rellich type, i.e.,  $\\mathsf v\\in L^{2}(\\mathbb{R}^{3})+L^{\\infty}(\\mathbb{R}^{3})$, equivalently,\n\\begin{equation}\n\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty} \\,,\\qquad \\mathsf v_{2}\\in L^{2}(  \\mathbb{R}^{3}) \\,,\\qquad \\mathsf v_{\\infty}\\in L^{\\infty}\n(\\mathbb{R}^{3})  \\,. \\label{K-R}%\n\\end{equation}\nWe use the same simbol $\\mathsf v$ to denote both the potential function and the corresponding multiplication operator $u\\mapsto \\mathsf v u$. \n\\par\nGiven $\\Omega\\subset\\mathbb{R}^{3}$, open and bounded with a Lipschitz boundary $\\Gamma$, we define $H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma)\\hookleftarrow H^{s}(\\mathbb{R}^{3})$ by  \n$$\nH^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H^{s}(\\Omega_{\\rm ex})\\,,\\qquad s\\ge 0\\,.\n$$\nWe refer to \\cite[Chapter 3]{McL} for the definition of the Sobolev spaces $H^{s}(\\mathbb{R}^{3})$, $H^{s}(\\Omega)$ and $H^{s}(\\Gamma)$. One has \n$$\nH^{s}(\\mathbb{R}^{3}\\backslash\\Gamma)=H^{s}(\\mathbb{R}^{3})\\,,\\qquad 0\\le s<1\/2\\,.\n$$\nSince (see \\cite[Theorems 3.29 and 3.30]{McL}),\n$$\nH^{s}(  \\mathcal O)   ^{\\ast}\n=H_{\\overline{\\mathcal O}}^{-s}(  \\mathbb{R}^{3})  \\,,\\qquad\ns\\in\\mathbb{R}\\,,\n$$\n$H_{\\overline{\\mathcal O}}^{-s}(  \\mathbb{R}^{3})$ denoting the set of distributions $H^{-s}(  \\mathbb{R}^{3})$ with support in ${\\overline{\\mathcal O}}$,\none has\n\\begin{equation}\nH^{s}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{\\ast}=H^{s}(  \\Omega)^{\\ast}\n\\oplus H^{s}(  \\mathbb{R}^{3}\\backslash\\overline\\Omega)^{\\ast}=\nH^{-s}_{\\overline\\Omega}(\\mathbb{R}^{3})\\oplus H^{-s}_{\\Omega^{c}}(\\mathbb{R}^{3})\\hookrightarrow H^{-s}(\\mathbb{R}^{3})\n\\,.\n\\label{dual}%\n\\end{equation}\nLet us notice that\n\\begin{equation}\\label{B}\n\\mathscr B(H^{s}(  \\mathbb{R}^{3}\\backslash\\Gamma),H^{t}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{\\ast})\n\\hookrightarrow\n\\mathscr B(H^{s}(  \\mathbb{R}^{3}),H^{-t}(  \\mathbb{R}^{3}))\\,,\\qquad s,t\\ge 0\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{B*}\n\\mathscr B(H^{-s}( \\mathbb{R}^{3}),H^{t}(  \\mathbb{R}^{3}))\\hookrightarrow\\mathscr B(H^{s}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{t}(  \\mathbb{R}^{3}\\backslash\\Gamma))\n\\,,\\qquad s,t\\ge 0\\,.\n\\end{equation}\n\\begin{lemma}\\label{v}\n\\begin{equation}\\label{v-sob}\n\\mathsf v\\in{\\mathscr B}(  H^{1+s}(  \\mathbb{R}^{3}\\backslash\\Gamma) ,H^{1-s}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,,\\qquad -1\\le s\\le 1  \\,. \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nGiven $u= u_{\\rm in}\\oplus u_{\\rm ex}\\in  H^{2}(  \\mathbb{R}^{3}\\backslash\\Gamma)    \n$ one has%\n\\[\n\\|\\mathsf v_{\\infty}u\\| _{L^{2}(\\mathbb{R}^{3})}\\leq\\|\\mathsf v\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\| u\\| _{L^{2}(\\mathbb{R}^{3})}\\leq\\|\\mathsf v\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\| u\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)  }\\,.\n\\]\nand\n\\begin{align*}\n\\|\\mathsf v_{2}u\\|_{L^{2}(\\mathbb{R}^{3})}=&\\|\\mathsf v_{2}\\|_{L^{2}(\\Omega)}\\|u_{\\rm in}\\|_{L^{\\infty}(\\Omega)}+\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\n\\|u_{\\rm ex}\\|_{L^{\\infty}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\\\\\n\\lesssim &\n\\|\\mathsf v_{2}\\|_{L^{2}(\\Omega)}\\|u_{\\rm in}\\|_{H^{2}(\\Omega)}+\n\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\n\\|u_{\\rm ex}\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\\\\\n\\lesssim &\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3})}\n\\|u\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\,.\n\\end{align*}\nHence $\\mathsf v\\in{\\mathscr B}(  H^{2}(  \\mathbb{R}^{3}\\backslash\\Gamma) ,L^{2}(  \\mathbb{R}^{3}))$. Then, for any $u,v\\in H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)$, one has\n\\begin{align*}\n&\\big|\\langle \\mathsf v u,v\\rangle_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big|\n=\\big|\\langle \\mathsf v u,v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\\\\\n=&\n\\big|\\langle u,\\mathsf v v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\n\\le \\|\\mathsf v\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma),L^{2}(\\mathbb{R}^{3})}\\|u\\|_{L^{2}(\\mathbb{R}^{3})}\n\\|v\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n\\end{align*}\nand so $u\\mapsto\\mathsf v u$ extends to a map in $\\mathscr B(L^{2}(  \\mathbb{R}^{3}),H^{2}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*} )$. The proof is then concluded by interpolation.\n\\end{proof}\nIn the following, $R_{z}$ denotes the resolvent of the free Laplacian, i.e.,\n\\begin{equation}\\label{free}\nR_{z}:=\\left(  -\\Delta+z\\right)  ^{-1}\\in\\mathscr B(H^{s}(\\mathbb{R}^{3}),H^{s+2}(\\mathbb{R}^{3}))\\,,\\quad s\\in\\mathbb{R}\\,.\n\\end{equation}\nSince $\\mathsf v$ is of Rellich-Kato type, one has (see, e.g., \\cite[Section 3, $\\S$5, Chap. V]{Kato}): \n\\begin{theorem}\\label{KR}\n$\\Delta+\\mathsf v:H^{2}(  \\mathbb{R}^{3})\\subset  L^{2}(  \\mathbb{R}^{3})\\to L^{2}(  \\mathbb{R}^{3}) $ is\nself-adjoint and semi-bounded from  above. Moreover, for $z\\in\\mathbb{C}$ sufficiently far away from $[0,+\\infty)$, $\\|\\mathsf v R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}<1$, \nand\n\\begin{equation}\nR_{z}^{\\mathsf v}:=(-(\\Delta+\\mathsf v)+z)^{-1}=R_{z}+R_{z}(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v R_{z}\n\\,,\\label{Rvz}\n\\end{equation}\n\\begin{equation}\n(  1-\\mathsf v R_{z})  ^{-1}=\\sum_{k=0}^{+\\infty}\\left(  \\mathsf v R_{z}\\right)  ^{k}\n\\in{\\mathscr B}(  L^{2}(\\mathbb{R}^{3}))\\,.\n\\label{L-v-est}%\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\\label{tpt} Let us notice that the Kato-Rellich theorem could be obtained by Corollary \\ref{cor1} by taking $\\tau_{1}u:=u$ and $B_{1}=\\mathsf v$. Hence, \\eqref{Rvz} holds for any $z$ in $\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]$ and $(  1+\\mathsf v R_{z})  ^{-1}\\in {\\mathscr B}(  L^{2}(\\mathbb{R}^{3}))$ there.\n\\end{remark}\n\\begin{remark}\\label{sa} \nBy \\eqref{free}, \\eqref{Rvz}, \\eqref{L-v-est}, \\eqref{v-sob} and \\eqref{B}, one has $R^{\\mathsf v}_{\\bar z}\\in\n\\mathscr B(L^{2}(  \\mathbb{R}^{3}),H^{2}(  \\mathbb{R}^{3}))$ and hence $(R_{\\bar z}^{\\mathsf v})^{*}\\in \\mathscr B(H^{-2}(  \\mathbb{R}^{3}),L^{2}(  \\mathbb{R}^{3}))$.  Since $(\\Delta+\\mathsf v)$ is self-adjoint in $L^{2}(\\mathbb{R}^{3})$, $(R_{\\bar z}^{\\mathsf v})^{*}|L^{2}(\\mathbb{R}^{3})=\nR^{\\mathsf v}_{z}$. Therefore, $R^{\\mathsf v}_{z}:L^{2}(\\mathbb{R}^{3})\\subset H^{-2}(\\mathbb{R}^{3})\\to L^{2}(\\mathbb{R}^{3})$ extends to an operator in $\\mathscr B(H^{-2}(  \\mathbb{R}^{3}),L^{2}(  \\mathbb{R}^{3}))$ which, by abuse of notation, we still denote by $R_{ z}^{\\mathsf v}$ and which coincides with $(R_{\\bar z}^{\\mathsf v})^{*}$. Then, by interpolation, one gets\n\\begin{equation}\\label{Rvz-int}\nR_{z}^{\\mathsf v}\\in \\mathscr B(H^{s-1}(  \\mathbb{R}^{3}),H^{s+1}(  \\mathbb{R}^{3}))\\,,\\qquad -1\\le s\\le 1\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark} By \\eqref{Rvz}, \n\\begin{equation}\\label{btr}\n(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v=(-\\Delta+z)R^{\\mathsf v}_{z}(-\\Delta+z)-(-\\Delta+z)\\,.\n\\end{equation}\nHence, by \\eqref{Rvz-int},  $(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3}))$ extends to  a map \n\\begin{equation}\\label{L-int}\n\\Lambda^{\\!\\mathsf v}_{z}\\in \\mathscr B(H^{s+1}(  \\mathbb{R}^{3}),H^{s-1}(  \\mathbb{R}^{3}))\\,,\n\\qquad -1\\le s\\le 1 \n\\end{equation}\nWith such a notation, $R_{z}^{\\mathsf v}$ in \\eqref{Rvz-int} has the representation\n\\begin{equation}\\label{RF-int}\nR_{z}^{\\mathsf v}=R_{z}+R_{z}\\Lambda^{\\!\\mathsf v}_{z}R_{z}\\,,\\qquad \n\\Lambda^{\\!\\mathsf v}_{z}|H^{2}(\\mathbb{R}^{3})=(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark}\\label{Lt} Since $\\|R_{z}\\mathsf v\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}=\\|(R_{z}\\mathsf v)^{*}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}=\\|\\mathsf v R_{\\bar z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\n<1$ whenever $z\\in\\mathbb{C}$ is sufficiently far away from $[0,+\\infty)$, one has\n\\begin{equation}\\label{Lt2}\n(  1-R_{z}\\mathsf v)  ^{-1}=\\sum_{k=0}^{+\\infty}\\left(  R_{z}\\mathsf v\\right)  ^{k}\n\\in{\\mathscr B}(  L^{2}(\\mathbb{R}^{3}))\n\\end{equation}\nand\n\\begin{equation}\\label{Lt-2}\n\\mathsf v(  1- R_{z}\\mathsf v)  ^{-1}\\in\\mathscr B(L^{2}(\\mathbb{R}^{3}),H^{-2}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nThen, \n$$\n\\big((  1- R_{z}\\mathsf v)  ^{-1}\\mathsf v\\big)^{*}=\n\\big(\\mathsf v(  1- R_{z}\\mathsf v)^{-1}\\big)^{*}=\\mathsf v((  1- R_{z}\\mathsf v)^{*})^{-1}=\\mathsf v(  1- R_{\\bar z}\\mathsf v)  ^{-1\n$$\nand so \n$$\n\\mathscr B(H^{-2}(  \\mathbb{R}^{3}),L^{2}(  \\mathbb{R}^{3}))\\ni(R^{\\mathsf v}_{z})^{*}=R_{\\bar z}+R_{\\bar z}\\mathsf v(  1- R_{\\bar z}\\mathsf v)^{-1}R_{\\bar z}=\nR^{\\mathsf v}_{\\bar z}=R_{\\bar z}+R_{\\bar z}\\Lambda_{\\bar z}^{\\!\\mathsf v}R_{\\bar z}\\,.\n$$\nTherefore \n\\begin{equation}\\label{LtL2}\n\\Lambda_{z}^{\\!\\mathsf v}|L^{2}(\\mathbb{R}^{3})=\\mathsf v(  1- R_{z}\\mathsf v)  ^{-1}\\,.\n\\end{equation}\n\\end{remark}\n\\begin{lemma}\n\\begin{equation}\\label{Lvz-int}\n\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}(  H^{1+s}(\\mathbb{R}^{3}\\backslash\\Gamma),\nH^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\n\\,, \\qquad -1\\le s\\le 1\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By Lemma \\ref{v} and by \\eqref{L-v-est}, one has $\\Lambda_{z}^{\\!\\mathsf v}=(  1+\\mathsf v R_{z})  ^{-1}\\mathsf v\\in{\\mathscr B}(  H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma),L^{2}(\\mathbb{R}^{3}))$. By Lemma \\ref{v}, \\eqref{Lt2} and \\eqref{LtL2},\n$\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}( L^{2}(\\mathbb{R}^{3}), H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$.\nThe proof is then concluded by interpolation.\n\\end{proof}\nBy $H^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\hookrightarrow H^{s-1}(\\mathbb{R}^{3})$ and \\eqref{free} one has\n\\begin{corollary}\n\\begin{equation}\\label{RLvz}\nR_{z}\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}(  H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma),\nH^{s}(\\mathbb{R}^{3}))\n\\,, \\qquad 0\\le s\\le 2\\,.\n\\end{equation}\n\n\\end{corollary}\nIn later proofs we will need the estimate provided in the following:\n\\begin{lemma}\nThere exist $c_{1}>0$, $c_{2}>0$ such that, for any $u\\equiv u_{\\rm in}\\oplus u_{\\rm ex}\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$ and for any $\\varepsilon>0$,\nthere holds\n\\begin{equation}\n\\big|\\langle \\mathsf v u,u\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big| \\leq\nc_{1}\\epsilon\\left(\\|\\nabla u_{\\rm in}\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|\\nabla u_{\\rm ex}\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\\right)+c_{2}(1+\\epsilon^{-3})\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,. \\label{V_est}%\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By $H^{1}(\\Omega_{\\rm in\/\\rm ex})  \\hookrightarrow H^{3\/4}(\\Omega_{\\rm in\/\\rm ex}) \\hookrightarrow L^{4}(\\Omega_{\\rm in\/\\rm ex})$, by the Gagliardo-Niremberg \ninequalities (see \\cite{BM} for the interior case and \\cite{CM} for the exterior one)\n$$\n\\|u_{\\rm in}\\|_{L^{4}(\\Omega_{\\rm in})}\\lesssim\\|u_{\\rm in}\\|_{H^{3\/4}(\\Omega_{\\rm in})}\\lesssim\\|u_{\\rm in}\\|^{3\/4}_{H^{1}(\\Omega_{\\rm in})}\\|u_{\\rm in}\\|^{1\/4}_{L^{2}(\\Omega_{\\rm in})}\\,,\n$$ \n$$\n\\|u_{\\rm ex}\\|_{L^{4}(\\Omega_{\\rm ex})}\\lesssim\\|\\nabla u_{\\rm ex}\\|^{3\/4}_{L^{2}(\\Omega_{\\rm ex})}\\|u_{\\rm ex}\\|^{1\/4}_{L^{2}(\\Omega_{\\rm ex})}\n$$ \nand by Young's inequality\n$$\nxy\\le \\frac1{\\alpha}\\left({\\epsilon}\\ x^{\\alpha}+{(\\alpha-1)}\\,{\\epsilon^{-1\/(\\alpha-1)}}\n\\ y^{\\frac{\\alpha-1}{\\alpha}}\\right)\\,,\\qquad x,y,\\epsilon>0,\\, \\alpha>1\\,,\n$$\none gets\n$$\n\\|u\\|_{L^{4}(\\mathbb{R}^{3})}^{2}\\lesssim\n\\epsilon\\left(\\|\\nabla u_{\\rm in}\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|u\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|\\nabla u_{\\rm ex}\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\\right)+\\frac{1}{3}\\,\\epsilon^{-3}\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n$$\nThe proof is then concluded by\n$$\n\\big|\\langle \\mathsf v u,u\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big| \\leq\n\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3})}\\|u\\|^{2}_{L^{4}(\\mathbb{R}^{3})}+\n\\|\\mathsf v_{\\infty}\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n$$\n\\end{proof}\n\\begin{lemma}\\label{vH-1} For any $z\\in\\mathbb{C}$ sufficiently far away from $[0,+\\infty)$, one has \n$$\\|\\mathsf v R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\!,H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}<1$$ and so\n\\begin{equation}\n(  1-\\mathsf v R_{z})  ^{-1}=\\sum_{k=0}^{+\\infty}\\left(  \\mathsf v R_{z}\\right)  ^{k}\n\\in{\\mathscr B}(  H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By \\eqref{V_est} and by the polarization identity, for any  $u\\in H^{2}(\\mathbb{R}^{3})$ and $v$ in $H^{1}(\\mathbb{R}^{3})$ one has\n\\begin{align*}\n\\big|\\langle \\mathsf v u,v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\n \\leq&\n\\frac14\\Big(c_{1}\\epsilon\\,\\big|\\langle \\nabla u,\\nabla v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\n+c_{2}(1+\\epsilon^{-3})\\big|\\langle u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\\Big) \\\\\n =&\n\\frac14\\Big(c_{1}\\epsilon\\,\\big|\\langle -\\Delta u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\n+c_{2}(1+\\epsilon^{-3})\\big|\\langle u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\\Big)\\,. \n\\end{align*}\nBy the density of $L^{2}(\\mathbb{R}^{3})$ in $H^{1}(\\mathbb{R}^{3})$, the above inequality (here we refer to the second line) holds for any $v\\in L^{2}(\\mathbb{R}^{3})$. Then, by considering the supremum over the set of functions in $H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$ with unitary norm, one gets\n\\begin{align*}\n\\|\\mathsf v u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\n\\le&\\frac14\\Big( c_{1}\\epsilon\\,\\|-\\Delta u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\\\\n\\le&\\frac14\\Big( c_{1}\\epsilon\\,\\|(-\\Delta+z) u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\,.\n\\end{align*}\nTherefore\n$$\n\\|\\mathsf v R_{z}u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\le\n\\frac14\\Big( c_{1}\\epsilon\\,\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|R_{z}u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\,.\n$$\nTo conclude the proof it suffices to show that  \n$$\n\\|R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\!,H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\lesssim{d^{-\\gamma}_{z}}\\,,\n$$\nequivalently\n\\begin{equation}\\label{eqv}\n\\|R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\lesssim\n{d^{-\\gamma}_{z}}\\,,\n\\end{equation}\nwhere $\\gamma>0$ and $d_{z}$ is the distance of $z$ from $[0,+\\infty)$.\n\\par\nLet $u\\equiv u_{\\rm in}\\oplus u_{\\rm ex}\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$; then $u= 1_{\\Omega_{\\rm in}}\\widetilde u_{\\rm in}+1_{\\Omega_{\\rm ex}}\\widetilde u_{\\rm ex}$, where $\\widetilde u_{\\rm in\/\\rm ex}\\in H^{1}(\\mathbb{R}^{3})$ is such that $\\widetilde u_{\\rm in\/\\rm ex}|\\Omega_{\\rm in\/\\rm ex}=u_{\\rm in\/\\rm ex}$. One has\n\\begin{align*}\n&\\|R_{z}u\\|^{2}_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}=\\|R_{z}u\\|^{2}_{H^{1}(\\mathbb{R}^{3})}\n=\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n \\|\\nabla R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n =\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n \\|R_{z}\\nabla  u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n\\\\\n\\le&\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n\\|R_{z}(1_{\\Omega_{\\rm in}}\\nabla\\widetilde u_{\\rm in}+1_{\\Omega_{\\rm ex}}\\nabla\\widetilde u_{\\rm ex}\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n+\\|R_{z}(\\widetilde u_{\\rm in}\\nabla 1_{\\Omega_{\\rm in}}+\\widetilde u_{\\rm ex}\\nabla 1_{\\Omega_{\\rm ex}}\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n\\le&\\|R_{z}\\|^{2}_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\\|u\\|^{2}_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n+\\|S\\!L_{z}\\nu[\\gamma_{0}]u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,,\n\\end{align*}\nwhere $\\nu$ is the exterior normal to $\\Gamma$ and $S\\!L_{z}$ is the single-layer operator, i.e., $S\\!L_{z}:=(\\gamma_{0}R_{\\bar z})^{*}=R_{z}\\gamma_{0}^{*}$ and $[\\gamma_{0}]$ denotes the jump of the Dirichlet trace across $\\Gamma$. Here we made use the distributional derivative of $\\nabla 1_{\\Omega_{\\rm in\/\\rm ex}}$, which, by the divergence theorem, gives, for any test function $\\varphi$,\n$$\n\\nabla 1_{\\Omega_{\\rm in\/\\rm ex}}(\\varphi)=\\mp\\int_{\\Omega_{\\rm in\/\\rm ex}}\\nabla\\varphi(x)\n\\,dx=\\mp\\int_{\\Gamma}\\nu(x)\\gamma_{0}\\varphi(x)\\,d\\sigma(x)=\\mp\\gamma_{0}^{*}\\nu\\gamma_{0}\\varphi\\,.\n$$\nSince \n$$\n\\|S\\!L_{z}\\nu[\\gamma_{0}]\\|_{L^{2}(\\mathbb{R}^{3})}\\le\n\\|S\\!L_{z}\\|_{H^{1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\n\\|[\\gamma_{0}]\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1\/2}(\\Gamma)}\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\,,\n$$\n\\begin{align*}\n&\\|S\\!L_{z}\\|_{H^{1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\\le\\|S\\!L_{z}\\|_{H^{-1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\n\\|S\\!L_{\\bar z}^{*}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\\\\\n=&\n\\|\\gamma_{0}R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\n\\le\\|\\gamma_{0}\\|_{H^{1}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\n\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n\\end{align*}\nand\n$$\n\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\\le d_{z}^{-1}\\,,\n\\qquad\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\\le d_{z}^{-1\/2}\\,,\n$$\n\\eqref{eqv} follows and the proof is concluded.\n\\end{proof}\n\\begin{remark} By Lemma \\ref{vH-1}, \n$$\n\\Lambda^{\\!\\mathsf v}_{z}|H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)=(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v\\,.\n$$\n\\end{remark}\n\\begin{remark} By the same kind of proof (indeed a shorter one) as in Lemma \\ref{vH-1}, one gets \n$$\n(  1-\\mathsf v R_{z})  ^{-1}=\\sum_{k=0}^{+\\infty}\\left(  \\mathsf v R_{z}\\right)  ^{k}\n\\in{\\mathscr B}(  H^{-1}(\\mathbb{R}^{3}))\n$$\nand, by \\eqref{v-sob}, $\\mathsf v\\in{\\mathscr B}(  H^{1}(\\mathbb{R}^{3}), H^{-1}(\\mathbb{R}^{3}))$. Therefore, $(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v\\in {\\mathscr B}(  H^{1}(\\mathbb{R}^{3}), H^{-1}(\\mathbb{R}^{3}))$ and, by \\eqref{L-int} and \\eqref{RF-int},\n\\begin{equation}\\label{LvHs1}\n\\Lambda^{\\!\\mathsf v}_{z}|H^{s}(\\mathbb{R}^{3})=(  1- \\mathsf v R_{z})  ^{-1}\\mathsf v\\,,\\qquad 1\\le s\\le 2\\,.\n\\end{equation}\nBy duality, similarly to Remark \\ref{Lt},  $(  1-R_{z}\\mathsf v)  ^{-1}\\in{\\mathscr B}(  H^{1}(\\mathbb{R}^{3}))$ and \n\\eqref{LtL2} improves to\n\\begin{equation}\\label{LvHs2}\n\\Lambda^{\\!\\mathsf v}_{z}|H^{s}(\\mathbb{R}^{3})=\\mathsf v(  1- \\mathsf v R_{z})  ^{-1}\\,,\\qquad 0\\le s\\le 1\\,.\n\\end{equation}\n\\end{remark}\n\\subsection{\\label{Sec_Layer}Boundary layer operators.}\nWe introduce the interior\/exterior Dirichlet and Neumann trace operators\n$$\n\\gamma_{0}^{\\rm in\/\\rm ex}:H^{s+1\/2}(\\Omega_{\\rm in\/ex})\\to B_{2,2}^{s}(\\Gamma)\\,,\\qquad s>0\\,, \n$$\n$$\n\\gamma_{1}^{\\rm in\/\\rm ex}:H^{s+3\/2}(\\Omega_{\\rm in\/ex})\\to B_{2,2}^{s}(\\Gamma)\\,,\\qquad s>0\\,,\n$$\nwhere $\\Omega_{\\rm in}\\equiv\\Omega$ and $\\Omega_{\\rm ex}:=\\Omega_{\\rm ex}$. The Besov-like trace spaces $B_{2,2}^{s}(  \\Gamma )  $ \nidentify with $H^{s}(\\Gamma)  $ when $|s|\\le k+1$ and\n$\\Gamma$ is of class $\\mathcal{C}^{k,1}$ (see \\cite{JoWa}). Then, we define  the bounded linear operators \n\\begin{equation}\\label{g0}\n\\gamma_{0}:H^{s+1\/2}(\\mathbb{R}^{3}\\backslash\\Gamma)\\to B_{2,2}^{s}(\\Gamma)\\,,\\quad\\gamma_{0}u:=\\frac12\\,(\\gamma_{0}^{\\rm in}(u|\\Omega_{\\rm in})+\\gamma_{0}^{\\rm ex}(u|\\Omega_{\\rm ex}))\\,,\\qquad s>0\\,,\n\\end{equation}\n\\begin{equation}\\label{g1}\n\\gamma_{1}:H^{s+3\/2}(\\mathbb{R}^{3}\\backslash\\Gamma)\\to B_{2,2}^{s}(\\Gamma)\\,,\\quad\\gamma_{1}u:=\\frac12\\,(\\gamma_{1}^{\\rm in}(u|\\Omega_{\\rm in})+\\gamma_{1}^{\\rm ex}(u|\\Omega_{\\rm ex}))\\,,\\qquad s>0\\,.\n\\end{equation}\nThe corresponding trace jump bounded operators are defined by\n\\begin{equation}\n[\\gamma_{0}]:H^{s+1\/2}(  \\mathbb{R}^3\\backslash\\Gamma )  \\rightarrow B_{2,2}^{s}(\\Gamma)\\,,\\quad[\n\\gamma_{0}]u:=\\gamma_{0}^{\\-}(u|\\Omega_{\\rm in})-\\gamma_{0}^{\\+}(u|\\Omega_{\\rm ex})\\,,\n\\end{equation}%\n\\begin{equation}\n[\\gamma_{1}]:H^{s+3\/2}(  \\mathbb{R}^3\\backslash\\Gamma )\\rightarrow B_{2,2}^{s}(\\Gamma)\\,,\\quad[\n\\gamma_{1}]u:=\\gamma_{1}^{\\-}(u|\\Omega_{\\rm in})-\\gamma_{1}^{\\+}(u|\\Omega_{\\rm ex})\\,.\n\\end{equation}\nBy \\cite[Lemma 4.3]{McL}, the trace maps $\\gamma_{1}^{\\-\/\\+}$ can be extended to the spaces $$H^{1}_{\\Delta}(\\Omega_{\\-\/\\+}):=\\{u_{\\-\/\\+}\\in H^{1}(\\Omega_{\\-\/\\+}):\\Delta_{\\Omega_{\\-\/\\+}}u_{\\-\/\\+}\\in L^{2}(\\Omega_{\\-\/\\+})\\}$$ \nas $H^{-1\/2}(\\Gamma)$-valued bounded operators:\n$$\n\\gamma_{1}^{\\-\/\\+}: H^{1}_{\\Delta}(\\Omega_{\\-\/\\+})\\to H^{-1\/2}(\\Gamma)\\,.\n$$\nThis gives the extensions of the maps $\\gamma_{1}$ and $[\\gamma_{1}]$ defined on $H^{1}_{\\Delta}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{1}_{\\Delta}(\\Omega_{\\-})\\oplus H^{1}_{\\Delta}(\\Omega_{\\+})$ with values in $H^{-1\/2}(\\Gamma)$. \\par\nThen, for any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$,\none defines the single and double-layer operators\n\\begin{equation}\\label{SL}\nS\\!L_{z}:=(\\gamma_{0}R_{\\bar z})^{*}=R_{z}\\gamma_{0}^{*}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{3\/2-s}(\\mathbb{R}^{3}))\\,,\n\\qquad s>0\\,,\n\\end{equation}\n\\begin{equation}\\label{DL}\nD\\!L_{z}:=(\\gamma_{1}R_{\\bar z})^{*}=R_{z}\\gamma_{1}^{*}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{1\/2-s}(\\mathbb{R}^{3}))\\,,\\qquad s>0\\,.\n\\end{equation}\nBy \\eqref{g0}, one has\n\\begin{equation}\\label{BSL}\nS_{z}:=\\gamma_{0}S\\!L_{z}\\in \\mathscr B((H^{s-1\/2}(\\Gamma),H^{s+1\/2}(\\mathbb{R}^{3})))\\,,\\qquad -1\/2<s<1\/2\\,.\n\\end{equation}\nBy the mapping properties of the double-layer operator, one gets\\footnote{here and below we can avoid the introduction of the cutoff funcion $\\chi$ appearing in \\cite[Theorems 6.11-6.13]{McL} since we are dealing with the constant coefficients strongly elliptic operator $-\\Delta+z$ (compare \\cite[Theorem 6.1]{McL} with \\cite[equation (6.10)]{McL})} (see \\cite[Theorem 6.11]{McL})\n\\begin{equation}\\label{map}\nD\\!L_{z}\\in \\mathscr B(H^{1\/2}(\\Gamma),H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n\\end{equation}\nHence,  by \n$$\n(-(\\Delta_{\\Omega_{\\-}}\\oplus\\Delta_{\\Omega_{\\+}})+z)D\\!L_{z}=0\\,,\n$$\none gets \n$$\nD\\!L_{z}\\in \\mathscr B(H^{1\/2}(\\Gamma),H^{1}_{\\Delta}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n$$\nThus\n$$\nD_{z}:=\\gamma_{1}D\\!L_{z}\\in \\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThese mapping properties can be extended to a larger range of Sobolev spaces (see \\cite[Theorem 6.12 and successive remarks]{McL}): \n\\begin{equation}\\label{SLext}\nS\\!L_{z}\\in \\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}))\\,,\\qquad -1\/2\\le s\\le 1\/2\\,,\n\\end{equation}\n\\begin{equation}\\label{BSLext}\nS_{z}\\in \\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1\/2}(\\Gamma))\\,,\\qquad\n-1\/2\\le s\\le 1\/2\\,, \n\\end{equation}\n\\begin{equation}\\label{DLext}\nD\\!L_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\\qquad \n-1\/2\\le s\\le 1\/2\\,, \n\\end{equation}\n\\begin{equation}\\label{BDLext}\nD_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s-1\/2}(\\Gamma))\\,,\\qquad \n-1\/2\\le s\\le 1\/2\n\\end{equation}\nand, whenever $s\\ge 0$ in \\eqref{SLext}, \\eqref{DLext} above, the following jump relations holds (see \\cite[Theorem 6.11]{McL})\n\\begin{equation}\\label{jump0}\n[\\gamma_{0}]S\\!L_{z}=0\\,,\\quad [\\gamma_{1}]S\\!L_{z}=-1\\,,\n\\quad \n\\end{equation}\n\\begin{equation}\\label{jump1}\n[\\gamma_{0}]D\\!L_{z}=1\\,,\\quad [\\gamma_{1}]D\\!L_{z}=0\\,.\n\\end{equation}\nWhenever the boundary $\\Gamma$ is of class ${\\mathcal C}^{1,1}$ one gets an improvement as regards the regularity properties of the single- and double-layer operators (see \\cite[Theorem 6.13 and Corollary 6.14]{McL}):\n\\begin{equation}\\label{SLext+}\nS\\!L_{z}\\in \\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\\qquad 1\/2<s\\le 1\\,,\n\\end{equation}\n\\begin{equation}\\label{DLext+}\nD\\!L_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\\qquad \n1\/2<s\\le 1\\,, \n\\end{equation}\nBy \\eqref{SL}, \\eqref{DL} and \\eqref{RLvz} one has\n\\begin{lemma} For any $z\\in \\varrho(\\Delta+\\mathsf v)\\cap (\\mathbb{C}\\backslash(-\\infty,0])$,\n\\begin{equation}\\label{SLv} \nS\\!L^{\\mathsf v}_{z}:=R^{\\mathsf v}_{z}\\gamma_{0}^{*}=S\\!L_{z}+R_{z}\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{3\/2-s}(\\mathbb{R}^{3}))\\,,\n\\qquad 0<s\\le 3\/2\\,,\n\\end{equation}\n\\begin{equation}\\label{DLv}\nD\\!L^{\\mathsf v}_{z}:=R^{\\mathsf v}_{z}\\gamma_{1}^{*}=D\\!L_{z}+R_{z}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{1\/2-s}(\\mathbb{R}^{3}))\\,,\\qquad 0<s\\le 1\/2\\,.\n\\end{equation}\n\\end{lemma}\nBy \\eqref{SLext}, \\eqref{DLext} and \\eqref{RLvz}, one has \n\\begin{lemma}\n\\begin{equation}\\label{SLvext}\nS\\!L^{\\mathsf v}_{z}\\in \\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3})\\,,\\qquad -1\/2\\le s\\le 1\/2\\,,\n\\end{equation}\n\\begin{equation}\\label{DLvext}\nD\\!L^{\\mathsf v}_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\n\\qquad -1\/2\\le s\\le 1\/2\\,.\n\\end{equation}\n\\end{lemma}\nBy \\eqref{SLext+}, \\eqref{DLext+} and \\eqref{RLvz}, one has \n\\begin{lemma} Let $\\Gamma\\in{\\mathcal C}^{1,1}$. Then\n\\begin{equation}\\label{SLvext+}\nS\\!L^{\\mathsf v}_{z}\\in \\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\\qquad 1\/2<s\\le 1\\,,\n\\end{equation}\n\\begin{equation}\\label{DLvext+}\nD\\!L^{\\mathsf v}_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\\qquad \n1\/2<s\\le 1\\,, \n\\end{equation}\n\\end{lemma}\nBy either \\eqref{SLv} or \\eqref{SLvext} one has\n\\begin{equation}\\label{BLv-}\n\\gamma_{0}S\\!L^{\\mathsf v}_{z}=S_{z}+\\gamma_{0}R_{z}\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\in\\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1\/2}(\\Gamma))\\,,\n\\qquad -1\/2<s<1\/2\\,.\n\\end{equation}\nSince $\\gamma_{0}R_{z}=(R_{\\bar z}\\gamma_{0}^{*})^{*}=S\\!L_{\\bar z}^{*}$, one gets the following improvement of \\eqref{BLv-}:\n\\begin{lemma}\n\\begin{equation}\\label{BLv}\nS^{\\mathsf v}_{z}:=S_{z}+S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\in\\mathscr B(H^{s-1\/2}(\\Gamma),H^{s+1\/2}(\\Gamma))\\,,\n\\qquad -1\/2\\le s\\le1\/2\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By \\eqref{SLext} and duality, $S\\!L_{\\bar z}^{*}\\in \\mathscr B(H^{-1-s}(\\mathbb{R}^{3}),H^{1\/2-s}(\\Gamma))$. The proof is then concluded by \\eqref{BSLext}, \\eqref{Lvz-int} and \\eqref{SLext} .\n\\end{proof}\nIf $\\Gamma\\in{\\mathcal C}^{1,1}$, then, by \\eqref{DLvext+},\n\\begin{equation}\\label{DBLvext-}\n\\gamma_{1}D\\!L_{z}^{\\mathsf v}=D_{z}+\\gamma_{1}R_{z}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s-1\/2}(\\Gamma))\\,,\\qquad 1\/2<s\\le 1\\,,\n\\end{equation}\nSince $\\gamma_{1}R_{z}=(R_{\\bar z}\\gamma_{1}^{*})^{*}=D\\!L_{\\bar z}^{*}$, one can improve \\eqref{DBLvext-} even without requiring $\\Gamma\\in{\\mathcal C}^{1,1}$:\n\\begin{lemma}\n\\begin{equation}\\label{DBLvext}\nD^{\\mathsf v}_{z}:=D_{z}+D\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}\\in \\mathscr B(H^{s+1\/2}(\\Gamma),H^{s-1\/2}(\\Gamma))\\,,\n\\qquad -1\/2\\le s\\le 1\/2\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} \nBy \\eqref{DLext} and duality, $D\\!L_{\\bar z}^{*}\\in \\mathscr B(H^{s+1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{-s-1\/2}(\\Gamma))$. The proof is then concluded by \\eqref{BDLext}, \\eqref{Lvz-int} and \\eqref{DLext} .\n\\end{proof}\nIn order to prove the jump relations of the double-layer operator relative to $\\Delta+\\mathsf v$ we  need a technical result:\n\\begin{lemma}\\label{tech} If $v\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}$, then $[\\gamma_{1}]R_{z}v=0$ in $H^{-1\/2}(\\Gamma)$ for any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$. \n\\end{lemma}\n\\begin{proof} At first let us notice that it suffices to show that the result holds for a single $z\\in\\mathbb{C}\\backslash(-\\infty,0]$. Indeed, by the resolvent identity $R_{w}v=R_{z}v+(z-w)R_{w}R_{z}v$, one gets $R_{w}R_{z}v\\in H^{3}(\\mathbb{R}^{3})\\subset\\text{\\rm ker}([\\gamma_{1}])$. In particular, we choose $z$ such that $\\text{\\rm ker}(S_{z})=\\{0\\}$ (see, e.g., Lemma \\eqref{coerc} below). \\par \nGiven $v\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}=\nH^{-1}_{\\overline\\Omega}(\\mathbb{R}^{3})\\oplus H^{-1}_{\\Omega^{c}}(\\mathbb{R}^{3})\\subseteq H^{-1}(\\mathbb{R}^{3})$ and $\\chi\\in{\\mathcal C}_{\\text{comp}}^{\\infty}(\\mathbb{R}^{3})$ such that $\\chi=1$ on a compact set containing an open  neighborhood of $\\overline\\Omega$, let us set $u:=\\chi R_{z}v$. Since $\\gamma_{1}^{\\rm in\/\\rm ex}u=\\gamma_{1}^{\\rm in\/\\rm ex}R_{z}v$, it suffices to show that $[\\gamma_{1}]u=0$. Let us define $u_{\\rm in\/\\rm ex}:=\\chi R_{z}v|\\Omega_{\\rm in\/\\rm ex}\\in H^{1}(\\Omega_{\\rm in\/\\rm ex})$, \n$f_{\\rm in\/\\rm ex}:=((-\\Delta+z)\\chi R_{z}v)|\\Omega_{\\rm in\/\\rm ex}\\in H^{1}(\\Omega_{\\rm in\/\\rm ex})$\n and $g_{\\rm in\/\\rm ex}:=\\gamma^{\\rm in\/\\rm ex}_{0}u_{\\rm in\/\\rm ex}\\in H^{1\/2}(\\Gamma)$. Then $u_{\\rm in\/\\rm ex}$ solves the Dirichlet boundary value problems\n\\begin{equation}\n\\begin{cases}(  -\\Delta_{\\Omega_{\\rm in\/\\rm ex}}+z)u_{\\rm in\/\\rm ex}  =f_{\\rm in\/\\rm ex}\\,,\\\\\n\\gamma^{\\rm in\/\\rm ex}_{0}u_{\\rm in\/\\rm ex}=g_{\\rm in\/\\rm ex}\n\\end{cases}\n\\end{equation}\nand so, by \\cite[Theorems 7.5 and 7.15]{McL} (notice that both $u_{\\rm ex}$ and $f_{\\rm ex}$ have a compact support; in particular the radiation condition ${\\mathcal M}u_{\\rm ex}=0$ there required is here satisfied),\n$\\psi_{\\rm in\/\\rm ex}:=\\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex}\\in H^{-1\/2}(\\Gamma)$ satisfy the equations\n\\begin{equation}\nS_{z}\\psi_{\\rm in\/\\rm ex}=\\frac{1}{2}\\,(1+D_{z})g_{\\rm in\/\\rm ex}-\\gamma_{0}R_{z}v\n\\end{equation}\nSince $u_{\\rm in}\\oplus u_{\\rm ex}=\\chi R_{z}v\\in H^{1}(\\mathbb{R}^{3})$, one has $g_{\\rm in}=g_{\\rm ex}$ and so \n$[\\gamma_{1}]R_{z}v=\\psi_{\\rm in}-\\psi_{ex}=0$ is consequence of $\\text{\\rm ker}(S_{z})=\\{0\\}$.\n\\end{proof}\n\\begin{lemma} If $s\\ge 0$ in \\eqref{SLvext} and \\eqref{DLvext}, then\n\\begin{equation}\\label{jumpv0}\n[\\gamma_{0}]S\\!L^{\\mathsf v}_{z}=0\\,,\\quad [\\gamma_{1}]S\\!L^{\\mathsf v}_{z}=-1\\,,\n\\quad \n\\end{equation}\n\\begin{equation}\\label{jumpv1}\n[\\gamma_{0}]D\\!L^{\\mathsf v}_{z}=1\\,,\\quad [\\gamma_{1}]D\\!L^{\\mathsf v}_{z}=0\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} $[\\gamma_{0}]S\\!L^{\\mathsf v}_{z}=0$ is consequence of $\\text{\\rm ran}(S\\!L^{\\mathsf v}_{z})\\subseteq H^{1}(\\mathbb{R}^{3})$. By \\eqref{RLvz}, $\\text{\\rm ran}(R_{z}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z})\\subseteq H^{1}(\\mathbb{R}^{3})$ and so $[\\gamma_{0}]D\\!L^{\\mathsf v}_{z}=[\\gamma_{0}]D\\!L_{z}+[\\gamma_{0}]R_{z}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}=[\\gamma_{0}]D\\!L_{z}=1$. \\par\nSince $\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\in\\mathscr B(H^{s-1\/2}(\\Gamma),H^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$, \n$\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}\\in\\mathscr B(H^{s+1\/2}(\\Gamma),\nH^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$ and $s\\ge 0$, by Lemma \\ref{tech} one gets \n$$\n[\\gamma_{1}]S\\!L^{\\mathsf v}_{z}=[\\gamma_{1}]S\\!L_{z}+[\\gamma_{1}]R_{z}\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}=\n[\\gamma_{1}]S\\!L_{z}=-1\\,,\n$$\n$$\n[\\gamma_{1}]D\\!L^{\\mathsf v}_{z}=[\\gamma_{1}]D\\!L_{z}+[\\gamma_{1}]R_{z}\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}=\n[\\gamma_{1}]D\\!L_{z}=0\\,.\n$$\n\\end{proof}\nWhen $\\mathsf v=0$, it is well known that the boundary layer operators have bounded inverses. This\nproperty is next extended to the operators relative to $\\Delta+\\mathsf v$.\n\\begin{lemma}\\label{coerc} There exist $Z^{\\circ}_{\\mathsf v,d}$ and $Z^{\\circ}_{\\mathsf v,n}$, not empty open subsets of $\\varrho(\\Delta+\\mathsf v)$, such that \n$$\n\\forall z\\in Z^{\\circ}_{\\mathsf v,d}\\,,\\quad (S_{z}^{\\mathsf v})^{-1}\\in{\\mathscr B}(  H^{1\/2}(\\Gamma)  ,H^{-1\/2}(\\Gamma) )\\,,\\qquad\\forall z\\in Z^{\\circ}_{\\mathsf v,n}\\,,\\quad\n(D_{z}^{\\mathsf v})  ^{-1}\\in{\\mathscr B}(  H^{-1\/2}(\\Gamma)  ,H^{1\/2}(\\Gamma) )\\,.\n$$\nIn particular, there exists $\\lambda_{\\mathsf v}>\\sup\\sigma(\\Delta+\\mathsf v)$ such that $[\\lambda_{\\mathsf v},+\\infty)\\subset Z^{\\circ}_{\\mathsf v,d}\\cap Z^{\\circ}_{\\mathsf v,n}$; moreover, $Z^{\\circ}_{\\mathsf v,d}\\cap Z^{\\circ}_{0,d}\\not=\\varnothing $, $Z^{\\circ}_{\\mathsf v,n}\\cap Z^{\\circ}_{0,n}\\not=\\varnothing $,  and both  $Z^{\\circ}_{\\mathsf v,d}$ and $Z^{\\circ}_{\\mathsf v,n}$ can be chosen to be symmetric with respect to the real axis.\n\\end{lemma}\n\\begin{proof} At first, let us notice that it suffices to show that the bounded inverses exist for any real $\\lambda \\ge \\lambda_{\\mathsf v}$ for some $\\lambda_{\\mathsf v}>\\sup\\sigma(\\Delta+\\mathsf v)$. Then, by the continuity of the maps $z\\mapsto S^{\\mathsf v}_{z}$ and $z\\mapsto D^{\\mathsf v}_{z}$, the bounded inverses exist in a complex open neighbourhood of $[\\lambda_{\\mathsf v},+\\infty)$.\\par\nWe proceed as in the proof of \\cite[Lemma 3.2]{JDE16}. By $(-(\\Delta+\\mathsf v)+\\lambda)S\\!L^{\\mathsf v}_{\\lambda}|\\Omega_{\\rm in\/\\rm ex}=0$, by Green's formula and by \\eqref{jumpv0}, one gets, for any $\\phi\\in H^{-1\/2}(\\Gamma)$,\n\\begin{align*}\n0=&\\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n-\\langle \\mathsf v S\\!L^{\\mathsf v}_{\\lambda}\\phi,S\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n+\\lambda\\,\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&+\\langle [  \\gamma_{1}] S\\!L^{\\mathsf v}_{\\lambda}\\phi ,\\gamma_{0}S\\!L_{\\lambda}\\phi\\rangle _{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\\\\n=&\\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n-\\langle \\mathsf v S\\!L^{\\mathsf v}_{\\lambda}\\phi,S\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n+\\lambda\\,\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&-\\langle \\phi ,S^{\\mathsf v}_{\\lambda}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\n\\end{align*}\nThen, by  \\eqref{V_est},\n\\[\n\\langle \\phi ,\\gamma_{0}S^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\geq(  1-c_{1}\\varepsilon) \\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n+(\\lambda-c_{2}(1+\\varepsilon^{-3}))\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n\\]\nChoosing $\\varepsilon>0$ such that $c_{1}\\varepsilon<1$ and then $\\lambda\\in\\varrho(\\Delta+\\mathsf v)$ such that $\\lambda>c_{2}(1+\\varepsilon^{-3})$ (this is always possible since $\\Delta+\\mathsf v$ in bounded from above), one gets\n\\[\n\\langle \\phi,S_{\\lambda}^{\\mathsf v}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim\\|S\\!L_{\\lambda}^{\\mathsf v}\\phi \\|_{H^{1}(  \\mathbb{R}^{3})  }^{2}\\,.\n\\]\nBy Green's formula again, one has, for any $u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\in H^{1}(\\Omega_{\\rm in\/\\rm ex})$, $(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\in L^{2}(\\Omega_{\\rm in\/\\rm ex})$,\n\\begin{align}\\label{Gf}\n&\\langle(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{L^{2},H^{1}(\\Omega_{\\rm in\/\\rm ex})}=\n\\langle\\nabla u_{\\rm in\/\\rm ex},\\nabla v_{\\rm in\/\\rm ex}\\rangle_{L^{2}(\\Omega_{\\rm in\/\\rm ex})}\\nonumber\\\\\n&-\\langle \\mathsf v u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{H^{-1}(\\Omega_{\\rm in\/\\rm ex}),\nH^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\lambda\\,\\langle u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{L^{2}(\\mathbb{R}^{3})}\\\\\n&\n\\pm \\langle \\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex},\\gamma^{\\rm in\/\\rm ex}_{0} v_{\\rm in\/\\rm ex}\n\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\\nonumber\n\\end{align}\nBy \n$$\n\\big|\\langle \\mathsf v u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{H^{-1}(\\Omega_{\\rm in\/\\rm ex}),\nH^{1}(\\Omega_{\\rm in\/\\rm ex})}\\big|\\lesssim \\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\n\\|v_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\\,,\n$$\n\\eqref{Gf} gives,\n\\begin{align*}\n&\\big|\\langle \\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex},\\gamma^{\\rm in\/\\rm ex}_{0} v_{\\rm in\/\\rm ex}\n\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\big|\\\\\n\\lesssim\n&\\big(\\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\|(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\|_{H^{-1}(\\Omega_{\\rm in\/\\rm ex})}\\big)\n\\|v_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\\,.\n\\end{align*}\nSince $\\gamma^{\\rm in\/\\rm ex}_{0}:H^{1}(\\Omega_{\\rm in\/\\rm ex})\\to H^{1\/2}(\\Gamma)$ is surjective, finally one gets \n\\begin{equation}\\label{cmp}\n\\|\\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex}\\|_{H^{-1\/2}(\\Gamma)}\\lesssim \n\\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\|(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\|_{H^{-1}(\\Omega_{\\rm in\/\\rm ex})}\\,.\n\\end{equation}\nThen, proceeding as in \\cite[Lemma 3.2]{JDE16} (compare (3.31) there with \\eqref{cmp} here), this\nyields\n\\begin{equation*}\n\\langle \\phi,S_{\\lambda}^{\\mathsf v}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim \\|\\phi\\|_{H^{-1\/2}(\\Gamma)}^{2}\\label{SL_coer}%\n\\end{equation*}\nand so $(S_{\\lambda}^{\\mathsf v})  ^{-1}\\in{\\mathscr B}(  H^{1\/2}(\n\\Gamma )  ,H^{-1\/2}(\\Gamma) )$ by the Lax-Milgram theorem.\\par\nAs regards $D_{\\lambda}^{\\mathsf v}$, the proof is almost the same. By $(-(\\Delta+\\mathsf v)+\\lambda)D\\!L^{\\mathsf v}_{\\lambda}|\\Omega_{\\rm in\/\\rm ex}=0$, by Green's formula and by \\eqref{jumpv1}, one gets, for any $\\phi\\in H^{1\/2}(\\Gamma)$,\n\\begin{align*}\n0=&\\|\\nabla D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\n\\|\\nabla D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\n-\\langle \\mathsf v D\\!L^{\\mathsf v}_{\\lambda}\\phi,D\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n+\\lambda\\,\\|D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&+\\langle D^{\\mathsf v}_{\\lambda}\\phi ,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\n\\end{align*}\nwhich leads to \n\\[\n-\\langle D_{\\lambda}^{\\mathsf v}\\phi,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim\\|D\\!L_{\\lambda}^{\\mathsf v}\\phi \\|_{H^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)  }^{2}\\,.\n\\]\nThen, proceeding as in \\cite[Lemma 3.2]{JDE16}, by  \\eqref{cmp}, this\nyields\n$$\n-\\langle D_{\\lambda}^{\\mathsf v}\\phi,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim \\|\\phi\\|_{H^{1\/2}(\\Gamma)}^{2}\n$$\nand so $(D_{\\lambda}^{\\mathsf v})  ^{-1}\\in{\\mathscr B}(  H^{-1\/2}(\n\\Gamma )  ,H^{1\/2}(\\Gamma) )$ by the Lax-Milgram theorem.\n\\end{proof}\n\n\n\\section{Laplacians with regular and singular perturbations}\nHere we apply the abstract results in Section \\ref{Sec_Krein}, presenting various examples were \nthe self-adjoint operator $A$ is the free Laplacian $\\Delta:H^{2}(\\mathbb{R}^{3})\\subset L^{2}(\\mathbb{R}^{3})\\to L^{2}(\\mathbb{R}^{3})$ and $A_{B_{1}}=\\Delta+\\mathsf v$. All over this section we consider a Kato-Rellich potential $\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}$ of short-range type (however, see next Remark \\ref{suff}), i.e.,\n\\begin{equation}\\label{short}\n\\mathsf v_{2}\\in L^{2}(\\mathbb{R}^{3}),\\quad  \\text{supp}(\\mathsf v_{2})\\ \\text{bounded}, \\qquad\n\\ |\\mathsf v_{\\infty}(x)|\\lesssim\\, (1+|x|\\,)^{-\\kappa(1+\\varepsilon)}\\,,\\quad\\kappa\\ge 1\\,,\\quad\\varepsilon>0\\,.\n\\end{equation}\nIn the next Lemmata we show that all our hypotheses but (H3) hold whenever $\\kappa=1$, while hypothesis (H3) holds whenever $\\kappa=2$; we conjecture that all our results hold true with $\\kappa=1$ and that the requirement $\\kappa=2$ is merely of technical nature. \\par\nWe take \n$$\n\\mathfrak h_{1}=H^{2}(\\mathbb{R}^{3})\\hookrightarrow \\mathfrak b_{1}=H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\n\\hookrightarrow\\mathfrak h_{1}^{\\circ}=L^{2}(\\mathbb{R}^{3})\n\\,,\n$$\nand, introducing the multiplication operator  $\\langle x\\rangle$ by $\\langle x\\rangle u: x\\mapsto(1+|x|^{2})^{1\/2}u(x)$, we define\n\\begin{equation}\\label{tau1}\n\\tau_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{2}(\\mathbb{R}^{3})\\,,\\quad \\tau_{1}u:=\\langle x\\rangle^{-s}u\\,,\\qquad\n1<2s< 1+\\varepsilon\\,,\n\\end{equation}\nand\n$$\nB_{1}u:=\\langle x\\rangle^{2s}\\mathsf v u\\,.\n$$\nBy our hypotheses on $\\mathsf v$ and $s$, one has $\\langle x\\rangle^{2s}\\mathsf v\\in L^{2}(\\mathbb{R}^{3})+L^{\\infty}(\\mathbb{R}^{3})$ and so, by Lemma \\ref{v},\n$$\nB_{1}\\in\\mathscr B(H^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma),H^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\nConsidering the weight $\\varphi(x)=(1+|x|^{2})^{w\/2}$, $w\\in\\mathbb{R}$, we use the notation $L^{2}_{\\varphi}(\\mathbb{R}^{3})\\equiv L^{2}_{w}(\\mathbb{R}^{3})$; $H^{k}_{w}(\\mathbb{R}^{3})$, $H^{k}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma)$ denotes the corresponding scales of weighted Sobolev spaces. \\par \nSince \n$$\n\\langle x\\rangle^{w}\\in\\mathscr B(H^{1}_{w'}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}_{w'+w}(\\mathbb{R}^{3}\\backslash\\Gamma))$$ and, by duality, $$\\langle x\\rangle^{w}\\in\\mathscr B(H^{1}_{w'}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}_{w'-w}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,,$$ \none gets \n\\begin{equation}\\label{v-w}\n\\langle x\\rangle^{-w-2s}B_{1}\\langle x\\rangle^{w}=\\mathsf v\\in \\mathscr B(H^{1}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}_{-w-2s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n\\end{equation}  \nSince \n\\begin{equation}\\label{R-w}\nR_{z}\\in\\mathscr B(H^{-1}_{w}(\\mathbb{R}^{3}),H^{1}_{w}(\\mathbb{R}^{3}))\\hookrightarrow \\mathscr B(H^{1}_{-w}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\n\\end{equation}\none has  \n$$\\tau_{1}G^{1}_{z}=\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}\\in\\mathscr B(H_{-w}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H_{w+2s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n$$ \nIn particular, one has  $\\tau_{1}G^{1}_{z}\\in\\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}),H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))$.\\par\nSince $(1-\\mathsf v R_{z})$ is invertible, \n$$\nM_{z}^{B_{1}}=1-B_{1}\\tau_{1}G^{1}_{z}=1-\\langle x\\rangle^{s}\\mathsf v R_{z}\\langle x\\rangle^{-s}=\n\\langle x\\rangle^{s}(1-\\mathsf v R_{z})\\langle x\\rangle^{-s\n$$ \nis invertible as well an\n\\begin{equation}\\label{LbLv}\n\\Lambda_{z}^{B_{1}}=(M_{z}^{B_{1}})^{-1}B_{1}=\\langle x\\rangle^{s}(1-\\mathsf v R_{z})^{-1}\\langle  x\\rangle^{s}\\mathsf v=\\langle x\\rangle^{s}\\Lambda_{z}^{\\!\\mathsf v}\\langle x\\rangle^{s}\n\\,.\n\\end{equation}\n\\begin{lemma} Let $\\mathsf v$ be as in \\eqref{short}, with $\\kappa=1$. Then, for any $s$ such that $1< 2s<1+\\varepsilon$,\n$$\n\\Lambda_{z}^{\\!\\mathsf v}\n\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\n\\end{lemma}\n\\begin{proof} By \\eqref{v-w} and by $(1-R_{z}\\mathsf v)^{-1}\\in \\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$ (see Lemma \\ref{vH-1}), one has\n$$\nR_{z}\\Lambda^{\\mathsf v}_{z}=(1-\\mathsf v R_{z}\\mathsf v)^{-1}\\mathsf v\\in \\mathscr B(H_{-2s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n$$\nBy interpolation (using \\cite{CE}), \n$$\nR_{z}\\Lambda^{\\mathsf v}_{z}\\in \\mathscr B(H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H_{s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\n$$\nand so \n$$\n\\Lambda^{\\mathsf v}_{z}\\in \\mathscr B(H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\n\\end{proof}\nBy the previous Lemma and \\eqref{LbLv}, \n$$\n\\Lambda_{z}^{B_{1}}\\in{\\mathscr B}(H^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma),H^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$ \nThen, by Theorem \\ref{KR} and Corollary \\ref{cor1} (see Remark \\ref{tpt}), $Z_{B_{1}}=\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]$ and\n\\begin{align*}\n(-A_{B_{1}}+z)^{-1}=&R_{z}^{B_{1}}=R_{z}+R_{z}\\langle x\\rangle^{-s}\\Lambda_{z}^{B_{1}}\\langle x\\rangle^{-s}R_{z}\\qquad z\\in\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]\\\\\n=&\nR_{z}+R_{z}\\Lambda_{z}^{\\mathsf v}R_{z}\\\\\n=&R^{\\mathsf v}_{z}=(-(\\Delta+\\mathsf v)+z)^{-1}\\,.\n\\end{align*}\n\\begin{remark}\\label{suff} By the above relation, the self-adjoint operator provided in Corollary \\ref{cor1} by the choice $\\tau_{1}u=\\langle x\\rangle^{-s}u$, $B_{1}=\\langle x\\rangle^{2s}\\mathsf v$, coincides with the one corresponding to $\\tau_{1}u=u$, $B_{1}=\\mathsf v$. The first choice is dictated by the need to obtain  LAP and a representation formula for the scattering couple $(A_{\\mathsf B}, A)$; whenever one is only interested in providing a resolvent formula for $A_{\\mathsf B}$, then the second choice is preferable: in this case, one does not need to work with a short-range potential $\\mathsf v$: a Kato-Rellich potential suffices.\n\\end{remark}\nNow, we prove the validity of the hypotheses (H1)-(H7); here and below we refer to the weighted spaces $L_{s}^{2}(\\mathbb{R}^{3})$ with $1<2s<1+\\varepsilon$. \\par\n\\begin{lemma} Let $\\mathsf v$ be short-range as in \\eqref{short}, with $\\kappa=1$.\nThen hypotheses (H1), (H2), (H6), (H7.1), (H7.2), (H7.3) hold true. \n\\end{lemma}\n\\begin{proof} By \\cite[Lemma 1, page 170]{ReSi IV}, $R_{z}=(-\\Delta+z)^{-1}\\in\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}))$ for any $z\\in \\mathbb{C}\\backslash(-\\infty,0]$. Therefore, by the resolvent identity $R^{\\mathsf v}_{z}=R_{z}(1-\\mathsf v R^{\\mathsf v}_{z})$, $z\\in\\varrho(-(\\Delta+\\mathsf v))$, and by $R^{\\mathsf v}_{z}\\in \\mathscr B(L_{s}^{2}(\\mathbb{R}^{3}), H^{2}(\\mathbb{R}^{3}))$, hypothesis (H1) is consequence of $\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}\\in\\mathscr B(H^{2}(\\mathbb{R}^{3}),L_{s}^{2}(\\mathbb{R}^{3}))$. Since $\\mathsf v_{2}$ has a compact support,  $\\mathsf v_{2}\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}), L_{s}^{2}(\\mathbb{R}^{3}))$ by Lemma \\ref{v}.  As regards $\\mathsf v_{\\infty}$, one has\n\\begin{align*}\n\\|\\mathsf v_{\\infty} u\\|^{2}_{L_{s}^{2}(\\mathbb{R}^{3})}=\\int_{\\mathbb{R}^{3}}|\\mathsf v_{\\infty} u|^{2}(1+|x|^{2})^{s}dx\\le c\\int_{\\mathbb{R}^{3}}(1+|x|)^{-2(1+\\varepsilon)}(1+|x^{2}|)^{s}\n|u|^{2}dx\n\\le \\,c\\,\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n\\end{align*}    \nBy \\cite[Theorem 4.1]{Agmon}, LAP holds for $A=\\Delta$; hence (H7.1) is satisfied. By the sort-range hypothesis on $\\mathsf v$ and by \\cite[Theorem 4.2]{Agmon}, LAP holds for $A_{B_{1}}\\equiv\\Delta+\\mathsf v$ as well and, by \\cite[Theorems 6.1 and 7.1]{Agmon} asymptotic completeness holds for the scattering couple $(A_{B_{1}},A)\\equiv(\\Delta+\\mathsf v,\\Delta)$. Hence hypotheses (H1), (H2)\\footnote{ here ${e}(A_{B_{1}})$ is a discrete set in $(-\\infty,0)$ by \\cite[Theorem 4.2]{Agmon}; it is given by the (possibly empty) set of negative (embedded) eigenvalues of $\\Delta+\\mathsf v$. It is known that there are no negative eigenvalue either if $|\\mathsf v(x)|\\lesssim (1+|x|)^{-(1+\\varepsilon)}$ (see \\cite{Vak}) or if $\\mathsf v\\in L^{3\/2}(\\mathbb{R}^{3})$, i.e., if $\\kappa=2$ in \\eqref{short} (see \\cite{IJ}).} and (H6) are verified.  \\par \nBy $R_{z}\\in\\mathscr B(L^{2}_{-s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$, one gets $G_{z}^{1*}=\\langle x\\rangle^{-s}R_{z}\\in\\mathscr B(L^{2}_{-s}(\\mathbb{R}^{3}),H^{2}(\\mathbb{R}^{3}))$ and so, by duality, $G_{z}^{1}\\in\\mathscr B(H^{-2}(\\mathbb{R}^{3}), L^{2}_{s}(\\mathbb{R}^{3}))$; moreover, by   $R^{\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$ and by a similar duality argument, one gets $G_{\\lambda}^{1,\\pm}\\in\\mathscr B(H^{-2}(\\mathbb{R}^{3}), L^{2}_{-s}(\\mathbb{R}^{3}))$. Thus hypothesis (H7.2) holds.\\par \nBy $\\Lambda_{z}^{\\!B_{1}}=\\langle x\\rangle^{s}\\Lambda_{z}^{\\!\\mathsf v}\\langle x\\rangle^{s}$, hypothesis (H7.3) is equivalent  to the existence in $\\mathscr B(H_{-s}^{2}(\\mathbb{R}^{3}),H_{s}^{-2}(\\mathbb{R}^{3}))$ of $\\lim_{\\epsilon\\searrow 0}\\Lambda_{\\lambda\\pm i\\epsilon}^{\\!\\mathsf v}=\\lim_{\\epsilon\\searrow 0}(1-\\mathsf v R_{\\lambda\\pm i\\epsilon})^{-1}\\mathsf v$. By \\eqref{short}, $\\mathsf v\\in \\mathscr B(H_{-s}^{2}(\\mathbb{R}^{3}),L_{s}^{2}(\\mathbb{R}^{3}))$. Then, $\\lim_{\\epsilon\\searrow 0}(1-\\mathsf v R_{\\lambda\\pm i\\epsilon})^{-1}$ exists in $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}))$ (see \\cite[proof of Theorem XIII.33, page 177]{ReSi IV}) and so (H7.3) holds.\n\\end{proof}\n\\begin{lemma}\\label{5.4} Let $\\mathsf v$ be short-range as in \\eqref{short}, with $\\kappa=2$.\nThen hypothesis (H3) holds true. \n\\end{lemma}\n\\begin{proof} The proof is the same as the one for \\cite[Lemma 4.5]{JDE18}, once one proves that \n\\begin{equation}\\label{once}\n\\mathsf v R^{\\mathsf v,\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{2s}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nSince $R^{\\mathsf v,\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{2s}(\\mathbb{R}^{3}), H^{2}_{-2s}(\\mathbb{R}^{3}))$, \n\\eqref{once} is consequence of \n\\begin{equation}\\label{vw}\n\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}\\in \\mathscr B(H^{2}_{-2s}(\\mathbb{R}^{3}), L^{2}_{2s}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nLemma \\ref{v} entails $\\mathsf v_{2}\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}), L^{2}(\\mathbb{R}^{3}))$ and so, since $\\mathsf v_{2}$ has a compact support, one gets that $\\mathsf v_{2}$ satisfies \\eqref{vw}. As regards $\\mathsf v_{\\infty}$, one has, by $1<2s<1+\\varepsilon$,\n\\begin{align*}\n\\|\\mathsf v_{\\infty} u\\|^{2}_{L^{2}_{2s}(\\mathbb{R}^{3})}=&\\int_{\\mathbb{R}^{3}}|\\mathsf v_{\\infty} u|^{2}(1+|x|^{2})^{2s}dx\\le c\\int_{\\mathbb{R}^{3}}(1+|x|)^{-4(1+\\varepsilon)}(1+|x|^{2})^{4s}\n|u|^{2}(1+|x|^{2})^{-2s}dx\\\\\n\\le& \\,c\\,\\|u\\|^{2}_{L^{2}_{-2s}(\\mathbb{R}^{3})}\\le c\\,\\|u\\|^{2}_{H^{2}_{-2s}(\\mathbb{R}^{3})}\n\\end{align*}    \nand so $\\mathsf v_{\\infty}$ satisfies \\eqref{vw} as well.\n\\end{proof}\nIn order to check the validity of the remaining hypotheses, we need to specify the map \n$\\tau_{2}$. In the following examples, according to the case, we take either \n\\begin{equation}\\label{td}\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to\\mathfrak h_{2}=B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=H^{s_{\\circ}}(\\Gamma)\\,,\\quad 0<s_{\\circ}\\le 1\/2\\,,\n\\end{equation}\nor\n\\begin{equation}\\label{tn}\n\\tau_{2}=\\gamma_{1}:H^{2}(\\mathbb{R}^{3})\\to\\mathfrak h_{2}=H^{1\/2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=H^{-1\/2}(\\Gamma)\\,.\n\\end{equation}\nHence, by what  is recalled in Subsection \\ref{Sec_Layer},  either $G^{2}_{z}=S\\!L_{z}$ or $G^{2}_{z}=D\\!L_{z}$\nand either\n$$\n\\tau G_{z}(u\\oplus\\phi)=\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}u+S_{z}\\phi\n$$\nor\n$$\n\\tau G_{z}(u\\oplus\\phi)=\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}u+D_{z}\\phi\\,.\n$$\nThus \\eqref{tauG} holds.\n\\par\n\\begin{remark} Notice that $\\gamma^{*}_{0}\\phi$ and $\\gamma_{1}^{*}\\phi$, whenever $\\phi\\in L^{2}(\\Gamma)$, identify with the tempered distributions which act on a test function $f$ respectively as\n$$\n(\\phi\\delta_{\\Gamma})f:=\\int_{\\Gamma}\\phi(x)f(x)\\,d\\sigma_{\\Gamma}(x)\\,,\\quad\n(\\phi\\delta'_{\\Gamma})f:=\\int_{\\Gamma}\\phi(x)\\nu(x)\\!\\cdot\\!\\nabla f(x)\\,d\\sigma_{\\Gamma}(x)\\,,\n$$\nwhere $\\nu$ is the exterior normal to $\\Gamma$. By a slight abuse of notation, in the following we set $\\gamma_{0}^{*}\\phi\\equiv \\phi\\delta_{\\Gamma}$ and $\\phi\\gamma_{0}^{*}\\equiv\\delta'_{\\Gamma}\\phi$ and so, either $\\tau^{*}(u\\oplus\\phi)=\\langle x\\rangle^{-s}u+\\phi\\delta_{\\Gamma}$ or $\\tau^{*}(u\\oplus\\phi)=\\langle x\\rangle^{-s}u+\\phi\\delta'_{\\Gamma}$.\n\\end{remark}\n\\begin{lemma} Let $\\mathsf v$ be short-range as in \\eqref{short}, with $\\kappa=1$ and \nlet $\\tau_{2}$ be either as in \\eqref{td} or as in \\eqref{tn}. Then hypotheses (H4.2), (H5) and (H7.4) hold true. \n\\end{lemma}\n\\begin{proof}\nBy the continuity of $z\\mapsto R^{\\pm}_{z}$ as a $\\mathscr B(H^{-1}_{s}(\\mathbb{R}^{3}), H_{-s}^{1}(\\mathbb{R}^{3}))$-valued map, one gets the continuity of $z\\mapsto G^{1,\\pm}_{z}=R^{\\pm}_{z}\\langle x\\rangle^{-s}$ as a $\\mathscr B(H^{-1}(\\mathbb{R}^{3}), H_{-s}^{1}(\\mathbb{R}^{3}))$-valued map. Hence, given $\\chi\\in{\\mathcal C}_{\\text{comp}}^{\\infty}(\\mathbb{R}^{3})$ such that $\\chi=1$ on a compact set containing an open  neighborhood of $\\overline\\Omega$, one gets the continuity of $z\\mapsto \\chi R^{\\pm}_{z}\\langle x\\rangle^{-s}$ as a $\\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}, H^{1}(\\mathbb{R}^{3}))$-valued map. Therefore, $z\\mapsto \\gamma_{0}G^{1,\\pm}_{z}=\\gamma_{0}R^{\\pm}_{z}\\langle x\\rangle^{-s}=\\gamma_{0}\\chi R^{\\pm}_{z}\\langle x\\rangle^{-s}$ is continuos as a $\\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}, H^{1\/2}(\\Gamma))$-valued map.  The continuity of \n$z\\mapsto \\gamma_{1}G^{1,\\pm}_{z}=\\gamma_{1}R^{\\pm}_{z}\\langle x\\rangle^{-s}=\\gamma_{1}\\chi R^{\\pm}_{z}\\langle x\\rangle^{-s}$ as a $\\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}, H^{-1\/2}(\\Gamma))$-valued map follows in an analogous way using the same reasoning as in the proof of Lemma \\ref{tech}. In conclusion,  hypothesis (H7.4) holds true.\\par\nSince $\\Gamma$ is compact, the embeddings $\\mathfrak h_{2}\\hookrightarrow\\mathfrak b_{2}$, where $\\mathfrak h_{2}$ and $\\mathfrak b_{2}$ are as in \\eqref{td} and \\eqref{tn}, are compact by standard results on Sobolev embeddings. \nSince $\\mathsf v\\in\\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),L^{2}_{1+\\varepsilon-(2s+\\gamma)}(\\mathbb{R}^{3}))$ and $(1+\\mathsf v R_{z})^{-1}\\in \\mathscr B(L^{2}(\\mathbb{R}^{3}))$, by taking $\\gamma=1+\\epsilon-2s>0$, one gets $\\Lambda^{\\!\\mathsf v}_{z}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3}))$. Hence, by the resolvent formula \\eqref{RF-int} and by $R_{z}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}))$, one gets $R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}))$. This entails $\\gamma_{0}R_{z}^{B_{1}}=\\gamma_{0}R_{z}^{\\mathsf v}=\\gamma_{0}\\chi R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),B^{2}_{2,2}(\\Gamma))$ and $\\gamma_{1}R_{z}^{B_{1}}=\\gamma_{1}R_{z}^{\\mathsf v}=\\gamma_{1}\\chi R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma))$. Then, by duality, one gets $G_{z}^{B_{1}}\\in \\mathscr B(\\mathfrak h_{2}^{*},L^{2}_{2s+\\gamma}(\\mathbb{R}^{3}))$. This shows that (H4.2) holds. \\par\nBy \\cite[Theorem 4.2]{Agmon}, the map $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}\\ni z\\mapsto R^{B_{1},\\pm}_{z}=R_{z}^{\\mathsf v,\\pm}\\in \\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$ is continuos. Hence, $z\\mapsto\\gamma_{0}R^{B_{1},\\pm}_{z}=\\gamma_{0}R_{z}^{\\mathsf v,\\pm}=\\gamma_{0}\\chi R_{z}^{\\mathsf v,\\pm}$ and  $z\\mapsto\\gamma_{1}R^{B_{1},\\pm}_{z}=\\gamma_{1}R_{z}^{\\mathsf v,\\pm}=\\gamma_{1}\\chi R_{z}^{\\mathsf v,\\pm}$ are continuos as  $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),B^{3\/2}_{2,2}(\\Gamma))$-valued and  $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma))$-valued maps respectively. Then, by duality, $z\\mapsto G^{B_{1},\\pm}$ is continuos on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ as a $\\mathscr B(\\mathfrak h_{2}^{*}, L^{2}_{-s}(\\mathbb{R}^{3}))$-valued map. Since both $\\gamma_{0}:H^{2}(\\mathbb{R}^{2})\\to B^{3\/2}_{2,2}(\\Gamma)$ and $\\gamma_{1}:H^{2}(\\mathbb{R}^{2})\\to H^{1\/2}(\\Gamma)$ are surjective,  $G^{B_{1},\\pm}_{z}\\in \\mathscr B(\\mathfrak h_{2}^{*}, L^{2}_{-s}(\\mathbb{R}^{3}))$ is the adjoint of a surjective map and hence is injective. Thus we proved that (H5) holds.\n\\end{proof}\nIn conclusion, we proved that all our hypotheses, except (H4.1), hold true without the need to specify the operators $B_{0}$ and $B_{2}$. The validity of hypothesis \n(H4.1), i.e. the semi-boundedness of $A_{\\mathsf B}$, will be checked case by case in the next examples. Before turning to such examples, we give a result that makes explicit the map $\\mathcal L_{\\lambda}$ appearing in Theorem \\ref{S-matrix}. \n\\begin{lemma}\\label{Llambda} Let $\\tau_{1}$ be as in \\eqref{tau1} and let $\\tau_{2}$ be either as in \\eqref{td} or as in \\eqref{tn}. \nThen $$L_{\\lambda}:=-\\mathcal L_{\\lambda}\\langle x\\rangle^{s}$$\nis $L^{2}({\\mathbb S}^{2})$-valued, where ${\\mathbb S}^{2}$ denotes the 2-dimensional unitary sphere in $\\mathbb{R}^{3}$, and\n$$\nL_{\\lambda}(u\\oplus\\phi)=\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\,\\left(L^{1}_{\\lambda}u+L^{2}_{\\lambda}\\phi\\right)\\,,\n$$\n$$\nL^{1}_{\\lambda}u(\\xi):=\\widehat{u}\\,(|\\lambda|^{1\/2}\\xi)\n$$\n$$\nL^{2}_{\\lambda}\\phi(\\xi):=\\frac1{(2\\pi)^{\\frac32}}\\,\\begin{cases}\\langle u^{\\xi}_{\\lambda}|\\Gamma,\\phi\\rangle_{H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma)}\\,,&\\tau_{2}=\\gamma_{0}\\,,\\\\\ni\\,|\\lambda|^{\\frac12}\\nu\\!\\cdot\\! \\xi\\,\\langle u^{\\xi}_{\\lambda}|\\Gamma,\\phi\\rangle_{L^{2}(\\Gamma)}\\,,&\\tau_{2}=\\gamma_{1}\\,.\n\\end{cases}\n$$\nHere  $u^{\\xi}_{\\lambda}$ is the plane wave with direction $\\xi\\in{\\mathbb S}^{2}$ and wavenumber $|\\lambda|^{\\frac12}$, i.e., $u^{\\xi}_{\\lambda}(x)=e^{i\\,|\\lambda|^{\\frac12}\\xi\\cdot x}$, $\\nu$ is the normal to $\\Gamma$ and $\\widehat u$ denotes the Fourier transform.\n\\end{lemma}  \n\\begin{proof} The unitary \nmap $F:L^{2}(\\mathbb{R}^{n})\\to \\int^{\\oplus}_{(-\\infty,0)}L^{2}({\\mathbb S}^{2})\\,d\\lambda\\equiv L^{2}((-\\infty,0);L^{2}({\\mathbb S}^{2}))$ diagonalizing $A=\\Delta$ is given by \n\\begin{equation}\\label{fourier}\n(Fu)_\\lambda(\\xi):=-\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\, \\widehat u(|\\lambda|^{1\/2}\\xi)\\,.\n\\end{equation} \nTherefore, by  $(\\mu-\\lambda)\\widehat{ R_{\\mu}f}(|\\lambda|^{1\/2}\\xi)=-\\widehat f(|\\lambda|^{1\/2}\\xi)$,\none gets \n\\begin{align*}\n(\\mu-\\lambda)(FR_{\\mu}\\tau_{1}^{*}\\langle x\\rangle^{s}u)_{\\lambda}(\\xi)=-\n\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\,\\widehat{u}\\,(|\\lambda|^{1\/2}\\xi)\n\\,.\n\\end{align*}\nAs regards $L^{2}_{\\lambda}$, the computation was given in \\cite[Theorem 5.1]{JMPA}. \n\\end{proof}\n\\begin{remark} Let us notice that, whenever $u\\in L^{2}_{s'}(\\mathbb{R}^{3})$, $s'>3\/2$,\n$$\nL^{1}_{\\lambda}u(\\xi)=\\frac1{(2\\pi)^{\\frac32}}\\,\\langle u^{\\xi}_{\\lambda},u\\rangle_{L^{2}_{-s'}(\\mathbb{R}^{3}),L^{2}_{s'}(\\mathbb{R}^{3})}\n$$\nand so $L^{1}_{\\lambda}$ and $L^{2}_{\\lambda}$ have a similar structure.\n\\end{remark}\n\\subsection{\\label{Sec_delta} Short-range potentials and semi-transparent boundary conditions\nof $\\delta_{\\Gamma }$-type} \nHere we take \n$$\n\\mathfrak h_{2}= B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=\\mathfrak b_{2,2}= H^{s_{\\circ}}(\\Gamma)\n\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\,,\\quad 0< s_{\\circ}<1\/2\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to  B^{3\/2}_{2,2}(\\Gamma)\\,,\\qquad\nB_{0}=1\\,\n\\qquad B_{2}=\\alpha\\,,\n$$\nwhere \n$$\n\\alpha\\in\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\,,\\quad \\alpha^{*}=\\alpha\\,.\n$$  \nLet us notice (see \\cite[Remark 2.6]{JDE18}) that in the case $\\alpha$ is the multiplication operator associated to a real-valued function \n$\\alpha$, then $\\alpha\\in L^{p}(\\Gamma)$, $p>2$, fulfills our hypothesis. \\par\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n\\begin{equation}\nM_{z}^{\\mathsf B\n=1-\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & \\alpha%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}  & \\langle x\\rangle^{-s}R_{z}\\gamma_{0}^{*}\\\\\n\\gamma_{0}R_{z} \\langle x\\rangle^{-s}& \\gamma_{0}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{M}_{z}^{\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{equation}\n\\begin{equation}\n{M}_{z}^{\\mathsf v,\\alpha}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v S\\!L_{z}\\\\\n-\\alphaS\\!L_{\\bar z}^{*} & 1-\\alpha S_{z}%\n\\end{bmatrix}\n\\,\n\\end{equation}\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,\\alpha}\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-s_{\\circ}}(\\Gamma))\\,.\n$$\nAccording to \n\\cite[Lemma 5.8]{JMPA}, for any $z\\in \\mathbb{C}\\backslash((-\\infty,0]\\cup\\sigma_{\\alpha})$, where $\\sigma_{\\alpha}\\subset (0,+\\infty)$ is finite, one has\n\\begin{equation}\n(M_{z}^{B_{0},B_{2}})^{-1}=(M_{z}^{\\alpha})^{-1}:=\n(  1-\\alpha S_{z})  ^{-1}\\in{\\mathscr B}(  H^{-s_{\\circ}}(\n\\Gamma )  )  \\,.\\label{S-alpha}%\n\\end{equation}\nThus $$\nZ_{B_{0},B_{2}}=Z_{\\alpha}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]:(M_{w}^{\\alpha})^{-1}\\in{\\mathscr B}(  H^{-s_{\\circ}}(\n\\Gamma )  ) ,\\ w=z,\\bar z\\}\\supseteq \\mathbb{C}\\backslash((-\\infty,0]\\cup\\sigma_{\\alpha})\n$$ \nand \n$$ \n\\Lambda_{z}^{B_{0},B_{2}}=(M_{z}^{B_{0},B_{2}})^{-1}B_{2}=\\Lambda_{z}^{\\!\\alpha}:=(1-\\alpha S_{z})^{-1}\\alpha\\in{\\mathscr B}( H^{s_{\\circ}}(\n\\Gamma ) ,  H^{-s_{\\circ}}(\\Gamma )  ) \\,.\n$$\nBy \\cite[Corollary 2.4]{JDE18}, for any $z\\in \\varrho(\\Delta+\\mathsf v)\\backslash\\sigma_{\\mathsf v,\\alpha}$, where $\\sigma_{\\mathsf v,\\alpha}\\subset\\mathbb{R}$ is finite, \n\\begin{equation}\n(\\widehat M_{z}^{\\mathsf B})^{-1}=(\\widehat M_{z}^{\\mathsf v,\\alpha})^{-1}:=\n(  1-\\alpha S_{z}^{\\mathsf v})  ^{-1}\n\\in{\\mathscr B}(  H^{-s_{\\circ}}(\\Gamma )  )  \n\\,.\\label{Sv-alpha}%\n\\end{equation}\nThus \n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,\\alpha}:=\\{z\\in \\varrho(\\Delta+\\mathsf v):(\\widehat M_{w}^{\\mathsf v,\\alpha})^{-1}\\in{\\mathscr B}(  H^{-s_{\\circ}}(\n\\Gamma )  ) ,\\ w=z,\\bar z\\}\\supseteq \\varrho(\\Delta+\\mathsf v)\\backslash\\sigma_{\\mathsf v,\\alpha}\n$$ \nand\n$$\n\\widehat\\Lambda^{\\mathsf B}_{z}=(\\widehat M_{z}^{\\mathsf B})^{-1}B_{2}=\n\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}:=(1-\\alpha S^{\\mathsf v}_{z} )^{-1}\\alpha\n\\in{\\mathscr B}(H^{s_{\\circ}}(\\Gamma ),   H^{-s_{\\circ}}(\\Gamma )  )\\,.\n$$\nHence,   \n$$\n\\Lambda_{z}^{\\mathsf B}=\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}({M}_{z}^{\\mathsf v,\\alpha})^{-1}\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & \\alpha%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n$$\nwhere, by \\eqref{LB-new} and \\eqref{LB-new2},\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}:=&\n\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}+\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& \\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\n\\\\\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & \n\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nS\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\,.\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbvalpha}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\n\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{s_{\\circ}}(\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-s_{\\circ}}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows \n\\begin{align}\nR_{z}^{\\mathsf v,\\alpha}=&R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\ \\alphaS\\!L_{\\bar z}^{*}\\end{bmatrix}\n\\label{Rv-alpha-0}\n\\\\ \n=&R_{z}+\n\\begin{bmatrix}R_{z}&S\\!L_{z}\\end{bmatrix}\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nS\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\ S\\!L_{\\bar z}^{*}\\end{bmatrix}\n\\label{Rv-alpha-1}\\\\ \n=&R_{z}^{\\mathsf v}+S\\!L_{z}^{\\mathsf v}{\\widehat\\Lambda}_{z}^{\\mathsf v,\\alpha}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}\\,.\n\\label{Rv-alpha-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,\\delta,\\alpha}$; \\eqref{Rv-alpha-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both\n\\eqref{Rv-alpha-1} and \\eqref{Rv-alpha-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\cap\\varrho(\\Delta+\\mathsf v)$. \n\\par\nBy Theorem \\ref{Th-add},\n$$\n\\Delta^{\\!\\mathsf v,\\delta,\\alpha}u=\\Delta u+\\mathsf v u+(\\alpha\\gamma_{0}u)\\delta_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $S\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\subseteq H^{3\/2-s_{\\circ}}(\\mathbb{R}^{3})\\,.\n$$ \nMoreover, by $R^{\\mathsf v}_{z}u\\in H^{2}(\\mathbb{R}^{3})$, so that $[\\gamma_{1}]R^{\\mathsf v}_{z}u=0$, and by \\eqref{jumpv0}, one gets $[\\gamma_{1}]R^{\\mathsf v,\\alpha}_{z}u=-{\\widehat\\Lambda}_{z}^{\\mathsf v,\\alpha}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}u=-\\widehat\\rho_{\\mathsf B}(R^{\\mathsf v,\\alpha}_{z}u)$. Hence, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\quad\\Longrightarrow\\quad\\alpha\\gamma_{0}u+[\\gamma_{1}]u=0\\,.\n$$\nSince $\\widehat Z_{\\mathsf v,\\alpha}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,\\delta,\\alpha}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,\\delta,\\alpha},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,\\alpha}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,\\alpha,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,\\alpha,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,\\alpha}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,\\alpha,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\begin{bmatrix}\nS\\!L^{+}_{\\lambda}\n(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha(S\\!L^{-}_{\\lambda})^{*}& S\\!L^{+}_{\\lambda}\n\\\\(S\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadS\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nS^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}S\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_dirichlet} Short-range potentials and Dirichlet boundary conditions.} \nHere we take \n$$\n\\mathfrak h_{2}= B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=H^{1\/2}(\\Gamma)\n\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow \\mathfrak b_{2,2}=\\mathfrak b_{2}^{*}= H^{-1\/2}(\\Gamma)\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to  B^{3\/2}_{2,2}(\\Gamma)\\,,\\qquad\nB_{0}=0\\,,\\qquad B_{2}=1\\,.\n$$\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n1 & 0\\\\\n0 & 0%\n\\end{bmatrix}-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}  & \\langle x\\rangle^{-s}R_{z}\\gamma_{0}^{*}\\\\\n\\gamma_{0}R_{z}\\langle x\\rangle^{-s} & \\gamma_{0}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}M_{z}^{\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,d}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v S\\!L_{z}\\\\\n-S\\!L_{\\bar z}^{*} & -S_{z}%\n\\end{bmatrix}\n\\,.\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,d}\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n$$\nBy Lemma \\ref{coerc} with $\\mathsf v=0$, for any $z\\in Z^{\\circ}_{0,d}\\not=\\varnothing $, \n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=\\Lambda_{z}^{B_{0},B_{2}}=\n(M_{z}^{d})^{-1}=\\Lambda_{z}^{\\!d}:=-S_{z}^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\nZ_{B_{0},B_{2}}=Z_{d}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]: (M_{z}^{d})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{0,d}\\,.\n$$\nBy Lemma \\ref{coerc} again, for any $z\\in Z^{\\circ}_{\\mathsf v,d}\\not=\\varnothing $, \n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat \\Lambda_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,d})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,d}:=-(S^{\\mathsf v}_{z})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,d}:=\\{z\\in \\varrho(\\Delta+\\mathsf v): (\\widehat M_{z}^{\\mathsf v,d})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{\\mathsf v,d}\\,.\n$$\nHence,\n\\begin{align*}\n\\Lambda^{\\! \\mathsf B}_{z}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}(M_{z}^{\\mathsf v,d})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{align*}\nwhere, by \\eqref{LB-new} and \\eqref{LB-new2},\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,d}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}-\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& -\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}\\\\\n-(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & -(S^{\\mathsf v}_{z})^{-1}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbv-dir}\n{\\Lambda}_{z}^{\\!\\mathsf v,d}\n\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{1\/2}(\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that\n\\begin{align}\n&R_{z}^{\\mathsf v,d}=R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v\\langle x\\rangle^{-s} R_{z}\\\\S\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-dir-0}\n\\\\\n=&R_{z}+\n\\begin{bmatrix}R_{z}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\S\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-dir-1}\n\\\\\n=&R_{z}^{\\mathsf v}-S\\!L^{\\mathsf v}(S_{z}^{\\mathsf v})^{-1}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}\n\\label{Rv-dir-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,d}$; \\eqref{Rv-dir-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,d})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both  \n\\eqref{Rv-dir-1} and \n\\eqref{Rv-dir-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,d})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par\nBy Theorem \\ref{Th-add} and by $[\\gamma_{1}]u=-\\widehat\\rho_{\\mathsf B}u$ for any $u\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})$,\n$$\n\\Delta^{\\!\\mathsf v,d}\\,u=\\Delta u+\\mathsf v u-([\\gamma_{1}]u)\\delta_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $S\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\subseteq H^{1}(\\mathbb{R}^{3})\\,.\n$$ \nMoreover, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\quad\\Longrightarrow\\quad\\gamma_{0}u=0\\,.\n$$\nTherefore, $\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\subseteq H_{0}^{1}(\\Omega_{\\rm in})\\oplus H_{0}^{1}(\\Omega_{\\rm ex})$.\nSince $\\widehat Z_{\\mathsf v,\\alpha}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,d}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,d},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,d}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,d,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,d}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,d,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,d}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,d,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(S^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\n-S\\!L^{+}_{\\lambda}\n(S^{\\mathsf v,+}_{\\lambda})^{-1}(S\\!L^{-}_{\\lambda})^{*}& S\\!L^{+}_{\\lambda}\n\\\\(S\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(S^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadS\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nS^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}S\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_neumann} Short-range potentials and Neumann boundary conditions.} \nHere we take \n$$\n\\mathfrak h_{2}=\\mathfrak b_{2}^{*}=\\mathfrak b_{2,2}= H^{1\/2}(\\Gamma)\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=\\mathfrak h^{*}_{2}=\\mathfrak b_{2,2}^{*}=H^{-1\/2}(\\Gamma)\n\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{1\/2}(\\Gamma)\\,,\\qquad\nB_{0}=0\\,,\\qquad B_{2}=1\\,.\n$$\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n1 & 0\\\\\n0 & 0%\n\\end{bmatrix}-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}  & \\langle x\\rangle^{-s}R_{z}\\gamma_{1}^{*}\\\\\n\\gamma_{1}R_{z}\\langle x\\rangle^{-s} & \\gamma_{1}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s}  & 0\\\\\n0 & 1\n\\end{bmatrix}M_{z}^{\\mathsf v,n}\n\\begin{bmatrix}\\langle x\\rangle^{s}  & 0\\\\\n0 & 1%\n\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,n}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v D\\!L_{z}\\\\\n-D\\!L_{\\bar z}^{*} & -D_{z}%\n\\end{bmatrix}\n\\,.\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,n}\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma),\nH_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n$$\nBy Lemma \\ref{coerc} with $\\mathsf v=0$, for any $z\\in Z^{\\circ}_{0,n}\\not=\\varnothing $, \n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=\\Lambda_{z}^{B_{0},B_{2}}=\n(M_{z}^{n})^{-1}=\\Lambda_{z}^{\\!n}:=-D_{z}^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\nZ_{B_{0},B_{2}}=Z_{n}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]: (M_{z}^{n})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{0,n}\\,.\n$$\nBy Lemma \\ref{coerc} again, for any $z\\in Z^{\\circ}_{\\mathsf v,n}\\not=\\varnothing $, \n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat \\Lambda_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,n})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,n}:=-(D^{\\mathsf v}_{z})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{n}:=\\{z\\in \\varrho(\\Delta+\\mathsf v): (\\widehat M_{z}^{\\mathsf v,n})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{\\mathsf v,n}\\,.\n$$\nHence, \n\\begin{align*}\n\\Lambda^{\\! \\mathsf B}_{z}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}(M_{z}^{\\mathsf v,n})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,n}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{align*}\nwhere, by \\eqref{LB-new} and by \\eqref{LB-new2}\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,n}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}-\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& -\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}\\\\\n-(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & -(D^{\\mathsf v}_{z})^{-1}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbv-neu}\n{\\Lambda}_{z}^{\\!\\mathsf v,n}\n\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that \n\\begin{align}\n&R_{z}^{\\mathsf v,n}\n=R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,n}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\ D\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-neu-0}\\\\ \n=&R_{z}+\n\\begin{bmatrix}R_{z}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\!\\!\\!-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\!\\!\\!-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\ D\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-neu-1}\\\\ \n=&R_{z}^{\\mathsf v}-D\\!L_{z}^{\\mathsf v}(D_{z}^{\\mathsf v})^{-1}{D\\!L^{\\mathsf v}_{\\bar z}}^{*}\n\\label{Rv-neu-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,n}$; \\eqref{Rv-neu-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,n})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both \\eqref{Rv-neu-1} and \\eqref{Rv-neu-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,n})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par \nBy Theorem \\ref{Th-add} and by $[\\gamma_{0}]u=\\widehat\\rho_{\\mathsf B}u$ for any $u\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})$,\n$$\n\\Delta^{\\!\\mathsf v,n}\\,u=\\Delta u+\\mathsf v u+([\\gamma_{0}]u)\\delta'_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $D\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})\\subseteq H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\\,.\n$$ \nMoreover, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})\\quad\\Longrightarrow\\quad\\gamma_{1}u=0\\,.\n$$\nSince $\\widehat Z_{\\mathsf v,n}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,n}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,n},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,n}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,n,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,n}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,n,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,n}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,n,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\n-D\\!L^{+}_{\\lambda}\n(D^{\\mathsf v,+}_{\\lambda})^{-1}(D\\!L^{-}_{\\lambda})^{*}& D\\!L^{+}_{\\lambda}\n\\\\(D\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadD\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}D\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nD^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{1}D\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_delta'} Short-range potentials and semi-transparent boundary conditions\nof $\\delta'_{\\Gamma }$-type} \nHere we take \n$$\n\\mathfrak h_{2}=\\mathfrak b_{2}^{*}=\\mathfrak b_{2,2}= H^{1\/2}(\\Gamma)\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow \\mathfrak b_{2}=\\mathfrak h^{*}_{2}=\\mathfrak b_{2,2}^{*}=H^{-1\/2}(\\Gamma)\n\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{1\/2}(\\Gamma)\\,,\\qquad\nB_{0}=\\theta\\,,\\qquad B_{2}=1\\,,\n$$\nwhere \n$$ \n\\theta\\in\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\,,\\quad \n0<s_{\\circ}<1\/2\\,,\\quad\\theta^{*}=\\theta\\,.\n$$\nLet us notice (see \\cite[Remark 2.6]{JDE18}) that in the case $\\theta$ is the multiplication operator associated to a real-valued function \n$\\theta$, then $\\theta\\in L^{p}(\\Gamma)$, $p>2$, fulfills our hypothesis. \nLet us also remark that $\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\subseteq\n\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))=\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})$.\\par\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n 1  & 0\\\\\n0 & \\theta%\n\\end{bmatrix}\n-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}  & \\langle x\\rangle^{-s}R_{z}\\gamma_{1}^{*}\\\\\n\\gamma_{1}R_{z}\\langle x\\rangle^{-s} & \\gamma_{1}R_{z}\\gamma_{1}^{*}%\n\\end{bmatrix}=\n\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \nM_{z}^{\\mathsf v,\\theta}\n\\begin{bmatrix}\\langle x\\rangle^{-s} & 0\\\\\n0 & 1\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,\\theta}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v D\\!L_{z}\\\\\n-D\\!L_{\\bar z}^{*} & \\theta-D_{z}\n\\end{bmatrix}\\,.%\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$,  one gets\n$$\nM_{z}^{\\mathsf v,\\theta}\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma),H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n$$\n\\begin{lemma}\\label{LLt} \n$$\n\\forall z\\in \\widehat Z^{\\circ}_{\\mathsf v,n}:=Z^{\\circ}_{\\mathsf v,n}\\cap\\mathbb{C}\\backslash\\mathbb{R}\\,,\\qquad (1-\\theta (D^{\\mathsf v}_{z})^{-1})^{-1}\\in \\mathscr B(  H^{-1\/2}(  \\Gamma)) \\,.\n$$\n\\end{lemma}\n\\begin{proof}\nWe follow the same the arguments as in the proof of \\cite[Lemma 5.4]{JMPA}. Since, by the compact embedding $H^{-s_{\\circ}}(\\Gamma)\\hookrightarrow H^{-1\/2}(\\Gamma)$, $\\theta(  D_{z}^{\\mathsf v})^{-1}\\in{\\mathscr B}(H^{-1\/2}(\\Gamma))$ is compact, \nby the Fredholm alternative, $1-\\theta(  D_{z}^{\\mathsf v})  ^{-1}$ has a bounded inverse if and only if it has trivial kernel. Let $\\varphi\\in H^{-1\/2}(\\Gamma)$ be such that $D_{z}^{\\mathsf v}\\varphi=\\theta\\varphi$; using the self-adjointness of $\\theta$, we get%\n\\[\n(  D_{z}^{\\mathsf v}-D_{\\bar z}^{\\mathsf v})  \\varphi=0\\,.\n\\]\nBy the resolvent identity,\n\\[\n\\text{Im}(z)\\gamma_{1}R_{\\bar z}^{\\mathsf v}R_{z}^{\\mathsf v}%\n\\gamma_{1}^{\\ast}\\varphi=0\\,.\n\\]\nThis gives \n\\begin{equation}\n\\|R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\varphi\\| _{L^{2}(\\mathbb{R}^{3})}=0\\,.\n\\end{equation}\nSince $(R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast})^{\\ast}=\\gamma_{1}R_{\\bar\n{z}}^{\\mathsf v}\\in{\\mathscr B}(  L^{2}(  \\mathbb{R}^{3}),H^{1\/2}(\\Gamma))  $ is surjective, then\n$R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\in{\\mathscr B}(  H^{-1\/2}(  \\Gamma)  ,L^{2}(  \\mathbb{R}^{3}))  $ has closed\nrange by the closed range theorem and, by \\cite[Theorem 5.2, p. 231]{Kato},\n\\[\n\\|R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\varphi\\|_{L^{2}(\n\\mathbb{R}^{3})  }\\gtrsim\\|\\varphi\\|_{H^{-1\/2}(\\Gamma)  }\\,.\n\\]\nThus $\\text{\\rm ker}(1-\\theta (D_{z}^{\\mathsf v})^{-1})=\\{0\\}$ and the proof is done.\n\\end{proof}\nAccording to Lemma \\ref{LLt} with $\\mathsf v=0$, for any $z\\in \\widehat Z^{\\circ}_{0,n}\\not=\\varnothing $,\n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=(M_{z}^{\\theta})^{-1}=\\Lambda_{z}^{\\!\\theta}:=(\\theta-D_{z})^{-1}=\n-D_{z}^{-1}(1-\\theta D_{z}^{-1})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus\n$$\nZ_{B_{0},B_{2}}=Z_{\\theta}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]:(M_{z}^{\\theta})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq\\widehat Z^{\\circ}_{0,n}\\,.\n$$\nAccording to Lemma \\ref{LLt} again, for any $z\\in \\widehat Z^{\\circ}_{\\mathsf v,n}\\not=\\varnothing $,\n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,\\theta})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,\\theta}:=(\\theta-D_{z}^{\\mathsf v})^{-1}=\n-(D^{\\mathsf v}_{z})^{-1}(1-\\theta (D^{\\mathsf v}_{z})^{-1})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,\\theta}:=\\{z\\in \\varrho(\\Delta+\\mathsf v):(\\widehat M_{z}^{\\mathsf v,\\theta})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq\\widehat Z^{\\circ}_{\\mathsf v,n}\\,.\n$$\nHence,  \n$$\n\\Lambda_{z}^{\\mathsf B}=\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \n(M_{z}^{\\mathsf v,\\theta})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s} & 0\\\\\n0 & 1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v  & 0\\\\\n0 & 1\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \n\\Lambda_{z}^{\\!\\mathsf v,\\theta}\n\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix}\\,,\n$$\nwhere, by \\eqref{LB-new} and by \\eqref{LB-new2},\n\\begin{align*}\n\\Lambda_{z}^{\\!\\mathsf v,\\theta}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v,\\theta}\\Lambda^{\\!\\mathsf v}_{z}+\\Lambda^{\\!\\mathsf v}_{z} D\\!L_{z}\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}D\\!L_{\\bar z}^{*}\\Lambda^{\\!\\mathsf v}_{z}\n& \\Lambda^{\\mathsf v}_{z} D\\!L_{z}\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}\\\\\n\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}D\\!L_{\\bar z}^{*}\\Lambda^{\\!\\mathsf v}_{z}& \\widehat\\Lambda^{\\mathsf v,\\theta}_{z}\\end{bmatrix}\\\\\n=&\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nD\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\,.\n\\end{align*}\nOne has \n\\begin{equation}\\label{Ltbbteta}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\theta}\n\\in{\\mathscr B}(H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}(  \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that\n\\begin{align}\nR_{z}^{\\mathsf v,\\theta}\n=&R_{z}+\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\theta}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\D\\!L^{*}_{\\bar z}\\end{bmatrix}\n\\label{Rv-teta-0}\\\\\n=&R_{z}+\n\\begin{bmatrix}R_{z}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nD\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\D\\!L^{*}_{\\bar z}\\end{bmatrix}\n\\label{Rv-teta-1}\\\\ \n=&R_{z}^{\\mathsf v}+D\\!L_{z}^{\\mathsf v}{\\widehat\\Lambda}_{z}^{\\mathsf v,\\theta}{D\\!L_{\\bar z}^{\\mathsf v}}^{*}\\,.\n\\label{Rv-teta-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}$; \\eqref{Rv-teta-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both \\eqref{Rv-teta-1} and \\eqref{Rv-teta-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par\nBy Theorem \\ref{Th-add},\n$$\n\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}u=\\Delta u+\\mathsf v u+(\\theta\\gamma_{1}u)\\delta'_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $D\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\subseteq H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\\,.\n$$ \nMoreover, by $R^{\\mathsf v}_{z}u\\in H^{2}(\\mathbb{R}^{3})$, so that $[\\gamma_{1}]R^{\\mathsf v}_{z}u=0$, and by \\eqref{jumpv1}, one gets $[\\gamma_{0}]R^{\\mathsf v,\\theta}_{z}u={\\widehat\\Lambda}_{z}^{\\mathsf v,\\theta}{D\\!L_{\\bar z}^{\\mathsf v}}^{*}u=\\widehat\\rho_{\\mathsf B}(R^{\\mathsf v,\\theta}_{z}u)$. Hence, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\quad\\Longrightarrow\\quad\\gamma_{1}u=\\theta[\\gamma_{0}]u\\,.\n$$\nProceeding as in \\cite[Subsection 5.5]{JMPA} (see the proof of Theorem 5.15 there), $\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}$ is bounded from above and so hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,\\theta}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,\\theta,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,\\theta,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,\\theta}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,\\theta,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\nD\\!L^{+}_{\\lambda}\n(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha(D\\!L^{-}_{\\lambda})^{*}& D\\!L^{+}_{\\lambda}\n\\\\(D\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadD\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nD^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}D\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere is a commonly accepted idea that the sunspot activity is produced\nby the large-scale toroidal magnetic field which is generated inside\nthe convection zone by means of the differential rotation \\citep{P55}.\nThe theory explains the 11-year solar cycle as a result of the large-scale\ndynamo operating in the solar interior, where, in addition to the\nmagnetic fields generated by the differential rotation, the helical\nconvective motions transforms the energy of the toroidal magnetic\nfields to poloidal. The effect of meridional circulation on the large-scale\ndynamo is not well understood. It is the essential part of the flux-transport\ndynamo model scenario \\citep{choud95,cd99} to explain the equatorward\ndrift of the toroidal magnetic field in the solar cycle. Here, it\nassumed that the toroidal field at the bottom of the convection zone\nforms sunspot activity. Feasibility of this idea can be questioned\nboth the observational and theoretical arguments \\citep{b05}. The\ndistributed dynamo models can be constructed with \\citep{2002AA...390..673B,2008AA...483..949J,2014AA563A18P}\nand without \\citep{moss00M,pip13M} effect of meridional transport\nof the large-scale magnetic field.\n\nRecent results of helioseismology reveal the double-cell meridional\ncirculation structure \\citep{Zhao13m,2017ApJ845.2B}. It demolishes\nthe previously accepted scenario of the flux-transport models \\citep{2014ApJ782.93H,2016MNRAS.456.2654W,2016ApJ832.9H}.\nContrary, \\citet{PK13} showed the distributed dynamo models can reproduce\nobservations with regards to the subsurface rotational shear layer\nand the double-cell meridional circulation. In their model, the double-cell\nmeridional circulation was modeled in following to results of helioseismology\nof \\citet{Zhao13m}. The effect of the multi-cell meridional circulation\non the global dynamo was also studied in the direct numerical simulations\n\\citep{kap2012,2016ApJ819.104G,2018AA609A..51W}. There were no attempts\nto construct the non-kinematic mean-field dynamo models with regards\nto the multi-cell meridional circulation.\n\nThe standard mean-field models of the solar differential rotation\npredict a one-cell meridional circulation per hemisphere. This contradicts\nto the helioseismology inversions and results of direct numerical\nsimulations. In the mean-field theory framework, the differential\nrotation of the Sun is explained as a result of the angular momentum\ntransport by the helical convective motions. Similarly to a contribution\nof the $\\alpha$ effect in the mean-electromotive force, i.e., \n\\[\n\\mathbf{\\mathcal{E}}=\\left\\langle \\mathbf{u}\\times\\mathbf{b}\\right\\rangle =\\hat{\\alpha}\\circ\\left\\langle \\mathbf{B}\\right\\rangle +\\dots,\n\\]\nwhere $\\mathbf{u}$ is the turbulent velocity $\\mathbf{u}$, and $\\mathbf{b}$\nis the turbulent magnetic field, the $\\Lambda$-effect, (e.g., \\citealp{1989drsc.book.....R})\nappears as the non-dissipative part of turbulent stresses\\textbf{\n\\[\n\\hat{T}_{ij}=\\left\\langle u_{i}u_{j}\\right\\rangle =\\Lambda_{ijk}\\Omega_{k}+\\dots\n\\]\n}where $\\boldsymbol{\\Omega}$ is the angular velocity. The structure\nof the meridional circulation is determined by directions of the non-diffusive\nangular momentum transport due to the $\\Lambda$ effect \\citep{2017ApJ835.9B}.\nIn particular, the vertical structure of the meridional circulation\ndepends on the sign of the radial effect. It was found that the double-cell\nmeridional circulation can be explained if the of $\\Lambda$-effect\nchanges sign in the depth of the convection zone. \\citet{2018ApJ854.67P}\nshowed that this effect can result from the radial inhomogeneity of\nthe convective turnover timescale. It was demonstrated that if this\neffect is taken into account then the solar-like differential rotation\nand the double-cell meridional circulation are both reproduced by\nthe mean-field model .\n\nIn this paper, we apply the meridional circulation profile, which\nis calculated from the solution of the angular momentum balance to\nthe nonkinematic dynamo models. Previously, the similar approach was\napplied by \\citet{1992AA...265..328B} and \\citet{2006ApJ...647..662R}\nin the distributed and the flux-transport models with one meridional\ncirculation cell as the basic stage in the non-magnetic case.\n\nOur main goal is to study how the double-cell meridional circulation\naffects the nonlinear dynamo generation of the large-scale magnetic\nfield. The magnetic feedback on the global flow can result in numerous\nphysical phenomena such as the torsional oscillations \\citep{1982SoPh...75..161L,2011JPhCS271a2074H},\nthe long-term variability of the magnetic activity \\citep{sok1994AA,2014JGRA119.6027F}\netc. The properties of the nonlinear evolution depend on the dynamo\ngoverning parameters such as amplitude of turbulent generation of\nthe magnetic field by the $\\alpha$ effect, as well as the other nonlinear\nprocesses involved in the dynamo, i.e., the dynamo quenching by the\nmagnetic buoyancy effect \\citep{kp93,tob98} and the magnetic helicity\nconservation \\citep{kleruz82}. We study if the long-term variation\nof magnetic activity can result from the increasing level of turbulent\ngeneration of magnetic field by the $\\alpha$ effect. The increasing\nof the $\\alpha$ effect results to an increase of the magnetic helicity\nproduction. This affects the large-scale magnetic field generation\nby means of the magnetic helicity conservation. Hence, the magnetic\nhelicity balance has to be taken into account. \n\n{It is hardly possible to consider in full all the goals within\none paper. From our point of view, the most important tasks includes:\nconstruction of the solar-type dynamo model with the multi-cell meridional\ncirculation and studying the principal nonlinear dynamo effects. The\nlatter includes the magnetic helicity conservation and the nonkinematic\neffects due to the magnetic feedback on the large-scale flow. Accordingly,\nthe paper is organized as follows.} Next Section describes the hydrodynamic,\nthermodynamic and magnetohydrodynamic parts of the model. Then, I\npresent an attempt to construct the solar-type dynamo model and discuss\nthe effect of the turbulent pumping on the properties of the dynamo\nsolution. {The next subsections consider result for the principal\nnonlinear dynamo effects. They deals with the global flows variations,\nthe Grand activity cycles and the magnetic cycle variations of the\nthermodynamic parameters in the model.} The paper is concluded with\na discussion of the main results using results of other theoretical\nstudies and results of observations.\n\n\\section{Basic equations.}\n\n\\subsection{The angular momentum balance\\label{subsec:am}}\n\nWe consider the evolution of the axisymmetric large-scale flow, which\nis decomposed into poloidal and toroidal components: $\\mathbf{\\overline{U}}=\\mathbf{\\overline{U}}^{m}+r\\sin\\theta\\Omega\\hat{\\mathbf{\\boldsymbol{\\phi}}}$,\nwhere $\\boldsymbol{\\hat{\\phi}}$ is the unit vector in the azimuthal\ndirection. The mean flow satisfies the stationary continuity equation,\n\\begin{equation}\n\\boldsymbol{\\nabla}\\cdot\\overline{\\rho}\\mathbf{\\overline{U}}=0,\\label{eq:cont}\n\\end{equation}\nDistribution of the angular velocity inside convection zone is determined\nby conservation of the angular momentum \\citep{1989drsc.book.....R}:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}\\overline{\\rho}r^{2}\\sin^{2}\\theta\\Omega & = & -\\boldsymbol{\\nabla\\cdot}\\left(r\\sin\\theta\\overline{\\rho}\\left(\\hat{\\mathbf{T}}_{\\phi}+r\\sin\\theta\\Omega\\mathbf{\\overline{U}^{m}}\\right)\\right)\\label{eq:angm}\\\\\n & + & \\boldsymbol{\\nabla\\cdot}\\left(r\\sin\\theta\\frac{\\overline{\\mathbf{B}}\\overline{B}_{\\phi}}{4\\pi}\\right).\\nonumber \n\\end{eqnarray}\nTo determine the meridional circulation we consider the azimuthal\ncomponent of the large-scale vorticity , $\\omega=\\left(\\boldsymbol{\\nabla}\\times\\overline{\\mathbf{U}}^{m}\\right)_{\\phi}$\n, which is governed by equation: \n\\begin{eqnarray}\n\\frac{\\partial\\omega}{\\partial t}\\!\\!\\! & \\negthinspace\\!=\\!\\!\\!\\! & r\\sin\\theta\\boldsymbol{\\nabla}\\cdot\\left(\\frac{\\hat{\\boldsymbol{\\phi}}\\times\\boldsymbol{\\nabla\\cdot}\\overline{\\rho}\\hat{\\mathbf{T}}}{r\\overline{\\rho}\\sin\\theta}-\\frac{\\mathbf{\\overline{U}}^{m}\\omega}{r\\sin\\theta}\\right)\\!\\!+r\\sin\\theta\\frac{\\partial\\Omega^{2}}{\\partial z}\\label{eq:vort}\\\\\n & +\\!\\!\\! & \\frac{1}{\\overline{\\rho}^{2}}\\left[\\boldsymbol{\\nabla}\\overline{\\rho}\\times\\boldsymbol{\\nabla}\\overline{p}\\right]_{\\phi}\\!\\!\\nonumber \\\\\n & + & \\!\\frac{1}{\\overline{\\rho}^{2}}\\left[\\!\\!\\boldsymbol{\\nabla}\\overline{\\rho}\\times\\left(\\!\\!\\boldsymbol{\\nabla}\\frac{\\overline{\\mathbf{B}}^{2}}{8\\pi}-\\frac{\\left(\\overline{\\mathbf{B}}\\boldsymbol{\\cdot\\nabla}\\right)\\overline{\\mathbf{B}}}{4\\pi}\\!\\right)\\!\\!\\right]_{\\phi},\\nonumber \n\\end{eqnarray}\nThe turbulent stresses tensor, $\\hat{\\mathbf{T}}$, is written in\nterms of small-scale fluctuations of velocity and magnetic field:\n\\begin{equation}\n\\hat{T}_{ij}=\\left(\\left\\langle u_{i}u_{j}\\right\\rangle -\\frac{1}{4\\pi\\overline{\\rho}}\\left(\\left\\langle b_{i}b_{j}\\right\\rangle -\\frac{1}{2}\\delta_{ij}\\left\\langle \\mathbf{b}^{2}\\right\\rangle \\right)\\right).\\label{eq:stres}\n\\end{equation}\nwhere ${\\partial\/\\partial z=\\cos\\theta\\partial\/\\partial r-\\sin\\theta\/r\\cdot\\partial\/\\partial\\theta}$\nis the gradient along the axis of rotation. The turbulent stresses\naffect generation and dissipation of large-scale flows, and they are\naffected by the global rotation and magnetic field. The magnitude\nof the kinetic coefficients in tensor $\\hat{\\mathbf{T}}$ depends\non the rms of the convective velocity, ${u}'$, the strength of the\nCoriolis force and the strength of the large-scale magnetic field.\nThe effect of the Coriolis force is determined by parameter $\\Omega^{*}=2\\Omega_{0}\\tau_{c}$,\nwhere $\\Omega_{0}=2.9\\times10^{-6}$rad\/s is the solar rotation rate\nand $\\tau_{c}$ is the convective turnover time. The effect of the\nlarge-scale magnetic field on the convective turbulence is determined\nby parameter $\\beta=\\left\\langle \\left|\\mathbf{B}\\right|\\right\\rangle \/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$.\n\nThe magnetic feedback on the coefficients of turbulent stress tensor\n$\\hat{\\mathbf{T}}$ was studied previously with the mean-field magnetohydrodynamic\nframework \\citep{rob-saw}. In our model we apply analytical results\nof \\citet{1994AN....315..157K}, \\citet{kit-rud-kuk} and \\citet{kuetal96}.\nThe analytical expression for $\\hat{\\mathbf{T}}$ is given in Appendix.\nIt was found that the standard components of the nondissipative of\n$\\hat{\\mathbf{T}}$ ($\\Lambda$-effect) are quenched with the increase\nof the magnetic field strength as $\\beta^{-2}$ and the magnetic quenching\nof the viscous parts is the order of $\\beta^{-1}$. Also, there is a non-trivial\neffect inducing the latitudinal angular momentum flux proportional\nto the magnetic energy \\citep{kit-rud-kuk,kuetal96}. This effect\nis quenched as $\\beta^{-2}$ for the case of the strong magnetic field.\nImplications of the magnetic feedback on the turbulent stress tensor\n$\\hat{\\mathbf{T}}$ were discussed in the models of solar torsional\noscillations and Grand activity cycles \\citep{kit-rud-kuk,kuetal96,p99,1999AA343.977K}.\nThe analytical results of the mean-field theory are in qualitative\nagreement with the direct numerical simulations \\citep{2007AN....328.1006K,kap2011,2017arXiv171208045K}.\n\nProfile of $\\tau_{c}$ (as well as profiles of $\\overline{\\rho}$\nand other thermodynamic parameters) is obtained from a standard solar\ninterior model calculated using the MESA code \\citep{mesa11,mesa13}.\nThe rms velocity, $u'$, is determined in the mixing length approximations\nfrom the gradient of the mean entropy, $\\overline{s}$, \n\\begin{equation}\nu'=\\frac{\\ell}{2}\\sqrt{-\\frac{g}{2c_{p}}\\frac{\\partial\\overline{s}}{\\partial r}},\\label{eq:uc}\n\\end{equation}\nwhere $\\ell=\\alpha_{MLT}H_{p}$ is the mixing length, $\\alpha_{MLT}=2.2$\nis the mixing length theory parameter, and $H_{p}$ is the pressure\nscale height. For a non-rotating star the ${u}'$ profile corresponds\nto results of the MESA code. The mean-field equation for heat transport\ntakes into account effects of rotation and magnetic field \\citep{2000ARep...44..771P}:\n\\begin{equation}\n\\overline{\\rho}\\overline{T}\\left(\\frac{\\partial\\overline{s}}{\\partial t}+\\left(\\overline{\\mathbf{U}}\\cdot\\boldsymbol{\\nabla}\\right)\\overline{s}\\right)=-\\boldsymbol{\\nabla}\\cdot\\left(\\mathbf{F}^{conv}+\\mathbf{F}^{rad}\\right)-\\hat{T}_{ij}\\frac{\\partial\\overline{U}_{i}}{\\partial r_{j}}-\\frac{1}{4\\pi}\\boldsymbol{\\mathcal{E}}\\cdot\\nabla\\times\\boldsymbol{\\overline{B}},\\label{eq:heat}\n\\end{equation}\nwhere, $\\overline{\\rho}$ and $\\overline{T}$ are the mean density\nand temperature, $\\boldsymbol{\\mathcal{E}}=\\left\\langle \\mathbf{u\\times b}\\right\\rangle $\nis the mean electromotive force. The Eq.(\\ref{eq:heat}) includes\nthe thermal energy loss and gain due to generation and dissipation\nof large-scale flows. The last term of the Eq.(\\ref{eq:heat}) takes\ninto account effect of thermal energy exchange because of dissipation\nand generation of magnetic field \\citep{2000ARep...44..771P}. In\nderivation of the mean-field heat transport equation (see, \\citealp{2000ARep...44..771P}),\nit was assumed that the magnetic and rotational perturbations of the\nreference thermodynamic state are small. Also the parameters of the\nreference state are given independently by the MESA code.\n\nFor the anisotropic convective flux we employ the expression suggested\nby \\citet{1994AN....315..157K} (hereafter KPR94), \n\\begin{equation}\nF_{i}^{conv}=-\\overline{\\rho}\\overline{T}\\chi_{ij}\\nabla_{j}\\overline{s}.\\label{conv}\n\\end{equation}\nThe further details about dependence of the eddy conductivity tensor\n$\\chi_{ij}$ from effects of both the global rotation and large-scale\nmagnetic field are given in Appendix. The diffusive heat transport\nby radiation reads, \n\\[\n\\mathbf{F}^{rad}=-c_{p}\\overline{\\rho}\\chi_{D}\\boldsymbol{\\nabla}T,\n\\]\nwhere \n\\[\n\\chi_{D}=\\frac{16\\sigma\\overline{T}^{3}}{3\\kappa\\overline{\\rho}^{2}c_{p}}.\n\\]\nBoth the eddy conductivity and viscosity are determined from the mixing-length\napproximation: \n\\begin{eqnarray}\n\\chi_{T} & = & \\frac{\\ell^{2}}{4}\\sqrt{-\\frac{g}{2c_{p}}\\frac{\\partial\\overline{s}}{\\partial r}},\\label{eq:chi}\\\\\n\\nu_{T} & = & \\mathrm{Pr_{T}}\\chi_{T},\\label{eq:nu}\n\\end{eqnarray}\nwhere $\\mathrm{Pr_{T}}$ is the turbulent Prandtl number. {Note,\nthat in Eq\\eqref{eq:chi} we employ factor $1\/2$ instead of $1\/3$.\nWith this choice the distribution of the mean entropy gradient, which\nresults from solution of the Eq\\eqref{eq:heat} for the nonrotating\nand nonmagnetic case is close to results of the MESA code.} It is\nassumed that $\\mathrm{Pr_{T}}={\\displaystyle 3\/4}$. {This\ncorresponds to the theoretical results of KPR94}. For this choice\nwe have the good agreement with solar angular velocity latitudinal\nprofile. We assume that the solar rotation rate corresponds to rotation\nrate of solar tachocline at 30$^{\\circ}$ latitude, i.e., $\\Omega_{0}\/2\\pi=430$nHz\n\\citep{1997SoPh170.43K}. We employ the stress-free boundary conditions\nin the hydrodynamic part of the problem. For the Eq(\\ref{eq:heat})\nthe thermal flux at the bottom is taken from the MESA code. At the\ntop, the thermal flux from the surface is approximated by the flux\nfrom a blackbody: \n\\begin{equation}\nF_{r}=\\frac{L_{\\odot}}{4\\pi r^{2}}\\left(1+4\\frac{T_{e}}{T_{eff}}\\frac{\\overline{s}}{c_{p}}\\right),\\label{eq:flx}\n\\end{equation}\nwhere where $T_{eff}$ is the effective temperature of the photosphere\nand $T_{e}$ is the temperature at the outer boundary of the integration\ndomain.\n\nFigure \\ref{fig:sun-flow} shows profiles of the angular velocity,\nstreamlines of the meridional circulations and the radial profiles\nof the angular velocity and the meridional flow velocity for a set\nof latitudes. The given results were discussed in details by \\citet{2018ApJ854.67P}.\nThe model shows the double-layer circulation pattern with the upper\nstagnation point at $r=0.88R_{\\odot}$. The amplitude of the surface\npoleward flow is about 15m\/s. The angular velocity profile shows a\nstrong subsurface shear that is higher at low latitudes and it is\nless near poles. Contrary to results of \\citet{Zhao13m} and model\nof \\citet{PK13} the double-cell meridional circulation structure\nextends from equator to pole. This is partly confirmed by the new\nresults of helioseismology by \\citet{2017ApJ849.144C} who also found\nthat the poleward flow at the surface goes close enough to pole. {It\nis important no mention that the current results of the helioseismic\ninversions for the meridional circulation remains controversial For\nexample, \\citet{2015ApJ813.114R} found that the meridional circulation\ncan be approximated by a single-cell structure with the return flow\ndeeper than 0.77R$_{\\odot}$. However, their results indicate an additional\nweak cell in the equatorial region, and contradict to the recent results\nof \\citet{2017ApJ845.2B} who confirmed a shallow return flow at 0.9R$_{\\odot}$.\nAlso, their results indicated that the upper meridional circulation\ncell extends close to the solar pole.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{fig1}\n\n\\caption{\\label{fig:sun-flow}a) angular velocity profile, $\\Omega\\left(r,\\theta\\right)\/2\\pi$,\ncontours are in range of 327-454 nHz; b) the radial profiles of the\nangular velocity for latitudes: $\\varphi=0^{\\circ}$, $30^{\\circ}$\nand $60^{\\circ}$; c) streamlines of the meridional circulation; d)\nradial profile of the meridional flow at $\\theta=45^{\\circ}$.}\n\\end{figure}\n\n\n\\subsection{Dynamo equations}\n\nWe model evolution of the large-scale axisymmetric magnetic field,\n$\\overline{\\mathbf{B}}$, by the mean-field induction equation \\citep{KR80},\n\\begin{equation}\n\\partial_{t}\\overline{\\mathbf{B}}=\\boldsymbol{\\nabla}\\times\\left(\\boldsymbol{\\mathcal{E}}+\\mathbf{\\overline{U}}\\times\\overline{\\mathbf{B}}\\right),\\label{eq:mfe-1}\n\\end{equation}\nwhere, $\\boldsymbol{\\mathcal{E}}=\\left\\langle \\mathbf{u\\times b}\\right\\rangle $\nis the mean electromotive force with $\\mathbf{u}$ and $\\mathbf{b}$\nstanding for the turbulent fluctuating velocity and magnetic field\nrespectively.\n\nSimilar to our recent paper (see, \\citep{2014ApJ_pipk,2017MNRAS.466.3007P}),\nwe employ the mean electromotive force in form: \n\\begin{equation}\n\\mathcal{E}_{i}=\\left(\\alpha_{ij}+\\gamma_{ij}\\right)\\overline{B}_{j}-\\eta_{ijk}\\nabla_{j}\\overline{B}_{k}.\\label{eq:EMF-1-1}\n\\end{equation}\nwhere symmetric tensor $\\alpha_{ij}$ models the generation of magnetic\nfield by the $\\alpha$- effect; antisymmetric tensor$\\gamma_{ij}$\ncontrols the mean drift of the large-scale magnetic fields in turbulent\nmedium, including the magnetic buoyancy; the tensor $\\eta_{ijk}$\ngoverns the turbulent diffusion. The reader can find further details\nabout the $\\boldsymbol{\\mathcal{E}}$ in the above cited papers.\n\nThe $\\alpha$ effect takes into account the kinetic and magnetic helicities\nin the following form: \n\\begin{eqnarray}\n\\alpha_{ij} & = & C_{\\alpha}\\eta_{T}\\psi_{\\alpha}(\\beta)\\alpha_{ij}^{(H)}+\\alpha_{ij}^{(M)}\\frac{\\overline{\\chi}\\tau_{c}}{4\\pi\\overline{\\rho}\\ell^{2}},\\label{alp2d-2}\\\\\n\\eta_{T} & = & \\frac{\\nu_{T}}{\\mathrm{Pm_{T}}}\n\\end{eqnarray}\nwhere $C_{\\alpha}$ is a free parameter which controls the strength\nof the $\\alpha$- effect due to turbulent kinetic helicity; tensors\n$\\alpha_{ij}^{(H)}$ and $\\alpha_{ij}^{(M)}$ express the kinetic\nand magnetic helicity parts of the $\\alpha$-effect, respectively;\n$\\mathrm{Pm_{T}}$ is the turbulent magnetic Prandtl number, and $\\overline{\\chi}=\\left\\langle \\mathbf{a}\\cdot\\mathbf{b}\\right\\rangle $\n($\\mathbf{a}$ and $\\mathbf{b}$ are the fluctuating parts of magnetic\nfield vector-potential and magnetic field vector). Both the $\\alpha_{ij}^{(H)}$\nand the $\\alpha_{ij}^{(M)}$ depend on the Coriolis number. Function\n$\\psi_{\\alpha}(\\beta)$ controls the so-called ``algebraic'' quenching\nof the $\\alpha$- effect where $\\beta=\\left|\\overline{\\mathbf{B}}\\right|\/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$,\n$u'$ is the RMS of the convective velocity. It is found that $\\psi_{\\alpha}(\\beta)\\sim\\beta^{-3}$\nfor $\\beta\\gg1$. The $\\alpha$- effect tensors $\\alpha_{ij}^{(H)}$\nand $\\alpha_{ij}^{(M)}$ are given in Appendix. \n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{fig2}\n\n\\caption{\\label{fig:alp} a) the radial profiles of the total (solid line)\nand anisotropic (dashed line) parts of the eddy diffusivity at $\\theta=45^{\\circ}$;\nb) radial profiles of the kinetic $\\alpha$-effect components at $\\theta=45^{\\circ}$;\nc) the equipartition strength of the magnetic field, $B_{eq}=\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$,\nwhere $u'$ is determined by the equatorial profile of the mean entropy,\nsee the Eq(\\ref{eq:uc}); the dashed line is from results of the reference\nmodel (MESA code) and the solid line is for the rotating convection\nzone,i.e., after solution of the Eq(\\ref{eq:heat}).}\n\\end{figure}\n\nContribution of the magnetic helicity to the $\\alpha$-effect is expressed\nby the second term in Eq.(\\ref{alp2d-2}). The evolution of the turbulent\nmagnetic helicity density, $\\overline{\\chi}=\\left\\langle \\mathbf{a}\\cdot\\mathbf{b}\\right\\rangle $,\nis governed by the conservation law \\citep{pip13M}:\n\n\\begin{eqnarray}\n\\frac{\\partial\\overline{\\chi}}{\\partial t} & = & -2\\left(\\boldsymbol{\\mathcal{E}}\\cdot\\overline{\\bm{B}}\\right)-\\frac{\\overline{\\chi}}{R_{m}\\tau_{c}}+\\boldsymbol{\\nabla}\\cdot\\left(\\eta_{\\chi}\\boldsymbol{\\nabla}\\bar{\\chi}\\right)\\label{eq:hel-1}\\\\\n &  & -\\eta\\overline{\\mathbf{B}}\\cdot\\mathbf{\\overline{J}}-\\boldsymbol{\\nabla}\\cdot\\left(\\boldsymbol{\\mathcal{E}}\\times\\overline{\\mathbf{A}}\\right),\\nonumber \n\\end{eqnarray}\nwhere $R_{m}=10^{6}$ is the magnetic Reynolds number and $\\eta$\nis the microscopic magnetic diffusion. In the drastic difference to\nanzatz of \\citet{kleruz82}, the Eq(\\ref{eq:hel-1}) contains the\nterm $\\left(\\boldsymbol{\\mathcal{E}}\\times\\overline{\\mathbf{A}}\\right)$.\nIt consists of the magnetic helicity density fluxes which result from\nthe large-scale magnetic dynamo wave evolution. The given contribution\nalleviates the catastrophic quenching problem \\citep{hub-br12,pip13M}.\nAlso the catastrophic quenching of the $\\alpha$-effect can be alleviated\nwith help of the diffusive flux of the turbulent magnetic helicity,\n$\\boldsymbol{\\boldsymbol{\\mathcal{F}}}^{\\chi}=-\\eta_{\\chi}\\boldsymbol{\\nabla}\\bar{\\chi}$\n\\citep{guero10,chatt11}. The coefficient of the turbulent helicity\ndiffusivity, $\\eta_{\\chi}$, is a parameter in our study. It affects\nthe hemispheric helicity transfer \\citep{mitra10}.\n\nIn the model we take into account the mean drift of large-scale field\ndue to the magnetic buoyancy, $\\gamma_{ij}^{(buo)}$ and the gradient\nof the mean density, $\\gamma_{ij}^{(\\Lambda\\rho)}$: \n\\begin{eqnarray}\n\\gamma_{ij} & = & \\gamma_{ij}^{(\\Lambda\\rho)}+\\gamma_{ij}^{(buo)},\\nonumber \\\\\n\\gamma_{ij}^{(\\Lambda\\rho)} & = & 3C_{pum}\\eta_{T}\\left(f_{1}^{(a)}\\left(\\mathbf{\\boldsymbol{\\Omega}}\\cdot\\boldsymbol{\\Lambda}^{(\\rho)}\\right)\\frac{\\Omega_{n}}{\\Omega^{2}}\\varepsilon_{inj}-\\frac{\\Omega_{j}}{\\Omega^{2}}\\varepsilon_{inm}\\Omega_{n}\\Lambda_{m}^{(\\rho)}\\right)\\label{eq:pump1}\\\\\n\\gamma_{ij}^{(buo)} & = & -\\frac{\\alpha_{MLT}u'}{\\gamma}\\beta^{2}K\\left(\\beta\\right)g_{n}\\varepsilon_{inj},\\nonumber \n\\end{eqnarray}\nwhere $\\mathbf{\\boldsymbol{\\Lambda}}^{(\\rho)}=\\boldsymbol{\\nabla}\\log\\overline{\\rho}$\n; functions $f_{1}^{(a)}$ and $K\\left(\\beta\\right)$ are given in\n\\cite{kp93,2017MNRAS.466.3007P}. The standard choice of the pumping\nparameter is $C_{pum}=1$. In this case the pumping velocity is scaled\nin the same way as the magnetic eddy diffusivity. In the presence\nof the multi-cell meridional circulation, the direction and magnitude\nof the turbulent pumping become critically important for the modelled\nevolution of the magnetic field. It is confirmed in the direct numerical\nsimulations, as well (see, \\cite{2018AA609A..51W}). For the standard\nchoice, the turbulent pumping is about an order of magnitude less than\nthe meridional circulation. For this case, explanation of the latitudinal\ndrift of the toroidal magnetic field near the surface faces a problem\n(cf., \\cite{PK13}). To study the effect of turbulent pumping we\nintroduce this parameter $C_{pum}$.\n\nFor the bottom boundary we apply the perfect conductor boundary conditions:\n$\\mathcal{E}_{\\theta}=0,\\,A=0$. The boundary conditions at the top\nare defined as follows. Firstly, following ideas of \\citet{1992AA256371M}\nand \\citet{pk11apjl} we formulate the boundary condition in the form\nthat allows penetration of the toroidal magnetic field to the surface:\n\\begin{eqnarray}\n\\delta\\frac{\\eta_{T}}{r_{e}}B+\\left(1-\\delta\\right)\\mathcal{E}_{\\theta} & = & 0,\\label{eq:tor-vac}\n\\end{eqnarray}\nwhere $r_{e}=0.99R_{\\odot}$, and parameter $\\delta=0.99$. The magnetic\nfield potential in the outside domain is \n\\begin{equation}\nA^{(vac)}\\left(r,\\mu\\right)=\\sum a_{n}\\left(\\frac{r_{e}}{r}\\right)^{n}\\sqrt{1-\\mu^{2}}P_{n}^{1}\\left(\\mu\\right).\\label{eq:vac-dec}\n\\end{equation}\n\nThe coupled angular momentum and dynamo equations are solved using\nfinite differences for integration along the radius and the pseudospectral\nnodes for integration in latitude. The number of mesh points in radial\ndirection was varied from $100$ to $150$. The nodes in latitude\nare zeros of the Legendre polynomial of degree ${N}$, where N was\nvaried from ${N=64}$ to ${N=84}$. The resolution with 64 nodes in\nlatitude and with 100 points in radius was found satisfactory. The\nmodel employed the Crank-Nicolson scheme, using a half of the time-step\nfor integration in the radial direction and another half for integration\nalong latitude.\n\nTo quantify the mirror symmetry type of the toroidal magnetic field\ndistribution relative to equator we introduce the parity index $P$:\n\\begin{eqnarray}\nP & = & \\frac{E_{q}-E_{d}}{E_{q}+E_{d}},\\label{eq:parity}\\\\\nE_{d} & = & \\int\\left(B\\left(r_{0},\\theta\\right)-B\\left(r_{0},\\pi-\\theta\\right)\\right)^{2}\\sin\\theta d\\theta,\\nonumber \\\\\nE_{q} & = & \\int\\left(B\\left(r_{0},\\theta\\right)+B\\left(r_{0},\\pi-\\theta\\right)\\right)^{2}\\sin\\theta d\\theta,\\nonumber \n\\end{eqnarray}\nwhere $E_{d}$ and $E_{q}$ are the energies of the dipole-like and\nquadruple-like modes of the toroidal magnetic field at $r_{0}=0.9R_{\\odot}$.\nAnother integral parameter is the mean density of the toroidal magnetic\nfield in the subsurface shear layer: \n\\begin{equation}\n\\overline{B^{T}}=\\sqrt{E_{d}+E_{q}}.\\label{eq:bt}\n\\end{equation}\nAnother parameter characterize the mean strength of the dynamo processes\nin the convection zone: \n\\begin{equation}\n\\overline{\\beta}=\\left\\langle \\left|\\overline{\\mathbf{B}}\\right|\/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}\\right\\rangle ,\\label{eq:bet}\n\\end{equation}\nwhere the averaging is done over the convection zone volume. The boundary\nconditions Eq(\\ref{eq:tor-vac}) provide the Poynting flux of the\nmagnetic energy out of the convection zone. Taking into account the\nEq(\\ref{eq:flx}) the variation of the thermal flux at the surface\nare given as follows: \n\\begin{eqnarray}\n\\delta F & = & \\delta F_{c}+\\delta F_{B}\\label{eq:dF}\\\\\n\\delta F_{c} & = & 4\\frac{T_{e}}{T_{eff}}\\frac{\\delta\\overline{s}}{c_{p}}\\label{eq:flxb}\\\\\n\\delta F_{B} & = & \\frac{1}{4\\pi}\\left(\\mathcal{E}_{\\phi}\\overline{B}_{r}-\\mathcal{E}_{\\theta}\\overline{B}_{\\phi}\\right),\\label{eq:fcfb}\n\\end{eqnarray}\nwhere $\\delta\\overline{s}$ is the entropy variation because of the\nmagnetic activity. The second term of the Eq(\\ref{eq:dF}) governs\nthe magnetic energy input in the stellar corona.\n\n\\section{Results}\n\nTo match the solar cycle period we put $\\mathrm{Pm_{T}}=10$ in all\nour models. {The theoretical estimations of \\citet{1994AN....315..157K}\ngives $\\mathrm{Pm_{T}}=4\/3$. This is the long standing theoretical\nproblem of the solar dynamo period \\citep{brsu05}. Currently, the\nsolar dynamo period can be reproduced for $\\mathrm{Pm_{T}}\\gg1$.\nThe issue exists both in the distributed and in the flux-transport\ndynamo. Moreover, the flux-transport dynamo can reproduce the observation\nonly with the special radial profile of the eddy diffusivity (see,\ne.g., \\citep{2006ApJ...647..662R}). In our models we employ the rotational\nquenching the eddy diffusivity coefficients and the high $\\mathrm{Pm_{T}}$.\nFigures \\ref{fig:alp}a and b show the radial profiles of the eddy\ndiffusivity coefficients and components of the $\\alpha_{ij}^{(H)}$\nat latitude $45^{\\circ}$ in our model for $\\mathrm{Pm_{T}}=10$.\nIn the upper part of the convection zone the magnitude of the turbulent\nmagnetic diffusivity is close to estimations of \\citet{1993AA...274..521M}\nbased on observations of the sunspot decay rate. The eddy diffusivity\nis an order of $10^{10}$cm\/s and less near the bottom of the convection\nzone. The diffusivity profile is the same as in our previous paper\n\\citet{2014ApJ_pipk}. }\n\n{The radial profiles of the $\\alpha$ effect for $C_{\\alpha}=C_{\\alpha}^{(cr)}$\nare illustrated in Figure \\ref{fig:alp}b. The $\\alpha$ effect (cf,\nthe above discussion about $\\Lambda$-effect) change the sign near\nthe bottom of the convection zone. This is also found in the direct\nnumerical simulations \\citep{2006AA455.401K}.}\n\nTable \\ref{tab:C} gives the list of our models, their control and\noutput parameters. We sort the models with respect to magnitude of\nthe $\\alpha$ effect using the ratio ${\\displaystyle \\frac{C_{\\alpha}}{C_{\\alpha}^{(cr)}}}$,\nthe magnitude of the eddy-diffusivity of the magnetic helicity density,\n${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$, where $\\eta_{T}=\\nu_{T}\/\\mathrm{Pm_{T}}$,\nand $\\nu_{T}$ is determined from Eq(\\ref{eq:nu}), and with respect\nof the magnetic feedback on the differential rotation.{ The\n$C_{pum}$ controls the pumping velocity magnitude (see, Eq\\eqref{eq:pump1});\nthe parameter ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$ show\nthe relative difference of the surface angular velocity between the\nsolar equator and pole; the strength of the dynamo is characterized\nby the range of the magnetic cycle variations of $\\overline{\\beta}$\n(see, Eq\\eqref{eq:bet}); the dynamo cycle period; the magnitude of\nthe surface meridional circulation. From the Table 1 we see that the\nnonkinematic runs show the magnetic cycle variations of ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$\nand the surface meridional circulation. }\n\nFigure \\ref{fig:alp}b shows the radial profiles of the equipartition\nstrength of the magnetic field, $B_{eq}=\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$\nin the solar convection zone for the reference model (non-rotating)\ngiven by MESA code and in the rotating convective zone. In the rotating\nconvection zone, the mean-entropy gradient is larger than in the nonrotating\ncase. {This is because of the rotational quenching of the eddy-conductivity.\nThe magnitude of the convective heat flux is determined by the boundary\ncondition at the bottom of the convection zone and it remains the\nsame for the rotating (our model) and nonrotating (MESA code) cases.\nAssuming that the convective turnover time is not subjected to the\nrotational quenching, the reduction of the eddy conductivity because\nof the rotational quenching is compensated by the increase of the\nmean-entropy gradient.} This results in the increase of the parameter\n$B_{eq}$.\n\n{The increase of the RMS convective velocity in case of the\nrotating convection zone seems to contradict the results of direct numerical\nsimulations of \\cite{2016AA596A.115W}. This is\nlikely because of inconsistent assumptions behind the MLT expression\nfor the RMS convective velocity, see Eq(\\ref{eq:uc}). The given issue\ncan affect the amplitude of the dynamo generated magnetic field near the\nbottom of the convection zone. Our models operate in regimes where\n$\\left|B\\right|\\le B_{eq}$, and the substantial part of the dynamo\nquenching is due to magnetic helicity conservation. Therefore the\ngiven issue does not much affect our results.}\n\n\n\n\\begin{table}\n\\caption{\\label{tab:C}Control and output parameters of the dynamo models.}\n\\begin{tabular}{>{\\centering}p{1cm}>{\\centering}p{2cm}>{\\centering}p{1cm}>{\\centering}p{2cm}>{\\centering}p{1.5cm}>{\\centering}p{2cm}>{\\centering}p{1cm}>{\\centering}p{1cm}}\n\\toprule \nModel  & $C_{pum}$  & ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$  & ${\\displaystyle \\frac{C_{\\alpha}}{C_{\\alpha}^{(cr)}}}$  & ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$  & $\\overline{\\beta}$  & \\begin{centering}\nPeriod \n\\par\\end{centering}\n{[}YR{]}  & $\\max U_{\\theta}$\n\n{[}M\/S{]}\\tabularnewline\n\\midrule \nM1  & 1  & $0.1$  & 1.1  & 0.279  & 0.05-0.1 & 6  & 15\\tabularnewline\n\\midrule \nM2  & Pm$_{T}$  & 0.1  & 1.1  & 0.279  & 0.13-0.24  & 10.5  & 15\\tabularnewline\n\\midrule \nM4  & -\/-  & 0.3  & -\/-  & 0.279  & 0.15-0.3  & 10.5,12.05  & 15\\tabularnewline\n\\midrule \nM5  & -\/-  & 0.01  & -\/-  & 0.279 & 0.14-0.26  & 9.3  & 15\\tabularnewline\n\\midrule \n &  &  & Nonkinematic & runs &  &  & \\tabularnewline\n\\midrule \nM3  & Pm$_{T}$  & 0.1 & 1.1  & 0.263-0.275  & 0.11-0.21  & 10.3  & 15.0\n\n$\\pm0.5$\\tabularnewline\n\\midrule \nM3a2  & -\/-  & -\/-  & 2  & 0.232-0.253  & 0.36-0.66  & 4.7,265  & 14.0\n\n$\\pm2.1$\\tabularnewline\n\\midrule \nM3a3  & -\/-  & -\/-  & 3  & 0.22-0.24  & 0.52-0.91  & 4.0  & 14.5 $\\pm2.5$\\tabularnewline\n\\midrule \nM3a4  & -\/-  & -\/-  & 4  & 0.21-0.245  & 0.68-1.05  & 3.4  & 14.7\n\n$\\pm3.$5\\tabularnewline\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Effects of turbulent pumping}\n\nAs the first step, we consider the kinematic dynamo model with the\nnonlinear $\\alpha$ effect. Results of the model M1 are shown in Figure\n\\ref{pumpa}. The model M1 roughly agree with results of \\citet{PK13}\n(hereafter, PK13). It employs the same mean electromotive force as\nin our previous paper. In particular, the maximum pumping velocity\nis the order of 1m\/s. The effective velocity drift due to the magnetic\npumping and meridional circulation is shown in Figures \\ref{pumpa}(a)\nand (b). Figures \\ref{pumpa}(d) and (e) show the time-latitude variations\nof the toroidal magnetic field at $r=0.9R$ and in the middle of the\nconvection zone. The agreement with the solar observations is worse\nthan in the previous model PK13 because of difference in the meridional\ncirculation structure. The model of PK13 employed the meridional circulation\nprofile provided by results of \\citet{Zhao13m}. In that profile,\nthe near-surface meridional circulation cell is more shallow and it\ndoes not touch the pole as it happens in the present model, see, Figure\n\\ref{fig:sun-flow}. By this reason, the polar magnetic field in model\nM1 is much larger than in results of PK13.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m1}\n\n\\caption{\\label{pumpa}a) Direction of pumping velocity of the toroidal magnetic\nfield in model M1; b) the effective velocity drift of the toroidal\nmagnetic field (pumping + meridional circulation); c) the snapshot\nof the toroidal magnetic field distribution (color image) and streamlines\nof the poloidal magnetic field in the Northern hemisphere of the Sun;\nd) the time-latitude diagram of the toroidal magnetic field evolution\n(contours in range of of $\\pm500$G at $r=0.9R$ and radial magnetic\nfield at the surface (color image).}\n\\end{figure}\n\nFor the purpose of our study, it is important to get the properties\nof the dynamo solution as close as possible to results of solar observations.\nTo solve the above issues we increase the turbulent pumping velocity\nmagnitude by factor $\\mathrm{Pm_{T}}$. The results are shown in Figure\n\\ref{pumpb}. The model has the correct time-latitude diagram of the\ntoroidal magnetic field in the subsurface shear layer. The surface\nradial magnetic field evolves in agreement with results of observations\n\\citep{2013AARv2166S}. The magnitude of the polar magnetic field\nis 10 G, which is in a better agreement with observations (e.g., \\cite{2007AAS...210.2405L})\nthan the model M1. Figures \\ref{pumpb} (a) and (b) show the effective\nvelocity drift of the large-scale toroidal magnetic field. The equatorward\ndrift with magnitude the  order of 1-2 m\/s operates in major part of the\nsolar convection zone from $0.75R$ to $0.91R$. Interesting that\nthe obtained results are similar to those from the direct numerical\nsimulation of \\citet{2018AA609A..51W}. Note that in the given model\nthe magnitude of the pumping velocity is about factor 2 less than\nin results of \\citet{2018AA609A..51W}. It seems that some of the\nissues in model M1 would be less pronounced if the meridional circulation\npattern was closer to results of helioseismology of \\citet{Zhao13m}\nor \\citet{2017ApJ849.144C}. However results of the direct numerical\nsimulations of \\citep{2016AA596A.115W,2018AA609A..51W} seem to show\nthat the evolution of the large-scale magnetic field inside convection\nzone does not depend much on the meridional circulation. This argues\nfor the strong magnetic pumping effects in the global dynamo. This\nimportant issue can be debated further. This is out of the main scopes\nof this paper. The rest of our models employ the same pumping effect\nas in the model M2 (see, Table \\ref{tab:C}). \n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m2}\n\n\\caption{\\label{pumpb}a) Direction of pumping velocity of the toroidal magnetic\nfield in model M2; b) the effective velocity drift of the toroidal\nmagnetic field (pumping + meridional circulation); c) the snapshot\nof the toroidal magnetic field distribution (color image) and streamlines\nof the poloidal magnetic field in the Northern hemisphere of the Sun;\nd) the time-latitude diagram of the toroidal magnetic field evolution\n(contours in range of of $\\pm1$kG at $r=0.9R$ and radial magnetic\nfield at the surface (color image).}\n\\end{figure}\n\n\n\\subsection{The global flows oscillations in magnetic cycle}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m3}\n\n\\caption{\\label{fig:s-bat}The model M3, a) Time-latitude butterfly diagram\nfor the toroidal field in the upper part of the convection zone (color\nimage) and the surface radial magnetic field shown by contours ($\\pm$5G);\nb) the surface variations of the azimuthal velocity (color image)\nand the meridional velocity (contours in the range of $\\pm0.5$ m\/s). }\n\\end{figure}\n\nFigure \\ref{fig:s-bat} show the time-latitude diagrams of the magnetic\nfield and the global flow variations for the model M3. The torsional\noscillations on the surface are about $\\pm2$m\/s. They are defined\nas follows, $\\delta U_{\\phi}=\\left(\\Omega\\left(r,\\theta t\\right)-\\overline{\\Omega\\left(r,\\theta,t\\right)}\\right)r\\sin\\theta$,\nwhere the averaging is done over the stationary phase of evolution.\nThe torsional wave has both the equator- and poleward branches. In\nthe equatorward torsional wave, the change from the positive to negative\nvariation goes about 2 years ahead of the maxima of the toroidal magnetic\nfield wave. This agrees with results of observations of \\citet{2011JPhCS271a2074H}\nand with direct numerical simulations of \\citet{2016ApJ828L.3G}.\nThe magnitude of the meridional flow variations agrees with results\nof \\citet{2014ApJ789L7Z}. Also, we see that on the surface the meridional\nvelocity variations converge toward the maximum of the toroidal magnetic\nfield wave. This is also in qualitative agreement with the observations.\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{m3s}\\caption{\\label{fig:M3}The model M3: a) snapshots of the magnetic field in\nfour phase of the magnetic cycle, the toroidal magnetic field strength\nis shown by color, contours show streamlines of the poloidal magnetic\nfield; b) color image show variations of the angular velocity, contours\n(range of $\\pm0.5$m\/s) show variations of the meridional flow; c)\ncontours show the azimuthal component of the total (kinetic and magnetic\nhelicity parts)$\\alpha$ -effect, the background image shows the part\nof the $\\alpha_{\\phi\\phi}$ induced by the magnetic helicity conservation\n(see, the second term of Eq(\\ref{alp2d-2})).}\n\\end{figure}\n\nFigure \\ref{fig:M3} shows snapshots of the magnetic field, the global\nflows variations and the azimuthal component of the total (kinetic\nand magnetic helicity parts)$\\alpha$ -effect for a half magnetic\ncycle. The Figure shows that a new cycle starts at the bottom of the\nconvection zone. The main part of the dynamo wave drifts to surface\nequatorward. There is a polar branch which propagates poleward along\nthe bottom of the convection zone. The torsional oscillations, as\nwell as, the meridional flow variations are elongated along the axis\nof rotation. This can be interpreted as a result of mechanical perturbation\nof the Taylor-Proudman balance \\citep{2006ApJ...647..662R}. We postpone\nthe detailed analysis of the torsional oscillation to another paper.\n{Figure \\ref{fig:M3}b shows that maxima of the meridional\nflow variations are located at the upper boundary of the dynamo domain.}\nThis is because the main drivers of the meridional circulation, which\nare the baroclinic forces, have the maximum near the boundaries of\nthe solar convection zone \\citep{rem2005ApJ,2015ApJ...804...67F,2017arXiv170202421P}.\nIn comparing Figures \\ref{fig:M3}c and \\ref{fig:alp} it is seen\nthat the dynamo wave affect the $\\alpha$ -effect. Also, in agreement\nwith our previous model \\citep{pip2013ApJ}, we find that the magnetic\nhelicity conservation results into increasing the $\\alpha$ -effect\nin the subsurface shear layer. It occurs just ahead of the dynamo\nwave drifting toward the top. The given effect support the equatorward\npropagation of the large-scale toroidal field in subsurface shear\nlayer \\citep{kap2012}.\n\nThe increasing the $\\alpha$-effect parameter results in a number\nof consequences for the non-linear evolution of the large-scale magnetic\nfield. The dynamo period is decreasing with the increase of the $\\alpha$-effect\n\\citep{pk11}. The magnitude of the dynamo wave increases with the\nincrease of the parameter $C_{\\alpha}$.{ Therefore, our models\nshow that in the distributed solar-type dynamo the dynamo period can\ndecrease with the increase of the magnetic activity level. This is\nin agreement with the results of the stellar activity observations\nof \\citet{1984ApJ...287..769N,2009AA_strassm,2017PhDT3E}. Here we\nfor the first time demonstrate this effect in the distributed dynamo\nmodel with the meridional circulation. }Figure \\ref{fig:m8} shows\nresults for the model M3a4 with ${\\displaystyle C_{\\alpha}=4C_{\\alpha}^{(cr)}}$.\nThe model shows the solar-like dynamo waves in the subsurface shear\nlayer. The toroidal magnetic field reaches the strength of 3kG in\nthe upper part of the convection zone. Simultaneously, the polar magnetic\nfield has the maximum strength of 100 G. Variations of the zonal and\nmeridional flows on the surface are about of factor 6 larger than\nin the model M3. The model M3a4 show a high level magnetic activity\nwith a strong toroidal magnetic field in the subsurface layer and\nvery strong polar field. {Results of stellar observations show\nthat this is expected on the young solar analogs and late K-dwarfs\nas well (e.g., \\citep{2005LRSP2.8B,2016MNRAS1129S}). However, the\ngiven results can not be consistent with those cases because in our\nmodel the rotation rate is much slower than for the young solar-type\nstars. Results of the linear models show that the internal differential\nrotation and meridional circulation change with increase of the stellar\nangular velocity \\citep{kit11}.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m8}\n\n\\caption{\\label{fig:m8}The model M3a4, a) Time-latitude butterfly diagram\nfor the toroidal field in the upper part of the convection zone (color\nimage) and the surface radial magnetic field shown by contours ($\\pm$100G);\nb) the surface variations of the azimuthal velocity (color image)\nand the meridional velocity (contours in the range of $\\pm3.5$ m\/s). }\n\\end{figure}\n\nFigure \\ref{fig:M3a4dr} shows snapshots of the global flows distributions\nin the solar convection zone. In drastic difference to the model M3,\nthe counter-clockwise meridional circulation cell in the upper part\nof the convection zone is divided into two parts. Also, the stagnation\npoint of the bottom cell is shifted equatorward.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\columnwidth]{m8dr}\n\n\\caption{\\label{fig:M3a4dr}The model M3a4: a) the snapshot of angular velocity\nprofile, $\\Omega\\left(r,\\theta\\right)\/2\\pi$, contours are in range\nof 347-450 nHz; b) the streamlines of the meridional circulation;\nc) the radial profile of the meridional flow at $\\theta=45^{\\circ}$;\nd) the profiles of the meridional flow at the specific depths of the\nsolar convection zone.}\n\\end{figure}\n\nFigure \\ref{fig:M3a4} shows the magnetic cycle variations of the\nglobal flows in the Northern segment of the solar convection zone\nfor the model M3a4. The patterns of these variations are qualitatively\nthe same as in the model M3. The model M3a4 show the strong magnetic\ncycle variations of the $\\alpha$ effect. Similar to the model M3\nwe see that the magnetic helicity conservation results into increasing\nthe $\\alpha$ -effect in the subsurface shear layer. It occurs just\nahead of the dynamo wave drifting toward the top. In the polar regions,\nthe $\\alpha$ -effect inverses the sign during inversion of the polar\nmagnetic field. This is different from the model M3 which has the smaller\nstrength of the polar magnetic field than the model M3a4.\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{mf6}\\caption{\\label{fig:M3a4}The same as Figure \\ref{fig:M3} for the model M3a4.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{f7a}\n\n\\caption{\\label{fig:avar}a) Variations of the equatorial symmetry (parity\nindex, see, Eq(\\ref{eq:parity})); b) the same as (a) for the mean\ndensity of the toroidal magnetic field flux in the subsurface shear\nlayer. The time series were smoothed to filter out the basic magnetic\ncycle.}\n\\end{figure}\n\n\n\\subsection{The long-term dynamo evolution}\n\nFigure \\ref{fig:avar} shows the smoothed time series of evolution\nof the global properties of the dynamo model, such as the equatorial\nsymmetry index, or the parity index $P$, (see, Eq(\\ref{eq:parity}))\nand the mean density of the toroidal magnetic field flux, $\\overline{B^{T}}$,\nin the subsurface shear layer, see, Eq(\\ref{eq:bt}). In each time\nseries, the basic magnetic cycle was filtered out. The set of models\nshown in Figure (\\ref{fig:avar}) illustrates the effect of variations\nof magnitude of the eddy-diffusivity of the magnetic helicity density,\n${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$. From results of \\citet{mitra10},\nit is expected that ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}<1$.\nIn our set of models it is $0.01<{\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}<0.3$.\nThe magnetic helicity diffusion affects the magnetic helicity exchange\nbetween hemispheres \\citep{2000JGR...10510481B,mitra10}. Therefore\nit affects an interaction of the dynamo waves through the solar equator.\nIt is found that the increasing of ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$\nresults into change of the parity index $P$. The model M4 show the\nsymmetric about equator magnetic field. The magnitude of $\\overline{B^{T}}$\nin the model M4 is larger than in the models M3 and M5. Both models\nM3 and M5 operate in a weak nonlinear regime with $0.13<\\overline{\\beta}<0.24$\n(see, Eq(\\ref{eq:bet}) and Table(\\ref{tab:C})). The model M4 has\na slightly higher $\\overline{\\beta}$. This means that the magnetic\nhelicity diffusion affects the strength of the dynamo. This is in\nagreement with \\citet{guero10}. The smoothed time series of $\\overline{B^{T}}$\nin the model M4 show oscillations at the end of the evolution. This\nis because the period of the symmetric dynamo mode ( $P=1$) is about\n12 years that is larger than the period of the basic magnetic cycle\nfor antisymmetric mode $\\left(P=-1\\right)$ . The latter is about\n10.5 years.\n\nIt is interesting to compare the kinematic and nonkinematic dynamo\nmodels, which are the models M2 and M3. The nonkinematic model M3\nhas the smaller parameters $\\overline{B^{T}}$and $\\overline{\\beta}$\nthan the model M2. Also, it is found that in the kinematic model M2\nthe stationary phase of evolution is the mix of the dipole-like and\nquadrupole-like parity, with the mean $P\\approx-0.9$. In the model\nM3 the mean $P\\approx-1$. Therefore the nonkinematic dynamo regimes\naffect the equatorial symmetry of the dynamo solution \\citep{bran89,p99}.\n{Figure \\ref{fig:avar}b shows another interesting difference\nbetween the nonlinear kinematic and nonkinematic runs. The kinematic\nmodels M2, M4 and M5 show a very slow evolution toward the stationary\nstage. The given effect was reported earlier by \\citet{pip2013ApJ}.\nThe high $R_{m}=10^{6}$ and small diffusivity $\\eta_{\\chi}$ result\nto the long time-scale of establishment of the nonlinear balance in\nmagnetic helicity density distributions.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{f7b}\n\n\\caption{\\label{fig:lt}The same as Figure (\\ref{fig:avar}) for the models\nwith different $C_{\\alpha}$, see Table(\\ref{tab:C}).}\n\\end{figure}\n\nFigure (\\ref{fig:lt}) shows results for the nonkinematic dynamo models\nin a range the $\\alpha$-effect parameter $C_{\\alpha}$. The increasing\nof the $C_{\\alpha}$ results into increasing the nonlinearity of the\ndynamo model. The parameter $\\overline{\\beta}$ grows from $0.2$\nto $1$ with the increasing of $C_{\\alpha}$by factor 4. The model\nM3a2 shows the long-term periodic variations of the parity index and\nthe magnitude of the toroidal magnetic field $\\overline{B^{T}}$.\nThese long-term cycles are likely due to the parity breaking because\nof the hemispheric magnetic helicity exchange. We made the separate\nrun where the magnetic helicity conservation was ignored and did not\nfind the long-term cycles solution. These cycles are not robust against\nchanges of $C_{\\alpha}.$ For the case $\\eta_{\\chi}=0.1\\eta_{T}$,\nthey exist in the range $1.5C_{\\alpha}^{(cr)}<C_{\\alpha}<3C_{\\alpha}^{(cr)}$.\nThe given range is likely to be changed with the change of the magnetic\nhelicity diffusion. We do not consider this case in our paper. Noticeably,\nthat the period of the long-cycle is likely changed with the variation\nof the $\\alpha$-effect. For the model M3a2, the period of the long\ncycle is about 265 years. In the model M3a3, the establishment of\nthe stationary stage proceeds with the long-term oscillations of about\n100 years period. We made the separate runs for the kinematic models\nwith the same $\\alpha$-effect and magnetic helicity diffusivity parameters\nas in the set shown in Figure (\\ref{fig:lt}). It was confirmed that\nthe long-term cycles are exists in the range $1.5C_{\\alpha}^{(cr)}<C_{\\alpha}<3C_{\\alpha}^{(cr)}$.\nTherefore, the magnetic helicity evolution is the most important parameter\nwhich governs the long-term periodicity in our model. The magnetic\nparity breaking and its effect to the Grand activity cycle are lively\ndebated in the literature \\citep{bran89,sok1994AA,1998MNRAS.297.1123K,2014JGRA119.6027F}.\n\n\\subsection{Magnetic cycle in the mean-field heat transport}\n\nThe global thermodynamic parameters are subjected to the magnetic\ncycle modulation because of energy loss and gain for the magnetic\nfield generation and dissipation. Also, the magnetic field quenches\nthe magnitude of the convective heat flux. This affects the mean entropy\ngradient. The Eq(\\ref{eq:heat}) takes these processes into account.\nWe remind that in derivation of the mean-field heat transport equation\n(see, \\citep{2000ARep...44..771P}), it was assumed that the magnetic\nand rotational perturbations of the reference thermodynamic state\nare small. Also, the basic parameters of the reference state, like\n$\\overline{\\rho}$, $\\overline{T}$, etc, are given by the reference\nstellar interior code (MESA). Another approach was suggested recently\nby \\citet{2015JPlPh..81e3904R}.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\columnwidth]{fig11a}\n\n\\caption{\\label{fig:m3h}Model M3, a) the time-latitude diagram for the near-surface\ntoroidal magnetic field (contours in range of $\\pm1kG$) and variation\nof the energy flux on the surface which is calculated relative to\nthe mean background convective flux; b) the same as (a) for the time-radius\ndiagram; c) the mean over the surface outflux of the convective energy\n(blue line), the magnetic energy and the total energy (black line).}\n\\end{figure}\n\nThe model calculates the mean entropy variations as an effect of the\nglobal flows and magnetic activity. Some results in this direction\nwere previously considered in the dynamo models by \\citet{1992AA...265..328B}\nand \\citet{2000ARep...44..771P}. Figures \\ref{fig:m3h} and \\ref{fig:m34}\nshow the time-latitude and time-radius diagrams for the magnetic field\nevolution and variations of the convective heat flux, as well as the\nmean heat and magnetic energy flux on the surface for the models M3\nand M3a4. Evolution of the thermodynamic perturbations goes in form\nof the cyclic modulations with the double frequency of the magnetic\ncycle. Like in our previous studies we can identify two major sources\nof the thermodynamic perturbations in the magnetic cycle. One is the\nso-called magnetic shadow, which is due to the magnetic quenching\nof the convective heat flux. This effect is quite small in comparison\nto the total convective flux. Another effect is the energy expenses\non the large-scale dynamo. This contribution is governed by the term\n${\\displaystyle -\\frac{1}{4\\pi}\\boldsymbol{\\mathcal{E}}\\cdot\\nabla\\times\\boldsymbol{\\overline{B}}}$in\nthe heat transport Eq(\\ref{eq:heat}). Note that our boundary conditions\n(see, Eq(\\ref{eq:tor-vac})), beside the penetration of the toroidal\nmagnetic field to the surface, allow the energy flux from the dynamo\nregion. This flux is related to the strength of the toroidal magnetic\nfield on the surface.\n\nThe model M3 shows that the perturbation of the heat energy flux on\nthe surface goes in phase with the evolution of the near-surface toroidal\nmagnetic field. The decreasing of the heat flux corresponds to the\nincrease of the toroidal magnetic field. In the growing phase of the\ncycle, the thermal energy is expended for the magnetic field generation.\nThe opposite process goes during the declining phase of the magnetic\ncycle. The magnitude of the thermal flux perturbation is about $10^{-4}$\nat the surface and it reaches $10^{-3}$ in the depth of the convection\nzone. The cyclic perturbations of the mean energy flux on the surface\nare about $10^{-4}$, which is an order of magnitude smaller than\nin the solar observations.\n\nThe mean Poynting flux on the surface has a maximum order of $10^{-5}$\nof the background heat energy flux. This flux represents the magnetic\nenergy input to the stellar corona. It can be considered as a part\nof energy source for the magnetic cycle variation of the solar X-ray\nluminosity. The solar observations show that variations the X-ray\nbackground flux is the order of $10^{-6}$ \\citep{2014ApJ793L.45W}.\n{Therefore the magnetic energy flux in the weakly nonlinear\nmodel M3 is enough to explain the solar X-ray luminosity variations\nassuming that all magnetic energy input is transformed into soft X-ray\nflux energy.}\n\nThe model M3a4 shows a different evolution of the thermal perturbations\nin the solar cycle. The growing phase of the magnetic cycle is much\nshorter than the declining phase. Also, the strength of the magnetic\nfield in the model M3a4 is about factor 3 higher than in the model\nM3. The effect of the magnetic shadow dominates contributions from\nthe heat energy expenses to the dynamo. In this case, the minima of\nthe thermal energy flux are roughly located in the extremes of the\ntoroidal magnetic field. By this reason, the mean surface heat energy\nflux reach minimum just after the maximum of the magnetic cycle. The\nmodel M3a4 shows the relative variations of the mean energy flux of\nthe order $2\\times 10^{-3}$, which is $0.2$ percent of the background\nheat flux.{ The strong magnetic feedback of the heat flux in\nthe model M3a4 likely results to the origin of the second near-equatorial\nmeridional circulation cell.} We postpone the analysis of this effect\nto another paper.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\columnwidth]{fig11b}\n\n\\caption{\\label{fig:m34}The same as Figure \\ref{fig:m3h} for the model M3a4.\nThe contours of the toroidal magnetic field show the range $\\pm3$kG. }\n\\end{figure}\n\nThe model M3a4 shows the mean Poynting flux the order of $4\\cdot10^{-4}$.\nThis value is comparable with the X-ray luminosity variations on the\nyoung solar analogs, e.g., $\\chi^{1}$Ori shows log$L_{x}\\sim29$\n\\citep{1998ASPC154.1041G}, which is about $10^{-4}$ of its bolometric\nluminosity.\n\n\\section{Discussion and conclusions}\n\nThe paper presents the new non-kinematic mean-field dynamo model which\ntakes into account the magnetic feedback on the angular momentum and\nheat transport inside the convection zone. For the first time, the\ndynamo model takes into account the complicated structure of the meridional\ncirculation cell in the convection zone. In following to \\citep{2018ApJ854.67P},\nthe double-cell meridional circulation structure results from inversion\nof the $\\Lambda$-effect sign in the lower part of the convection\nzone. The similar results are suggested by the helioseismology inversions.\nNote that the recent results of \\citet{2017ApJ849.144C} showed the\nmuch smaller magnitude of the meridional circulation at the bottom\nin comparison to the previous results of \\citet{Zhao13m}. Despite\nthe qualitative similarity of our results with the helioseismology\ninversions, there are some contradictions. The precise determination\nof the solar meridional circulation profile remains a matter of the\ntheoretical and observational progress. \n\n{The propagation direction of the dynamo wave has been a major\nproblem plaguing distributed turbulent dynamos since helioseismology\nrevealed the solar rotation profile \\citep{choud95,b05,brsu05}. The\nissue is further sharpen by uncertainty of the meridional circulation\ndistribution inside the solar convection zone. There is a common misconception\nin the dynamo community that turbulent dynamos cannot reproduce solar-like\nactivity. Results of Subsection 3.1 show that it is possible to construct\nthe distributed dynamo model with the solar-type magnetic cycles.\nThis requires some tuning of the turbulent pumping coefficients. The\nrequired magnitude of the equatorward pumping velocity is about 3\nm\/s. Also, in the subsurface layer the negative gradient of the angular\nvelocity and the positive sign of the $\\alpha$-effect provide the\nequatorward diffusive drift of the dynamo wave in following to the\nParker-Yoshimura law \\citep{1955ApJ...121..491P,yosh1975}. In addition,\nthe nonlinear models show the increase of the $\\alpha$-effect ahead\nof the dynamo wave, which amplifies the equatorward drift of the toroidal\nmagnetic field, see, the second column of Figures \\ref{fig:M3} and\n\\ref{fig:M3a4}. It was found that the resulted pumping velocity distribution\nand the effective velocity drift of the large-scale magnetic field\nare similar to the recent results of direct numerical simulations\nby \\citet{2018AA609A..51W}. For the given choice of the turbulent\npumping magnitude, the simulated dynamo wave pattern remains robust\nin a range of the surface meridional flow variations $\\pm3.5$m\/s. }\n\nThe dynamo model, which is constructed Subsection 3.1, is applied\nto for the numerical study the effect of the magnetic field on the\nglobal flows and heat transport in the solar convection zone. The\nlarge-scale magnetic field affects the global flow by means of both\nthe mechanical and thermal effects \\citep{2006ApJ...647..662R}, who\nshowed that, in the presence of the meridional circulation, the ``mechanical''\nperturbations of the angular momentum transport (associated with the\nlarge- or small-scale Lorentz force) result in the axial distribution\nof the torsional oscillation's magnitude inside the convection zone.\nThis is a consequence of the Taylor-Proudman balance. The results\nof the helioseismology show deviations of the torsional oscillation\nfrom the axial distribution. This was interpreted as results magneto-thermal\nperturbations. This fact was confirmed in our results. The distribution\nof magnitude of the zonal flow variations in the subsurface layer\ndeclines toward the equator in following the dynamo wave. In the model\nM3a4 the effect is stronger than in the weakly nonlinear model M3.\nThe proof requires the further analysis like that made in the paper\nby \\citet{2016ApJ828L.3G}. This is postponed to another paper.\n\nHere, I would like to stress two facts. Firstly, the numerical model\nsuggests the pattern of the torsional oscillation which is in qualitative\nagreement with observations of \\citet{2011JPhCS271a2074H}. Secondly,\nthe increasing magnetic activity level, for example, due to the $\\alpha$\n- effect, results in the increasing of the torsional oscillation magnitude\nand complication of the meridional circulation structure. Comparing\nthe model M3 and M3a4, we find that the stagnation point of the bottom\ncell is moved to the equator and the upper meridional circulation\ncell is divided into two cells. The two stagnation points of the upper\nmeridional circulation cell result in to two maxima of the meridional\ncirculation at the surface. There is a similarity between our results\nand the recent results of \\citet{2017ApJ849.144C} who also found\nthat the upper circulation cell consists of two. Our model suggest\nthat it may be a result of the strong toroidal magnetic field in the\nsubsurface shear. At the surface, the variations of the meridional\ncirculation in the magnetic cycle show the effective periodical flow\ntowards the maximum of the toroidal magnetic field. In the model M3\nthe magnitude of this flow is about $\\pm0.5$m\/s. The increasing magnetic\nactivity level in the model M3a4 result in the magnitude of the effective\nflow the order of $\\pm3.5$m\/s and the separation into the equatorial\nand polar meridional circulation cells. A similar effect was also\nseen in results of \\citet{1992AA...265..328B}. Their model has a\ndifferent background differential rotation and meridional circulation\n(one-cell structure) distributions. Note that in the subsurface layer\nthe model M3a4 operates at the sub-equipartition level, $\\beta\\sim0.6-1$,\nwhich can be confronted with Figure \\ref{fig:alp}b.\n\nThe dynamo model shows the decreasing dynamo period with the increasing\n$\\alpha$-effect and the increasing level of the magnetic activity.\nThis is in agreement with results of \\citet{pk11} and \\citet{pipea2012AA}.\nObservations, also, show that, in general, the high solar cycle is\nshorter than the low solar cycle \\citep{soon93,hath09}. Also, the\nshape of the high cycle is different in compare with the low cycles.\nThis can be deduced in comparison of the time-latitude diagrams of\nthe model M3 and M3a4, for example, in confronting Figures \\ref{fig:m3h}a\nand \\ref{fig:m34}a, we see that in the model M3a4 the growing phase\nof the dynamo wave is much shorter than the decaying phase. In the\nmodel M3, the given asymmetry is less than in the model M3a4. For\nthe first time, we find these relationships (period-amplitude and\nshape-amplitude) in the mean-field model with the meridional circulation.\nThe flux-transport models fail to explain the relationship between\nthe solar cycle period and cycle amplitude. In particular, the kinematic\nmodels of \\cite{2014ApJ...791...59K} show the increasing dynamo\nperiod with the increasing level of the magnetic activity. This contradicts\nto solar and stellar observations (see, e.g., \\cite{2009AA_strassm,2017PhDT3E}).\nOn another hand, the distributed dynamo models are well fitted both\nin the solar and stellar observations. Our current results show that\nincluding the meridional circulation in the model does not necessarily\nresult in the increasing magnetic cycle period with the increasing\nlevel of magnetic field strength. Therefore the issue of the flux-transport\nscenario is likely connected with the localized character of the magnetic\nfield generation effects in the dynamo model.\n\n{For the first time we demonstrate that the Grand activity\ncycles can result from effect of the meridional circulation and the\nequatorial parity breaking because of the hemispheric magnetic helicity\nexchange. The solar data show that hemispheric magnetic helicity exchange\nis a real phenomenon \\citep{2000JGR...10510481B,2010ApJ719.1955Z,2012ApJ...758...61Y}.\nIn our model, the process is controlled by diffusion of the magnetic\nhelicity density and by the magnetic helicity transport with the meridional\ncirculation.} \\citet{mitra10} found this phenomenon in the direct\nnumerical simulations. In the weakly nonlinear regimes, the strength\nof the magnetic helicity diffusion affects the parity of solution.\nWe find that the symmetric about equator toroidal magnetic field is\ngenerated when the diffusion coefficient is high enough. This is consistent\nwith the observed increase of the $\\alpha$-effect ahead of the dynamo\nwave. {The Grand activity cycle regime is not robust and it\ndisappears when $C_{\\alpha}>3C_{\\alpha}^{(cr)}$}. This coincides\nwith the formation of the second meridional circulation cell near\nthe equator. Currently, it is not clear if both phenomena are tightly\nrelated or this is an accident. This will be studied further.\n\nWe show the first results about effects of the large-scale magnetic\nactivity on the heat transport and the heat energy flux from the dynamo\nregion. This was previously discussed in papers of \\citet{1992AA...265..328B}\nand \\citet{2000ARep...44..771P}. In the mean-field framework, the\nmajor contributions of the large-scale magnetic field on the heat\nenergy balance inside the convection zone are caused by the magnetic\nquenching of the eddy heat conductivity and the energy expenses (associating\nwith the heat energy loss and gain) on the large-scale dynamo. These\nprocesses are modeled by the mean-field heat transport equations.\nThe magnetic perturbations of the heat flux in the model M3 are an\norder of $10^{-3}$ of the background value. It is an order of magnitude\nless at the surface because of the screening effect and the smaller\nstrength of the large-scale magnetic field in the upper layer of the\nconvection zone. The heat perturbation screening effect is due to\nthe huge heat capacity of the solar convection zone \\citep{stix:02}.\nResults of the model M3a4 illustrate it better than the model M3.\nThe model M3a4 shows the strong toroidal magnetic field in the bulk\nof the convection zone (see Figure \\ref{fig:m34}b). In the upper\nlayer of the convection zone, the strength of the toroidal field exceeds\nthe equipartition level. Besides this, the heat flux perturbations\nare efficiently smoothed out toward the top of the dynamo region.\nAnother interesting feature is that the weakly nonlinear model M3\nshows the increasing mean heat flux at the maximum of the magnetic\ncycle. In the model with the overcritical $\\alpha$ effect, we find\nthe opposite situation. The solar observations show the increasing\nluminosity during the maximum of the solar cycles \\citep{1999GeoRL26.3613W}.\nThe variation of the photometric brightness of solar-type stars tends\nto inverse the sign with the increasing level of the magnetic activity\n\\citep{2016ASPC504.273Y}. From the point of view of our model, this\nmeans that the effect of the magnetic shadow become dominant when\nthe total magnetic activity is increased. This is a preliminary conclusion.\nAlso, the relationship between the magnetic shadow effect in the large-scale\ndynamo and the stellar surface darkening because of starspots is not\nstraightforward.\n\nFinally, our results can be summarized as follows: \n\\begin{enumerate}\n\\item We constructed the nonkinematic solar-type dynamo model with the double-cell\nmeridional circulation. The role of the turbulent pumping in the dynamo\nmodel should be investigated. This requires a better theoretical and\nobservational knowledge of the solar meridional circulation. \n\\item The torsional oscillations are explained as a result of the magnetic\nfeedback on the angular momentum transport by the turbulent stresses,\nthe effect of the Lorentz force and the magneto-thermal perturbations\nof the Taylor-Proudman balance. The increasing level of the magnetic\nactivity results in separation of the upper meridional circulation\ncell for two parts. \n\\item The model shows the decrease of the dynamo period with the increase\nof the magnetic cycle amplitude. The shape of the strong magnetic\ncycle is more asymmetric than the shape of the weak cycles. \n\\item The magnetic helicity density diffusion and the increase turbulent\ngeneration of the large-scale magnetic field results in the increasing\nhemispheric magnetic helicity exchange, the magnetic parity breaking\nand the Grand activity cycles. The Grand activity cycles exists in\nthe intermediate range of the $\\alpha$ - effect parameter, when $1.5C_{\\alpha}^{(cr)}<C_{\\alpha}<3C_{\\alpha}^{(cr)}$.\nIt seems that the Grand activity cycles disappear together with the\nformation of the second meridional circulation cell near the equator. \n\\item The increasing turbulent generation of the large-scale magnetic field\nchanges of the relationship between the magnetic cycle phase and the\nmean sign of the heat flux perturbation at the surface. For the high\nlevel of the magnetic activity, the heat flux is reduced in the maximum\nof the magnetic cycle. \n\\end{enumerate}\n{Acknowledgments.} This work was conducted as a part of FR\nII.16 of ISTP SB RAS. Author thanks the financial support from of\nRFBR grant 17-52-53203. \n\\begin{center}\n{REFERENCES} \n\\par\\end{center}\n\n\\bibliographystyle{elsarticle-harv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nOne of the main purposes of the $B$ factories and current LHCb is to study CP violation (CPV), \nwhich is important for us to understand the puzzle of the matter-antimatter asymmetry\nin the Universe. As the observables, the (in)direct CP-violating asymmetries (CPAs) require \nboth weak and strong phases~\\cite{Geng:2016kjv,Hsiao:2014mua,Geng:2006jt},\nwhereas the T-violating triple momentum product correlations (TPCs), such as\n$\\vec{p}_1\\cdot (\\vec{p}_2\\times \\vec{p}_3)$ in a four-body decay,\ndo not necessarily need a strong phase~\\cite{Bensalem,Geng:2005wt}.\nFor example, \nthe LHCb Collaboration has provided the first evidence for CPV from the TPCs \nin $\\Lambda_b\\to p\\pi^-\\pi^+\\pi^-$~\\cite{Aaij:2016cla}, and measured TPCs\nin $\\Lambda_b\\to p K^-\\mu^+\\mu^-$~\\cite{Aaij:2017mib}.\nAs the similar baryonic cases, \nthe four-body baryonic $B$ decays can also provide TPCs.\n\nFor a long time, the $B^-\\to \\Lambda \\bar p \\pi^+\\pi^-$ decay\nwas the  only observed decay mode \nin $B\\to{\\bf B_1\\bar B_2}M_1M_2$~\\cite{Chen:2009xg}. \nUntil very recently,  more four-body baryonic  B decays have been \nobserved by the LHCb~\\cite{BtoBBMM_LHCb},\nwhich motivate us to give  theoretical estimations on the corresponding decay branching ratios.\nThe experimental measurements for the branching ratios of \n$\\bar B^0\/B^-\\to{\\bf B_1\\bar B_2}M_1M_2$ \nat the level of $10^{-6}$ are given by~\\cite{Chen:2009xg,BtoBBMM_LHCb}\n\\begin{eqnarray}\\label{br_data}\n{\\cal B}(\\bar B^0\\to p \\bar p \\pi^+\\pi^-)\n&=&(3.0\\pm 0.2\\pm 0.2\\pm 0.1)\\times 10^{-6}\\,,\\nonumber\\\\\n{\\cal B}(\\bar B^0\\to p \\bar p K^\\mp\\pi^\\pm)\n&=&(6.6\\pm 0.3\\pm 0.3\\pm 0.3)\\times 10^{-6}\\,,\\nonumber\\\\\n{\\cal B}(B^-\\to \\Lambda \\bar p \\pi^+\\pi^-)\n&=&(5.92^{+0.88}_{-0.84}\\pm 0.69)\\times 10^{-6}\\,,\n\\end{eqnarray}\nwhere the resonant ${\\cal B}(B^-\\to \\Lambda \\bar p$ \n$(\\rho^0,f_2(1270)\\to)\\pi^+\\pi^-)$\nhave been excluded from the data~\\cite{Chen:2009xg}.\nIn comparison with\n${\\cal B}(\\bar B^0\\to p \\bar p K^+ K^-)\n\\simeq (1.3\\pm 0.3)\\times 10^{-7}$ and \n${\\cal B}(\\bar B^0_s\\to p\\bar p \\pi^+ \\pi^-)\n<7.3\\times 10^{-7}$ (90\\% C.L.)~\\cite{BtoBBMM_LHCb},\nthe decays with ${\\cal B}\\sim10^{-6}$ in Eq.~(\\ref{br_data}) \nare recognized to have the same theoretical correspondence,\nwhere $\\bar B^0\/B^-\\to{\\bf B_1\\bar B_2}M_1M_2$\nproceed through the $B\\to M_1M_2$ transition along with \nthe $\\bf B_1\\bar B_2$ production, as depicted in Fig.~\\ref{dia}.\nNote that\nthe $\\bar B^0_s$ decays of $\\bar B^0_s\\to p\\bar p K^\\pm\\pi^\\mp$ and $p\\bar p K^+ K^-$\nwith $\\bar s$ being replaced by $\\bar d$ \nin $\\bar B^0\\to p \\bar p \\pi^+\\pi^-$ and $p \\bar p K^\\mp\\pi^\\pm$\n have also been found with the branching ratios of order $10^{-6}$~\\cite{BtoBBMM_LHCb}, respectively.\n\nIn this report, \nwe will calculate the four-body baryonic $B$ decays in accordance with\nthe decaying processes in Fig.~\\ref{dia},\nwith the extraction of \nthe $B\\to M_1 M_2$ transition form factors from the $B\\to D^{(*)}M_1M_2$ and \n$B\\to M_1M_2M_3$ decays and the adoption of the timelike baryonic form factors \nfrom the two-body and three-body baryonic $B$ decays.\nOur theoretical approach will be useful for the estimations of \nTPCs in $B\\to{\\bf B_1\\bar B_2}M_1M_2$ to be compared to\n future measurements by the LHCb.\n\n\\vspace*{0.3cm}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.1in]{4b_BtoBBMM_A.eps}\n\\includegraphics[width=2.1in]{4b_BtoBBMM_B.eps}\n\\includegraphics[width=2.1in]{4b_BtoBBMM_C.eps}\n\\includegraphics[width=2.1in]{4b_BtoBBMM_D.eps}\n\\includegraphics[width=2.1in]{4b_BtoBBMM_E.eps}\n\\caption{Feynman diagrams for the charmless four-body baryonic $B$ decays, where\n(a,b,c) depict the $\\bar B^0_{(s)}\\to p\\bar p M_1M_2$ decays,\nwhile (d,e) the $B^-\\to \\Lambda\\bar p M_1M_2$ decays.}\\label{dia}\n\\end{figure}\n\\vspace*{0.3cm}\n\\section{Formalism}\nIn terms of the quark-level effective Hamilontion for the charmless $b\\to q_1\\bar q_2 q_3$ transition, \nthe amplitudes of the four-body baryonic $B$ decays \nby the generalized factorization approach are derived as~\\cite{ali} \n\\begin{eqnarray}\\label{amp1}\n&&{\\cal A}_1(\\bar B^0_{(s)}\\to p\\bar p M_1M_2)=\n\\frac{G_F}{\\sqrt 2}\\bigg\\{\\bigg[\n\\langle p\\bar p|\\alpha_+^q(\\bar u u)_V-\\alpha_-^q(\\bar u u)_A|0\\rangle+\n\\langle p\\bar p|\\beta_+^q(\\bar dd)_V-\\beta_-^q(\\bar dd)_A|0\\rangle\\nonumber\\\\\n&&+(\\alpha_4^q-\\alpha_{10}^q\/2)\\langle p\\bar p|(\\bar qq)_{V-A}|0\\rangle\n\\bigg]\\langle M_1M_2|(\\bar q b)_{V-A}|\\bar B^0_{(s)}\\rangle\\nonumber\\\\\n&&+\\alpha_6^q\\langle p\\bar p|(\\bar qq)_{S+P}|0\\rangle\n\\langle M_1M_2|(\\bar qb)_{S-P}|\\bar B^0_{(s)}\\rangle\\bigg\\}\\,,\n\\nonumber\\\\\n&&{\\cal A}_2(B^-\\to\\Lambda\\bar p M_1 M_2)\n\\frac{G_F}{\\sqrt 2}\\bigg\\{\n(\\alpha_1^s+\\alpha_4^s)\\langle \\Lambda\\bar p|(\\bar s u)_{V-A}|0\\rangle \n\\langle M_1M_2|(\\bar u b)_{V-A}|B^-\\rangle\\nonumber\\\\\n&&+\\alpha_6^s\\langle \\Lambda\\bar p|(\\bar s u)_{S+P}|0\\rangle\n\\langle M_1 M_2|(\\bar u b)_{S-P}|B^-\\rangle\\bigg\\}\\,,\n\\end{eqnarray}\nwhere  $G_F$ is the Fermi constant, $V_{ij}$ are the CKM matrix elements,\nand $(\\bar q_1 q_2)_{V(A)}$ and $(\\bar q_1 q_2)_{S(P)}$ stand for \n$\\bar q_1 \\gamma_\\mu(\\gamma_5) q_2$ and $\\bar q_1(\\gamma_5) q_2$, respectively.\nThe parameters $\\alpha^q_{\\xi}$ and $\\beta^q_\\eta$ \nin Eq.~(\\ref{amp1})\nare given by\n\\begin{eqnarray}\n\\alpha^q_\\pm&=&\\alpha_2^q+\\alpha_3^q\\pm\\alpha_5^q+\\alpha_9^q\\,,\n\\beta^q_\\pm=\\alpha_3^q\\pm\\alpha_5^q-\\alpha_9^q\/2\\,,\\nonumber\\\\\n\\alpha_{1,2}^q&=&V_{ub}V_{uq}^* a_{1,2} \\,,\n\\alpha_j^q=-V_{tb}V_{tq}^*a_j\\,,\n\\alpha_6^q=V_{tb}V^*_{tq}2a_6\\,,\n\\end{eqnarray}\nwith $q=(d,s)$ and $j=(3,4,5,9,10)$,\nwhere $a_i\\equiv c^{eff}_i+c^{eff}_{i\\pm 1}\/N_c^{eff}$ for $i=$ odd (even)\nwith the effective color number $N_c^{eff}$ and\nWilson coefficients $c_{i}^{eff}$ in Ref.~\\cite{ali}.\nFrom \n${\\cal A}_1(\\bar B^0_{(s)}\\to p\\bar p M_1M_2)$ and \n${\\cal A}_2(B^-\\to\\Lambda\\bar p M_1 M_2)$ in Eq.~(\\ref{amp1}), \nthe allowed decays are\n\\begin{eqnarray}\n&&\n\\bar B^0\\to p\\bar p \\pi^+\\pi^-,\\,\n\\bar B^0_s\\to p\\bar p K^+\\pi^-\\,,\\;\\text{(q=d)}\\nonumber\\\\ \n&&\n\\bar B^0\\to p\\bar p \\pi^+ K^-,\\,\n\\bar B^0_s\\to p\\bar p K^+ K^-\\,,\\;\\text{(q=s)}\\nonumber\\\\ \n&&\nB^-\\to \\Lambda\\bar p \\pi^+\\pi^-,\\;B^-\\to\\Lambda\\bar p K^+ K^-\\,.\n\\end{eqnarray}\nNote that  \nthe $\\bar B^0\\to p\\bar p \\pi^+ K^-$ and $\\bar B^0_s\\to p\\bar p K^+ K^-$ decays\nhave the matrix elements of\n$\\langle p\\bar p|(\\bar s s)_{V,A,S,P}|0\\rangle$ with the $\\bar ss$ quark currents,\nwhich eventually cause the terms of $\\alpha_{4,6,10}^s$\nto give nearly zero contributions\ndue to the OZI suppression of $\\bar s s\\to p\\bar p$~\\cite{Hsiao:2014tda}.\n\n\nFor the matrix elements in Eq.~(\\ref{amp1}), the baryon-pair productions\nfrom the quark currents are given by~\\cite{Chua:2002yd,Geng:2005wt} \n\\begin{eqnarray}\\label{FFactor1}\n\\langle {\\bf B_1\\bar B_2}|\\bar q_1\\gamma_\\mu q_2|0\\rangle\n&=&\n\\bar u\\bigg[F_1\\gamma_\\mu+\\frac{F_2}{m_{\\bf B_1}+m_{\\bf \\bar B_2}}i\\sigma_{\\mu\\nu}q_\\mu\\bigg]v\\;,\\nonumber\\\\\n\\langle {\\bf B_1\\bar B_2}|\\bar q_1\\gamma_\\mu \\gamma_5 q_2|0\\rangle\n&=&\\bar u\\bigg[g_A\\gamma_\\mu+\\frac{h_A}{m_{\\bf B_1}+m_{\\bf \\bar B_2}}q_\\mu\\bigg]\\gamma_5 v\\,,\\nonumber\\\\\n\\langle {\\bf B_1\\bar B_2}|\\bar q_1 q_2|0\\rangle &=&f_S\\bar uv\\;,\n\\langle {\\bf B_1\\bar B_2}|q_1\\gamma_5 q_2|0\\rangle =g_P\\bar u \\gamma_5 v\\,,\n\\end{eqnarray}\nwhere $q=p_{\\bf B_1}+p_{\\bf\\bar B_2}$, $t\\equiv q^2$,\n$u$($v$) is the (anti-)baryon spinor, and  \n$(F_{1,2},g_A,h_A,f_S,g_P)$ are the timelike baryonic form factors.\nOn the other hand, the $B\\to M_1 M_2$ transition matrix elements \nare parameterized as~\\cite{Lee:1992ih}\n\\begin{eqnarray}\\label{BtoMM}\n&&\\langle M_1 M_2|\\bar q_1\\gamma_\\mu(1-\\gamma_5)b|B\\rangle\\nonumber\\\\\n&=&\nh\\epsilon_{\\mu\\nu\\alpha\\beta}p_B^\\nu p^\\alpha (p_{M_2}-p_{M_1})^\\beta\n+irq_\\mu+iw_+ p_\\mu+iw_-(p_{M_2}-p_{M_1})\\,,\n\\end{eqnarray}\nwhere $p=p_{M_2}+p_{M_1}$ and $(h,r,w_{\\pm})$ are the form factors. \nSubsequently, one can also get \n$\\langle M_1 M_2|\\bar q_1(\\gamma_5)b|B\\rangle$ from Eq.~(\\ref{BtoMM}) \nbased on equations of motion. In terms of the approach of pQCD counting rules,\nthe momentum dependences for\nthe $0\\to {\\bf B_1\\bar B_2}$ and $B\\to M_1M_2$\ntransition form factors are \ngiven by~\\cite{Brodsky:1973kr,Brodsky:2003gs,Chua:2002pi,Chua:2004mi}\n\\begin{eqnarray}\\label{timelikeF2}\n&&F_1=\\frac{\\bar C_{F_1}}{t^2}\\,,\\;g_A=\\frac{\\bar C_{g_A}}{t^2}\\,,\\;\nf_S=\\frac{\\bar C_{f_S}}{t^2}\\,,\\;g_P=\\frac{\\bar C_{g_P}}{t^2}\\,,\\;\\nonumber\\\\\n&&h=\\frac{C_h}{t^2}\\,,\\;w_-=\\frac{D_{w_-}}{t^2}\\,,\n\\end{eqnarray}\nwhere $\\bar C_i=C_i [\\text{ln}({t}\/{\\Lambda_0^2})]^{-\\gamma}$\nwith $\\gamma=2.148$ and $\\Lambda_0=0.3$ GeV.\nWe note that since\n$F_2$ is derived to be $F_2=F_1\/(t\\text{ln}[t\/\\Lambda_0^2])$~\\cite{Belitsky:2002kj},\nwhich is much less than $F_1$, while \nthe small value of ${\\cal B}(\\bar B^0\\to p\\bar p)=\n(1.5^{+0.7}_{-0.5})\\times 10^{-8}$~\\cite{pdg,Aaij:2013fta}\ncauses a tiny $C_{h_A}$~\\cite{Hsiao:2014zza} in $h_A=C_{h_A}\/t^2$,\nwe may not consider the effects from $F_2$ and $h_A$.\nIn addition, by following Ref.~\\cite{Chua:2002pi},\nwe have neglected the terms related to\n$r$ and \n$w_+$ in Eq.~(\\ref{BtoMM}) \ndue to the wrong parity~\\cite{Drutskoy:2002ib}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.8in]{4b_BtoBBMM_1.eps}\n\\caption{Three angles of $\\theta_{\\bf B}$, $\\theta_{\\bf M}$,\nand $\\phi$ in the phase space for the four-body $B\\to {\\bf B_1\\bar B_2}M_1M_2$ decays.\n}\\label{4body}\n\\end{figure}\nThe integration over the phase space of the four-body \n$B(p_B)\\to {\\bf B_1}(p_{\\bf B_1}){\\bf\\bar B_2}(p_{\\bf \\bar B_2})$\n$M_1(p_{M_1}) M_2 (p_{M_2})$ decay relies on\nthe five kinematic variables, that is, $s\\equiv p^2$, \n$t$ and the three angles of $\\theta_{\\bf B}$, $\\theta_{\\bf M}$ and $\\phi$.\nIn Fig.~\\ref{4body},\nthe angle $\\theta_{\\bf B(M)}$ is between $\\vec{p}_{\\bf B_1}$  ($\\vec{p}_{M_1}$)\nof the $\\bf B_1\\bar B_2$ ($M_1 M_2$) rest frame and \nthe line of flight of the $\\bf B_1\\bar B_2$ ($M_1 M_2$) system in the $B$ meson rest frame, \nwhile the angle $\\phi$ is from the $\\bf B_1\\bar B_2$ plane \nto the $M_1 M_2$ plane,\ndefined by the momenta of the $\\bf B_1\\bar B_2$ and $M_1 M_2$ pairs\nin the $B$ rest frame, respectively.\nThe partial decay width reads~\\cite{Geng:2011tr,Geng:2012qn}\n\\begin{eqnarray}\nd\\Gamma=\\frac{|\\bar {\\cal A}|^2}{4(4\\pi)^6 m_B^3}X\n\\alpha_{\\bf B}\\alpha_{\\bf M}\\, ds\\, dt\\, d\\text{cos}\\,\\theta_{\\bf B}\\, d\\text{cos}\\,\\theta_{\\bf M}\\, d\\phi\\,,\n\\end{eqnarray}\nwhere $X$, $\\alpha_{\\bf B}$ and $\\alpha_{\\bf M}$ are given by\n\\begin{eqnarray}\nX&=&\\bigg[\\frac{1}{4}(m_B^2-s-t)^2-st\\bigg]^{1\/2}\\,,\\nonumber\\\\\n\\alpha_{\\bf B}&=&\\frac{1}{t}\\lambda^{1\/2}(t,m_{\\bf B_1}^2,m_{\\bf \\bar B_2}^2)\\,,\\nonumber\\\\\n\\alpha_{\\bf M}&=&\\frac{1}{s}\\lambda^{1\/2}(s,m_{M_1}^2,m_{M_2}^2)\\,,\n\\end{eqnarray}\nrespectively,\nwith $\\lambda(a,b,c)=a^2+b^2+c^2-2ab-2bc-2ca$, while\nthe allowed ranges of the five variables are given by\n\\begin{eqnarray}\n&&(m_{M_1}+m_{M_2})^2\\leq s\\leq (m_{B}-\\sqrt{t})^2\\,,\\;\\;\n(m_{\\bf B_1}+m_{\\bf \\bar B_2})^2\\leq t\\leq (m_B-m_{M_1}-m_{M_2})^2\\,,\\nonumber\\\\\n&&0\\leq \\theta_{\\bf B},\\,\\theta_{\\bf M}\\leq \\pi\\,,\\;\\;0\\leq \\phi\\leq 2\\pi\\,.\n\\end{eqnarray}\n \n\n\\section{Numerical Results and Discussions }\nFor the numerical analysis, \nthe CKM matrix elements in the Wolfenstein parameterization \nare presented as\n\\begin{eqnarray}\n&&(V_{ub},\\,V_{tb})=(A\\lambda^3(\\rho-i\\eta),1)\\,,\\nonumber\\\\\n&&(V_{ud},\\,V_{td})=(1-\\lambda^2\/2,A\\lambda^3)\\,,\\nonumber\\\\\n&&(V_{us},\\,V_{ts})=(\\lambda,-A\\lambda^2),\n\\end{eqnarray}\nwith $(\\lambda,\\,A,\\,\\rho,\\,\\eta)=(0.225,\\,0.814,\\,0.120\\pm 0.022,\\,0.362\\pm 0.013)$~\\cite{pdg}.\nTo estimate the non-factorizable effects\nin the generalized factorization approach~\\cite{ali},\n$N_c^{eff}$ ranges from 2 to $\\infty$. In Table~\\ref{alpha_i},\nwe show the values of $a_i$\nfor the $b\\to d$ and $b\\to s$ transitions with $N_c^{eff}=(2, 3,\\infty)$, respectively.\n\\begin{table}[b]\n\\caption{The parameters $a_i$ with $N_c^{eff}=2,\\,3$, and $\\infty$\nto estimate the non-factorizable effects in the generalized factorization.}\\label{alpha_i}\n{\\footnotesize\n\\begin{tabular}{|c||ccc|ccc|}\n\\hline\n&\\multicolumn{3}{c|}{($b\\to d$ transition)}&\\multicolumn{3}{c|}{($b\\to s$ transition)}\\\\\n$a_i$& $N_c^{eff}=2$ &  $3$ & $\\infty$ &$N_c^{eff}=2$&$3$&$\\infty$\\\\%\\hline\n\\cline{2-7}\n$a_1$ \n& ----- &  ----- & -----\n&$0.98$&$1.05$&$1.17$ \\\\\n$a_2$ \n&$0.22$&$0.02$&$-0.37$\n&$0.22$&$0.02$&$-0.37$ \\\\\n$10^{4}a_3$ \n&$-10.4- 6.9i$  &  $72.4$ & $237.9+ 13.9i$\n&$-13.1-15.6i$ &  $72.4$ & $243.2+31.2i$\n\\\\\n$10^{4}a_4$ \n& $-377.6-34.7i$ &  $-417.2-37.0i$ & $-496.5-41.6i$ \n& $-391.0-77.9i$ &  $-431.6-83.1i$ & $-512.6-93.5i$\n\\\\\n$10^{4}a_5$ \n& $-171.4 - 6.9i$ &  $-65.8$ & $145.3+13.9i$\n& $-174.1-15.6i$ &  $-65.8$ & $150.7+31.2i$\n\\\\\n$10^{4}a_6$ \n& $-560.7-34.7i$ &  $-584.9-37.0i$ & $-633.4-41.6i$\n& $-574.1-77.9i$ &  $-599.3-83.1i$ & $-649.5-93.5i$\n\\\\\n$10^{4}a_9$ \n& $-93.3-1.4i$ & $-99.5-1.4i$ & $-112.0-1.4i$\n& $-93.5-2.2i$ & $-99.8-2.2i$ & $-112.3-2.2i$\n\\\\\n$10^{4}a_{10}$ \n&$-18.5-0.7i$&$0.18-0.46i$&$37.5$\n& ----- &  ----- & -----\\\\\n\\hline\n\\end{tabular}\n}\n\\end{table}\n\nAccording to the extractions of $(C_h,C_{w_-})$ in Refs.~\\cite{Chua:2002pi,Chua:2004mi},\nwe fit the $B\\to \\pi\\pi$ transition form factors with the branching ratios of \n$\\bar B^0\\to D^{(*)0} \\pi^+ \\pi^-$, \n$B^-\\to \\pi^-\\pi^+\\pi^-$ and\n$B^-\\to K^{*-}\\pi^+\\pi^-$,\nand the $B\\to (K\\pi,KK)$ ones with those of \n$B^-\\to D^{(*)0} K^- K^0$, \n$\\bar B^0\\to D^0 K^- \\pi^+$ and\n$B^-\\to K^{*-}K^+ K^-$. Note that \nthe contributions from the resonant\n$B\\to D^{(*)}(M_0\\to)M_1M_2$, and\n$B^-\\to K^{*-} (M_0\\to)M_1 M_2$ decays\nwith $\\rho^0,f_2(1270)\\to \\pi^+\\pi^-$ or $\\phi\\to K^+K^-$\nhave been excluded from the data. \nUnfortunately, the current observations of \n${\\cal B}(\\bar B^0_s\\to M_1 M_2 M_3)$ are not sufficient\nfor us to extract the $\\bar B^0_s\\to M_1 M_2$ transition form factors.\nAs a result, we obtain\n\\begin{eqnarray}\n(C_h,C_{w_-})|_{B\\to \\pi\\pi}&=&(3.6\\pm 0.3,0.7\\pm 0.2)\\,GeV^3\\,, \\nonumber\\\\\n(C_h,C_{w_-})|_{B\\to KK(K\\pi)}&=&(-38.9\\pm 3.3,14.2\\pm 2.3)\\,GeV^3\\,.\n\\end{eqnarray}\nThe timelike baryonic form factors in Eq.~(\\ref{FFactor1})\ncan be related with the $SU(3)$ flavor and $SU(2)$ spin symmetries,\nsuch that $(C_{F_1},C_{g_A},C_{f_S},C_{g_P})$ are recombined \nby a new set of constant parameters\nas~\\cite{Brodsky:1973kr,Chua:2002yd,Geng:2016fdw,Hsiao:2016amt,ang_BtopLpi}\n\\begin{eqnarray}\\label{0toBB}\n&&\nC_{F_1}=\\frac{5}{3}C_{||}+\\frac{1}{3}C_{\\overline{||}}\\,,\\;\nC_{g_A}=\\frac{5}{3}C_{||}^*-\\frac{1}{3}C_{\\overline{||}}^*\\,,\\;\\;\n\\text{(for $\\langle p\\bar p|\\bar u\\gamma_\\mu(\\gamma_5)u|0\\rangle$)}\\nonumber\\\\\n&&\nC_{F_1}=\\frac{1}{3}C_{||}+\\frac{2}{3}C_{\\overline{||}}\\,,\\;\nC_{g_A}=\\frac{1}{3}C_{||}^*-\\frac{2}{3}C_{\\overline{||}}^*\\,,\\;\\;\n\\text{(for $\\langle p\\bar p|\\bar d\\gamma_\\mu(\\gamma_5)d|0\\rangle$)}\\nonumber\\\\\n&&\nC_{f_S}=\\frac{1}{3}\\bar C_{||}\\,,\\;C_{g_P}=\\frac{1}{3}\\bar C_{||}^*\\,,\\;\\;\n\\text{(for $\\langle p\\bar p|\\bar d(\\gamma_5)d|0\\rangle$)}\\nonumber\\\\\n&&\nC_{F_1}=\\sqrt\\frac{3}{2}C_{||}\\,,\\;\nC_{g_A}=\\sqrt\\frac{3}{2}C_{||}^*\\,,\\;\\;\n\\text{(for $\\langle \\Lambda\\bar p|\\bar s\\gamma_\\mu(\\gamma_5)u|0\\rangle$)}\\nonumber\\\\\n&&\nC_{f_S}=-\\sqrt\\frac{3}{2}\\bar C_{||}\\,,\nC_{g_P}=-\\sqrt\\frac{3}{2}\\bar C_{||}^*\\,,\\;\\;\n\\text{(for $\\langle \\Lambda\\bar p|\\bar s(\\gamma_5)u|0\\rangle$)\n\\end{eqnarray}\nwith  \n$C_{||(\\overline{||})}^*\\equiv C_{||(\\overline{||})}+\\delta C_{||(\\overline{||})}$ and \n$\\bar C_{||}^*\\equiv \\bar C_{||}+\\delta \\bar C_{||}$, in which \n$\\delta C_{||(\\overline{||})}$ and $\\delta \\bar C_{||}$ have been added \nto explain the large and unexpected \nangular distributions in $\\bar B^0\\to \\Lambda\\bar p\\pi^+$ and \n$B^-\\to \\Lambda\\bar p\\pi^0$~\\cite{ang_BtopLpi,Wang:2007as},\nto account for the fact that the $SU(3)$ flavor and $SU(2)$ spin symmetries \nat large $t$ ($t\\to \\infty$)~~\\cite{Brodsky:1973kr} should be broken \nat $t\\simeq m_B^2$~\\cite{ang_BtopLpi}. \nThe extractions of the form factors by the data of \n$\\bar B^0\\to n\\bar p D^{*+}$, \n$\\bar B^0\\to \\Lambda\\bar p D^{(*)+}$,\n$\\bar B^0\\to \\Lambda\\bar p \\pi^+$, \n$B^-\\to \\Lambda\\bar p  (\\pi^0,\\rho^0)$, \n$\\bar B^0_{(s)}\\to p\\bar p$ and\n$B^-\\to \\Lambda\\bar p $\ngive~\\cite{Geng:2016fdw} \n\\begin{eqnarray}\\label{fitC2}\n&&(C_{||},\\,\\delta C_{||})=(154.4\\pm 12.1,\\,19.3\\pm 21.6)\\;{\\rm GeV}^{4}\\,,\\nonumber\\\\\n&&(C_{\\overline{||}},\\,\\delta C_{\\overline{||}})=(18.1\\pm 72.2,\\,-477.4\\pm 99.0)\\;{\\rm GeV}^{4}\\,,\\nonumber\\\\\n&&(\\bar C_{||},\\,\\delta\\bar C_{||})=(537.6\\pm 28.7,\\,-342.3\\pm 61.4)\\;{\\rm GeV}^{4}\\,,\n\\end{eqnarray}\nwhere the added constants for the broken effects have \nbeen approved by the excellent agreement for \n${\\cal B}(\\bar B^0_s\\to \\Lambda \\bar p K^+ +\\bar \\Lambda p K^-)$~\\cite{LHCb:2017khw}.\nSubsequently, \nwe evaluate the branching ratios of $B\\to {\\bf B_1\\bar B_2}M_1 M_2$\nas shown in Table~\\ref{br_BtoBBMM}, and draw the distributions vs. $m_{\\bf B_1\\bar B_2}$\nin Fig.~\\ref{spec}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.3in]{diagramA.eps}\n\\includegraphics[width=2.3in]{diagramB.eps}\n\\caption{Invariant dibaryon mass spectra for \n$B^-\\to \\Lambda\\bar p M_1M_2$ (left panel) and \n$\\bar B^0\\to p\\bar p M_1M_2$ (right panel), respectively.}\\label{spec}\n\\end{figure}\n\n\\begin{table}[b\n\\caption{The branching ratios of $B\\to{\\bf B_1\\bar B_2}M_1M_2$,\nwhere the errors come from the non-factorizable effects, \nCKM matrix elements, and form factors, respectively.}\\label{br_BtoBBMM}\n\\begin{tabular}{|c|c|c|}\n\\hline\nbranching ratios&our results&data\\\\\\hline\n$10^6{\\cal B}(B^-\\to \\Lambda\\bar p \\pi^+\\pi^-)$\n&$3.7^{+1.2}_{-0.5}\\pm 0.1\\pm 0.9$&$5.9\\pm 1.1$\\\\\n$10^6{\\cal B}(B^-\\to\\Lambda\\bar p K^+ K^-)$\n&$3.0^{+1.1}_{-0.5}\\pm 0.1\\pm 0.7$&-----\\\\\n$10^6{\\cal B}(\\bar B^0\\to p\\bar p \\pi^+\\pi^-)$\n&$3.0^{+0.5}_{-0.3}\\pm 0.3\\pm 0.7$&$3.0\\pm 0.3$\\\\\n$10^6{\\cal B}(\\bar B^0\\to p\\bar p \\pi^\\pm K^\\mp)$\n&$6.6\\pm 0.5\\pm 0.0\\pm 2.3$&$6.6\\pm 0.5$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nAs seen in Table~\\ref{br_BtoBBMM}, although the predicted result of\n${\\cal B}(B^-\\to \\Lambda\\bar p \\pi^+\\pi^-)=(3.7^{+1.5}_{-1.0} )\\times 10^{-6}$\nis a little lower, it is consistent with the data in Eq.~(\\ref{br_data}) \nby taking the uncertainties into account. \nWith the replacement of $B^-\\to \\pi^+\\pi^-$ by $B^-\\to K^+ K^-$,\nthe $B^-\\to \\Lambda\\bar p \\pi^+\\pi^-$ and $\\Lambda\\bar p K^+K^-$ decays\nshare the same decaying configuration. We hence predict that \n${\\cal B}(B^-\\to\\Lambda\\bar p K^+ K^-)=(3.0^{+1.3}_{-0.9})\\times 10^{-6}$,\nwhich is accessible to the LHCb and BELLE experiments.\nUnlike the $B^-\\to \\Lambda\\bar p M_1M_2$ decays, where \n$a_{1,4,6}$ are stable by ranging $N_c^{eff}$ from 2 to $\\infty$,\nthe tree-level dominant $\\bar B^0\\to p\\bar p \\pi^+\\pi^-$ decay has\n$\\alpha^d_\\pm \\simeq V_{ub}V_{ud}^* a_2$ in Eq.~(\\ref{amp1}) \nto be sensitive to the non-factorizable effects.\nSince the non-factorizable effects are uncomputable, according to the data of \n${\\cal B}(\\bar B^0\\to p\\bar p \\pi^+\\pi^-)=(3.0\\pm 0.3)\\times 10^{-6}$~\\cite{BtoBBMM_LHCb} \nin Eq.~(\\ref{br_data}), we obtain \n${\\cal B}(\\bar B^0\\to p\\bar p \\pi^+\\pi^-)=(3.0\\pm 0.9)\\times 10^{-6}$,\nwhere $a_2=0.26\\pm 0.01$ \n with the tiny value of $\\delta a_2=0.01$ from the new data is compatible to\n${\\cal O}(0.2-0.3)$ from \nthe two-body  $B$ and $\\Lambda_b$ and  three-body baryonic $B$\ndecays~\\cite{Neubert:2001sj,Hsiao:2015cda,Hsiao:2016amt}.\nFor the measured branching ratio of $\\bar B^0\\to p\\bar p \\pi^+ K^- +p\\bar p \\pi^- K^+$,\nit is found that the contribution is mainly from \nthe penguin-level dominant $\\bar B^0\\to p\\bar p \\pi^+ K^-$ mode.\nNote that\n$a_{3,5}$ from $\\alpha^s_\\pm \\simeq \\beta^s_\\pm=-V_{tb}V_{ts}^*(a_3\\pm a_5+a_9)$\nare also sensitive to the non-factorizable effects. With $N_c^{eff}=3$, \nwe obtain ${\\cal B}(\\bar B^0\\to p\\bar p \\pi^+ K^-)=(6.6\\pm 2.4)\\times 10^{-6}$,\nwhich suggests that the decay is free from the non-factorizable effects. \nIn Table~\\ref{br_BtoBBMM}  we have included the data to constrain the non-factorizable effects, \nwhich results in $\\delta N_c^{eff}=0.06$.\nWe note that the two spectra in Fig.~\\ref{spec} for \n$B^-\\to \\Lambda\\bar p M_1M_2$ and $\\bar B^0\\to p\\bar p M_1M_2$\npresent the threshold effects as the peaks around the threshold areas of \n$m_{\\Lambda\\bar p}\\simeq m_\\Lambda +m_{\\bar p}$ and \n$m_{p\\bar p}\\simeq m_p+m_{\\bar p}$, respectively, \nwhich are commonly observed \nin the three and four-body baryonic $B$ decays~\\cite{BtoBBMM_LHCb,Wang:2007as}. \n\nFinally, we remark that \nwe cannot explain the \ndata of ${\\cal B}(\\bar B^0_s\\to p\\bar p K^\\pm \\pi^\\mp, p\\bar p K^+ K^-)\n=(1.5\\pm 0.7,4.6\\pm 0.6)\\times 10^{-6}$\nmeasured by the LHCb~\\cite{BtoBBMM_LHCb}\ndue to the lack of the information for the transition form factors of $\\bar B^0_s\\to (K^+\\pi^-,K^+ K^-)$.\nThis calls for \nthe theoretical and experimental studies of \nthe three-body mesonic $\\bar B^0_s$ decays that could proceed\nwith the $\\bar B^0_s\\to M_1 M_2$ transitions, \nsuch as the \n$\\bar B^0_s\\to D_s^{*-}\\pi^+ K^0$,\n$\\bar B^0_s \\to D^{*0}\\pi^+ K^-(K^+K^-)$ and \n$\\bar B^0_s\\to \\rho^- \\pi^+ K^0$ decays with one of the mesons \nto be a vector one, \nin order to extract both $(h,w_-)$ in Eq.~(\\ref{BtoMM}). On the other hand, \nthe observed \n$\\bar B^0_s\\to D^0 K^+ \\pi^-$ and $\\bar B^0_s\\to D^0 K^+ K^-$ decays~\\cite{pdg}\nare also important as they relate to $w_-$.\n\n\n\\section{Conclusions}\nIn sum, we have studied the charmless four-body baryonic $B\\to {\\bf B_1 \\bar B_2}M_1 M_2$ decays,\nwhere the primary decaying processes are regarded as \nthe $B\\to M_1M_2$ transitions along with the baryon-pair productions.\nAccording to the new extractions of the $B\\to M_1 M_2$ transition form factors from\nthe three-body $B\\to D^{(*)}M_1 M_2$ and $B\\to M_1M_2M_3$ decays,\nwe have shown that  \n${\\cal B}(B^-\\to \\Lambda\\bar p \\pi^+\\pi^-)=(3.7^{+1.5}_{-1.0} )\\times 10^{-6}$ and \n${\\cal B}(\\bar B^0\\to p\\bar p \\pi^+\\pi^-,p\\bar p \\pi^+ K^-)=\n(3.0\\pm 0.9,6.6\\pm 2.4)\\times 10^{-6}$, which agree with the data.\nWe have also predicted \n${\\cal B}(B^-\\to\\Lambda\\bar p K^+ K^-)=(3.0^{+1.3}_{-0.9})\\times 10^{-6}$\nto be accessible to \n the LHCb and BELLE experiments.\nThe study of $B\\to {\\bf B_1 \\bar B_2}M_1 M_2$ benefits the future test of T violation,\nas the T-odd triple momentum product correlation of $\\vec{p}_1\\cdot (\\vec{p}_2\\times \\vec{p}_3)$\ncan be directly constructed.\n\n\\section*{ACKNOWLEDGMENTS}\nWe would like to thank Dr. Eduardo Rodrigues for useful discussions.\nThis work was supported in part by National Center for Theoretical Sciences,\nMoST (MoST-104-2112-M-007-003-MY3), and\nNational Science Foundation of China (11675030).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Description of the problem and motivation} \n\nFor a set $A$ of integers, $\\gcd(A)$ and ${\\mathrm {lcm}}(A)$  denote the \ngreatest common divisor ($\\gcd$) and the least common multiple \n(${\\mathrm {lcm}}$) of the elements of $A$, respectively. \nWe consider some questions of how $\\gcd$ and ${\\mathrm {lcm}}$ \nbehave on various subsets $S$ of the original set $A$.\n\n\nWe are interested in both designing algorithms to construct  \nsuch sets $S$ with prescribed properties of  $\\gcd(S)$ and ${\\mathrm {lcm}}(S)$ \nand also in upper and lower bounds on what one can possibly achieve.\n\nWe consider the question of finding a subset\n$S\\subseteq A$ of the smallest possible cardinality with minimal\n$\\gcd$, namely, $\\gcd(S)= \\gcd(A)$, or with maximal ${\\mathrm {lcm}}$, namely,\n${\\mathrm {lcm}}(S) = {\\mathrm {lcm}}(A)$.  We also consider a modification of this question\nwhere we impose that a specific set $B$ of integers be contained in\n$S$. This $B$ may contain elements of $A$. This question arises in the\ntheory of {\\it circulant graphs} and is a special case of graph {\\it\n  editing problems\\\/}, see~\\cite{DaMo12}, \\cite{Gol13}, \\cite{Math10}\nand~\\cite{MaSz12} for the background and further references.\n\nTo explain this connection we recall that an (undirected)\n\\emph{circulant graph} $G(A,m)$ on $m$ nodes, labelled $0, 1, \\ldots,\nm-1$, is defined by a set $A$ of integers\ncalled {\\it links\\\/}, where the nodes $i$ and $j$ are connected if and\nonly if $|i-j| \\equiv a \\bmod m$ for some $a \\in A$.  Clearly,\n$G(A,m)$ is connected if and only if $\\gcd\\(A\\cup \\{m\\}\\) = 1$. Thus\nit is natural to ask how many links can at most be removed from $A$ so\nthat the new circulant graph is still connected. This leads to\nthe above question with $B = \\{m\\}$.\n\n\nThe above can be generalized as follows:\n\n\\begin{question} \n  \\label{ques:gcd subset} Given two sets $A$ and $B$ of positive integers, find\n  a subset $S \\subseteq A$ of the smallest possible size with $\\gcd(S\n  \\cup B) = \\gcd(A \\cup B)$.\n\\end{question}\n\nSimilarly, we  also ask:\n\n\\begin{question} \n  \\label{ques:lcm subset} Given two sets $A$ and $B$ of positive integers, find\n  a subset $S \\subseteq A$ of the smallest possible size with \n${\\mathrm {lcm}} ( S \\cup B) = {\\mathrm {lcm}} (A \\cup B)$.\n\\end{question}\n\nWe first formalize these questions as decision problems.\n\n\\begin{prblm} {\\sl Minimum subset with minimal $\\gcd$}, \\textsc{MinGcd}\n\\begin{description}\n\\item[Input] Sets $A$ and $B$ of positive integers, positive integer\n  $k$.\n\\item[Question] Does $A$ contain a subset $S$ with $\\#S \\leq k$ and\n  $\\gcd(S \\cup B) = \\gcd(A \\cup B)$?\n\\end{description}\n\\end{prblm}\n\n\\begin{prblm} {\\sl Minimum subset with maximal ${\\mathrm {lcm}}$}, \\textsc{MaxLcm}\n\\begin{description}\n\\item[Input] Sets $A$ and $B$ of positive integers, positive integer\n  $k$.\n\\item[Question] Does $A$ contain a subset $S$ with $\\#S \\leq k$ and\n  ${\\mathrm {lcm}}(S \\cup B) = {\\mathrm {lcm}}(A \\cup B)$?\n  \\end{description}\n\\end{prblm}\n\nThe input size of an instance $(A,B)$ for both  \\textsc{MinGcd} and \n \\textsc{MaxLcm} is naturally defined as\n$$\nI(A,B) = \\sum_{a\\in A\\cup B} \\rf{\\log(a+1)}\n$$\nwhere $\\log z$ denotes the binary logarithm of $z\\ge 1$. \n\n\nFor each of these (and other similar)  problems \\textsc{X}, we denote as \n\\textsc{OPT-X} the\ncorresponding optimization problem, where one has to find subsets as\ndescribed with minimal $k$.\n\n\\subsection{Main Results}\nWe can now formulate our main results. \n\n\\begin{thm}\n\\label{thm:GCDLCM-NP}\n\\textsc{MinGcd} and \\textsc{MaxLcm} are $\\mathrm{NP}$-complete.\n\\end{thm}\n\nFurthermore, a combination of the classical greedy approximation \nalgorithm of~\\cite[Theorem~4]{John74}\nand known inapproximability results, see, for example,~\\cite[Theorem~7]{AMS06},\nyield the following.\n\n\\begin{thm}\n\\label{thm:GCDLCM-Appr}\n\\textsc{OPT-MinGcd} and \\textsc{OPT-MaxLcm}  can be approximated in polynomial\ntime within a factor $O(\\log I(A,B))$, but not within a factor $o(\\log\nI(A,B))$ if $\\mathrm{P \\neq NP}$.\n\\end{thm}\n\n\n\n\n\\section{Reductions Between Various Problems} \n\\label{sec:Prob}\n\n\\subsection{Reduction to $B = \\varnothing$}\n\nWe start by constructing from $A,B \\subseteq \\mathbb{Z}$ a set $A_B  \\subseteq\n\\mathbb{Z}$ so that\n\\begin{align*}\n\\text{\\textsc{OPT-MinGcd}} (A,B) = \\text{\\textsc{OPT-MinGcd}} (A_B , \\varnothing).\n\\end{align*}\nThis reduces the general case to the special situation where $B =\n\\varnothing$. Moreover, given a minimum solution set $S$ for one of\nthe two problems, one can easily find a solution for the other\none. \n\nFor any integer $a$, we define the nonnegative integer\n$$\na_{B} = \\gcd(\\{a\\} \\cup B),\n$$\nand apply this element-wise to any $S \\subseteq \\mathbb{Z}$:\n$$\nS_{B} = \\{a_{B} \\colon a \\in S\\}.\n$$\nWe claim that for any $S \\subseteq A$ we have\n\\begin{align}\\label{al:SA}\n\\gcd(S \\cup B) = \\gcd(S_{B}).\n\\end{align}\nFor any $c \\in \\mathbb{Z}$, we have\n\\begin{align*}\n  c \\; | \\gcd (S \\cup B)& \\Longleftrightarrow \\forall a \\in S \\;\n  \\forall b  \\;  \\in B \\quad c \\;| \\;a \\text{ and } c \\;| \\;b\\\\\n  & \\Longleftrightarrow (\\forall a \\in S \\quad c \\;| \\;a) \\text{ and } c \\;| \\gcd(B)\\\\\n  & \\Longleftrightarrow \\forall a \\in S \\quad c \\;| \\;a_{B}\n  \\Longleftrightarrow c \\; | \\gcd (S_{B}).\n\\end{align*}\nIn particular, we have $\\gcd(A \\cup B) = \\gcd(A_{B})$.\n\nDistinct $ a \\in A$ may yield the same $a_{B}$.\nHowever, if $S \\subseteq A$ has minimal size with $\\gcd(S \\cup B) = \\gcd(A\n\\cup B)$, then $a \\mapsto a_{B}$ is injective on $S$, and $\\#S =\n\\#S_{B}$. Thus\n$$\n\\text{\\textsc{OPT-MinGcd}} (A,B) \\geq \\text{\\textsc{OPT-MinGcd}} (A_B , \\varnothing).\n$$\n\n\nFor the reverse direction, we take a section $\\sigma$ of $a \\mapsto\na_{B}$ on $A$, so that $\\sigma(b) \\in A$ and $(\\sigma(b))_{B} = b$ for $b\n\\in A_{B}$.\nFor any $T \\subseteq A_{B}$ of minimal size with $\\gcd(T) = \\gcd\n(A_{B})$, we claim that\n$$\n\\gcd (\\sigma(T) \\cup B) = \\gcd (A \\cup B).\n$$\nThis follows from (\\ref{al:SA}), since $(\\sigma(T))_{B}= T$ and \n$$\n\\gcd (A \\cup B) = \\gcd (A_{B}) = \\gcd(T) = \\gcd (\\sigma(T) \\cup B).\n$$\n\nWe have $\\#\\sigma(T) \\leq \\#T$ and $\\gcd((\\sigma(T))_{B}) =\n\\gcd(A_{B})$. The minimality of $\\#T$ implies that $\\#\\sigma (T) =\n\\#T$ and thus\n$$\n\\text{\\textsc{OPT-MinGcd}} (A,B) \\leq \\text{\\textsc{OPT-MinGcd}} (A_B , \\varnothing).\n$$\n\n\nOverall, it follows that the minimal solution sizes for $(A, B)$ and\n$A_{B}$ are equal, and that the solution sets are related by the above\ncorrespondence. Clearly the set $A_{B}$ can be constructed in time \npolynomial in $I(A,B)$.\nThus both the decision and the optimization versions\nof the general and the special cases are polynomial-time equivalent.\n\nSo from now on we assume that the input consists of one set $A$ and denote \nby $I(A) = I(A,\\emptyset)$ the input size. \n\n\n\n\\subsection{Minimum Cover Problem} \n\nWe present polynomial time reductions between \\textsc{MinGcd}, \\textsc{MaxLcm} and the\nfollowing problem, which is well studied in complexity theory.\n\n\n\n\n\n\\begin{prblm} {\\sl Minimum cover}, \\textsc{MinCover}\n\\begin{description}\n\\item[Input] List ${\\mathcal C}$ of subsets of a finite set $X$, positive integer\n  $k$.\n\\item[Question] Does ${\\mathcal C}$ contain a cover for $X$ of size $k$ or less,\n  that is, a subset ${\\mathcal D} \\subseteq {\\mathcal C}$ with $\\#{\\mathcal D} \\leq k$ such that\n  every element of $X$ belongs to at least one member of ${\\mathcal D}$?\n  \\end{description}\n\\end{prblm}\n\nFurthermore, let $n$ be the input size, usually about $\\#{\\mathcal C} \\cdot \\log m$ if $X =\n\\{1, \\ldots, m\\}$. Then \\textsc{OPT-MinCover} can be approximated in\npolynomial time with\\-in a factor of $O(\\log n)$, but no smaller\nfactor (unless $P=NP$), see~\\cite{AMS06}. \n\nIt is well known that \\textsc{MinCover} is NP-complete, see, for\nexample,~\\cite[Problem~SP5, Section~A.3.1]{garjoh79}.\nIn the next subsections, we present various reductions between\n\\textsc{MinCover} and our problems. The latter are trivially in NP, and their\nreduction to \\textsc{MinCover} transfers approximation algorithms for the latter\nto approximation algorithms for our problems.\nOn the other hand, the reductions from \\textsc{MinCover} to our problems\nshow that the latter cannot be approximated too well.\n\n\\subsection{Reduction from  \\textsc{MaxLcm} to \\textsc{MinCover}}\n\\label{sec:L2MC}\n\nLet us take an instance $(A,k)$\nof \\textsc{MaxLcm}. We compute a \\emph{coprime basis} \n($B,e$) of $A$, where $B$ consists of pairwise coprime integers $b \\geq\n2$ and $e  \\colon~ A \\times B \\longrightarrow \\mathbb{N}$ is such that $a =\n\\prod_{b \\in B} b^{e(a,b)}$ for all $a \\in A$. By dropping the unneeded\nelements $b$ where $e(a,b)=0$ for all $a\\in A$ from $B$, \nwe may assume that\n\\begin{equation}\\label{eq:BA}\n\\forall b \\in B ~\\exists ~a \\in A \\colon~e(a,b) \\geq 1. \n\\end{equation}\nWe recall that \\cite[Section~4.8]{bacsha96} discuss coprime bases (under the\ndesignation of \\emph{gcd-free basis}) and show that one can be computed with\n$O(I(A)^{2})$ bit operations, where, as before, $I(A)$ is the input size. They\nuse classical arithmetic. According to~\\cite{bern05}, fast\narithmetic yields an algorithm using $I(A) (\\log I(A))^{O(1)}$\noperations. \n\nBy the above, the size of $B$ is polynomial in that of $A$.\nWe note that the size of $B$ can actually be much smaller than that of $A$:\nTake the first $m$ primes, all exponent vectors\n$e$ in $\\{1,2\\}^m$, and then all $2^m$ values $ a_e = \\prod_{1\\leq i \\leq m} p_i^{e_i}$.\nThen the coprime basis $B$ consists of just these\n$m$ primes and  $\\text{size}(A)$ is only logarithmic in $\\text{size}(B)$.\nThat is no worry, since we only use this reduction to derive good approximations for \n\\textsc{MinCover} (which do not exist by the hardness result mentioned above) from\ngood approximations to our problems; hence the latter do not exist either.\n\nFor $b \\in B$, we let $d(b) = \\max \\{e(a,b)  \\colon~ a \\in A\\}$. Thus $d(b) \\geq\n1$ by~\\eqref{eq:BA}, and ${\\mathrm {lcm}} (A) = \\prod_{b \\in B} b^{d(b)}$; see \nalso~\\cite[Corollary~4.8.2]{bacsha96}. For $a \\in A$, we set\n$$\nC_{a} = \\{b \\in B  \\colon~ e(a,b)= d(b)\\}.\n$$\nWe now take a subset $E \\subseteq A$ such that\n$\\{C_a\\colon a \\in A \\} = \\{C_a\\colon a \\in E \\} $\nand the $C_a$ in the latter set are pairwise distinct.\n Clearly this can be \ndone in time polynomial in $I(A)$. It is also clear that \n${\\mathrm {lcm}}( E) = {\\mathrm {lcm}} (A)$. \n\nNow we consider the  \\textsc{MinCover}\ninstance with $X=B$ and ${\\mathcal C}=\\{C_{a}  \\colon~ a \\in E \\}$.\nFor $S \\subseteq E$, we consider ${\\mathcal D}= \\{C_{a}  \\colon~ a \\in S\n\\}$. Then $\\#{\\mathcal D} = \\#S$, and \n\\begin{align*}\n{\\mathrm {lcm}} (S) = {\\mathrm {lcm}}( E) = {\\mathrm {lcm}} (A)  &\\Longleftrightarrow \\forall b \\in B \\quad  b^{d(b)} \\mid\n{\\mathrm {lcm}} (S)\\\\\n& \\Longleftrightarrow \\forall b \\in B ~ \\exists a \\in S \\quad b^{d(b)} \\mid\na\\\\\n& \\Longleftrightarrow \\forall b \\in B ~ \\exists a \\in S \\quad e(a,b) =\nd(b)\\\\\n& \\Longleftrightarrow \\forall b \\in B = X ~ \\exists a \\in S \\quad b \\in\nC_{a}\\\\ \n&\\Longleftrightarrow {\\mathcal D} \\text{ covers } X.\n\\end{align*}\nThus a solution $S$ of \\textsc{MaxLcm} with $\\#S \\leq k$ implies one\nof \\textsc{MinCover} with $\\#{\\mathcal D} \\leq k$. \n\nConversely,  given a cover  ${\\mathcal D}= \\{C_{a}  \\colon~ a \\in S\n\\}$ of $X$ with $\\#{\\mathcal D} \\leq k$ we conclude that $\\# S \\le k$, since the sets\n$C_{a}$ for $a \\in E$  are pairwise distinct. \n \nThus the size of the smallest set $S \\subseteq A$ \nwith ${\\mathrm {lcm}}(S) = {\\mathrm {lcm}}(A)$ and the size of the smallest cover \n${\\mathcal D}$ of $X$ coincide. \nThis concludes the reduction. \n\n\n\\subsection{Reduction from  \\textsc{MinGcd} to \\textsc{MinCover}}\n\\label{sec:G2MC}\nWe replace $d(b)$ in the previous reduction by $g(b) = \\min \\{e(a,b)\n\\colon~ a \\in A\\}$.\nThen\n$$\n\\gcd(A) = \\prod_{b \\in B}b^{g(b)}.\n$$\nWe see that $\\gcd (E) = \\gcd(A)$ divides $\\gcd(S)$\nand in the argument for $\\gcd(E) = \\gcd(S)$,\n a divisibility $b^{d(b)} \\mid u$ has to be replaced by $b^{g(b)+1}\n\\nmid u$. Otherwise the argument goes through unchanged.\n\n\n\\subsection{Reduction from \\textsc{MinCover} to \\textsc{MaxLcm}}\n\\label{sec:MC2L}\n\nWe are given a list ${\\mathcal C}$ of sets $C_{1}, \\ldots, C_{l} \\subseteq\nX$, where $X = \\{1, \\ldots, m \\}$, and $k \\geq 1$. We may assume that  $X= \\bigcup_{i \\leq l} ~\nC_{i}$ and $k \\leq l$, otherwise the \\textsc{MinCover} problem is trivial.\nWe let $p_{1} < p_{2} < \\cdots < p_{m}$ be the first\n$m$ prime numbers, $a= \\prod_{j \\in X} p_{j}$,\n$$\na_{i} = \\prod_{j \\in C_{i}} p_{j},\n$$\nfor $i \\leq l$ and $A=\\{a_{1}, \\ldots, a_{l} \\}$. Thus $a = {\\mathrm {lcm}}\n(A)$. We use the same value of $k$ for both problems. Since $p_{m}\n\\leq (1+o(1)) \\, m \\ln m$ as $m \\to \\infty$, the bit size of $(A,k)$ is in\n$O(l m \\log m)$. The set $A$ can be computed in\ntime polynomial in $lm$, using the sieve of Eratosthenes for generating\nthe primes.\n\nSuppose that $I \\subseteq \\{1, \\ldots, m \\}$ is such that $\\#I \\leq k$\nand ${\\mathrm {lcm}} (S) = {\\mathrm {lcm}} (A)$, where $S = \\{a_{i} \\colon~ i \\in I \\}$. Let\n$$\n{\\mathcal D} = \\{C_{i}  \\colon~ i \\in I \\}.\n$$\nThen $\\#{\\mathcal D} \\leq k$. Furthermore, for any $j \\in X$, $p_{j}$ divides\n${\\mathrm {lcm}} (A) = {\\mathrm {lcm}} (S)$ and hence $a_{i}$ for some $i \\in I$. It follows\nthat $j \\in C_{i} \\in {\\mathcal D}$. Thus ${\\mathcal D}$ covers $X$.\n\n\nOn the other hand, suppose that $I \\subseteq \\{1, \\ldots, m\\}$ is such\nthat $\\#I \\leq k$ and ${\\mathcal D} = \\{C_{i}  \\colon~ i \\in I$\\}\n covers $X$. Then $S = \\{a_{i}  \\colon~ i \\in I \\}$ satisfies $\\#S \\leq\n k$ and ${\\mathrm {lcm}} (S) = a = {\\mathrm {lcm}} (A)$. \n\n\n\\subsection{Reduction from \\textsc{MinCover} to \\textsc{MinGcd}}\n\\label{sec:MC2G}\nFor an analogous reduction to \\textsc{MinGcd}, we replace\n$a_{i}$ by $a\/p_{i}$ in the above.\n\n\n\\section{Proofs of Main Results}\n\nWe start with the upper bounds claimed in Theorems\n\\ref{thm:GCDLCM-NP} and \\ref{thm:GCDLCM-Appr}.\nThe fact that \\textsc{MaxLcm} and \\textsc{MinGcd} are NP-complete is trivial.\nFurthermore, the reductions of Sections~\\ref{sec:L2MC} and~\\ref{sec:G2MC}\nshow that the know approximation algorithms for \\textsc{MinCover}\nalso yield ones for our problems.\n\nFurthermore, our claimed lower bounds (NP-hardness and inapproximability)\nfollow from the reductions in\nSections~\\ref{sec:MC2L} and~\\ref{sec:MC2G}\nfrom our problems to\n\\textsc{MinCover}, together with the NP-hardness and inapproximability\nof \\textsc{MinCover}, as cited above.\n\n\n\\section*{Acknowledgments}\n\nThe authors are grateful to Fedor Fomin for some references on \nthe approximability and inapproximability of the set-cover problem. \n\nThe first author's work was supported by the B-IT Foundation and the\nLand Nordrhein-Westfalen. The second author's work was supported by the \nAustralian Research Council grants DP110100628 and DP140100118.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe approach to producing unconditional simulations for arbitrary surfaces (sphere, ellipsoid, geoid, torus, etc.) will be described in this paper. The approach is also applicable to nonsmooth surfaces. This paper is based on ideas from a turning band approach (\\cite{TwoDimensionalSimulationByTurningBands}, \\cite{TurningBandTwoDimensional}, \\cite{ClosedFormSolutionsOfTheTwoDimensionalTurningBandEquation}, and \\cite{TurningBandSimulationsInThreeDimensions}) and kernel convolution (\\cite{KernelConvolution1996}, \\cite{KernelConvolution1998}, and \\cite{KernelConvolutionFFT}). Similar to the turning band, all processing is done on a set of random lines. The difference is in using a random set of points covering an area of interest and projecting them onto the set of random lines. For example, the set of uniformly distributed points on the surface of a sphere is projected onto the set of random lines. The kernel convolution is applied to each line, and the reconstruction of the process is performed similarly to the turning band approach. Using different kernels will produce a nonstationary solution.\n\n\n\n\n\\section\n{\n  Kernel Convolution by Turning Band\n  \\label{sec:KernelConvolutionByTurningBand}\n}\n\nHere are the steps of the algorithm:\n\\begin{enumerate}\n  \\item \\label{en:UniformDistributionOnSurface} From a uniform distribution, simulate $N$ points $\\vec{s_i}$ on a surface. \\footnote{Replacing uniform distribution on the surface by pseudorandom uniform distribution will produce an even coverage and guarantee the stability of the algorithm.}\n  \\item From a standard normal distribution, simulate value $\\xi_i$ for each point.\n  \\item Simulate $N_d$ random or pseudorandom directions (the algorithm to simulate pseudorandom directions can be found in \\cite{RandomDirections} and \\cite{TurningBandSimulationsInThreeDimensions}).\n  \\item Cycle over all directions.\n  \\begin{enumerate}\n    \\item Project points to a direction.\n    \\item Apply kernel convolution (this is usually done using FFT).\n  \\end{enumerate}\n  \\item For each location where the result is desired, project that location to all directions and average the results of kernel convolutions.\n\\end{enumerate}\n\nThe applied one-dimensional kernel $k_1{\\left( \\bullet \\right)}$ will correspond to a three-dimensional kernel $k_3{\\left( \\bullet \\right)}$ through integral\n\\begin{equation}\n  k_3{\\left( \\vec{h} \\right)}\n  =\n  \\oiint\n  {\n    k_1{\\left( \\left( \\vec{h}, \\hat{n} \\right) \\right)}\n    \\dif{\\hat{n}}\n  }\n  ,\n  \\label{eq:IntegralKernel}\n\\end{equation}\nwhere $\\left( \\vec{x}, \\vec{y} \\right)$ is a scalar product.\n\nThe kernel distance is the direct distance in three-dimensional space.\n\nThis is similar to the turning band method, and the relationship between three-dimensional or two-dimensional covariance with one-dimensional covariance can be found in \\cite{TwoDimensionalSimulationByTurningBands}, \\cite{TurningBandTwoDimensional}, and \\cite{ClosedFormSolutionsOfTheTwoDimensionalTurningBandEquation}. The same relationship holds true for the kernel functions. In a three-dimensional case, the formula is\n\\begin{equation}\n  k_1(h) = (h \\cdot k_3{\\left( h \\right)})'\n  .\n  \\label{eq:RelationshipBetweenKernelsThree}\n\\end{equation}\n\nIn two dimensions for kernel $k_2{\\left( \\bullet \\right)}$\n\\begin{equation*}\n  k_1(h)\n  =\n  k_2{\\left( 0 \\right)}\n  +\n  h\n  \\int\\limits_{0}^{h}\n  {\n    \\dfrac\n    {\n      k_2'{\\left( x \\right)}\n    }\n    {\n      \\left(\n        h^2 - x^2\n      \\right)\n      ^\n      {\n        \\frac{1}{2}\n      }\n    }\n    \\dif{x}\n  }\n  =\n  k_2{\\left( 0 \\right)}\n  +\n  h\n  \\int\\limits_{0}^{\\frac{\\pi}{2}}\n  {\n    k_2'\n    {\n      \\left(\n        h\n        \\sin{\\left( \\Theta \\right)}\n      \\right)\n    }\n    \\dif{\\Theta}\n  }\n  .\n\\end{equation*}\n\n\\section{Kernel Convolution for Unconditional Simulation}\n\nApplying the algorithm from the section ``\\nameref{sec:KernelConvolutionByTurningBand}'' for any location $\\vec{s}$ will produce a random field approximately equal to\n\\begin{equation}\n  \\sum_{i = 0}^{N}\n  {\n    \\left(\n      k_3\n      {\n        \\left( \\vec{s_i} - \\vec{s} \\right)\n      }\n      \\cdot\n      \\xi_i\n    \\right)\n  }\n  .\n  \\label{eq:KernelApplied}\n\\end{equation}\n\nThe approximation is due to the finite number of directions.\n\nThe variance at location $\\vec{s}$ is approximately equal to\n\\begin{equation}\n  \\sum_{i = 0}^{N}\n  {\n    k_3\n    ^2\n    {\n      \\left( \\vec{s_i} - \\vec{s} \\right)\n    }\n  }\n  .\n  \\label{eq:KernelVariance}\n\\end{equation}\n\nFor the sphere, the estimate of \\eqref{eq:KernelVariance} does not depend on location $\\vec{s}$. This does not hold true for other surfaces. However, for any surface with finite sampling $N$, it is not a constant. Calculating this value is important in standardizing unconditional simulations. The estimate of \\eqref{eq:KernelVariance} can be found by applying the algorithm from the section ``\\nameref{sec:KernelConvolutionByTurningBand}'' for the square of the kernel for the same set of points with values of one. Dividing the approximation of \\eqref{eq:KernelApplied} by the square root of the approximation of \\eqref{eq:KernelVariance} will produce an approximation of the Gaussian process with unit variance.\n\nAnother way to estimate \\eqref{eq:KernelVariance} is by producing many unconditional simulations and estimating the variance from them. This will also allow the estimation of covariances.\n\nCompared to the turning band approach, using this algorithm is significantly slower. The slowest part is projecting simulated points to directions. Notice that the simulated points do not depend on the kernel. Therefore, to speed up the approach, it is possible to simulate points and project them to directions in advance. Combining them by bins will significantly reduce the amount of memory necessary to store precalculated results.\n\n\\section{Conditional Simulations}\n\nTo perform conditional simulations, it is necessary to know the covariance function. For a sphere, the covariance function is isotropic and does not depend on location. Therefore, it can be derived directly from the kernel as described in \\cite{CompactCovarianceModelOnASphere} and \\cite{KernelConvolutionForRingsOnPlane}. For arbitrary surfaces, it is possible to make many unconditional simulations and from them estimate the covariance function. Conditional simulations are obtainable from unconditional simulations through solutions of the Kriging system \\cite{ConditionalSimulationsGeostatisticsForConditionalSimulationOfOreBodies}, \\cite{ConditionalSimulationsBook}, and \\cite{ConditionalSimulationsSpatialPredictionSpatialSamplingAndMeasurementError}. \\cite{GeostatisticalMappingWithContinuousMovingNeighborhood} describes the approach to construct conditional simulations without abrupt changes.\n\n\\section{Class of Exponential Kernels}\n\nLet the kernel function have the form\n\\begin{equation}\n  k_3{\\left( h \\right)}\n  =\n  \\sum_{i}\n  {\n    \\left(\n      P_i{\\left( h \\right)}\n      \\cdot\n      \\mathsf{e}^{-\\Theta_i \\cdot h}\n    \\right)\n  }\n  ,\n  \\label{eq:Form}\n\\end{equation}\nwhere $P_i$ is a polynomial.\n\nThe result of the convolution can be obtained by the sequential application of the kernel. The technique is described in \\cite{Smoothing} and \\cite{SmoothingNetwork} with the difference being that the integration is replaced by summation. This reduces the complexity of the kernel convolution from $ O{\\left( n \\cdot \\ln{n} \\right)} $ to $ O{\\left( n \\right)} $, where $n$ is the number of bins on the line.\n\nFor finite kernel\n\\begin{equation}\n  k_3{\\left( h \\right)}\n  =\n  \\left\\{\n  {\n    \\begin{aligned}\n      &\\sum_{i}\n      {\n        \\left(\n          P_i{\\left( h \\right)}\n          \\cdot\n          \\mathsf{e}^{-\\Theta_i \\cdot h}\n        \\right)\n      }\n      ,\n      &\\text{ if }\n      h < T,\\\\\n      &0,\n      &\\text{ otherwise},\n    \\end{aligned}\n  }\n  \\right.\n  \\label{eq:FormFinite}\n\\end{equation}\nwhere $T$ is the kernel width. To be able to evaluate $k_1{\\left( h \\right)}$ from \\eqref{eq:RelationshipBetweenKernelsThree}, it is necessary to satisfy $\\lim\\limits_{h \\rightarrow T^{-}}{k_3{\\left( h \\right)}} = 0$. Otherwise, $k_1{\\left( h \\right)}$ will not be finite.\n\nThere are several advantages of using the finite kernel:\n\\begin{itemize}\n  \\item Ability to produce unconditional simulations for a large area by partitioning it into small pieces (tiles) for independent processing.\n  \\item Ability to have a flexible class of kernels in form \\eqref{eq:FormFinite}.\n\\end{itemize}\n\nThe ability to have a flexible class of kernels is important for nonstationary simulations. It allows not only changes in kernel bandwidth but in their shape as well. One way to construct a flexible class of kernels in form \\eqref{eq:FormFinite} is by changing the polynomial part. It is reasonable to use decreasing kernels because the majority of physical processes gradually decreases dependence over distance. From~\\cite{MonotonicFunction}, a Bernstein density (a weighted sum of beta probability distribution functions defined on the unit interval\n$ \\left[ 0 , 1 \\right] $) is defined as\n\\newcommand{\\myfunctionbeta}[1]\n{\n  Beta\n  {\n    \\left(\n      #1\n      \\mid\n      j , m - j + 1\n    \\right)\n  }\n}\n\\newcommand{\\myfunctionbetacorrection}[2][]\n{\n  \\sum _{j = 1} ^{m}\n    \\left(\n      \\myfunctionwi[#1]{j}\n      \\cdot\n      \\myfunctionbeta{#2}\n    \\right)\n}\n\\newcommand{\\myfunctionbetacorrectionintegrated}[2][]\n{\n  \\sum _{j = 1} ^{m}\n    \\left(\n      \\myfunctionwi[#1]{j}\n      \\int\\limits_{x}^{1}\n      {\n        \\myfunctionbeta{#2}\n        \\dif{y}\n      }\n    \\right)\n}\n\\begin{equation*}\n  p{\\left( x \\mid m , \\vec{w} \\right)} = \\myfunctionbetacorrection{x} \\; , \\;\n  \\myfunctionwi{j} \\geq 0 \\; , \\;\n  \\sum _{j = 1} ^{m} {\\myfunctionwi{j} = 1} \\; ,\n\\end{equation*}\nwhere\n$ m $ is the number of distribution functions,\n$ \\vec{w} $ are weights,\n$ Beta{\\left( \\bullet \\right)}$\nis the beta probability density function\n\\begin{equation*}\n  \\myfunctionbeta{x}\n  =\n  \\dfrac{m !}{\\left( j - 1 \\right) ! \\cdot \\left( m - j \\right) !}\n  \\cdot\n  x ^{j - 1}\n  \\cdot\n  \\left( 1 - x \\right) ^{m - j}\n  \\; .\n\\end{equation*}\n\nForming\n\\begin{equation*}\n  P{\\left( x \\right)}\n  =\n  \\int\\limits_{x}^{1}\n  {\n    p{\\left( x \\mid m , \\vec{w} \\right)}\n  }\n  =\n  \\myfunctionbetacorrectionintegrated{y}\n\\end{equation*}\nand multiplying by $\\mathsf{e}^{-\\frac{h}{\\Theta}}$ produces a flexible family of kernels. The choice of $\\Theta$ is arbitrary. If the value is too large, the floating point evaluation will lead to a high round off error. If the value is too small, it will not produce good family of kernels. Let's define $\\Theta = 3$.\n\nIn the case of $m = 2$, this becomes\n\\begin{equation}\n  \\left\\{\n    \\begin{aligned}\n      &\n      \\left(\n        \\myfunctionwi{1} \\left( 1 - h \\right)^2\n        +\n        \\myfunctionwi{2} \\left( 1 - h^2 \\right)\n      \\right)\n      \\cdot\n      \\mathsf{e}^{-\\frac{h}{3}}\n      ,\n      &\n      \\text{ if }\n      h < 1,\n      \\\\\n      &\n      0\n      ,\n      &\n      \\text{otherwise},\n    \\end{aligned}\n  \\right.\n  \\label{eq:KernelsPolynomialExponentTwo}\n\\end{equation}\nwhere $0 \\leq \\myfunctionwi{i} \\leq 1$, $i = 1, 2$, and $\\myfunctionwi{1} + \\myfunctionwi{2} = 1$. They are shown in Figure~\\ref{fig:KernelsPolynomialExponent}a.\n\n\\begin{figure} [htb]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[width = 8 cm, keepaspectratio]{KernelsPolynomialExplonent.png} &\n    \\includegraphics[width = 8 cm, keepaspectratio]{KernelsPolynomial.png} \\\\\n    a) &\n    b)\n  \\end{tabular}\n  \\caption\n  {\n    Example of kernels \\eqref{eq:KernelsPolynomialExponentTwo}.\n    $\\myfunctionwi{1} = \\dfrac{i}{15}$, $\\myfunctionwi{2} = 1 - \\myfunctionwi{1}$, $i = \\overline{0..15}$. a) With exponent. b) Without exponent.\n  }\n  \\label{fig:KernelsPolynomialExponent}\n\\end{figure}\n\nIn the case of $m = 3$\n\\begin{equation}\n  \\left\\{\n    \\begin{aligned}\n      &\n      \\left(\n        \\myfunctionwi{1} \\left( 1 - h \\right)^3\n        +\n        \\myfunctionwi{2} \\left( 1 - h \\right)^2 \\left( 1 + 2 h \\right)\n        +\n        \\myfunctionwi{3} \\left( 1 - h^3 \\right)\n      \\right)\n      \\cdot\n      \\mathsf{e}^{-\\frac{h}{3}}\n      ,\n      &\n      \\text{ if }\n      h < 1,\n      \\\\\n      &\n      0\n      ,\n      &\n      \\text{otherwise},\n    \\end{aligned}\n  \\right.\n  \\label{eq:KernelsPolynomialExponentThree}\n\\end{equation}\nwhere $0 \\leq \\myfunctionwi{i} \\leq 1$, $i = 1, 2, 3$, and $\\myfunctionwi{1} + \\myfunctionwi{2} + \\myfunctionwi{3} = 1$. They are shown in Figure~\\ref{fig:KernelsPolynomialExponentThree}a.\n\n\\begin{figure} [htb]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[width = 8 cm, keepaspectratio]{KernelsPolynomialExplonentThree.png} &\n    \\includegraphics[width = 8 cm, keepaspectratio]{KernelsPolynomialThree.png} \\\\\n    a) &\n    b)\n  \\end{tabular}\n  \\caption\n  {\n    Example of kernels \\eqref{eq:KernelsPolynomialExponentThree}.\n    $\\myfunctionwi{1} = \\dfrac{i}{7}$, $\\myfunctionwi{2} = \\dfrac{j}{7}$, $\\myfunctionwi{3} = 1 - \\myfunctionwi{1} - \\myfunctionwi{2}$, $i = \\overline{0..7}$, $j = \\overline{0..7 - i}$. a) With exponent. b) Without exponent.\n  }\n  \\label{fig:KernelsPolynomialExponentThree}\n\\end{figure}\n\nFrom \\eqref{eq:RelationshipBetweenKernelsThree}, it follows that $k_1{\\left( h \\right)}$ has the same form as \\eqref{eq:FormFinite}. Notice that $k_3^2{\\left( h \\right)}$ also has the same form, and the one-dimensional kernel corresponding to $k_3^2{\\left( h \\right)}$ will be of the same form.\n\nIt is possible to extend this approach for anisotropic kernels by adjusting weights and one-dimensional kernel shape in each direction or by stretching the space.\n\n\\section{Examples}\n\nFigure~\\ref{fig:UnconditionalSimulationTorus} shows unconditional simulation on a torus.\n\n\\begin{figure} [htb]\n  \\centering\n  \\includegraphics[width = 12 cm, keepaspectratio]{UnconditionalSimulationTorus.jpg}\n  \\caption{Unconditional simulation on a torus with inner radius 1, outer radius 3, and kernel $\\mathsf{e}^{-10 h}$. Number of simulated points $N = 67,108,864$. Number of random directions $N_d = 1,024$.}\n  \\label{fig:UnconditionalSimulationTorus}\n\\end{figure}\n\nFigure~\\ref{fig:UnconditionalSimulatiOnEllipsoid} shows nonstationary unconditional simulation on a spheroid with eccentricity equals $\\frac{3}{4}$. The kernel changes smoothly in shape from left to right. $1,024$ unconditional simulations were used to estimate variance and standardize unconditional simulations. Due to changes in the kernel, the right and left parts of the spheroid have different covariance structures. This is visible as a smoother random field on the right compared to the random field on the left.\n\n\\begin{figure} [htb]\n  \\centering\n  \\includegraphics[width = \\columnwidth, keepaspectratio]{NonstationarySimulationsSpheroid.png}\n  \\caption\n  {\n    Unconditional simulation on an oblate spheroid with semimajor axis equals $2$, and semiminor axis equals $1$. The surface is shown as elevation to depict changes in smoothness. The kernel has the form of \\eqref{eq:KernelsPolynomialExponentTwo}. It changes smoothly from the kernel on the left\n    $\n      \\left( 1 - \\left( 3h \\right)^2 \\right)\n      \\cdot\n      \\mathsf{e}^{-h}\n    $\n    to the kernel on the right\n    $\n      \\left( 1 - 3h \\right)^2\n      \\cdot\n      \\mathsf{e}^{-h}\n    $\n    with bandwidth equals $\\frac{1}{3}$.\n    Number of simulated points $N = 262,144$. Number of random directions $N_d = 1,024$.\n  }\n  \\label{fig:UnconditionalSimulatiOnEllipsoid}\n\\end{figure}\n\n\\section{Convergence Issue}\n\nWhen the kernel bandwidth tends to be small, the convergence of \\eqref{eq:KernelApplied} to the true kernel \\eqref{eq:IntegralKernel} becomes slow. The number of required directions makes this method inefficient. This is because the distribution of the number of points that are farther away from the center of the kernel (location $\\vec{s}$ in \\eqref{eq:KernelApplied}) in three dimensions increases squarely. This is even true for some directions when points are only located on the surface.\n\n\\section{Tile Approach}\n\nFinite kernel and overlapping tiles overcome the issue described in the previous section. For example, in two dimensions, if all tiles are $2$ by $2$ and start from integer numbers (corners can be rounded), then the kernel with a bandwidth not exceeding $\\frac{1}{2}$ will always be completely inside at least one tile; see Figure~\\ref{fig:OverlappingTiles}. This is applicable in any number of dimensions.\n\n\\begin{figure} [htb]\n  \\centering\n  \\begin{tikzpicture}[scale = 2.0]\n   \n    \\fill [cyan] (2.5, 1) -- (3.5, 1) arc (-90:0:0.5) -- (4, 2.5) arc (0:90:0.5) -- (2.5, 3) arc (90:180:0.5) -- (2, 1.5) arc (180:270:0.5) -- cycle;\n   \n   \n   \n    \\fill [red] (2.7, 1.8) circle (0.5);\n    \\tkzInit[xmin = 0, ymin = 0, xmax = 5, ymax = 4]\n    \\tkzGrid\n    \\tkzAxeXY\n  \\end{tikzpicture}\n  \\caption\n  {\n    The red circle is the area covered by a finite kernel. The blue rounded square is the tile completely covering the finite kernel.\n  }\n  \\label{fig:OverlappingTiles}\n\\end{figure}\n\nThe bandwidth should not be significantly smaller than the tile width, and the exponential part in the kernel can be dropped. The flexible family of kernels without the exponential part is shown in Figures~\\ref{fig:KernelsPolynomialExponent}b and \\ref{fig:KernelsPolynomialExponentThree}b.\n\nFigure~\\ref{fig:UnconditionalSimulatiOnPlane}a shows the effect of using the tile approach. The small discontinuities are visible in the horizontal and vertical lines passing through the center. Increasing the number of bands will reduce this effect. Figure~\\ref{fig:UnconditionalSimulatiOnPlane}b shows the result of evaluating \\eqref{eq:KernelApplied} and \\eqref{eq:KernelVariance} directly.\n\n\\begin{figure} [htb]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[width = 8.5 cm, keepaspectratio]{UnconditionalSimulationsUsingTiles.png} &\n    \\includegraphics[width = 8.5 cm, keepaspectratio]{UnconditionalSimulationsExact.png} \\\\\n    a) &\n    b)\n  \\end{tabular}\n  \\caption\n  {\n    Unconditional simulation in two dimensions by kernel\n    $\n      k_3{\\left( h \\right)}\n      =\n      \\left( 1 - 2h \\right)^2\n    $\n    with bandwidth equals $\\frac{1}{2}$.\n    Number of simulated points $N = 65,536$. a)~The tile approach with the number of random directions $N_d = 1,024$ (uniformly distributed in three dimensions). b)~Direct calculation.\n  }\n  \\label{fig:UnconditionalSimulatiOnPlane}\n\\end{figure}\n\nIf a continuous result is required, overlapped tiles can be joined smoothly. This will increase calculations by the number of tiles necessary to process. One way to reduce the number of tiles is to use a triangular grid (use nonequilateral triangles for arbitrary surfaces). For each node of the triangular grid, define a tile that covers all neighboring triangles and at least a kernel bandwidth. For each triangle, the barycentric coordinate system will give weights for the tiles. There will only be a maximum of three tiles with nonzero weights; therefore, the complexity is increased three times.\n\n\\section{Using Integer Arithmetic}\n\nIn the turning band approach, the directions are chosen randomly or pseudorandomly. In this section, another approach will be described to make directions proportional to integer numbers. Using such directions will remove errors from applying fixed steps (bins) in each direction.\n\nLet's take a sphere of radius $R = \\sqrt{S}$, where $S$ is a natural number. If the radius $R$ approaches infinity, all integer vectors inside the sphere (all integer vertices except the center of a coordinate system) tend to be uniformly distributed. However, the unique directions are not uniformly distributed (see example in Figure~\\ref{fig:IntegerDirections}a). But assigning a proper weight to each direction will resolve it. The appropriate weights can be assigned as proportionate to the number of vectors with the same direction or to the area of the spherical Voronoi diagram cells.\n\n\\begin{figure} [htb]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[width = 8.5 cm, keepaspectratio]{IntegerDirections.png} &\n    \\includegraphics[width = 8.5 cm, keepaspectratio]{IntegerUniformDirections.png} \\\\\n    a) &\n    b)\n  \\end{tabular}\n  \\caption\n  {\n    a) All integer directions are shown as dots on the sphere for $S = 100$. The number of unique directions equals $1,729$ (counting $\\vec{v}$ and $-\\vec{v}$ once). b) A uniform set of integer directions satisfying that any two directions are at least $3 \\degree$ apart. The number of directions equals $1,405$.\n  }\n  \\label{fig:IntegerDirections}\n\\end{figure}\n\nSelecting only directions separated by some degree will make a uniform coverage (see Figure~\\ref{fig:IntegerDirections}b). The list of integer directions is shown in the \\hyperref[SectionAppendix]{appendix}. This is similar to using a set of pseudorandom directions as described in \\cite{RandomDirections} and \\cite{TurningBandSimulationsInThreeDimensions}.\n\nBecause each direction is represented as $\\alpha \\cdot \\left( x, y, z \\right)$, where $\\alpha$ is a real number (square root of a natural number) and $x$, $y$, and $z$ are integers, any coordinate with integer components will be projected to a position $\\alpha \\cdot i$, where $i$ is an integer number.\n\nIf random points are placed on an integer grid, there are no errors due to binning. If the resultant location also has an integer coordinate, there is no additional error. Notice that the kernel \\eqref{eq:Form} or \\eqref{eq:FormFinite} can be calculated for any position on direction, see \\cite{Smoothing} and \\cite{SmoothingNetwork}. The turning band approach \\cite{TwoDimensionalSimulationByTurningBands}, \\cite{TurningBandTwoDimensional}, \\cite{ClosedFormSolutionsOfTheTwoDimensionalTurningBandEquation}, and \\cite{TurningBandSimulationsInThreeDimensions} might also benefit from using integer arithmetic.\n\n\\section{Conclusion}\n\nThe new approach to producing unconditional nonstationary simulations on arbitrary surfaces is equivalent to the kernel convolution approach with the kernel shape depending on location (\\cite{KernelConvolution1996}, \\cite{KernelConvolution1998}, and \\cite{KernelConvolutionFFT}). Applying a kernel to the arbitrary surfaces produces a larger class of valid covariance functions than the class of valid covariance functions in three-dimensional space.\n\nPrecalculating projections of random points in all directions for all tiles might be necessary for the efficiency of the approach. Nevertheless, even with precalculations, this approach is not efficient. The better approach is based on kernel convolution using FFT as described in \\cite{ConvolutionFFT}.\n\nNew flexible classes of kernels constructed from polynomials are described in this paper. Because they are linear combinations of some basis kernels, constructing covariances for all pairs of basis kernels is sufficient to calculate covariance between any two kernels. Unconditional simulation for a nonstationary kernel is a weighted sum of unconditional simulations corresponding to basis kernels for the same set of random points.\n\nUsing integer arithmetic, an error introduced in the turning band approach due to binning along each direction can be avoided.\n\n\\section{Acknowledgment}\n\nThe author would like to thank David Burrows for help with area preserving transformation from authalic sphere to spheroid (authalic sphere is a sphere were the surface area equals the spheroid surface area). This transformation was used in simulating uniformly distributed random points on the surface of a spheroid for the creation of Figure~\\ref{fig:UnconditionalSimulatiOnEllipsoid}.\n\n\n\\newcommand{\\doi}[1]{\\textsc{doi}: \\href{http:\/\/dx.doi.org\/#1}{\\nolinkurl{#1}}}\n\n\n\\begingroup\n\\raggedright\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}