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+{"text":"\\section{Introduction}\nBetavoltaic effect refers to the electric power production by a p-n junction bombarded by beta-particles that ionize the semiconductor material. Among the advantages of beta-batteries are their long service duration, amounting to years or even decades, and the possibility to use in the hard-to-reach areas. Betavoltaics and photovoltaics are related disciplines. In both cases, electric power results from the separation of electron-hole pairs produced by beta-electrons or photons by a p-n junction in the presence of a load in the circuit. In comparison to photovoltaics, publications in the field of the basic principles and applications of betavoltaic elements have been less numerous initially (see, e.g., Refs.~\\cite{Rap54, Pfa54, Rap56, Fli64, Ols73, Ols74, Olstech}), but started to attract the attention of the researchers in the recent years \\cite{And00, Bow02, Ada12, Ols12}. \n\nThe main task in betavoltaic design is the choice of a beta-source\/semiconductor combination, which should meet certain requirements. In particular, the beta-particles produced by the source must be absorbed efficiently by the semiconductor. Within the semiconductor, the diffusion length of the electron-hole pairs generated by the beta-flux should be large enough to allow them to reach the p-n junction with as little losses as possible. Because only the relatively low-energy beta-electrons are utilized effectively (with energies varying between 5 and 70 keV) for the realistic semiconductor thicknesses, three main beta-sources are presently employed in betavoltaic applications: Tritium \\Tr, Nickel $^{63}$Ni, and Promethium \\Pm. The respective mean energies of the electrons produced by these sources are 5.7, 18, and 62 keV.\n\nThe efficiency, $\\eta$, of a betavoltaic converter is proportional to the collection coefficient, $Q$, of the electron-hole pairs generated by the beta-flux.  In Refs.~\\cite{Pfa54, Olstech}, $Q$ was calculated under the assumption that the generation function of electron-hole pairs by a beta-flux $g(x) \\propto \\exp(-\\alpha x)$. In reality, the generation function is close to zero within the so-called ``dead layer'' under the front surface, and exhibits a maximum at some distance $x_m$ from the surface \\cite{Dmi78}. This implies that this exponential approximation is correct starting from some $x$-value greater than $x_m$. The emergence of the maximum in the $g(x)$ curve is due to the fact that, initially, the primary electrons pass through the semiconductor with only weak scattering. The dead layer thickness $x_m$ increases with the energy of the incident beta-electrons. For GaAs, $x_m$ is in the range 0.1 -- 1 $\\mu$m \\cite{Dmi78}.\n\nAlthough the works \\cite{Pfa54, Olstech} do report analytical expressions for $Q$ (obtained under the assumption of the absence of the dead layer), the values of $Q = 1$ and 0.7 were used in the calculations of beta-conversion efficiency \\cite{Olstech, Ols12}. While the value $Q = 1$ corresponds to the limiting conversion efficiency that is maximal in principle, the choice $Q = 0.7$ was not explained in \\cite{Olstech, Ols12}.\n\nIn this work, we derive an expression for $Q$ taking the dead layer into account, and also using the realistic values of the nonradiative Shockley-Reed-Hall (SRH) recombination lifetime, $\\tau_{SR}$, for direct-bandgap semiconductors. In such materials, the values of $\\tau_{SR}$ are usually short, and are in the range of $10^{-9}-10^{-7}$ s. We use the so obtained collection coefficient to derive the expression for the realistically attainable beta-conversion efficiency $\\eta$ of various combinations of beta-sources and direct-bandgap semiconductors. When calculating the efficiency, we focus on GaAs as a typical example. We show that decreasing $\\tau_{SR}$ and increasing the dead layer thickness leads to a strong reduction of $Q$ below 1, and to the corresponding reduction of the beta-conversion efficiency.\n\n\\section{Analysis of the collection coefficient}\nWe assume that the electron-hole pairs are generated only weakly within the dead layer, $x < x_m$, while for $x > x_m$, the generation function has the form $g(x) = I_0\\,\\exp(-\\alpha (x-x_m))$, where $I_0$ is the electron-hole pair generation rate in the $x_m$-plane, and $\\alpha^{-1}$ is the characteristic decay length. Furthermore, we assume that $d_p < x_m$ and $S_d \\ll D\/L$, $d_p$ being the junction depth, $S_d$ the recombination rate on the back surface of the base, and $L$ and $D$ the diffusion length and coefficient of the excess electron-hole pairs generated in the base region. The sketch of our structure is summarized in Fig.~\\ref{fig1}.\n\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.4]{fig1.eps}\n\\caption{Schematic illustration of a p-n junction of thickness $d = d_p + d_b$, where $d_p$ is emitter depth and $d_b$ is base thickness. The dead layer of thickness $x_m$ extends into the base region.}\n\\label{fig1}\n\\end{figure}\n\nApart from the SRH mechanism with the lifiteme $\\tau_{SR}$, the electron-hole pairs in GaAs also recombine radiatively; the characteristic time of this process is $\\tau_r = (AN_d)^{-1}$, where $A$ is the radiative recombination coefficient, and $N_d$ is the base doping concentration. Therefore, the diffusion length can be written as\n\\begin{equation}\nL = (D\\tau_b)^{1\/2}\\ ,\n\\label{1}\n\\end{equation}\nwith $\\tau_b = \\left(\\tau_{SR}^{-1} + \\tau_r^{-1}\\right)^{-1}$ being the effective lifiteme in the neutral base region.\n\nContinuity equation for the excess concentration of the electron-hole pairs, $\\Delta p_1$, within the dead layer (i.e., for $x < x_m$, region 1), where generation is negligible, has the form\n\\begin{equation}\n\\frac{d^2\\Delta p_1}{dx^2} - \\frac{\\Delta p_1}{L^2} = 0\\ ,\n\\label{2}\n\\end{equation}\nIn the rest of the semiconductor ($x > x_m$, region 2), the continuity equation for the excess electron-hole pair density, $\\Delta p_2$, is\n\\begin{equation}\n\\frac{d^2\\Delta p_2}{dx^2} - \\frac{\\Delta p_2}{L^2} = -\\frac{\\alpha I_0\\,e^{-\\alpha (x-x_m)}}{D}\\ .\n\\label{3}\n\\end{equation}\nThe equations (\\ref{2}) and (\\ref{3}) are supplemented by the boundary conditions\n\\begin{eqnarray}\n&&\\Delta p_1(x = d_p) = 0\\ ,\\ \\ \\frac{d\\Delta p_2}{dx}(x = d) = 0\\ , \\nonumber \\\\ \n&&\\Delta p_1(x = x_m) = \\Delta p_2(x = x_m)\\  ,\\nonumber \\\\\n&&\\frac{d\\Delta p_1}{dx}(x = x_m) = \\frac{d\\Delta p_2}{dx}(x = x_m)\\ .\n\\label{4}\n\\end{eqnarray}\nThe first condition reflects the fact that the electron-hole pairs are separated at the junction depth. The second one indicates the absence of surface recombination at the back of the base. The remaining two expressions are the usual continuity conditions for $\\Delta p(x)$ and $d\\Delta p(x)\/dx$ at $x = x_m$. The collection coefficient is then defined as the ratio of the current at the junction depth, $d_p$, to the pair generation rate in the plane of highest generation at $x = x_m$:\n\\begin{equation}\nQ = \\frac{D}{I_0}\\frac{d\\Delta p_1}{dx}(x = d_p)\\ .\n\\label{5}\n\\end{equation}\nThe solution of (\\ref{2}) and (\\ref{3}) that satisfies the first two conditions (\\ref{4}) can be written as\n\\begin{eqnarray}\n&&\\Delta p_1(x) = C\\sinh\\frac{x - d_p}{L}\\ ,\\nonumber \\\\\n&&\\Delta p_2(x) = C'\\cosh\\frac{x - d}{L} \\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + B\\left(e^{-\\alpha(x - x_m)} - \\beta\\,e^{-x\/L}\\right)\\ ,\\nonumber \\\\\n&&B = \\frac{\\alpha\\,I_0\\,L^2}{D\\left(1 - \\alpha^2L^2\\right)}\\ ,\\ \\ \\beta = \\alpha L\\exp{\\left[\\left(\\frac{1}{L} - \\alpha\\right)d\\right]}\n\\end{eqnarray}\nwith constants $C$, $C'$ to be determined from the remaining two conditions (\\ref{4}). This procedure yields:\n\\begin{eqnarray}\n&&Q = \\alpha L\\,\\times \\nonumber \\\\\n&&\\frac{\\alpha L\\left(\\cosh\\frac{d - x_m}{L} - e^{-\\alpha(d - x_m)}\\right) -\\sinh\\frac{d - x_m}{L}}{\\left[(\\alpha L)^2-1\\right]\\cosh\\frac{d - d_p}{L}}\\ .\n\\label{8}\n\\end{eqnarray}\nIf $d - x_m \\gg L$ and $\\alpha(d - x_m) \\gg 1$, this expression simplifies to\n\\begin{equation}\nQ = \\frac{\\alpha L}{1 + \\alpha L}e^{(d_p - x_m)\/L}\\ .\n\\label{8a}\n\\end{equation}\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig2.eps}\n\\caption{(a) Collection coefficient, $Q$, as a function of the diffusion length, $L$,  for different absorption coefficients, $\\alpha$, in the limit $\\alpha (d - x_m) \\gg 1$, $d - x_m \\gg L$, see Eq.~(\\ref{8a}). The values used, $\\alpha = 10^5, 6\\cdot 10^3$, and $6\\cdot 10^2$\\,cm$^{-1}$, approximately correspond to the respective mean beta-energies of $5.7, 20$, and $60$\\,keV for GaAs-based p-n junction \\cite{Tri67}. The dashed curves are calculated for different dead layer thicknesses, $x_m$, and $d_p = 10^{-5}$ cm. The solid curves are from the standard relation $Q = \\alpha L\/(1 + \\alpha L)$, valid in the absence of the dead layer. (b) Collection coefficient (\\ref{8a}) for different junction depth values for $x_m = 10^{-5}$\\,cm and $\\alpha = 10^5$\\,cm$^{-1}$, corresponding to the beta-particle energy of about 5.7\\,eV in the \\Tr\/GaAs combination.}\n\\label{fig2}\n\\end{figure}\n\nFig.~\\ref{fig2} shows the dependence of the collection coefficient $Q$ on the diffusion length from Eq.~(\\ref{8a}). As seen in this figure, the strongest reduction of $Q$ due to the presence of the dead layer is for the case of the \\Tr\\  beta-source. The smallest discrepancy in the $Q$-values obtained with and without taking into account the dead layer  is found for the curves corresponding to $\\alpha = 6\\cdot 10^2$\\,cm$^{-1}$, realized in the case of the \\Pm-source. In this case, to obtain $Q > 1\/2$, one would need the diffusion length $L > 35\\ \\mu$m. The values $Q \\approx 1$ can be achieved only in Si p-n junctions with long minority carrier lifetimes \\cite{Gor00}.\n\nIn Fig.~\\ref{fig2}(b), the junction depth was varied at a fixed electron energy (and thus constant $\\alpha$) and dead layer thickness. As seen in this figure, the collection coefficient increases not only upon increasing $L$, but also upon approaching the junction depth to the $x_m$-value. This effect is especially important for small diffusion length $L$.\n\nA further conclusion from Fig.~\\ref{fig2} is that collection of the electron-hole pairs generated by the electron flux will be quite efficient in the case when the diffusion length exceeds the dead layer thickness, $L > x_m$. An alternative way to increase $Q$ is to use deeper junctions with $d_p \\approx x_m$.\n\nLet us find the relation between the diffusion length and SHR lifetime $\\tau_{SR}$ for the case of GaAs. The radiative recombination coefficient $A$ in GaAs is an effective parameter defined by the relation $A = A_0(1 - \\gamma_r)$ \\cite{Din11}, where $A_0 \\approx 6\\cdot 10^{-10}$ cm$^3$\/s \\cite{Sach14}, and $\\gamma_r$ is the photon re-absorption coefficient. In our calculations, we assumed the value $A = 2\\cdot 10^{-10}$\\,cm$^3$\/s, as can be derived for poorly reflecting GaAs-based plane-parallel p-n structures without multiple reflection using the approach from \\cite{Din11}. In the work \\cite{Sach14}, it was shown that for realistic lifetimes  $\\tau_{SR}$, the open-circuit voltage $V_{OC}$ of GaAs-based p-n junctions increases with the base doping level, $N_d$, and, taking into account the interband Auger recombination, it has a maximum at $N_d \\approx 10^{17}$\\,cm$^{-3}$.\n\nLet us first assume that the GaAs p-n junction base is of p-type, and the diffusion coefficient of electron-hole pairs is 50 cm$^2$\/s. Then, for $A \\approx 2\\cdot 10^{-10}$\\,cm$^3$\/s, $N_d = 10^{17}$ cm$^{-3}$, and lifetimes $\\tau_{SR} = 10^{-9}, 10^{-8}$, and $10^{-7}$ s, diffusion length $L$ has the respective values of 2.2, 6.45, and 12.9 $\\mu$m. \n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig3.eps}\n\\caption{Collection coefficient $Q$ of a \\Tr\/GaAs betavoltaic pair as a function of the junction depth for (a) p-type base and (b) n-type base.}\n\\label{fig3}\n\\end{figure}\n\nFig.~\\ref{fig3}(a) shows the dependence of the collection coefficient, $Q$, of a pair \\Tr\/GaAs as a function of the junction depth, $d_p$, for these three values of $\\tau_{SR}$ at $x_m = 0.15\\,\\mu$m \\cite{Dmi78} and junction thickness $d = 10\\,\\mu$m. As seen in this figure, $Q$ is close to 1 for $\\Delta x = x_m - d_p < 0.1\\,\\mu$m. For $\\Delta x > 0.1\\,\\mu$m, the $Q$-value decreases with $\\Delta x$, but remains rather large.\n\nPresented in Fig.~\\ref{fig3}(b) is the collection coefficient vs. $d_p$ for the case when the base region of the p-n junction is of the n-type. In this case, for $\\tau_{SR} = 10^{-9}, 10^{-8}$, and $10^{-7}$\\,s, and $A \\approx 2\\cdot 10^{-10}$\\,cm$^3$\/s and $N_d = 10^{17}$\\,cm$^{-3}$, and taking into account that $D = 7$\\,cm$^2$\/s, the diffusion length $L = 0.83, 2.41$, and $4.83\\,\\mu$m, respectively. As seen in the figure, in this case $Q$ is also quite large. For $\\tau_{SR} = 10^{-7}$ and $10^{-8}$\\,s, $Q$ is still close to 1, while for $\\tau_{SR} = 10^{-9}$\\,s, $Q$ exceeds 0.75 even for small $\\Delta x$.\n\nIt should be noted that, because of rather strong absorption of the electrons emitted by the \\Tr-source by the auxiliary layers of a betavoltaic element (such as protection coating or contact layers), additional reduction of the beta-generated current can take place, leading to the efficiency reduction.\n\nLet us now analyze the collection coefficient for the \\Pm\/GaAs pair. In this case, according to \\cite{Tri67}, $\\alpha \\approx 600$\\,cm$^{-1}$, i.e., excess electron-hole density decays much more slowly than in the \\Tr\/GaAs case. For this the inequality $\\alpha L \\gg 1$ is alway satisfied even for the shortest lifeteme of $10^{-9}$\\,s. In contrast, for \\Pm\\ source, $\\alpha L = 1.5$ for $L = 25\\,\\mu$m, while $\\alpha L = 0.06$ for $L = 1\\,\\mu$m, so that $Q$ is always notably smaller than 1.\n\nBut this is not the only reason for the reduction of $Q$ in realistic \\Pm\/GaAs structures. When manufacturing solar cells based on the direct-bandgap semiconductors, such as GaAs, full thicknesses of p-n junctions are chosen rather small (of the order of a few $\\mu$m). Such structures were used in \\cite{And00}. In contrast, for the \\Pm\/GaAs pair used in betavoltaics, the situation might be very different, especially for large values of $L$. In this case, the product $\\alpha d$ will be small, so that for full absorption of beta-flux much thicker p-n junctions are required compared to those typically used in photovoltaics.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig4.EPS}\n\\caption{Collection coefficient of a \\Pm\/GaAs betavoltaic element as a function of junction depth for different Shockley-Reed lifetimes and element thicknesses for the case of (a) p-type base and (b) n-type base. }\n\\label{fig4}\n\\end{figure}\n\nShown in Fig.~\\ref{fig4} is the collection coefficient as a function of $d_p$ for a \\Pm\/GaAs pair calculated for different lifetimes $\\tau_{SR}$ and junction thicknesses $d$ of 10 and 100\\,$\\mu$m. In this case, according to \\cite{Dmi78}, $x_m = 3\\cdot 10^{-4}$\\,cm$^3$\/s. Panels (a) and (b) correspond to the cases of p- and n-base conduction types, respectively. As seen in the figure, rather high values of $Q \\ge 0.4$ for the \\Pm\/GaAs pair can be achieved only for the junction thickness $d \\approx 100\\,\\mu$m. Also, collection coefficient decreases dramatically as $\\tau_{SR}$ decreases.\n\nIt should be noted that similar results for the attainable $Q$ are expected for other direct-bandgap  A$_3$B$_5$ semiconductors, in particular, the ones based on the three-component compounds.\n\n\\section{Open-circuit voltage analysis}\nWhen estimating the limiting efficiency value \\cite{Olstech, Ada12}, we used the Shockley-Queisser approach \\cite{Sho61}, in which not only the current density, but also the open-circuit voltage, $V_{OC}$, is assumed to be maximal. Therefore, our next task is to calculate the open-circuit voltage, $V_{OC}$, with realistic values of $\\tau_{SR}$. It is given by the standard expression\n\\begin{equation}\nV_{OC} = \\frac{k_BT}{q}\\ln\\frac{N_d\\Delta p^*}{n_i^2}\\ ,\n\\label{11}\n\\end{equation}\nwhere $\\Delta p^* = \\Delta p(x = d_p + w)$ is the excess minority carrier density in the base at the boundary between the space-charge region and quasilinear region of thickness $w$, $N_d$ is the equilibrium density of the majority carriers in the quasineutral base region, and $n_i$ is the intrinsic charge carrier density. It is related to the effective densities of states in the conduction and valence bands, $N_c$ and $N_v$, as\n\\begin{equation}\nn_i = \\sqrt{N_cN_v}\\exp\\left(-\\frac{E_g}{2k_BT}\\right)\\ .\n\\label{17}\n\\end{equation}\n\nWe assume that both $d_p$ and $w$ are much smaller than the diffusion length $L$. This allows us to approximate \n\\begin{equation}\n\\Delta p(x = 0) \\approx \\Delta p^*\\ .\n\\end{equation} Such an approximation introduces a negligible error into $V_{OC}$ from Eq.~(\\ref{11}) in view of its logarithmic dependence on $\\Delta p^*$.\n\nWe will assume that recombination dominates in the quasineutral base region and in the space-charge region. Then, $V_{OC}$ can be obtained using the approach from \\cite{Sach14}. Taking into account the generation-recombination processes, we first write the continuity equation for the excess carrier density supplemented by the boundary conditions:\n\\begin{eqnarray}\n&&\\frac{d^2\\Delta p}{dx^2} - \\frac{\\Delta p}{L^2} - r(x)\\,\\Delta p(x)  + g(x) = 0\\ , \\nonumber \\\\\n&&\\frac{d\\Delta p}{dx}(x = d) = 0\\ ,\\nonumber\\\\\n&& D\\frac{d\\Delta p}{dx}(x = 0) = S_0\\,\\Delta p^*\\ ,\n\\label{9}\n\\end{eqnarray}\nwhere the third term describes recombination processes in the space-charge region of the abrupt junction, and the last one corresponds to the beta-induced generation. The first boundary condition is consistent with our assumption $S_d \\ll D\/L$ from the beginning of the previous section, and the second one is responsible for recombination effects in the $x = d_p + w$ plane.\n\nIntegration of the continuity equation results in the balance equation for the generation-recombination currents, according to which the current density for electronic excitation is proportional to the integral of the generation term,\n\\begin{equation}\nJ_\\beta = q\\,\\int_0^d dx\\,\\frac{\\Delta p(x)}{\\tau_b} + q\\left(S_0 + R_{SC}\\right)\\,\\Delta p^*\\ ,\n\\label{10}\n\\end{equation}\nwhere $q$ is the elementary charge. The right-hand side in (\\ref{10}) is responsible for the recombination in the bulk and on the front side of the emitter and within the space-charge region. The space-charge region recombination rate is given by \\cite{Sze}\n\\begin{eqnarray}\n&&R_{SC}(\\Delta p^*) = \\frac{L_D}{\\sqrt{2}\\tau_{SR}}\\int_{y_{pn}}^{-1} dy\\,N_d\\,\\left(1 - y + e^y\\right)^{-1\/2}\\times \\nonumber\\\\\n&&\\Big[N_d e^y + n_i e^{E_r\/k_BT} + b\\left(\\frac{n_i^2}{N_d} + \\Delta p^*\\right)e^{-y} \\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ + b n_i e^{-E_r\/k_BT}\\Big]^{-1}\\ ,\n\\nonumber\n\\end{eqnarray} \nwhere $b = \\sigma_p\/\\sigma_n$ is the ratio of the capture cross-sections of holes and electrons by a recombination level, $E_r$ is the recombination level energy measured from the middle of the bandgap,  $y_{pn}$ is the dimensionless potential at the p-n boundary, $L_D$ is the Debye length.\n\nTo evaluate the first integral in (\\ref{10}), we have employed the following approximative procedure. First, we write the solution of the continuity equation (\\ref{9}) as a sum of homogeneous and inhomogeneous parts,\n\\begin{equation}\n\\Delta p(x) = \\frac{e^{-x\/L} + e^{(x-2d)\/L}}{1 + e^{-2d\/L}}\\Delta p^* + \\Delta p_i(x)\\ ,\n\\end{equation}\nwhere the homogeneous term satisfies the first boundary condition in (\\ref{9}) and gives the value $\\Delta p(x = 0) = \\Delta p^*$. The inhomogeneous contribution $\\Delta p_i(x)$, with $\\Delta p_i(x = 0) = 0$, is notably different from zero only within a relatively thin layer below the front surface of the emitter, where the generation-recombination processes take place. Therefore, the contribution to the integral of the second term can be neglected in comparison to the integral of the homogeneous term, allowing us to write\n\\begin{equation}\n\\int_0^d dx\\Delta p(x) \\approx \\Delta p^*L\\tanh(d\/L)\\ .\n\\end{equation}\nThis approximation should produce a negligible error in $V_{OC}$ in view of its logarithmic dependence on $\\Delta p^*$. Substitution of this result into Eq.~(\\ref{11}) taking into account that $L^2 = D\\tau_b$ yields\n\\begin{equation}\nJ_\\beta = q\\Delta p^*\\left[\\frac{D}{L}\\tanh\\left(\\frac{d}{L}\\right) +S_0 + R_{SC}(\\Delta p^*)\\right]\\ .\n\\label{14}\n\\end{equation}\n\nThe current density $J_\\beta$ is inversely proportional to the energy required to create one electron-hole pair, $\\varepsilon$, which is approximately related to the bandgap $E_g$ as \\cite{Klein68}\n\\begin{equation}\n\\varepsilon = 2.8\\,E_g + 0.5\\,\\text{eV}\\ .\n\\label{10a}\n\\end{equation}\nDenoting is the current density in the case of Si ($E_g = 1.12$\\,eV) by $J_0$, the current density in the case of arbitrary bandgap can be approximated as \n\\begin{equation}\nJ_\\beta =J_0\\,Q\\cdot 3.64\\,\\text{eV}\/\\varepsilon\\ .\n\\end{equation}\nWe note that, usually, $J_0$ is in the $1$ -- $10^2\\,\\mu$A\/cm$^2$ range \\cite{Olstech}. The value of $\\Delta p^*$ found from Eq.~(\\ref{14}) should be substituted into Eq.~(\\ref{11}) to obtain the open-circuit voltage $V_{OC}$.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig5.EPS}\n\\caption{Open-circuit voltage as a function of the base doping level for different Shockley-Reed lifetimes in the case of (a) p-type base and (b) n-type base for $E_g$ = 1.43\\,eV, $T = 300$\\,K, and $J_0 = 10$\\,$\\mu$A\/cm$^2$.}\n\\label{fig5}\n\\end{figure}\n\nFig.~\\ref{fig5} shows the dependence of $V_{OC}$ of a GaAs-based p-n junction on the base doping level, $N_d$, neglecting the surface recombination, that is, $S_0 \\approx 0$. As seen in Fig.~\\ref{fig5}, $V_{OC}$ increases with $N_d$. On the one hand, the values of $V_{OC}$ for the pair \\Tr\/GaAs is notably smaller than in the solar cells \\cite{Sach14}, because the beta-produced current densities are at least two order of magnitude smaller than the short-circuit current densities in photovoltaic cells. On the other hand, the open-circuit voltages in Fig.~\\ref{fig5} exceed the values obtained experimentally in \\cite{And00}. The reason is that, in \\cite{And00}, the current density $J_0$ was of the order of $1\\,\\mu$A\/cm$^2$, whereas in our calculations, we have taken $J_0 = 10\\,\\mu$A\/cm$^2$. If the values $J_0 = 1\\,\\mu$A\/cm$^2$, $N_d = 5\\cdot 10^{16}$\\,cm$^{-3}$, and $\\tau_{SR} = 10^{-9}$\\,s are used, we obtain $V_{OC} = 0.44$\\,V, which practically coincides with the value given in \\cite{And00}.\n\n\n\\section{Refined calculation of the limiting betaconversion efficiency}\nAccording to Olsen \\cite{Olstech}, the efficiency of a betavoltaic element, $\\eta$, is\n\\begin{equation}\n\\eta = \\eta_\\beta\\,\\eta_C\\,\\eta_S\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\eta_\\beta = N_\\beta\/N_0\n\\end{equation}\nis the fraction of beta-flux that reaches the semiconductor,\n\\begin{equation}\n\\eta_C = (1 - r)\\,Q\n\\end{equation}\nis the coupling efficiency, given by the product of absorption probability of a beta-particle ($r$ is the electron reflection coefficient from the semiconductor surface) and collection efficiency $Q$ of electron-hole pairs, and, finally, the semiconductor efficiency\n\\begin{equation}\n\\eta_S = q\\,V_{OC}\\,FF\/\\varepsilon\\ ,\n\\end{equation}\nwhere $q$ is the elementary charge, $V_{OC}$ is the open-circuit voltage, $FF$ is the fill factor, $\\varepsilon$ is the energy necessary to generate one electron-hole pair from Eq.~(\\ref{10a}).\n\nLet us obtain $V_{OC}$ within the Shockley-Queisser approximation, where $\\tau_{SR} \\to \\infty$, $S_0 and R_{SC} \\to 0$, and the only recombination mechanism present is radiative recombination, characterised by the coefficient $A$. In this case, $V_{OC}^{lim}$ can be found analytically from (\\ref{10}) and (\\ref{14}):\n\\begin{equation}\nV_{OC}^{lim} = \\frac{k_BT}{q}\\ln\\frac{J_\\beta}{qAdn_i^2}\\ , \\\\\n\\label{16}\n\\end{equation}\n\nThe fill factor can be found using the expression from \\cite{Olstech}\n\\begin{equation}\nFF = \\left[v_{OC} - \\ln(v_{OC} + 0.72)\\right]\/(v_{OC} + 1)\\ ,\n\\label{18}\n\\end{equation}\nwhere $v_{OC} = V_{OC}\/k_BT$.\n\nTo calculate the limiting beta-conversion efficiency, we take $Q = 1$, $r = 0$, $\\eta_\\beta = 1$, corresponding to the bidirecional source in the terminology of \\cite{Olstech}. In this case\n\\begin{equation}\n\\eta_{lim} = \\frac{q\\,V_{OC}^{lim}\\,FF_{lim}}{2.8\\,E_g + 0.5}\\ ,\n\\label{19}\n\\end{equation}\nwhere $V_{OC}^{lim}$ is given by (\\ref{16}). \n\nWhen calculating $\\eta_{lim}$, several issues may arise. First, the parameters $N_c$, $N_v$, and $A$ are material-specific in every semiconductor. Second, when evaluating $V_{OC}^{lim}$ and $FF_{lim}$, Olsen had used, for each source, concrete current density $J_0$ of the order of $10^2\\,\\mu$A\/cm$^2$ for \\Pm\\ and $1\\,\\mu$A\/cm$^2$ for \\Tr. Finally, $V_{OC}^{lim}$ depends on the p-n junction thickness $d$. Therefore, all parameters in (\\ref{19}) must be specified. Since such key parameters as $A$, $N_c$, and $N_v$ are known only for concrete semiconductors and concrete bandgap values $E_g$, in the best-case scenario, the dependence $\\eta_{lim}(E_g)$ can be found as a set of support points for the known semiconductors with different $E_g$. Fitting this with a smooth curve might not be accurate enough.\n\nIn this work, we calculated $\\eta_{lim}$ only for the case of GaAs using Eq.~(\\ref{19}). For $A = 2\\cdot 10^{-10}\\,cm^3$\/s and $d = 10\\,\\mu$m gives for $J_0 = 10^2\\,\\mu$A\/cm$^2$ the value $\\eta_{lim} \\approx 17$\\,\\%, and for $J_0 = 1\\,\\mu$A\/cm$^2$, $\\eta_{lim} \\approx 14$\\,\\%. Note that the values of $\\eta_{lim}$ obtained here notably exceed the ones obtained by Olsen in \\cite{Olstech, Ols12}. In the rest of this work, we will use the values obtained for the \\Pm\/GaAs and \\Tr\/GaAs combinations, respectively.\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig6.EPS}\n\\caption{Beta-conversion efficiency of a \\Tr\/GaAs pair as a function of junction depth for different Schokley-Reed lifetimes for the case of (a) p-type base and (b) n-type base.}\n\\label{fig6}\n\\end{figure}\n\n\\begin{figure}[t!] \n\\includegraphics[scale=0.25]{fig7.EPS}\n\\caption{Beta-conversion efficiency of a \\Pm\/GaAs element vs. junction depth for different Schokley-Reed lifetimes and element thcknesses for (a) p-type base and (b) n-type base.}\n\\label{fig7}\n\\end{figure}\n\n\\section{Calculation of the attainable betaconversion efficiency}\nFig.~\\ref{fig6} shows the attainable efficiency as a function of $d_p$ for the \\Tr\/GaAs combination, obtained from\n\\begin{equation}\n\\eta = \\eta_{lim}Q\\frac{V_{OC}}{V^{lim}_{OC}}\\ ,\n\\label{20}\n\\end{equation}\nwhere $\\eta_{lim} \\approx 14$\\,\\%, $Q$ is given by Eq.~(\\ref{8}), $V_{OC}$ is found from Eq.~(\\ref{11}), and $V_{OC}^{lim}$  from Eq.~(\\ref{16}).\n\nWhen plotting Fig.~\\ref{fig6}, we varied the lifetime at a constant $d = 10\\,\\mu$m. Panels (a) and (b) correspond to the base of the p- and n-type, respectively. As seen in Fig.~\\ref{fig6}, the attainable efficiency values are rather high and are in the range of (6.4 - 12.5)\\%.\n\nIt should be noted that our results for \\Tr\/GaAs pair agree well with those given in the review \\cite{Ols12} citing Refs.~\\cite{And00, Bow02, Ada12}, namely, $\\eta =$ (4 - 7) \\%. In these works, a \\Tr-source was used with the $A_3B_5$-based semiconductors. But, as evident from the figures shown, the possibilities of increasing the efficiency of \\Tr\/$A_3B_5$ betaconversion are far from being exhausted.\n\nShown in Fig.~\\ref{fig7} is the attainable beta-efficiency (\\ref{20}) as a function of $d_p$ for \\Pm\/GaAs pair with $\\eta_{lim}$ = 17\\,\\%. The $\\tau_{SR}$ values used were $10^{-9}, 10^{-8}$, and $10^{-7}$\\,s, and GaAs thicknesses were 10 and 100 $\\mu$m. Fig.~\\ref{fig7}(a) and (b) correspond to the p- and n-types of the base conductivity. As seen in this figure, $\\eta$ reduces rather strongly as $\\tau_{SR}$ is decreased. For the highest $\\tau_{SR} = 10^{-7}$\\,s, $\\eta$ decreases with decreasing $d$. The highest efficiency attainable, $\\eta = 7.25$\\,\\%, is achieved for $\\tau_{SR} = 10^{-7}$ s and $d = 100\\,\\mu$m, and the lowest value of $0.51$ \\% is realized for $\\tau_{SR} = 10^{-9}$ s and $d = 10\\,\\mu$m.\n\nThus, we conclude that a \\Pm\/GaAs-based betaconverter is not as efficient as a \\Tr\/GaAs-based one. Perhaps, the very small efficiency of the \\Pm\/GaAs battery obtained in \\cite{Fli64} is due to the small thickness of GaAs and small lifiteme $\\tau_{SR}$. The same applies also to the cases when, instead of GaAs, other direct-bandgap semiconductors are used.\n\n\\section{Conclusions}\nOur analysis, focusing on the attainable collection coefficient $Q$ and open-circuit voltage values $V_{OC}$, has revealed the following features of current collection of the GaAs-based beta-elements.\n\nEfficient collection of the electron-hole pairs generated by a beta-flux can be achieved when the diffusion length exceeds the dead layer thickness, $L > x_m$. An alternative way to increase collection coefficient is to use deep junctions, for which $d_p \\simeq x_m$.\n\nAdditional mechanisms responsible for the reduction of current generated by beta-electrons are possible, leading to smaller betaconversion efficiency. They may be due, for instance, to the strong absorption of the beta-electrons by auxiliary layers of a betavoltaic element.\n\nUsing the Shockley-Queisser approximation, we have derived the limiting betaconversion efficiency, $\\eta_{lim}(E_g)$. Our analysis has shown that, because the main parameters affecting the efficiency are very different for different semiconductors, the $\\eta_{lim}(E_g)$ curve can be build as a set of support points for semiconductors with different bandgaps, and not as a smooth curve.\n\n\\Pm\\ beta-source performs more poorly than \\Tr-source, because the electron-hole pair generation depth in the case of \\Pm-source is large, whereas the diffusion length of GaAs is small. Therefore, the majority of electron-hole pairs generated in the base recombine before reaching the p-n junction.\n\nIn the case of \\Tr-source, the picture is different. The collection coefficient is rather high, because of the small generation depth of electron-hole pairs. Therefore, the realistic betaconversion efficiency for the \\Tr\/GaAs pair will be rather high for relevant parameters (lifitemes and diffusion coefficients) of the semiconductor.\n\nSimilar results are expected also in the case, when other direct-bandgap semiconductors are used instead of GaAs.\n\n\\acknowledgments\nM.E. would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe cost of lattice QCD simulations with dynamical fermions is dominated by the solution of the Dirac equation in both the ensemble generation phase, where configurations of gauge fields are generated, and the measurement phase, where expectation values of physical observables are measured. \nThe Dirac matrix, which is the gauge field dependent discretization of the fermionic part of the continuous QCD action, is a large sparse linear system and inverting the corresponding Dirac equation poses tremendous numerical difficulty. \nFor domain wall fermions(DWF) the conjugate gradient(CG) algorithm proves to be a stable algorithm to solve the Dirac equation but the convergence rate is limited by the condition number of the Dirac matrix, which is typically large in simulations with physical pion mass.  \n\nFor the measurement phase various eigen-space methods, including EigCG\\cite{Stathopoulos2010} and implicitly restarted Lanczos algorithm with Chebyshev polynomial\\cite{YSaad1980}, have been developed successfully to speed up the inversion.\nLow-lying eigenvectors(eigenvectors corresponding to small eigenvalues) of the Dirac matrix are generated and the previously large condition number is effectively reduced to improve the convergence rate of CG.\nIn this phase for one gauge field configuration typically a large number of Dirac equations with the same Dirac matrix but different right hand sides(RHS, or sources) are solved. The large number of sources amortizes the cost of eigenvector generation and the total computation time is reduced.\n\nThis is not the case for the ensemble generation phase. During a typical hybrid Monte Carlo(HMC) evolution of a gauge field as few as one Dirac equation is solved for a single Dirac matrix. This renders it not worthwhile to generate the low-lying eigenvectors for a particular Dirac matrix.\n\nThe development of supercomputers has greatly increased the number of floating point operations per second(flops) that can be performed on each processor(node).\nModern lattice simulations usually divide the gauge field and pseudo-fermion fields into sub-fields that are stored and computed locally on different processors of a large parallel computer.\nThis increases the total theoretical floating point operation capability.\nInter-processor data transfer(communication), however, is needed to perform coherent operations, including the Dirac matrix multiplication. Computations locally performed on one processor require contents of the sub-fields that are stored and updated on other processors.\nFor a specific operation if the rate of communication could not keep up with the local flops then communication becomes the bottleneck and the high flops are not utilized. \n\nFor standard CG solver with DWF one Dirac matrix multiplication is performed for each iteration.  The precise requirement varies with the size of the lattice and processor grid, but roughly this requires one byte of communication for each local floating point operation.\nOn some of the newest machines, for example the SUMMIT machine at Oak Ridge National Laboratory(ORNL), inter-processor communication speed is much less than the  requirement set by their high local floating point operation capability.\n\nIn \\cite{Luscher2004} a domain decomposition algorithm is proposed for Dirac equation with Wilson fermion. Local inversions are performed on two halves of the lattice iteratively. However, attempts to apply the same or similar algorithms to the inversion of the DWF Dirac equation have not been successful. \n\n\nIn this work we report on our investigation into a preconditioned CG solver for solving the DWF Dirac equation for the ensemble generation phase of the simulation. We find a preconditioner that decreases the number of CG iterations needed for a solution, while increasing the local computation required per iteration, thus changing the balance of local computation to off-processor communication.\n\n\n\n\\section{Method}\n\\subsection{Multisplitting Algorithm}\nIn \\cite{OLeary1985} a \\textit{multisplitting} algorithm is proposed for solving generic large linear systems distributed across a parallel computer.\nCompared to the domain decomposition algorithm in \\cite{Luscher2004}, it does not require checkerboarding.\nBefore each iteration the boundary content of the solution field on each of the processors is communicated to its neighbors.\nDuring each iteration, the algorithm uses this communicated neighboring solution field as the Dirichlet boundary condition to perform the inversion of a local matrix on each processor. \nAfter each iteration, the updated boundary content is again communicated to prepare for the next iteration.\n\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{ms_dec.pdf}\n\t\\caption{Decomposition of the matrix $A$, the solution vector $x$ and the right-hand-side(RHS) vector $b$ into local parts on each node.}\\label{fig:ms_dec}\n\\end{figure}\nFollowing \\cite{Jezequel2012}, suppose the equation to be solved is $Ax=b$. For a \\textit{particular} processor the matrix $A$ and vectors $x$ and $b$ are decomposed according to figure \\ref{fig:ms_dec}, where $x_s$ and $b_s$ are the part that is locally stored on this processor. On each processor the original equation turns into\n\\begin{equation}\\label{local}\n\tA_sx_s+A_lx_l+A_rx_r=b_s.\n\\end{equation} \nThe $A_lx_l+A_rx_r$ part involves off-processor content and is calculated before each iteration via communication. $A_s$ is the part of the matrix that requires only the locally stored part of $x$ on a certain processor $s$, i.e. $x_s$. Then for each iteration the algorithm solves the equation \n\\begin{equation}\\label{eq:ms}\n\tA_sx_s=b_s-A_lx_l-A_rx_r\n\\end{equation}\nlocally for $x_s$ on this processor. The updated solution $x_s$ will then be communicated to the neighboring processors. This whole procedure can be done concurrently on all nodes once the communication work to calculate $A_lx_l+A_rx_r$ is done.\n\n\n\n\\subsection{Domain Wall Fermions}\nThe domain wall fermion(DWF)\\cite{Jansen1996} formulation is based on Wilson fermion and a fictitious fifth dimension. Modern numerical implementations of DWF utilize the fact that only the matrix elements that connect the \\textit{even} sites to \\textit{odd} sites and those connecting \\textit{odd} sites to \\textit{even} sites depend the gauge field. The matrix entries that connect \\textit{even} sites to \\textit{even} sites and those connect \\textit{odd} sites to \\textit{odd} sites are constant. Here the even-odd parity is defined by the 4D components of a site:\n\\begin{equation}\n\\mathrm{parity}\\equiv (x+y+z+t)\\mod 2.  \n\\end{equation}\nIn the 4D even-odd preconditioning form the M\\\"obius DWF Dirac equation can be written as, \n\\begin{equation}\n \\begin{pmatrix}\n  M_5 & M^4_{eo} \\\\\n  M^4_{oe} & M_5 \\\\\n \\end{pmatrix}\n  \\begin{pmatrix}\n  \\psi_e \\\\\n  \\psi_o\n  \\end{pmatrix}\n  =\n  \\begin{pmatrix}\n  \\phi_e \\\\\n  \\phi_o\n  \\end{pmatrix},  \n\\end{equation}\nwhere the subscript $e\/o$ refer to even and odd sites. This is equivalent to solving the following even-odd preconditioned equation,\n\\begin{equation}\\label{dirac_equation}\nD_{PC}\\psi_e=\\hat\\phi_e,\\ D_{PC}\\equiv M_5-M^4_{eo}M_5^{-1}M^4_{oe}, \\hat\\phi_e\\equiv\\phi_e-M^4_{eo}M_5^{-1}\\phi_o. \n\\end{equation}\nHere $M^4_{eo\/oe}$ includes the Wilson hopping term $D^w_{x,y}$ that connects 4D space-time sites to their nearest neighbors,\n\\begin{equation}\nM^4_{oe\/eo}=D^w_{x,y}M_\\phi,\\ D^w_{x,y}\\equiv\\sum_\\mu\\left[(1+\\gamma_\\mu)U^\\dagger_{x-\\hat{\\mu},\\mu}\\delta_{x-\\hat\\mu,y}+(1-\\gamma_\\mu)U^\\dagger_{x,\\mu}\\delta_{x+\\hat{\\mu},y}\\right],\n\\end{equation}\nand $M_5$ and $M_\\phi$ are constant matrices that are diagonal in the four Euclidean space-time dimensions. Details of these matrices can be found in \\cite{Brower2014}.\n\nThe CG algorithm requires the matrix to be hermitian and positive definite. A common practice is to multiply both sides of (\\ref{dirac_equation}) with $D^\\dagger_{PC}$ and solve the equation with the normal operator $D^\\dagger_{PC}D_{PC}$ and the new RHS $D_{PC}^\\dagger\\hat\\phi_e$ instead,\n\\begin{equation}\\label{normal_equation}\nD^\\dagger_{PC}D_{PC}\\psi_e=D_{PC}^\\dagger\\hat\\phi_e.  \n\\end{equation}\n\n\\subsection{Dirichlet Boundary Condition on the 4-Hop Normal Operator}\nThere are four Wilson hopping terms, one in each $M_{eo\/oe}^4$, in the normal operator $D^\\dagger_{PC}D_{PC}$,\n\\begin{equation}\\label{eq:DdagD}\n  D^\\dagger_{PC}D_{PC}=\\big[M_5-\\textcolor{red}{M^4_{eo}}M_5^{-1}\\textcolor{red}{M^4_{oe}}\\big]^\\dagger\\big[M_5-\\textcolor{red}{M^4_{eo}}M_5^{-1}\\textcolor{red}{M^4_{oe}}\\big].\n\\end{equation}\n\nTo apply the multisplitting algorithm to equation (\\ref{normal_equation}) Dirichlet boundary conditions are to be enforced on the normal operator $D^\\dagger_{PC}D_{PC}$, i.e. the local part(the $A_s$ in (\\ref{local})) of this normal operator needs to be constructed. As the vector content is distributed across the processors according to its 4D space-time location, this local part for $D^\\dagger_{PC}D_{PC}$ includes \\textit{snake} terms that hop out of the boundary and hop back in as the various components in (\\ref{eq:DdagD}) are evaluated. Figure \\ref{fig:snake} illustrates this and gives some examples of the snake terms. These terms are truncated if Dirichlet boundary conditions are enforced on each of the four $M^4_{eo\/oe}$ hopping terms sequentially. Our simulation results show that the inclusion of these snake terms is crucial to the convergence.\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{snake.pdf}\n\t\\caption{The normal operator $D^\\dagger_{PC}D_{PC}$ has as many as $4$ Wilson hopping terms. Enforcing Dirichlet boundary condition on it requires the inclusion of the \\textit{snake} terms, e.g. the black arrows.}\\label{fig:snake}\n\\end{figure}\n\n\\subsection{Multisplitting Algorithm as a Preconditioner of CG}\nIn \\cite{Luscher2004} to achieve faster convergence the domain decomposition algorithm is eventually used as a preconditioner of GCR. In this work we use the multisplitting algorithm as a preconditioner of CG.\n\nPseudocode for a generic preconditioned CG is shown below, where we are solving $Ax=b$ and $M$ is the preconditioning matrix. The preconditioning step is marked with blue background. The overall convergence rate of preconditioned CG is estimated by the condition number of $AM^{-1}$. If the condition number of $AM^{-1}$ is smaller then that of the original matrix $A$, faster convergence rate is achieved.\n\\begin{algorithm}\n\\setstretch{1.15}\n\\caption{Preconditioned Conjugate Gradient $Ax=b$}\n\\begin{algorithmic}\n\\State ${r}_0 = {b} - {A x}_0$\n\\State ${z}_0 = {M}^{-1} {r}_0$ \n\\State ${p}_0 = {z}_0$ \n\\State $k = 0$  \n\\While {have not converged}\n\\State $\\alpha_k = {\\langle{r}_k,{z}_k\\rangle}\/{\\langle{p}_k,{A p}_k \\rangle}$ \n\\State ${x}_{k+1} = {x}_k + \\alpha _k {p}_k$ \n\\State ${r}_{k+1} = {r}_k - \\alpha _k {A p}_k$ \n\\State \\colorbox{blue!30}{${z}_{k+1} = {M}^{-1} {r}_{k+1}$\n\\State $\\beta _k = {\\langle {z}_{k+1}, {r}_{k+1}\\rangle}\/{\\langle {z}_k,{r}_k \\rangle}$ \n\\State ${p}_{k+1} = {z}_{k+1} + \\beta _k {p}_k$ \n\\State $k = k + 1$ \n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nNow for this preconditioning step we use the multisplitting algorithm to solve for $z_{k+1}$ in\n\\begin{equation}\n  Az_{k+1}=r_{k+1}.\n\\end{equation}\nTo avoid inter-processor communication, a zero initial guess($x_l=x_r=0$) is used in (\\ref{eq:ms}) and only the first iteration is performed. With $r_{k+1}$ as the RHS and $z_{k+1}$ the solution,\n\\begin{equation}\n  A_s x_s = b_s -A_lx_l - A_r x_r \\rightarrow A_s z_{k+1,s} = r_{k+1,s}.\n\\end{equation}\nThis is equivalent to using the local part of the matrix $A$, $A_s$, on each processor as the preconditioner $M$ in the preconditioned CG,\n\\begin{equation}\n  M=\\bigoplus_s A_s,\\ s=\\mathrm{node\\ index}.\n\\end{equation}\nThe local nature of $A_s$ makes it possible to perform the preconditioning step concurrently on all the processors without communication. We refer to this as multisplitting preconditioned CG(MSPCG).\n\n\n\\section{Results}\n\nThe multisplitting preconditioned CG is applied to solve Dirac equations on three 2+1 flavor lattice ensembles generated with M\\\"obius domain wall fermions, all with physical input quark masses. Standard CG is used to perform the inversion in the preconditioning step. Instead of adopting a precision based stopping condition, a fixed number of CG iterations, which will be referred as \\textit{inner iterations}, are performed for these preconditioning solves. The iterations performed in the overall preconditioned CG will be referred as \\textit{outer iterations}. In table \\ref{table:result} the numbers of outer iterations needed for the preconditioned CG to converge are reported on the different lattice ensembles, together with the stopping condition for the outer CG(precision) and the processor grid size used. The numbers of iterations to reach the same precision with standard CG are also included for comparison, where the inner iteration number is marked with \\textit{plain}.\n\nTypically on these ensembles with $6$ inner iterations the preconditioned CG reduces the outer iteration count by a factor of $3$. More inner iterations reduce the outer iteration count more but the reduction saturates as the inner iteration count increases: with large number of inner iterations the inner CG solves the preconditioning inversion completely and no further numerical benefit can be exploited.\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.5}\n\\centering\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\nlattice size & $a^{-1}[\\mathrm{GeV}]$ & precision & processor grid size & inner iterations & outer iterations \\\\\n\\hline\n\\hline\n\\multirow{4}{*}{$32^3\\times 64$} & \\multirow{4}{*}{$1.37$} & \\multirow{4}{*}{$10^{-8}$} & $-$ & plain & $13594$ \\\\\n&& & $2^3\\times 4$ & $3$ & $9106$ \\\\\n&& & $2^3\\times 4$ & $4$ & $6020$ \\\\\n&& & $2^3\\times 4$ & $6$ & $5126$ \\\\\n\\hline\n\\hline\n\\multirow{4}{*}{$64^3\\times 128$} & \\multirow{4}{*}{$2.36$} & \\multirow{4}{*}{$10^{-10}$} & $-$ & plain & $18092$ \\\\\n&&& $4^3\\times 8$ & $6$ & $6008$ \\\\\n&&& $4^3\\times 8$ & $12$ & $5083$ \\\\\n&&& $4^3\\times 8$ & $18$ & $4948$ \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{$80^2\\times96\\times 192$} & \\multirow{2}{*}{$3.00$} & \\multirow{2}{*}{$10^{-10}$} & $-$ & plain & $16783$ \\\\\n&&& $4^2\\times 8^2$ & $6$ & $5719$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Number of outer iterations need to converge the multisplitting preconditioned  CG for the lattice ensembles tested in this work. \\textit{Inner iterations} refers to the fixed number of CG iterations performed for the preconditioning inversion.  Rows marked with \\textit{plain} indicate the iteration count for the same standard CG to converge.}\\label{table:result}\n\\end{table}\n\n\\section{Conclusion}\nOur results show the MSPCG reduces the number of outer iterations needed to solve the DWF Dirac equation, reducing the inter-processor communication at the expense of performing more local inner iterations. \nWe observe that executing a fixed number of inner CG iterations for the preconditioning inversion, instead of using a precision based stopping condition, does not jeopardize the convergence of the outer CG. \nThis is true even when as few as $3$ inner iterations are performed.\nAs a consequence the inner iteration count is a parameter that can be tuned to achieve maximum speed up in the trade-off between inter-processor communication and local computation. \n\nWe note that while the multisplitting algorithm can split the general matrix $A$ in a variety of ways, the splitting presented here, used as a preconditioner in CG, makes it equivalent to the additive Schwarz algorithm. (The additive Schwarz algorithm has been used for the Dirac equation inversion for the  fermions\\cite{Osaki2010, Babich2011}.) We use the name MSPCG, as it is through the process of applying the multisplitting algorithm to the DWF Dirac equation that we realize the necessity of including the snake terms in the local matrix.\n\n\n\n\\urlstyle{tt}\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCoagulation processes arise in various areas of physics; one may think of\npolymerisation, growth of ordered domains in non-equilibrium magnetic systems \n\\cite{bray}, dynamics of droplets when water condenses on non-wetting \nsurfaces \\cite{dgy}, etc. \nThe substance, or ``mass'' that aggregates is very frequently not conserved\nduring the process: for example, \nagglomerating insoluble inclusions in molten metal\nmay be lost from the melt by attachment to the wall of the vessel \\cite{wmg}.\nTherefore the theoretical investigation of the kinetics of such non-conserving\ncoagulation processes is of great importance.  \nMoreover, the models developed for the description of such systems may \nshow interesting behaviour: the Smoluchowsky equation with certain coagulation\nkernels exhibits gelation transition and, in general, even the simplest\nmodels with conserved mass may have non-trivial solutions, see\ne.g. Ref. \\cite{wmg} and references therein. \nBeside quite realistic ones, \nthere is a special class of (possibly non-conserving) coagulation models  \nwhere only the actually smallest one among the masses is active \nwhile the other masses are temporarily inert.   \nThis type of {\\it extremal dynamics} can be regarded as a rough \napproximation for models where the reaction rates are \ndecreasing functions of the mass of particles.  \nIn what follows, we shall survey three processes with extremal\ndynamics in detail. \nWe mention, however, that models of this type have also been \nintroduced in the\ncontext of dynamics of growing and coalescing droplets \\cite{dgy} or\nmultispecies pair annihilation reactions \\cite{dht}.  \n\nIn the one-dimensional Glauber-Ising model started from a random initial state\nat zero temperature, the domain walls move as independent random walkers and\nannihilate upon meeting. While the closest pairs of walls come together and\nannihilate, the other domain walls hardly move. A simplified model of\nevolution of distances $X_i$ between adjacent walls can be formulated as\nfollows \\cite{nagai,kon}. The shortest interval $X_m$ is eliminated\ntogether with the two adjacent intervals $X_1$ and $X_2$ and replaced by \n$\\tilde X=X_1+X_2+X_m$. \nAs the density of walls tends to zero,\nthe distributions of intervals at different times become self-similar,\ndepending on a single time-dependent length scale, and the corresponding\nscaling function can be calculated exactly \\cite{nagai,bdg,rutenberg}.  \nAnother quantity of interest is the fraction of space which has never been\ntraversed by a domain wall. The length $Y_i$ of such parts of intervals\ntransforms in the way $\\tilde Y=Y_1+Y_2$ when the shortest interval is\neliminated. \nThe characteristic value of $X$ depends on\nthe fraction $c$ of the initial intervals that have not yet been eliminated \nas $X\\sim c^{-\\alpha}$, obviously, with $\\alpha=1$, while \nit has been found that $Y\\sim c^{-\\beta}$, where the persistence \nexponent $\\beta=0.824924 12\\dots$ is the zero of a \ntranscendental equation \\cite{bdg}.   \nIn addition to this, the autocorrelation exponent has also been exactly\ncalculated in this model \\cite{bd}. To obtain this quantity, the overlap $Z_i$\nof an interval with its initial state that transforms as $\\tilde\nZ=Z_1+Z_2-Z_m$ had to be considered.\nLater, a generalisation of persistence has been studied in the\nsame model, which required to introduce an auxiliary variable transforming \nas $\\tilde Y=Y_1+Y_2+pY_m$ \\cite{mb}. Here, \nthe generalised persistence exponent has\nbeen found to vary monotonically with the partial survival factor $p$\nin the range $-1\\le p\\le 1$.\n\nThe next example is the strong disorder renormalisation group transformation \nof inhomogeneous quantum spin chains \\cite{mdh}. Here, the degrees of freedom\nrelated to the largest coupling (a bond between neighbouring spins or a local\nexternal field) are eliminated one after the other. \nIn terms of logarithmic couplings, $X_i$, the\nrenormalisation rule generally reads as $\\tilde X=X_1+X_2-X_m$, \nwhere $\\tilde X$ is a newly formed effective variable and $X_1$,$X_2$ \nare variables adjacent to the smallest one, $X_m$.\nFor the relation between these variables and the couplings in the particular \n Hamiltonians we refer the reader to Ref. \\cite{fisher}.   \nA variable $Y_i$ that transforms according to  the rule \n$\\tilde Y=Y_1+Y_2$ under such a\nrenormalisation step can be interpreted in the case of a particular model, the\ntransverse field Ising chain, as the magnetic moment of a spin. \nFor this process with i.i.d. random initial variables\n$X_i$, which corresponds to critical spin chains, the distribution of $X$\nflows again to a fixed point where it shows scaling behaviour. \nThe characteristic value of $X$ increases in the course of the process \nas $X\\sim c^{-\\alpha}$ with \n$\\alpha=1\/2$, while the variable $Y$ grows as $Y\\sim c^{-\\beta}$ with\n$\\beta=(1+\\sqrt{5})\/4=0.809016\\dots$ \\cite{fisher}.         \nNote that the coagulation rules in the above two models \ndiffer only in the sign of $X_m$, \nwhich leads to different exponents $\\alpha$ and $\\beta$.\n \nOur third example is a random graph where three edges emanate from each\nnode, and which is built on a regular one-dimensional lattice by adding long\nedges in the following way. To each edge of the one-dimensional\nlattice that we call short edges, a random weight $X_i$ is assigned. \nDefining the length of a path as the sum\nof weights of the edges it contains, the closest pair of nodes of degree 2\nwith respect to this metric is chosen and connected by an edge of unit\nweight. This step is then iterated until all nodes become of degree 3 \n\\cite{juhasz}. \nFor this graph, a renormalisation procedure can be formulated \nwhere loops are eliminated \nstep by step in reversed order compared to the construction procedure. \nFormally, the short edge with the minimal weight $X_m$ is eliminated\ntogether with the nodes it connects, as well as with the \nneighbouring short edges with \nweights $X_1$, $X_2$ and a new effective short edge is formed with a weight\ncalculated asymptotically as $\\tilde X=X_1+X_2$.  \nAccording to numerical results, \nthe characteristic value of effective weights grows as \n$X\\sim c^{-\\alpha}$ with $\\alpha=0.826(1)$ \\cite{juhasz}.\nThis exponent characterises at the same time the \ndiameter of finite graphs with $N$ nodes with respect \nto the above metric via $D(N)\\sim N^{\\alpha}$. \n\nAs can be seen, these seemingly different problems can be treated in a common\nframework and can be interpreted as coagulation\nprocesses with extremal dynamics. \nIn the first example, the total sum of the variables $X_i$ is\nconserved while in the latter two cases it is not. \nWe will study in this work a coagulation model \ncontrolled by a parameter $\\omega$ that interpolates \ncontinuously between the first two models and incorporates the third one as a\nspecial case, as well. \nWe are interested in the exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$\nfor intermediate values of the parameter $\\omega$ and shall\nprovide accurate estimates for $\\alpha_{\\omega}$ that is obtained \nas the root of a \ntranscendental equation while $\\beta_{\\omega}$ is accurately determined \nby the numerical analysis of a system of non-linear differential equations. \nWe shall see that $\\alpha_{\\omega}$ varies monotonically between \nthe corresponding values of\nthe two marginal models, while, \nunlike the generalised persistence exponent of the model with\npartial survival mentioned above \\cite{mb}, the exponent $\\beta_{\\omega}$\nshows a maximum when $\\omega$ is varied. \nAs can be seen, the transformation rule of the variable \n$Y$ does not depend directly on the\nparameter $\\omega$ but it is influenced indirectly via the correlations \nemerging between $X$ and $Y$, the strength of which is controlled\nby $\\omega$. Therefore our results may contribute to the\nunderstanding of the role of correlations in such models.       \nMoreover, these investigations provide an accurate estimate for the diameter\nexponent of the graph quoted above, for which we obtain $\\alpha=0.82617561$ in\nagreement with the previous numerical result.   \n\nThe rest of the paper is organised as follows. In Section \\ref{model},\nthe model and its continuum description is introduced. \nIn Sections \\ref{asec} and \\ref{bsec}, the way of approximative\ndetermination of the exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$\nis presented. Some calculations are given in the Appendix. Finally,\nresults are discussed in Section \\ref{disc}. \n\n\\section{The model and its continuum formulation}\n\\label{model}\n\n\\subsection{Definition of the model}\n\nLet us consider a finite set of positive \nvectors $V_i=(X_i,Y_i)$ indexed by the\nintegers $i=1,2,\\dots,N$. We assume, moreover, that $N$ is odd. \nThe vectors are independent, identically distributed random variables \ndrawn from a continuous distribution $\\rho(X,Y)dXdY$, for which we require \nthat all moments exist.\nThe first components $X_i$ and the second components $Y_i$ are called\nprimary and secondary variables, respectively.  \nAssume, furthermore, that $\\omega\\in [-1,1]$ is a fixed real number. \nNow, the following procedure is considered on this set. \nThe vector $V_m$ with the smallest\nprimary variable is chosen and, at the same time, two further vectors $V_i$ and\n$V_j$ are chosen at random from the set. These three vectors are removed\nand a new vector $\\tilde V$ with components\n\\begin{eqnarray}\n\\tilde X=X_i+X_j+\\omega X_m \\nonumber \\\\\n\\tilde Y=Y_i+Y_j \n\\label{rules}\n\\end{eqnarray}\nis added to the set.\nThereby the number of vectors in the set is reduced by\ntwo. Note that the vectors remain independent after such an operation and\nthat \n\\begin{equation}\n\\tilde X\\ge X_i,X_j,X_m\n\\label{ineq}\n\\end{equation}\neven for $\\omega=-1$.   \nThis step is then iterated until a single vector $V_N=(X_N,Y_N)$ is left in\nthe set.\nIn this general formulation, the cases $\\omega=1,-1,0$ correspond to\nthe three models in the order as they were quoted in the Introduction.  \nBased on the known asymptotical behaviour of $X_N$ and $Y_N$ for \nlarge $N$ in the marginal cases $\\omega=-1,1$, \nwe expect\n\\begin{equation}\nX_N\\sim N^{\\alpha_{\\omega}} \\quad {\\rm and} \\quad Y_N\\sim N^{\\beta_{\\omega}}\n\\label{powerlaw}\n\\end{equation} \nto hold also for intermediate parameter values $-1<\\omega<1$\nwith some exponents $\\alpha_{\\omega}$ and $\\beta_{\\omega}$ that may \ndepend on $\\omega$. \n\n\\subsection{Continuum formulation}\n\nNow, we consider the continuum limit $N\\to\\infty$ and introduce \nthe probability density $P_{\\Gamma}(X)$ of the primary variable \nthat has the support \n$\\Gamma\\le X<\\infty$ and that depends on the lower boundary \n$\\Gamma$ as a parameter. \nThe function $P_{\\Gamma}(X)$ is normalised as $\\int_{\\Gamma}^{\\infty}P_{\\Gamma}(X)dX=1$ for any $\\Gamma$.\nFollowing Ref. \\cite{bdg}, we consider, furthermore, the expected value \n$\\overline{Y}_{\\Gamma}(X)$ of the secondary\nvariable under the condition that the primary variable is $X$.        \nIn the continuum limit, the system is described by these two\nfunctions of $X$, which depend on the lower boundary of the support \n$\\Gamma$ as a parameter.\nThe inequality (\\ref{ineq}) implies that,\nas the fraction of vectors $c_{\\Gamma}$ that have not yet been eliminated decreases in\nthe course of the coagulation process, the lower edge $\\Gamma$ \nof the distribution continuously increases.          \nAs it is shown in the Appendix, one may write \nthe following differential equation for $P_{\\Gamma}(X)$:\n\\begin{equation}\n\\frac{\\partial P_{\\Gamma}(X)}{\\partial \\Gamma}=P_{\\Gamma}(\\Gamma)\n\\Theta[X-(2+\\omega)\\Gamma]\\int_{\\Gamma}^{X-(1+\\omega)\\Gamma}P_{\\Gamma}(X')P_{\\Gamma}(X-X'-\\omega\\Gamma)dX',\n\\label{Pdiff}\n\\end{equation}\nwhere $\\Theta(X)$ is the Heaviside step function. \nThe fraction $c_{\\Gamma}$ \nis related to $\\Gamma$ as \n$dc_{\\Gamma}\/c_{\\Gamma}=-2P_{\\Gamma}(\\Gamma)d\\Gamma$ or, equivalently,\n\\begin{equation} \n\\frac{dc_{\\Gamma}}{d\\Gamma}=-2P_{\\Gamma}(\\Gamma)c_{\\Gamma}.\n\\label{cdiff}\n\\end{equation}\nThe function $Q_{\\Gamma}(X)$ defined as \n\\begin{equation}\nQ_{\\Gamma}(X)\\equiv P_{\\Gamma}(X)\\overline{Y}_{\\Gamma}(X),\n\\label{Qfunc}\n\\end{equation}\ncan be shown to obey the differential equation  \n\\begin{equation}\n\\frac{\\partial Q_{\\Gamma}(X)}{\\partial \\Gamma}=2P_{\\Gamma}(\\Gamma)\\Theta[X-(2+\\omega)\\Gamma]\\int_{\\Gamma}^{X-(1+\\omega)\\Gamma}Q_{\\Gamma}(X')P_{\\Gamma}(X-X'-\\omega\\Gamma)dX'.\n\\label{Qdiff}\n\\end{equation} \nThe derivation of this equation is given again in the Appendix. \n\n\\subsection{Fixed point solution}\n\nIn the marginal cases $\\omega=-1,1$, it is known that, for any well-behaving\ninitial distributions $\\rho(X,Y)$ with finite moments, the \nsolutions of Eqs. (\\ref{Pdiff}) and (\\ref{Qdiff}) tend to a universal \nfixed point solution $P^*_{\\Gamma}(X)$, $Q^*_{\\Gamma}(Y)$ \nin the limit $\\Gamma\\to\\infty$ that has the scaling property \n\\begin{eqnarray} \nP^*_{\\Gamma}(X)=\\Gamma^{-1}f(X\/\\Gamma)  \\nonumber \\\\\nQ^*_{\\Gamma}(X)=\\Gamma^{\\delta_{\\omega}-1}g(X\/\\Gamma),\n\\label{fp} \n\\end{eqnarray}\nwith some number $\\delta_{\\omega}$ that is related to the growth\nexponents as\\footnote{This can be seen from the equation \n$\\overline{Y}^*_{\\Gamma}(\\Gamma)\\equiv\nQ^*_{\\Gamma}(\\Gamma)\/P^*_{\\Gamma}(\\Gamma)=\\Gamma^{\\delta_{\\omega}}g(1)\/f(1)$\nthat indicates the asymptotical relation $Y\\sim X^{\\delta_{\\omega}}$ between the\ntypical values of primary and secondary variables.} \n\\begin{equation} \n\\delta_{\\omega}=\\beta_{\\omega}\/\\alpha_{\\omega}.\n\\label{delta}\n\\end{equation} \nTherefore we expect this to hold also for intermediate parameter values  \n$-1<\\omega<1$ with some (a priori unknown) exponent $\\delta_{\\omega}$ \nthat may depend on $\\omega$. \nIndeed, the functions in Eq. (\\ref{fp}) solve Eqs. (\\ref{Pdiff}) and\n(\\ref{Qdiff}) provided that the universal \nscaling functions $f(x)$ and $g(x)$\nsatisfy the following differential equations:\n\\begin{eqnarray}\n\\frac{d[xf(x)]}{dx}=-f_1\\Theta(x-2-\\omega)\\int_1^{x-1-\\omega}f(x')f(x-x'-\\omega)dx'\n\\label{fdiff}\n\\\\\n\\frac{d[x^{1-\\delta_{\\omega}}g(x)]}{dx}x^{\\delta_{\\omega}}=-2f_1\\Theta(x-2-\\omega)\\int_1^{x-1-\\omega}g(x')f(x-x'-\\omega)dx',\n\\label{gdiff}\n\\end{eqnarray}\nwhere the notation $f_1\\equiv f(1)$ has been used. \nFor an alternative derivation of these equations in the case $\\omega=1$, \nsee Ref. \\cite{bdg}.\nUsing the fixed point solution, Eq. (\\ref{cdiff}) can be integrated yielding\nthe asymptotic relation in the large $\\Gamma$ limit: \n\\begin{equation} \n\\Gamma\\sim c_{\\Gamma}^{-\\frac{1}{2f_1}}.\n\\end{equation} \nComparing this with Eq. (\\ref{powerlaw}), we obtain the relation:\n\\begin{equation} \n\\alpha_{\\omega}=\\frac{1}{2f_1}.\n\\label{exprel}\n\\end{equation}\n\n\\section{Approximative determination of $\\alpha_{\\omega}$} \n\\label{asec}\n\nAs can be seen, Eq. (\\ref{fdiff}) does not contain $g(x)$ and together\nwith Eq. (\\ref{exprel}) it constitutes an autonomous problem for the calculation of the exponent $\\alpha_{\\omega}$. \nFor the special case $\\omega=-1$, the solution of Eq. (\\ref{fdiff}) is of\nsimple form: $f(x)=e^{-x+1}$; this yields $\\alpha_{-1}=\\frac{1}{2}$. \nIn the other marginal case, $\\omega=1$, \nthe Laplace transform of the solution is\nknown \\cite{nagai,rutenberg} and $\\alpha_{1}=1$. \nIn the case $-1<\\omega<1$, where Eq. (\\ref{fdiff}) is not soluble, we shall\nconstruct an approximative solution that enables us to give an \naccurate estimate\nof $\\alpha_{\\omega}$. An alternative way related to the numerical analysis of\nthe Laplace transforms is presented in the next section. \n\nSome properties of the scaling function $f(x)$ can be easily established\nby investigating Eq. (\\ref{fdiff}) without knowing the exact solution.\nApparently, the r.h.s. of Eq. (\\ref{fdiff}) and, as a consequence, \n$f(x)$ is non-analytical at $x=x_1\\equiv 2+\\omega$. \nBut, as $f(x)$ itself appears on the r.h.s. as a convolution with a shifted\nargument $x-1-\\omega$, the r.h.s. as well as $f(x)$ \nmust be non-analytical also at \n$x=x_2\\equiv x_1+1+\\omega$. Iterating this argument, it turns out that there\nare infinitely many points where $f(x)$ is non-analytical.\nTo be precise, one can show by recursion that the \n$2n$th derivative of $f(x)$ is discontinuous at\\footnote{For a more direct way to this result in the case $\\omega=1$, where the explicit\nform of the scaling function $f(x)$ is available, see Ref. \\cite{rutenberg}.} \n\\begin{equation} \nx_n=1+(1+\\omega)n, \\qquad n=0,1,2,\\dots.\n\\end{equation} \nFurthermore, the function value of $f(x)$ at some $x'$ is determined \nby $f(x)$ in the restricted domain $(1,x'-1-\\omega)$. \nDue to this property, $f(x)$ can be constructed \nin the intervals $[x_n,x_{n+1}]$ step by step \nstarting with $n=0$. However, the solution is more and more complicated for\nincreasing $n$ as it contains multiple integrals that \ncannot be evaluated analytically. \nIn the domains $[x_n,x_{n+1}]$, $n=1,2,\\dots$, the function $f(x)$ can be written in the following form:\n\\begin{equation} \nf(x)=\\frac{1}{x}\\sum_{i=0}^nf_1^{2i+1}C^{(2i+1)}_{\\omega}(x),\n\\quad  x_n\\le x\\le x_{n+1},   \n\\label{series}\n\\end{equation}\nwhereas $f(x)=0$ if $x<x_0$.\nThe functions $C^{(2i+1)}_{\\omega}(x)$ are independent of $f_1$ and the first\nthree of them read as \n\\begin{eqnarray} \nC^{(1)}_{\\omega}(x)=1, \\\\ \\nonumber \nC^{(3)}_{\\omega}(x)=-2\\int_{x_1}^x\\frac{\\ln (x'-\\omega\n  -1)}{x'-\\omega}dx', \\\\ \\nonumber\nC^{(5)}_{\\omega}(x)=-2\\int_{x_2}^x\\int_{x_0}^{x'-x_2+1}\\frac{C^{(3)}_{\\omega}(x'-x''-\\omega)}{x''(x'-x''-\\omega)}dx''dx'.\n\\end{eqnarray}\nSubstituting an exponential trial function in Eq. (\\ref{fdiff}), we obtain that\nthe asymptotical solution $f_{\\infty}(x)$ in the limit $x\\to\\infty$ is \nof the form  \n\\begin{equation} \nf_{\\infty}(x)=\\frac{a}{f_1}e^{-a(x+\\omega)},\n\\label{asymp}\n\\end{equation}\nwith the some number $a$ that is not fixed by this substitution. \nThe graph of $f(x)$ for $\\omega=0$ is shown in Fig.\\ref{fig1}.\n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig1.ps}\n\\caption{\\label{fig1} Graphs of the functions $f(x)$, $g(x)$ and $f_{\\infty}(x)$ for $\\omega=0$. The former two are obtained by\nnumerical integration of Eqs. (\\ref{fdiff}) and (\\ref{gdiff}), whereas the\nlatter is given in Eq. (\\ref{asymp}).}  \n\\end{figure}\nAs can be seen, $f(x)$ tends rapidly to $f_{\\infty}(x)$ for\nincreasing $x$. \nThis suggests an approximation for $f(x)$ in which $f(x)$ is replaced by\n the simple asymptotical function $f_{\\infty}(x)$ for large $x$. \nTo be precise, the $n$th ($n=0,1,2,\\dots$) approximant $f^{(n)}(x)$ is defined as \n\\begin{eqnarray}\nf^{(n)}(x)=f(x), \\qquad {\\rm if} \\quad x\\le x_n \\nonumber \\\\\nf^{(n)}(x)=f_{\\infty}(x), \\qquad {\\rm otherwise}.\n\\end{eqnarray}      \nThe two unknown parameters $f_1$ and $a$ are determined by the requirements\nthat $f^{(n)}(x)$ is continuous at $x=x_n$, i.e. \n\\begin{equation} \nf(x_n)=f_{\\infty}(x_n),\n\\end{equation}\nand that it is normalised as \n\\begin{equation} \n\\int_{x_0}^{x_n}f(x)dx+\\int_{x_n}^{\\infty}f_{\\infty}(x)dx=1.\n\\end{equation}\nUsing the expression in Eq. (\\ref{series}), straightforward calculations\nresult in \nthat the $n$th approximant  $f_1^{(n)}$ ($n>0$) is the root of the\nfollowing transcendental equation: \n\\begin{eqnarray} \n\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+1}C^{(2i+1)}_{\\omega}(x_n)+ \\nonumber \\\\\n+\\frac{x_n}{x_n+\\omega}\n\\left[1-\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+1}N^{(2i+1)}_{\\omega}(x_n)\\right]\n\\ln\\left[f_1^{(n)}-\\sum_{i=0}^{n-1}[f_1^{(n)}]^{2i+2}N^{(2i+1)}_{\\omega}(x_n)\\right]=0,\n\\nonumber \\\\\n\\label{trans}\n\\end{eqnarray}\nwhere the function $N_{\\omega}^{(2i+1)}(x)$ has been introduced as\n\\begin{equation} \nN_{\\omega}^{(2i+1)}(x)\\equiv\\int_{x_i}^{x}\\frac{C^{(2i+1)}_{\\omega}(x')}{x'}dx'.\\label{nint}\n\\end{equation}\nWe have numerically calculated the root of Eq. (\\ref{trans}) \nand the $n$th approximant $\\alpha_{\\omega}^{(n)}$ of $\\alpha_{\\omega}$ \nby using Eq. (\\ref{exprel}) for $n=1,2,3$ and\nfor several values of $\\omega$. This has necessitated \nthe numerical evaluation of\nthe integrals in Eq. (\\ref{nint}) for $n>1$. \nResults are shown in Fig. \\ref{fig2} and some numerical values are given in\nTable I. As can be seen, the approximants $\\alpha_{\\omega}^{(n)}$ converge\nrapidly with increasing $n$ and they increase monotonically with $\\omega$. \nThe best estimate for the diameter exponent of the graph cited in the\nIntroduction is $\\alpha_0^{(3)}=0.82617561$. \n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig2.ps}\n\\caption{\\label{fig2} The third approximant $\\alpha_{\\omega}^{(3)}$ \nof the exponent $\\alpha_{\\omega}$ plotted against $\\omega$.}\n\\end{figure}\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{|r||c|c|c|c|c|}\n\\hline $\\omega$ & $\\alpha_{\\omega}^{(1)}$ &$\\alpha_{\\omega}^{(2)}$\n&$\\alpha_{\\omega}^{(3)}$ & $\\delta_{\\omega}$ \n& $\\beta_{\\omega}=\\delta_{\\omega}\\alpha_{\\omega}^{(3)}$\\\\\n\\hline\n\\hline -0.9 & 0.54752760 & 0.54752815 & 0.54752815 & 1.48973578 & 0.81567227\\\\\n\\hline -0.8 & 0.59036797 & 0.59037862 & 0.59037860 & 1.38841226 & 0.81968889\\\\\n\\hline -0.7 & 0.62906729 & 0.62911723 & 0.62911708 & 1.30687751 & 0.82217896\\\\\n\\hline -0.6 & 0.66421085 & 0.66434418 & 0.66434376 & 1.23995279 & 0.82375490\\\\\n\\hline -0.5 & 0.69632781 & 0.69659263 & 0.69659189 & 1.18399594 & 0.82476197\\\\\n\\hline -0.4 & 0.72586754 & 0.72630756 & 0.72630667 & 1.13643762 & 0.82540221\\\\\n\\hline -0.3 & 0.75320237 & 0.75385266 & 0.75385199 & 1.09543864 & 0.82579860\\\\\n\\hline -0.2 & 0.77863838 & 0.77952410 & 0.77952421 & 1.05965756 & 0.82602872\\\\\n\\hline -0.1 & 0.80242716 & 0.80356405 & 0.80356560 & 1.02809664 & 0.82614309\\\\\n\\hline  0.0 & 0.82477635 & 0.82617193 & 0.82617561 & 0.99999999 & 0.82617561\\\\\n\\hline 0.1 &  0.84585830 & 0.84751339 & 0.84751989 & 0.97478484 & 0.82614953\\\\\n\\hline 0.2 &  0.86581708 & 0.86772721 & 0.86773715 & 0.95199472 & 0.82608118\\\\\n\\hline 0.3 &  0.88477397 & 0.88693068 & 0.88694457 & 0.93126697 & 0.82598219\\\\\n\\hline 0.4 &  0.90283179 & 0.90522369 & 0.90524198 & 0.91230963 & 0.82586098\\\\\n\\hline 0.5 &  0.92007834 & 0.92269199 & 0.92271501 & 0.89488491 & 0.82572374\\\\\n\\hline 0.6 &  0.93658910 & 0.93940970 & 0.93943769 & 0.87879701 & 0.82557503\\\\\n\\hline 0.7 &  0.95242937 & 0.95544130 & 0.95547441 & 0.86388324 & 0.82541834\\\\\n\\hline 0.8 &  0.96765597 & 0.97084323 & 0.97088154 & 0.85000684 & 0.82525595\\\\\n\\hline 0.9 &  0.98231868 & 0.98566518 & 0.98570869 & 0.83705175 & 0.82508918\\\\\n\\hline 1.0 &  0.99646128 & 0.99995110 & 0.99999976 & 0.82492447 & 0.82492427\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table1} Approximants of the exponents $\\alpha_{\\omega}$, \n$\\delta_{\\omega}$ and $\\beta_{\\omega}$ for different values of $\\omega$.} \n\\end{table}\n\n\n\\section{Numerical determination of $\\beta_{\\omega}$} \n\\label{bsec}\n\nNext, we turn to the determination of the exponent $\\delta_{\\omega}$ (and, at\nthe same time, $\\beta_{\\omega}$ through Eq. (\\ref{delta})), which\nrequires the analysis of the full problem, i.e. the system of differential\nequations (\\ref{fdiff}) and (\\ref{gdiff}). \nPrior to this, \na few remarks concerning the scaling function $g(x)$ are in order. \nFirst, as a consequence of the definition in Eq. (\\ref{Qfunc}), $g(x)$ apparently \ninherits the singularity properties of $f(x)$ discussed in the previous\nsection. Furthermore, it can be written in a form analogous to\nEq. (\\ref{series}). In the domain $[x_0,x_1]$, it has a simple form:\n\\begin{equation} \ng(x)=g(1)x^{\\delta_{\\omega}-1}, \\qquad  x_0\\le x\\le x_1.\n\\label{power}\n\\end{equation}\nSecond, the differential equation (\\ref{gdiff}) gives the scaling function\n$g(x)$ only up to a multiplicative constant. \nThis non-universal constant depends\non the initial distribution $\\rho(X,Y)$ and it is fixed in a \nnon-trivial way by the original equations (\\ref{Pdiff}) and \n(\\ref{Qdiff}) that are valid for any $\\Gamma$.  \nThird, the equation (\\ref{gdiff}) contains the a priori unknown \nparameter $\\delta_{\\omega}$ that must be fixed by physical\nconsiderations about the solution that depends on $\\delta_{\\omega}$.\nNamely, the physically acceptable solution must be nonnegative and must have\nthe only reasonable asymptotics allowed by Eq. (\\ref{gdiff}): \n\\begin{equation}\ng_{\\infty}(x)\\simeq const\\cdot xe^{-ax},\n\\end{equation}\nwhere the number $a$ is the same as that appears in Eq. (\\ref{asymp}). \nNumerical analysis of Eq. (\\ref{gdiff}) shows that these requirements are\nfulfilled only for a single value of the parameter $\\delta_{\\omega}$. \n\nFollowing Ref. \\cite{bdg}, it is, however, simpler to analyse \nthe Laplace transform of\nthe equations (\\ref{fdiff}) and (\\ref{gdiff}). \nIntroducing the functions \n\\begin{equation}\n\\phi(p)=\\int_1^{\\infty}e^{-px}f(x)dx, \\qquad \n\\psi(p)=\\int_1^{\\infty}e^{-px}g(x)dx,\n\\label{laplace}\n\\end{equation}\nthe equations (\\ref{fdiff}) and (\\ref{gdiff}) transform to  \n\\begin{eqnarray}\np\\phi'(p)=f_1[e^{-\\omega p}\\phi^2(p)-e^{-p}], \n\\label{phidiff} \\\\\np\\psi'(p)=-\\delta_{\\omega}\\psi(p) -g_1e^{-p} + 2f_1e^{-\\omega p}\\psi(p)\\phi(p),\n\\label{psidiff}\n\\end{eqnarray}\nwhere the prime denotes derivation by $p$ and $g_1\\equiv g(1)$. \nThese equations are not soluble in the parameter range $-1<\\omega<1$ but\nasymptotical expressions of the solution can be established. \nThe functions $\\phi(p)$ and $\\psi(p)$ have the small-$p$ expansions:\n\\begin{equation} \n\\phi(p)=\\sum_{n=0}^{\\infty}a_np^n, \\qquad \n\\psi(p)=g_1\\sum_{n=0}^{\\infty}b_np^n. \n\\label{psiseries}\n\\end{equation}\nSubstituting these into Eqs. (\\ref{phidiff}) and (\\ref{psidiff}), we obtain  \nthat the expansion coefficients for $-1<\\omega<1$ are given by \n$a_0=1$, $b_0=\\frac{1}{2f_1-\\delta_{\\omega}}$ and by the following recursion\nrelations for $n>0$\\footnote{These series expansions are also valid for $\\omega=-1$ with \n$a_2=5\/2$, and for $\\omega=1$ with  $a_1=-2e^{\\gamma}$ \\cite{bdg}, where $\\gamma$ is Euler's constant, given by $\\gamma=-\\int_0^{\\infty}\\ln t e^{-t}dt=0.577215\\dots$.}: \n\\begin{eqnarray}\na_n=\\frac{\\frac{(-1)^n}{n!}(\\omega^n-1) + \n\\sum_{0\\le i,j,k<n;~ i+j+k=n}\\frac{(-\\omega)^i}{i!}a_ja_k}{\\frac{n}{f_1}-2} \\\\\nb_n=\\frac{\\frac{(-1)^n}{n!}\\left(\\frac{\\omega^n}{2f_1-\\delta_{\\omega}}-\\frac{1}{2f_1}\\right)\n  + \\frac{1}{2f_1-\\delta_{\\omega}}a_n +\\sum_{0\\le i,j,k<n;~\n    i+j+k=n}\\frac{(-\\omega)^i}{i!}a_jb_k}{\\frac{n+\\delta_{\\omega}}{2f_1}-1}. \n\\end{eqnarray}\n\n\nNext, we discuss the large-$p$ behaviour of $\\psi(p)$. \nThe differential equation (\\ref{psidiff}) can have two kinds of \nasymptotical solutions depending on the parameter $\\delta_{\\omega}$.\nIf the second term on the r.h.s. dominates, we obtain \n\\begin{equation} \n\\psi'(p)\\simeq -g_1\\frac{e^{-p}}{p},\n\\label{a1}\n\\end{equation}  \nwhile, if the first term dominates, we obtain\n\\begin{equation} \n\\psi(p)\\simeq const\\cdot p^{-\\delta_{\\omega}}.\n\\label{a2}\n\\end{equation}\nOn the other hand, it follows from Eq. (\\ref{laplace}) that \n$\\psi(p)$ must have the large-$p$ asymptotics:\n$\\psi(p)\\simeq g_1e^{-p}\/p[1+O(1\/p)]$. Using that the function $g(x)$ is\nexplicitely known in the domain $1\\le x\\le 2+\\omega$ (see \nEq. (\\ref{power})), we can obtain a more accurate asymptotical form \nfor $1\/p\\ll 1+\\omega$. Replacing $g(x)$ by the function in Eq. (\\ref{power})\nin the entire domain $x\\ge 1$, the integral in Eq. (\\ref{laplace}) can be\nevaluated, yielding\n\\begin{equation} \n\\psi_{\\infty}(p)= g_1e^{-p}\\sum_{k=0}^{\\infty}\n\\left(\\begin{array}{c}\n\\delta_{\\omega}-1 \\\\ \nk\n\\end{array}\\right)\n\\frac{k!}{p^{k+1}}, \\qquad p\\gg \\frac{1}{1+\\omega}.\n\\label{asympfull}\n\\end{equation}\nThis shows that the physically acceptable asymptotics is that given \nin Eq. (\\ref{a1}) whereas that in Eq. (\\ref{a2}) is non-physical. \nNumerical analysis of the differential equations (\\ref{phidiff}) and\n(\\ref{psidiff}) with the correct value of $f_1$ \nshows the following behaviour of the solution when the parameter $\\delta_{\\omega}$ is varied: \nFor small enough $\\delta_{\\omega}$, the function \n$\\psi(p)$ is non-monotonous and tends to zero from below \nfor increasing $p$ as given in\nEq. (\\ref{a2}); for large enough $\\delta_{\\omega}$, the function $\\psi(p)$  \ndecays monotonically to zero again with the asymptotics given in\nEq. (\\ref{a2}). These parameter regimes are separated by a ``critical'' value\nof $\\delta_{\\omega}$. At this value, the solution decays monotonically to zero\nwith the physically acceptable asymptotics given in Eq. (\\ref{a1}).\n  \nThe numerical estimation of $\\delta_{\\omega}$ is based on this scenario: \nThe differential equations (\\ref{phidiff}) and\n(\\ref{psidiff}) are integrated from $p=0$ to some large $p$ \nand the true value of\n$\\delta_{\\omega}$ is selected by the condition that $\\psi(p)$ has the correct\nasymptotics. \nWe have assumed here that $f_1$ is already at our disposal. This can be\nobtained either by the approximative procedure described in the \nprevious section or, analogous to the above method, from the condition that\n$\\phi(p)$ has the correct asymptotics given by Eq. (\\ref{asympfull}) \nwith $\\delta_{\\omega}=0$.           \n\nBefore presenting numerical results on $\\delta_{\\omega}$, we show that,  \nin the case $\\omega=0$, the assumption on the uniqueness of the value\n$\\delta_{\\omega}$ that corresponds to the correct asymptotics \nimplies that $\\delta_0=1$.\nFor $\\omega=0$, the primary and the secondary variables coagulate according to\nthe same rules, see Eq. (\\ref{rules}). If these variables are initially\nperfectly correlated, i.e. $X_i=bY_i$ with some common constant \n$b$ for all $i$,  it is obvious that $\\alpha_0=\\beta_0$. \nNevertheless, this equality holds for general initial distributions, as well. \nIndeed, it is easy to check that for $\\omega=0$, the function\n\\begin{equation} \n\\psi(p)=-\\frac{g_1}{f_1}\\phi'(p)\n\\label{psi0}\n\\end{equation}\nsolves Eq. (\\ref{psidiff}) provided that\n$\\delta_0=1$ and $\\phi(p)$ is the solution of\nEq. (\\ref{phidiff}). \nIn terms of the scaling functions, equation (\\ref{psi0}) reads as \n$g(x)=xf(x)\\frac{g_1}{f_1}$. \nAs the asymptotics of the solution in Eq. (\\ref{psi0}) is \nphysically acceptable, we conclude that \n\\begin{equation} \n\\delta_0=1.\n\\end{equation}  \n\nThe details of the numerical determination of $\\delta_{\\omega}$ are the\nfollowings. The best approximant that we have, $f_1^{(3)}$, has been\nsubstituted in Eqs. (\\ref{phidiff}) and (\\ref{psidiff}) and a trial value \nfor $\\delta_{\\omega}$ has been chosen.  \nAs the derivatives $\\phi'(p)$ and $\\psi'(p)$ calculated from these equations \nare of the form $0\/0$ at $p=0$, the functions $\\phi(p)$ and $\\psi(p)$ \nwere first calculated at $p=0.05$ by using the small-$p$ expansion in\nEq. (\\ref{psiseries}) in order to avoid numerical uncertainties of the\nintegration in the vicinity of $p=0$. \nThen, starting from $p=0.05$, the differential equations were integrated by\nthe Bulirsch-Stoer method \\cite{numrec} to some $p_f$, where $\\psi(p_f)$ is\ncompared to the asymptotical form in Eq. (\\ref{asympfull}). \nIn practice, we have monitored the derivative $\\psi'(p_f)$ rather than\n$\\psi(p_f)$ and $p_f=12$ was sufficiently large so that the asymptotical\nvalue (at the critical $\\delta_{\\omega}$) is reached within the numerical\naccuracy of the integration. \nThe true $\\delta_{\\omega}$ was then\nselected by the condition $\\psi'(p_f)=\\psi_{\\infty}'(p_f)$. \nThe exponent $\\delta_{\\omega}$ determined in this way is plotted against \n$\\omega$ in Fig. \\ref{fig3} whereas \n$\\beta_{\\omega}=\\delta_{\\omega}\\alpha^{(3)}_{\\omega}$ is plotted against\n$\\omega$ in Fig. \\ref{fig4}.\nSome numerical values can be found in Table \\ref{table1}. \nAs can be seen, $\\delta_{\\omega}$ decreases monotonically with $\\omega$ but \n$\\beta_{\\omega}$ has a maximum at $\\omega=0$. The latter observation will\nbe explained in the next section. \n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig3.ps}\n\\caption{\\label{fig3} Numerically calculated exponent $\\delta_{\\omega}$ \nplotted against $\\omega$.}\n\\end{figure}\n\\begin{figure}[h]\n\\includegraphics[width=0.6\\linewidth]{fig4.ps}\n\\caption{\\label{fig4} Numerically calculated exponent \n$\\beta_{\\omega}=\\delta_{\\omega}\\alpha^{(3)}_{\\omega}$ \nplotted against $\\omega$. The horizontal lines indicate known values in the\nmarginal cases $\\omega=-1,1$.}\n\\end{figure}\n \n\\section{Discussion}\n\\label{disc}\n\nWe have shown that two problems, the renormalisation group procedure of certain\ndisordered quantum spin chains and a simple model describing coarsening in\nthe Glauber-Ising model can be deformed into each other by varying a single\nparameter. \nThe interpolating model is a special type of non-conserving \ncoagulation process where only the actually smallest mass is active.   \nIn the range $-1\\le\\omega\\le 1$, the exponents $\\alpha_{\\omega}$ and \n$\\beta_{\\omega}$ that characterise the\ngrowth of the primary and the secondary variables, respectively,  \nvary continuously with $\\omega$. \nThe latter exponent exhibits a maximum, which contrasts with the \nmonotonous dependence of the generalised persistence exponent on \nthe partial survival factor that appears directly \nin the transformation rule of the secondary variable \\cite{mb}.  \nAlthough, we have focused on the range $-1\\le\\omega\\le 1$,\nthe equations written down in this work are valid also for $\\omega >1$. \nIn that case, the growth of the primary variable becomes super-linear, meaning\nthat $\\alpha_{\\omega}>1$. \n\nAn intriguing feature of the process studied in this work is the universality\nwith respect to the initial distribution of the variables:\nFor a fixed $\\omega$,\nany sufficiently rapidly decaying initial distribution tends \nat late times to a\nuniversal distribution that displays scaling.\nAlthough, the process is universal in this sense, we have pointed out that\nit is sensitive to the variations of the reaction rules parameterised by \n$\\omega$. \nThe dependence of $\\alpha_{\\omega}$ on $\\omega$ is obvious since the\ntransformation rule of the primary variable contains $\\omega$ explicitely.\nThe growth of the secondary variable is, however, \naffected by $\\omega$ in a more subtle way.    \nFocusing on the secondary variables, the difference to the process\nof primary variables with $\\omega=0$ \nis that, here, not exactly the smallest variable is\nremoved from the set. \nThis is the reason for \nthat $\\beta_{\\omega}$ is unequal to $\\alpha_0$ for $\\omega\\neq 0$. \nNevertheless, for any $\\omega$, the removed secondary variable\nis typically relatively small since $X_i$ and $Y_i$ become\npositively correlated in the course of the process. \nDue to these correlations, the strength of which is controlled by \n$\\omega$, the variation of $\\beta_{\\omega}$ is relatively slight. \nIndeed, it is by an order of magnitude\nsmaller than that of $\\alpha_{\\omega}$.\n\nFor $\\omega=0$, we have shown that $\\alpha_0=\\beta_0$ even if the primary and\nthe secondary variables are initially not perfectly correlated. This can be\nunderstood also on a microscopic level since, in this case,  \nthe vectors in the set are sums of an increasing number of \ninitial vectors. Thus, the ratios $\\tilde X_i\/\\tilde Y_i$ tend\nstochastically to a common constant in the limit $\\Gamma\\to\\infty$ \nfor all $i$. In words, the two types of variables become asymptotically\nperfectly correlated for $\\omega=0$.      \nNow, we are in a position to understand why the exponent $\\beta_{\\omega}$ \nis maximal at $\\omega=0$.\nAt that point, the correlations are (at least asymptotically) perfect and\nalmost always the smallest one among the secondary variables is removed. \nFor $\\omega\\neq 0$, however, the correlations are no longer perfect and, \nas a consequence, not strictly the smallest secondary variables are eliminated.\nTherefore the fastest growth of $Y$ is realized at $\\omega=0$.\n\nIn a general aspect, the benefit of the analysis carried out in this\nwork is that\nthe numerical technique developed here for obtaining accurate estimates of \nthe growth exponents may also apply to other\nnon-soluble coagulation processes with extremal dynamics. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the challenging problems in quantum gravity is to understand the\nmicroscopic properties of black holes, in particular, the statistical\norigin of the Bekenstein-Hawking entropy.\nNew idea for an explanation of the \norigin of the Bekenstein-Hawking entropy has been provided by recent\ndevelopment in our understanding of non-perturbative superstring theory.\nIt is based on the D-brane description of black\nholes\\cite{Strominger-Vafa} and the AdS\/CFT \ncorrespondence\\cite{Maldacena1,Witten,GKP}.\nThese are much related with each other under\nthe Maldacena duality\\cite{Maldacena1}.\n\nThree-dimensional Einstein equation with negative\ncosmological constant has the solutions called \nthe BTZ black holes\\cite{BTZ1,BTZ2}.\nThese black holes have locally ${\\rm AdS}_{3}$ geometry.\nVia the ${\\rm AdS}_{3}\/{\\rm CFT_{2}}$ correspondence, one can also \nhope to be able to analyze the microscopic properties of \nthe BTZ black holes based on a local field theory on\nthe boundary. Infinite dimensional algebra of two-dimensional conformal\nsymmetry, that is, the Virasoro algebra, provides an important clue\nfor our understanding the correspondence and the Maldacena duality.\nThe pioneering work is Strominger's counting of microscopic states\nof the BTZ black holes\\cite{Strominger}.\nBut the qualitative aspects of this counting still remain obscure .\n\nIn this paper, we will discuss the three-dimensional extremal BTZ \nblack holes in the context of the Maldacena duality.\nAlthough this duality has been conjectural yet, various checks have been\ncarried out. (See \\cite{Maldacena2} and references therein.)\nIn this perspective, the extremal BTZ black holes can be identified\nwith the primary states which are 1\/2 BPS states in the N=(4,4)\ntwo-dimensional supersymmetric $\\sigma$-model.\nThis $\\sigma$-model has a quantity called elliptic genus \nconvenient to count the degeneracy of these states.\nWe explicitly count the microscopic states of the extremal BTZ black\nholes with this identification by using the elliptic genus and\nthe unitary representation theory of the N=4 superconformal\nalgebra. The microscopic entropy of these black holes obtained by\nthis counting agrees with the entropy \\'a la Bekenstein-Hawking.\n\nThis paper is organized as follows.\nIn section 2, we will summarize the\nprevious results about the BTZ black holes from the perspective of the\n${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence in a pure quantum gravity and\nin non-perturbative superstring theory, i.e., the Maldacena duality.\nIn section 3, after a brief introduction of N=4 superconformal algebra,\nblack hole states are discussed in the unitary representation\ntheory.\nIn section 4, some facts about the elliptic\ngenus of the N=(4,4) supersymmetric $\\sigma$-model are reviewed.\nIn section 5, we count the number of 1\/8 BPS states in the D1-D5 brane\nsystem in IIB supergravity via the elliptic genus of this\n$\\sigma$-model and then finally count the microscopic states of the\nextremal BTZ black holes.\nIn section 6, some other related topics are discussed.\n  \n\n\\section{BTZ black holes and ${\\bf AdS_{3}\/CFT_{2}}$\ncorrespondence}\n\n\\subsection{BTZ black holes in a three-dimensional pure quantum \ngravity}\n\nThe BTZ black holes\n\\footnote{Exact solutions of the vacuum \nEinstein equation with a negative cosmological \nconstant $\\Lambda=-1\/l^2$.}\nare three-dimensional black holes specified \nby their mass $M$ and angular momenta $J$, where $|J| \\leq Ml$.   \nIn terms of the Schwarzschild coordinates \n$(t,\\phi,r)$, with the ranges \n$-\\infty<t<+\\infty$, \n$0\\leq \\phi <2\\pi$ \nand $0<r<+\\infty$, \nthe black hole metric $ds_{{\\rm BTZ}_{(J,M)}}^2$ \nhas the form \n\\begin{eqnarray}\nds^{2}_{{\\rm BTZ}_{(J,M)}} \n\\equiv \n-N^2(dt)^2+N^{-2}(dr)^2\n+r^2(d\\phi+N^{\\phi}dt)^2, \n\\label{dsX(J,M)}\n\\end{eqnarray}\nwhere $N$ and $N^{\\phi}$ are the functions of the radial coordinate $r$ \n\\begin{eqnarray}\nN^2 =\n\\frac{(r^2-r_+^2)(r^2-r_-^2)}{l^2r^2},~~~~\nN^{\\phi} =  \n\\left\\{ \n\\begin{array}{cc}\n\\frac{r_+r_-}{lr^2} & \\mbox{when}~~J \\geq 0, \n\\\\ \n-\\frac{r_+r_-}{lr^2} & \\mbox{when}~~J < 0.  \n\\end{array}\n\\right. \n\\label{N-Nphi}\n\\end{eqnarray}\nThe outer and inner horizons are located respectively \nat $r=r_+$ and $r=r_-$. Information about the mass \nand angular momentum is encoded in $r_{\\pm}$ by  \n\\begin{eqnarray} \nr_{\\pm}^2 \\equiv \n4GMl^2 \\left( \n1\\pm \\sqrt{1-\\frac{J^2}{M^2l^2}} \n\\right), \n\\label{rpm}\n\\end{eqnarray}\nwhere $G$ is Newton constant. \nIn the case of $Ml=|J|$,  the black hole is called extremal. \nIt holds $r_+=r_-$  \nand then the outer and inner horizons coincide  \nwith each other. \nWhen $Ml>|J|$, it is called non-extremal.\nAnd in the case of $J = 0$ and $Ml = -l\/8 G$,\nthe geometry corresponds to the global ${\\rm AdS}_{3}$.\n\nThe outer horizon of these solutions has finite area.\nThe semiclassical argument leads to\nthe finite Bekenstein-Hawking entropy:\n\\begin{eqnarray}\nS \\equiv \\frac{A}{4 G} = \\frac{2 \\pi r_{+}}{4G}. \\qquad (A : {\\rm area\n\\ of \\ the \\ outer \\ horizon})  \\label{eq:entropy}\n\\end{eqnarray}\n\n\nQuantization of three-dimensional\npure gravity with negative cosmological constant is discussed in \n\\cite{Nakatsu}.\nIt is prescribed, through the detailed analysis of \nBrown-Henneaux's asymptotic Virasoro symmetry\\cite{Brown-Henneaux}, \nas the geometric quantization of the Virasoro coadjoint orbits\nof the Virasoro central charge $c=3 l\/2 G$.\n\nThe BTZ black holes and the ${\\rm AdS}_{3}$ correspond to the primary\nstates (highest weight states) \nof the Virasoro algebra of \nBrown-Henneaux\\footnote{Strictly speaking,\nthese states correspond to the geometry of the exterior of outer\nhorizon of the BTZ black holes and the geometry without the origin of \nthe ${\\rm AdS_{3}}$ respectively}:\n\\begin{eqnarray}\n{\\rm BTZ}_{(J, M)} &\\Longleftrightarrow& |J, M\\rangle \\equiv \n|h\\rangle \\otimes |\\tilde{h}\\rangle,  \\label{eq:pribtz}\\\\  \n{\\rm AdS}_{3} &\\Longleftrightarrow& |vac\\rangle\n\\equiv |0\\rangle\\otimes|0\\rangle, \n\\label{eq:priads}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nh = \\frac{1}{16Gl}(r_{+}+r_{-})^2 + \\frac{c}{24}, \\quad {\\tilde h} = \n\\frac{1}{16Gl}(r_{+}-r_{-})^2 + \\frac{c}{24}.\n\\end{eqnarray}\nThe extremal BTZ black holes correspond to\n\\begin{eqnarray}\n{\\rm BTZ}_{(Ml, M)} \\Longleftrightarrow |Ml, M\\rangle \\equiv \n|h \\rangle \\otimes |\\frac{c}{24} \\rangle. \\label{eq:priext}\n\\end{eqnarray}\n \nThe total Hilbert space of the theory, which includes \nexcited states (secondary states), \nis obtained by the tensor products ${\\cal V}_{h}\\otimes{\\tilde {\\cal\nV}_{{\\tilde h}}}$ of the Verma modules of the Virasoro algebra.\n(${\\cal V}_{h}$ and ${\\tilde {\\cal V}}_{{\\tilde h}}$ are respectively\nthe Verma modules of the left-moving and right-moving sectors.) \nThese Verma modules constitute the unitary irreducible \nrepresentations of the Virasoro algebra. \nWe can identify the states excited by $L_{-n}$ in the Verma module\nwith massive gravitons on the corresponding background geometry.\n\nIn view of the ${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence,\nthis Hilbert space should be realized by the corresponding boundary CFT. \nIn fact it was done\\cite{Nakatsu} based on \nthe Liouville field $X$ with a specific\nbackground charge. The action is given by\n\\begin{eqnarray}\nS[X]=\\frac1{4\\pi i}\\int_{{\\bf P}^1} \n\\bar{\\partial} X \\wedge \\partial X\n     + \\frac{\\alpha_0}{2\\pi}\\int_{{\\bf P}^1} RX, \n\\qquad \\left( \\alpha_0 \\equiv \\sqrt{\\frac{l}{8G}} \\right)\n\\end{eqnarray}\nwhere $R$ is the Riemann tensor of \na fixed K\\\"ahler metric on ${\\bf P}^1$.\nThe stress tensor $T(z)$ has the form\n\\begin{eqnarray}\nT(z)=-\\frac{1}{2} \n\\partial X \\partial X(z)+\\alpha_0\\partial^2 X(z), \n\\end{eqnarray}\nand provides the generators \nof the Virasoro algebra  \nwith the central charge $1+12\\alpha_0^2 = 1+3l\/2G$.\nThis central charge is the same as that of Virasoro algebra of \nBrown-Henneaux in the semiclassical limit, i.e.,  $l\/G \\gg 1$.\nThe Fock space ${\\cal F}_{k}$\\footnote{Similar\narguments hold for the anti-holomorphic (right-moving) part.} is built on \nthe Fock vacuum $|k\\rangle$, which is introduced as \nthe state obtained from the ordinary $SL_{2}({\\bf C})$-invariant vacuum \n$|0\\rangle$ by the relation\n$|k\\rangle = \\lim_{z\\rightarrow 0} e^{i k X(z)}|0\\rangle$.\n\nThe BTZ black hole states (\\ref{eq:pribtz}) can be \nidentified with the following Fock vacuum:\n\\begin{eqnarray}\n{\\rm BTZ}_{(J, M)} \\Longleftrightarrow |J, M\\rangle \\equiv \n  |k_{(J, M)}\\rangle \\otimes |\\tilde{k}_{(J, M)}\\rangle, \n\\label{black hole state in 2d}\n\\end{eqnarray}\nwhere $k_{(J,M)}$ and $\\tilde{k}_{(J,M)}$ are given by \n\\begin{eqnarray}\nk_{(J, M)} &\\equiv& \n-i\\sqrt{\\frac l{8G}}+\\frac{r_+ +r_-}{\\sqrt{8Gl}},\n\\nonumber \\\\ \n\\tilde{k}_{(J, M)} &\\equiv& \n -i\\sqrt{\\frac l{8G}}+\\frac{r_+ -r_-}{\\sqrt{8Gl}}.   \n\\label{k(J,M)}\n\\end{eqnarray}\n${\\rm AdS}_{3}$ state (\\ref{eq:priads}) can be  \nidentified with the $SL_{2}({\\bf C})$-invariant vacuum:\n\\begin{eqnarray}\n{\\rm AdS}_{3} \\Longleftrightarrow |vac\\rangle\n\\equiv |0\\rangle\\otimes|0\\rangle.\n\\end{eqnarray}\n\nThe Fock spaces ${\\cal F}_{k} \\otimes {\\tilde {\\cal\nF}}_{{\\tilde k}}$ built on these primary\nstates give the unitary irreducible representations of the Virasoro\nalgebra with $c=1+3l\/2G$, and coincide with the physical Hilbert\nspace of the previous quantization of three-dimensional pure\ngravity.\n\nTo summarize, in this correspondence of three-dimensional pure gravity\nand the boundary CFT, the BTZ black holes appear as the primary\nstates of the Virasoro algebra with $c = 3l\/2G$ in both descriptions.\nThis result may not be desirable for the counting of microscopic states of\nthe BTZ black holes. We cannot count in principle the\ndegeneracy of these primary states with this boundary theory,\nsince this Liouville field theory has continuum\nspectrum of primary states.\n\n\\subsection{BTZ black holes and Maldacena duality}\n\nNext, we consider the ${\\rm AdS}_{3}\/{\\rm CFT}_{2}$ correspondence \nin superstring theory.\nThrough the analysis\nof the near horizon limit of the BPS solitonic solution of \nIIB supergravity, which describes\nthe bound state of $Q_{1}$ D1-branes and $Q_{5}$ D5-branes,\nMaldacena has conjectured in \\cite{Maldacena1},\n\\begin{center}\nIIB superstring theory on (${\\rm AdS}_{3} \\times {\\rm S}^{3})_{Q_{1}Q_{5}} \n\\times M_{4}$ \\quad ($M_{4}$ = $K3$ or $T^4$)\\\\ \n$\\Updownarrow$ dual  \\\\ two-dimensional N=(4,4) supersymmetric\n$\\sigma$-model \\\\ on the Higgs branch of world volume\ntheory of the D1-D5 system.\n\\end{center}\nHere, we indicated the dependence of the radius of \n${\\rm AdS}_{3}$ and ${\\rm S}^{3}$ on $Q_{1}Q_{5}$ \n(see below). We call this duality simply the Maldacena duality.\n\nWe will discuss mainly the case of $M_{4}$ = $K3$ in the following.\nThe N=(4,4) $\\sigma$-model can be regarded as\nthe $\\sigma$-model on the target space of the $k$-th symmetric product of\n$K3$ \\cite{Vafa1,Dijkgraaf}, where\n\\begin{eqnarray}\nk = Q_{1}Q_{5} + 1.\n\\end{eqnarray} \nSince the symmetric product \nis 4$k$-dimensional hyper-K\\\"ahler manifold, it\nhas automatically the N=(4,4) superconformal symmetry.\nThe Virasoro subalgebra and zero mode of the\nSU(2) current algebra may\nbe identified with the Virasoro algebra of \nBrown-Henneaux on the boundary of ${\\rm AdS}_{3}$ and \nthe isometry of ${\\rm S}^{3}$, respectively. We will discuss \nthe N=4 superconformal algebra in more detail in section 3.\n\nThe extremal BTZ black holes can also be obtained\nas the near horizon limit of the similar BPS solitonic solutions of \nIIB supergravity.\nTherefore we can expect that the extremal BTZ black holes \ncan be analyzed by this N=(4,4) $\\sigma$-model.\n\nWe summarize some related facts of IIB supergravity here.\nIIB supergravity on $S^{1} \\times K3$ whose radius and volume are $R$\nand $(2 \\pi)^4 \\alpha^{'2} v$ has the BPS solitonic solution\n(see for example \\cite{Skenderis} and references therein):\n\\begin{eqnarray}\nds_{10}^{2} &=& f_{1}^{-\\frac{1}{2}}f_{5}^{-\\frac{1}{2}}\\{ - dt^2 +\ndx_{5}^{2} + f_{N}(dt+dx_{5})^2\\} \\nonumber \\\\ && + f_{1}^{\\frac{1}{2}}\nf_{5}^{\\frac{1}{2}}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}) + \nf_{1}^{\\frac{1}{2}}f_{5}^{-\\frac{1}{2}} ds_{K3}^{2}, \\label{eq:10dim} \n\\end{eqnarray}\nwith periodic identification $x_{5} \\sim x_{5} + 2 \\pi R$\nin string frame and\n\\begin{eqnarray}\ne^{-2 (\\phi - \\phi_{\\infty})} &=& f_{5}f_{1}^{-1}, \n\\quad C_{05}^{(R)} = \\frac{1}{2}(f_{1}^{-1} - 1), \\nonumber \\\\\nH_{ijk}^{(R)} &=& (*_{6} \\ dC^{(R)})_{ijk} =\n\\frac{1}{2}\\epsilon_{ijkl}\\partial_{l}f_{5} \\qquad (i,j,k,l =\n1,2,3,4),\n\\end{eqnarray}\nwhere $C^{R}$ is Ramond-Ramond 2-form and $*_{6}$ is Hodge dual in\n6-dimension ($t,x_{1},\\cdots,x_{5}$).\nAnd $f_{1}$, $f_{5}$ and $f_{N}$ are following functions with respect\nto radial coordinate, \\\\\n$r = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$:\n\\begin{eqnarray}\nf_{1} &=& 1 + \\frac{{\\tilde Q}_{1}}{r^2}, \\qquad {\\tilde Q}_{1} =\n\\frac{\\alpha^{'} g_{st}}{v} Q_{1}, \\\\\nf_{5} &=& 1 + \\frac{{\\tilde Q}_{5}}{r^2}, \\qquad {\\tilde Q}_{5} = \n\\alpha^{'} g_{st}Q_{5}, \\\\\nf_{N} &=& \\frac{{\\tilde N}}{r^2}, \\qquad \\qquad {\\tilde N} =\n\\frac{\\alpha^{'2} g_{st}^{2}}{R^{2} v} N.\n\\end{eqnarray}\n\nThis solution corresponds to the configuration\nof the bound state of $Q_{5}$ D5-branes wrapping on $K3 \\times S^{1}$\nand $Q_{1}$ D1-branes wrapping on $S^{1}$ with $N$ units of KK momenta\nalong $S^{1}$, i.e., $x_{5}$-direction, and preserves four\nsupercharges. So it is a 1\/8 BPS state\\cite{Strominger-Vafa}.\n\nHere we can get the extremal BTZ black hole as the near horizon limit of\nthe geometry (\\ref{eq:10dim})\\cite{Maldacena-Strominger,Skenderis}.  \nThe near horizon limit is defined as\n\\begin{eqnarray*}\n\\alpha{'} \\rightarrow 0, \\qquad {\\rm with} \\qquad  \nU \\equiv \\frac{r}{\\alpha^{'}}, \\  R  \\  {\\rm and}  \\   v  \\\n  {\\rm fixed}.\n\\end{eqnarray*}\nIn this limit, the metric (\\ref{eq:10dim})\ndescribes $({\\rm BTZ}_{(Ml,M)} \\times S^{3})_{Q_{1}Q_{5}}\n\\times K3$ with $Ml = J = N$.\nThe radius of ${\\rm AdS}_{3}$ and ${\\rm S}^{3}$ coincide and become\n$l = g_{st}^{1\/2} \\alpha^{' 1\/2} (Q_{1}Q_{5}\/v)^{1\/4}$. \\\\\nAnd the three-dimensional effective Newton\nconstant on ${\\rm BTZ}_{(N,N\/l)}$ is given by\n$G_{{\\rm eff}}^{(3)} = l\/(4 Q_{1} Q_{5})$.\nThere exists on\nthis background the asymptotic Brown-Henneaux's Virasoro symmetry\nwith central charge,\n\\begin{eqnarray}\nc = \\frac{3l}{2G_{{\\rm eff}}^{(3)}} = 6 Q_{1}Q_{5}.\n\\end{eqnarray}\nThis is the same as the central charge of the N=(4,4) $\\sigma$-model \nat the semiclassical limit $Q_{1}Q_{5} \\gg 1$. \nThe Bekenstein-Hawking entropy becomes\n\\begin{eqnarray}\nS = \\frac{2 \\pi r_{+}}{4 G_{{\\rm eff}}^{3}} = 2 \\pi\n\\sqrt{Q_{1}Q_{5}N}, \\label{eq:extentropy}\n\\end{eqnarray}\nwhich is valid in the semiclassical region $N \\gg Q_{1}Q_{5} \\gg 1$.\n\nIf one accepts the Maldacena duality, the extremal black hole should be\nidentified with the primary state\n\\begin{eqnarray}\n|N+\\frac{c}{24}\\rangle \\otimes |\\frac{c}{24} \\rangle, \\label{eq:extprimary}\n\\end{eqnarray}\nof the N=(4,4) $\\sigma$-model.\nOne can ask whether the entropy (\\ref{eq:extentropy}) can be regarded as \nthe degeneracy of the primary state (\\ref{eq:extprimary}) of this N=(4,4)\n$\\sigma$-model. In the sequel, we will discuss this question and\nanswer in the affirmative. \n\n\\section{N=4 superconformal symmetry}\nN=(4,4) $\\sigma$-model is known to be finite to all orders of\nperturbation and to be conformally invariant at the quantum\nlevel. Thus the states of the $\\sigma$-model on the $k$-th symmetric\nproduct\\footnote{We will\ndenote $k$-th symmetric product of $K3$ as $S^k K3 \\equiv\nK3^{\\otimes k}\/S_{k} \\ (S_{k}$ is a $k$-dimensional symmetric\ngroup).}, $S^k K3$, constitute the unitary\nirreducible representations of the underlying N=4 superconformal\nalgebra (N=4 SCA).\n\n\\subsection{Basics of N=4 superconformal algebra}\nN=4 SCA is generated by $L_{n},\\ J_{n},\\ G_{r}^{i}$ and ${\\bar\nG}_{r}^{i}$ with \n\\begin{eqnarray*}\n[L_{m},L_{n}] &=& (m-n) L_{m+n}+\\frac{k}{2}m(m^2-1)\\delta_{n+m,0}, \\quad\n\\{G_{r}^{i},G_{s}^{j}\\} = \\{\\bar{G}_{r}^{i},\\bar{G}_{s}^{j}\\} = 0, \\\\\n\\{G_{r}^{i},{\\bar G}_{s}^{j}\\} &=& 2\n\\delta^{ij}L_{r+s}-2(r-s)\\sigma_{ij}^{a}J_{r+s}^{a} + \\frac{k}{2}(4\nr^2-1) \\delta_{r+s,0}, \\\\\n\\left[ J_{m}^{a},J_{n}^{b} \\right] &=& i \\epsilon^{abc} J_{m+n}^{c} + \n\\frac{k}{2}m\\delta_{m+n,0}, \\\\\n\\left[ J_{m}^{a},G_{r}^{i} \\right] &=&\n-\\frac{1}{2}\\sigma_{ij}^{a}G_{m+r}^{j}, \\quad \n\\left[ J_{m}^{a},{\\bar G}_{r}^{i} \\right] =\n\\frac{1}{2}\\left(\\sigma_{ij}^{a}\\right)^{*}{\\bar G}_{m+r}^{j}, \\\\\n\\left[ L_{m},G_{r}^{i} \\right] &=&\n(\\frac{m}{2}-r)G_{m+r}^{i}, \\quad \\left[ L_{m},{\\bar G}_{r}^{i} \\right] =\n(\\frac{m}{2}-r){\\bar G}_{m+r}^{i}, \\\\\n\\left[ L_{m},J_{n}^{a} \\right] &=& -n J_{m+n}^{a}. \n\\end{eqnarray*}\n($\\sigma_{ij}^{a}$ is Pauli matrix and $J_{n}^{(\\pm)} \\equiv \nJ_{n}^{1}\\pm i J_{n}^{2}$.) $L_{n},J_{n}^{a}$ and \n$G_{r}^{i}({\\bar G}_{r}^{i})$ represent the\nFourier components of the energy momentum tensor, SU(2) current and four\nsupercurrents, respectively. \n$G_{r}^{i} ({\\bar G}_{r}^{i})$ transforms as SU(2) doublet (its conjugate) \nunder the global SU(2) symmetry which is generated by $J_{0}^{a}$.\nThe level $k$ of the SU(2) current algebra ($\\widehat{{\\rm SU}(2)}_{k}$)\nmust be positive integer for \nunitary representations. The central charge $c$ of the Virasoro subalgebra is \n$6k$. \n\nTwo different boundary conditions of the supercurrents\nprovide two different sectors of this algebra. \nIf $G_{r}^{i}({\\bar G}_{r}^{i})$ has $r \\in\n{\\bf Z}+ \\frac{1}{2} $, it is called the Neveu-Scwarz (NS) sector \nand if $r \\in {\\bf Z}$, it is called the Ramond (R) sector.\nThese sectors are related by the automorphism of the algebra\ncalled spectral flow.\nWe will discuss the R-sector in the following.\n\nUnitary irreducible representations of N=4 SCA have\ntwo distinct types\\cite{Eguchi1}.\nThey are built on highest weight states $|h,l\\rangle$ called massive\nprimary and massless primary in the R-sector.\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item Massive primary state\n      \\begin{eqnarray}\n      L_{n}|h,l\\rangle &=& G_{n}^{i}|h,l\\rangle = \n      {\\bar G}_{n}^{i}|h,l\\rangle = J_{n}^{a}|h,l\\rangle = 0, \\qquad n\n      \\geq 1 \\nonumber \\\\\n      J_{0}^{(+)}|h,l\\rangle &=& G_{0}^{2} |h,l\\rangle = \n      {\\bar G}_{0}^{1}|h,l\\rangle = 0, \\nonumber \\\\\n      L_{0}|h,l\\rangle &=& h |h,l\\rangle, \\quad J_{0}^{3} |h,l\\rangle =\n      l |h,l\\rangle, \\nonumber \\\\\n      h &>& \\frac{k}{4} = \\frac{c}{24}, \n      \\qquad l = \\frac{1}{2},1,\\cdots,\\frac{k}{2}-\\frac{1}{2},\\frac{k}{2}. \n      \\label{eq:massive}\n      \\end{eqnarray}\n\\item Massless primary state\n      \\begin{eqnarray}\n      L_{n}|h,l\\rangle &=& G_{n}^{i}|h,l\\rangle = \n      {\\bar G}_{n}^{i}|h,l\\rangle = J_{n}^{a}|h,l\\rangle = 0, \\qquad n\n      \\geq 1 \\nonumber \\\\\n      J_{0}^{(+)}|h,l\\rangle &=& G_{0}^{i} |h,l\\rangle = \n      {\\bar G}_{0}^{i}|h,l\\rangle = 0, \\qquad i=1,2 \\nonumber \\\\\n      L_{0}|h,l\\rangle &=& h |h,l\\rangle, \\quad J_{0}^{3} |h,l\\rangle =\n      l |h,l\\rangle, \\nonumber \\\\\n      h&=&\\frac{k}{4}=\\frac{c}{24},\\qquad l =0,\\frac{1}{2},\\cdots,\n      \\frac{k}{2}-\\frac{1}{2},\\frac{k}{2}. \\label{eq:massless} \n      \\end{eqnarray}\n\\end{enumerate}\nThese representations are called massive representation ${\\cal M}^{k}_{(h,l)}$ \nand massless representation ${\\cal M}^{k}_{0 (l)}$ respectively.\n\nThe massive representations have the same number of bosonic and\nfermionic states at each level and the Witten index is equal to \nzero. \nThese representations correspond to\nthe representations which have spontaneously broken supersymmetry.\nThe Witten index of the massless representations is non-zero. \nThese representations are the representations which have\nunbroken supersymmetry. \nThe primary states of massless representations have dimension \n$h=k\/4=c\/24$. These are the ground states of the R-sector.\n\nCharacter of the representation is introduced by\n${\\rm ch}^{(R)}(\\tau,z) = \n{\\rm Tr} (q^{L_{0}-\\frac{c}{24}} y^{2 J_{0}^{3}})$.\n($q=e^{2 \\pi i \\tau}$ and $y=e^{2 \\pi i z}$.)  \nTheir explicit form is given in \\cite{Eguchi2}.     \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item The character of the massive representation ${\\cal M}^{k}_{(h,l)}$: \n      \\begin{eqnarray}\n      {\\rm ch}^{(R) k}(h,l ; \\tau,z) =\n      q^{h-\\frac{k}{4}-\\frac{l^2}{k+1}}\n      \\frac{\\theta_{2}(\\tau,z)^2}{\\eta(\\tau)^3}\n      \\chi_{k-1}^{l-\\frac{1}{2}}(\\tau,z),  \\label{eq:charmass}\n      \\end{eqnarray}\n      where $\\chi_{k}^{l}(\\tau,z)$ is the character of \n      $\\widehat{{\\rm SU}(2)}_{k}$ of isospin $l$,\n      \\begin{eqnarray}\n      \\chi_{k}^{l}(\\tau,z) &=& \\frac{q^{\\frac{(l+\\frac{1}{2})^2}{k+2}\n      -\\frac{1}{8}}}{\\prod_{n=1}^{\\infty}(1-q^n)(1-y^2\n      q^n)(1-y^{-2}q^{n-1})} \\nonumber \\\\\n      &&\\times \\sum_{m=0}^{\\infty}q^{(k+2)m^2 + (2 l + 1)}\\left(y^{2\n      \\{(k+2)m + l\\}}-y^{-2 \\{ (k+2)m+l+1 \\}}\\right) \\nonumber \\\\\n      && = \\frac{\\Theta_{2l+1, k+2}(\\tau, 2z) - \\Theta_{-2l-1, k+2}(\\tau,\n        2z)}{\\Theta_{1,2}(\\tau,2z) - \\Theta_{-1,2}(\\tau,2z)}.\n      \\label{eq:charaffine}\n      \\end{eqnarray}\n      $\\Theta_{l,k}(\\tau,z) = \\sum_{n=-\\infty}^{\\infty} q^{k\n      (n+\\frac{l}{2k})^2} y^{k (n+\\frac{l}{2k})}$ is the theta\n      function associated with $\\widehat{{\\rm SU}(2)}_{k}$ of isospin $l$.\n\\item The character of the massless representation ${\\cal M}^{k}_{0 (l)}$:\n      \\begin{eqnarray}\n      {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z) &=&\n      q^{-\\frac{1}{8}}\\frac{\\theta_{2}(\\tau,z)^2}{\\eta(\\tau)^3} \n      \\frac{1}{\\prod_{n=1}^{\\infty}(1-q^n)(1-y^2\n      q^n)(1-y^{-2}q^{n-1})} \\nonumber \\\\\n      \\times \\sum_{m=-\\infty}^{\\infty}&& \\hskip-1cm q^{(k+1)m^2 + 2 l m} \n      \\left( \\frac{y^{2 \\{(k+2)m+l-\\frac{1}{2}\\}}}{(1+y^{-1}q^{-m})^2} - \n      \\frac{y^{-2 \\{ (k+2)m+l+\\frac{1}{2}\\}}}{(1+y q^{-m})^2} \\right).\n      \\label{eq:charless}\n      \\end{eqnarray}\n\\end{enumerate}\n\nThese characters enjoy the following properties.\nThe Witten index of the representation \ncan be obtained, if one sets $z=1\/2$, i.e., $y=-1$:\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h,l;\\tau,z=\\frac{1}{2}) &=& 0, \\\\\n{\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z=\\frac{1}{2}) &=& \n(-1)^{2 l}(2 l+1).\n\\end{eqnarray}\nThe characters of massive and massless\nrepresentations are related by\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h=\\frac{k}{4},l;\\tau,z) &=&  \n{\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l;\\tau,z)  \n+ 2 \\ {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l-\\frac{1}{2};\\tau,z)\n\\nonumber \\\\\n&&+ \\ {\\rm ch}_{0}^{(R) k}(h=\\frac{k}{4},l-1;\\tau,z).\n\\label{eq:relation}\n\\end{eqnarray}\n\n\\subsection{Identification of the black hole states}\n\nAs argued in section 2.2, the extremal BTZ black hole will correspond\nto the primary state (\\ref{eq:extprimary}) of the N=(4,4)\n$\\sigma$-model.\nIt is a Virasoro primary state of the underlying N=4\nSCA. The fact that the extremal BTZ black holes are the 1\/2 \nBPS states with respect to the Poincare supersymmetry in \nthree dimensions\\footnote{These correspond to the 1\/4\nBPS states in Anti-de Sitter supersymmetry in 3-dimension.} implies\nthat this primary state is in the tensor\nproduct of massive and massless representations ${\\cal\nM}^{k}_{(h,l)} \\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$ which\nis built on \n\\begin{eqnarray}\n|h=N+\\frac{k}{4},l\\rangle \\otimes |{\\tilde h}=\\frac{k}{4},{\\tilde\n l}\\rangle. \\label{eq:prisigma}\n\\end{eqnarray}\n\nActually we can proceed further. Since the extremal \nBTZ black holes do not have the conserved \ncharge corresponding to isospin $l$ and ${\\tilde l}$,\nthe primary state (\\ref{eq:extprimary}) may be identified with the\nVirasoro primary state having vanishing isospins $l={\\tilde\nl}=0$ in \n${\\cal M}^{k}_{(h,l)} \\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$.\nThe primary states with $l=0$ \nin ${\\cal M}^{k}_{(h,l)}$\nare as follows:\nwhen $l \\in {\\bf Z}$, they are given by \n\\begin{eqnarray}\n&&J_{0}^{(-) \\ l} |h=N+\\frac{k}{4},l\\rangle, \\nonumber \\\\\n{\\rm and} && J_{0}^{(-) \\ l-1} ({\\bar G}_{0}^{2} G_{0}^{1} - \n\\frac{h-k\/4}{l}) |h=N+\\frac{k}{4},l \\rangle, \\label{eq:integer}\n\\end{eqnarray}\nand when $l \\in {\\bf Z}+\\frac{1}{2}$, they are \n\\begin{eqnarray}\n&&J_{0}^{(-) \\ l-\\frac{1}{2}} G_{0}^{1} |h=N+\\frac{k}{4},l\\rangle,\n\\nonumber \\\\\n\\hskip-1.4cm{\\rm and} \\qquad \n&&J_{0}^{(-) \\ l-\\frac{1}{2}}{\\bar\nG}_{0}^{2}|h=N+\\frac{k}{4},l\\rangle.\n\\label{eq:halfint} \n\\end{eqnarray}\nThe primary states with ${\\tilde l}=0$ in\n${\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$ is identified with\n${\\bar J}_{0}^{(-) \\ {\\tilde l}} |{\\tilde h}=k\/4,{\\tilde\nl}\\rangle.$\nThe primary state (\\ref{eq:extprimary}) can be identified with the\ntensor product of these states. So, the degeneracy of the state is\nalmost same as the degeneracy of the representation ${\\cal M}^{k}_{(h,l)} \n\\otimes {\\tilde {\\cal M}}^{k}_{0 ({\\tilde l})}$. \n  \n\\section{Elliptic genus for $\\sigma$-model on symmetric product\nof K3}\nTo count the number of 1\/2 BPS states in the N=(4,4)\n$\\sigma$-model,\nthe so-called ``elliptic genus'' is a convenient tool.\nWe summarize some properties of the elliptic genus\nemphasizing its modular transform and examine it from the perspective\nof N=4 SCA. \n\\subsection{Elliptic genus as a weak Jacobi form}\nThe elliptic genus of target space $M$ \nis defined by the following trace in the R-R sector of the underlying\nN=(2,2) superconformal field theory\\footnote{One can obtain the following\ntopological indices of the target space $M$, if one sets $z$ to be\nspecific value.\n\\begin{eqnarray}\nZ[M](\\tau,0) &:& {\\rm Elliptic \\ extension \\ of \\ Euler \\ number}\n\\nonumber \\\\\nZ[M](\\tau,\\frac{1}{2})&:& {\\rm Elliptic \\ extension \\ of \\\nHirzebruch \\ signature} \\nonumber \\\\\nq^{\\frac{c}{24}}Z[M](\\tau,\\frac{\\tau+1}{2})&:& \n{\\rm Elliptic \\ extension \\ of \\ Dirac \\ genus} \\nonumber\n\\end{eqnarray}}.\n\\begin{eqnarray}\nZ[M](\\tau,z) = {\\rm Tr}_{{\\rm R}\\textrm{-}{\\rm R}}(-1)^{J_{0}-{\\bar\nJ}_{0}} q^{L_{0}-\\frac{c}{24}} {\\bar q}^{{\\bar\nL}_{0}-\\frac{c}{24}} y^{J_{0}}, \\label{eq:genus}\n\\end{eqnarray}\nwhere $J_{0}$ and ${\\bar J}_{0}$ are the integral N=2 U(1) charges\nof the left-moving and the right-moving sectors\\footnote{N=2 \nSCA can be embedded into N=4 SCA by\n$G_{r} = G_{r}^{1} + {\\bar G}_{r}^{2}$, ${\\bar G}_{r} = \nG_{r}^{2} + {\\bar G}_{r}^{1}$ and $J_{n}=2 J_{n}^{3}$.}.   \nThe elliptic genus is\nindependent of ${\\bar \\tau}$ by virtue of supersymmetry of the R-sector.\nThe contribution of the right-moving sector is only from the\nground states.\nBut all states in the left-moving sector contribute to $Z[M](\\tau,z)$.\nSo the elliptic genus $Z[M](\\tau,z)$ is a useful quantity for \ncounting of the 1\/2 BPS states.\n\nThe following theorem is known about this elliptic genus. (See\n \\cite{Kawai2} for detail.)\n\n\\noindent\n{\\bf Theorem 1.} If the target space $M$ of the $\\sigma$-model is an\neven-dimensional Calabi-Yau manifold, then the elliptic genus\n$Z[M](\\tau,z)$ is a weak Jacobi form of weight 0 and index\n$d\/2 (d=\\dim_{{\\bf C}} M)$ without character.\n\nWeak Jacobi form in the above theorem is defined as\nfollows\\cite{Eichler}.\n\n\\noindent \n{\\bf Definition.} A function \n$\\phi(\\tau,z)$ is called a weak Jacobi form of weight $k \\in {\\bf Z}$ \nand index $m \\in {\\bf Z}_{>0}\/2$ without character, if it satisfies\n(i)$\\sim$(iv): \n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item $\\phi(\\tau,z)$ is a holomorphic function with respect to $\\tau\n      \\in {\\bf H}^{+} \\ ({\\bf H}^{+}:$ upper half plane) and $z \\in\n      {\\bf C}$.\n\\item $\\phi(\\frac{a \\tau + b}{c \\tau + d},\\frac{z}{c \\tau + d}) = \n      (c \\tau + d)^{k}e^{\\frac{2 \\pi i m c z^{2}}{c \\tau +\n      d}}\\phi(\\tau,z). \\quad (a,b,c,d \\in {\\bf Z} \\ {\\rm and} \\ ad-bc=1)$\n\\item $\\phi(\\tau,z + \\lambda \\tau + \\mu) = e^{- 2\\pi i m (\\lambda^2\n      \\tau + 2 \\lambda z)} \\phi(\\tau,z). \\quad (\\lambda, \\mu \\in {\\bf Z})$\n\\item $\\phi(\\tau,z)$ has the Fourier expansion of the form \\\\\n      \\begin{eqnarray*}\n      \\phi(\\tau,z) = \\sum_{n=0}^{\\infty} \\sum_{r = - \\infty}^{\\infty}\n      c(n,r) q^n y^r \\quad (q= e^{2 \\pi i \\tau},y = e^{2 \\pi i z}).\n      \\end{eqnarray*}\n\\end{enumerate}\n\nWhen $M$ is $K3$, the elliptic genus $Z[K3](\\tau,z)$\nbecomes a weak Jacobi form of weight 0 and index 1 without character.\nAn actual calculation of the N=(4,4) supersymmetric\n$\\sigma$-model on $K3$\\cite{Eguchi4,Kawai1} determines it explicitly as\n\\begin{eqnarray}\nZ[K3](\\tau,z) = 24 \\wp(\\tau,z)K^2(\\tau,z), \\label{eq:pfnK3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\wp(\\tau,z) &:& {\\rm Weierstrass's} \\ \n\\wp {\\rm -function}, \\nonumber \\\\\nK(\\tau,z) &=& i \\frac{\\theta_{1}(\\tau,z)^2}{\\eta(\\tau)^3}.\n\\label{eq:kfn} \n\\end{eqnarray} \n\nWe need the following theorem about weak Jacobi form\\cite{Kawai2,Eichler}. \n\n\\noindent\n{\\bf Theorem 2}. If we assign weights 4, 6 and 2 respectively to\n$E_{4}(\\tau), E_{6}(\\tau)$\\footnote{$E_{4}(\\tau)$ and $E_{6}(\\tau)$\nare the Eisenstein series.} and $\\wp(\\tau,z)$, any weak Jacobi\nform of weight $2l$ ($l \\in {\\bf Z}_{\\geq 0}$) and index $k$ ($k \\in {\\bf\nZ}_{\\geq 0}$) can be expressed as \n\\begin{eqnarray*}\n{\\cal G}_{2 l + 2 k}(E_{4}(\\tau), E_{6}(\\tau), \\wp(\\tau,z))\nK^{2k}(\\tau,z),\n\\end{eqnarray*}\nwhere ${\\cal G}_{2l + 2k}(E_{4},E_{6},\\wp)$ is a homogenious \npolynomial of weight (2$l$ + 2$k$) and its degree as a\npolynomial in $\\wp$ is at most $k$.\n\n$S^k K3$ is 2$k$-dimensional Calabi-Yau manifold. \nThe elliptic genus\n$Z[S^{k}K3](\\tau,z)$ becomes a weak Jacobi form of weight 0 and index $k$.   \nAccording to theorem 2, it has the\nfollowing form:\n\\begin{eqnarray}\nZ[S^k K3](\\tau,z) = {\\cal G}_{2k}(E_{4}(\\tau), E_{6}(\\tau),\n\\wp(\\tau,z))K^{2k}(\\tau,z). \\label{eq:generalgenus}\n\\end{eqnarray}  \n\nThe homogenious polynomial ${\\cal G}_{2k}$ is determined for lower\nvalues of $k$\\cite{Kawai2}\\footnote{It is worth commenting that \nthe coefficient of the first term in the\nbracket is the Euler number of $S^k K3$, $\\chi(S^k K3)$.}.\n\\begin{eqnarray}\nk=1 &:& 24\\wp K^{2} \\nonumber \\\\\nk=2 &:& (324 \\wp^{2} + \\frac{3}{4} E_{4})K^{4} \\nonumber \\\\\nk=3 &:& (3200 \\wp^{3} + \\frac{64}{3}E_{4}\\wp +\n\\frac{10}{27}E_{6})K^{6}\n\\label{eq:pfnprod}\n\\end{eqnarray}\n\n\\subsection{Elliptic genus of $S^k K3$ and characters of N=4\nsuperconformal algebra}\nThe N=(4,4) $\\sigma$-model on the target space of $K3$ has been\nanalyzed in detail by Eguchi et.al.\\cite{Eguchi4} in the context of \na compactification of string theory on $K3$. The elliptic extension\nof the Hirzebruch signature of $K3$ \ncan be represented by the characters of the N=4\nSCA as\n\\begin{eqnarray}\n&&\\hspace{-2cm} Z[K3](\\tau,\\frac{1}{2}) \\nonumber \\\\\n&=&-2 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0) +\n20 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=0;\\tau,0) \\nonumber \\\\\n&&+ F(\\tau) \\ {\\rm ch}^{(R) k=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0)\n\\nonumber \\\\\n&=& 24 \\ {\\rm ch}_{0}^{(R) k=1}(h=\\frac{1}{4},l=0;\\tau,0) +\n {\\tilde F}(\\tau) \\ {\\rm ch}^{(R)\nk=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0),\n\\label{eq:genusK3}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\tilde F}(\\tau) = -2 + F(\\tau) = \\sum_{n=0}^{\\infty}a_{n}q^{n} \\qquad\n(a_{0} = -2, \\ a_{n} \\in {\\bf Z}_{\\geq 0} (n>0)).\n\\end{eqnarray} \nWe have used eq.(\\ref{eq:relation}) to obtain the last equality in \n(\\ref{eq:genusK3}).\nThe degeneracy of the massive primary states in the left-moving sector\nis encoded in ${\\tilde F}(\\tau)$. The coefficient $a_{n}$ is the\ndegeneracy of the massive primary states of $h=n+1\/4$.\n\nThe function ${\\tilde F}(\\tau)$ can be\ndetermined by combining\ntwo expressions of elliptic genus (\\ref{eq:pfnK3}) and (\\ref{eq:genusK3}),\n\\begin{eqnarray}\nZ[K3](\\tau,\\frac{1}{2}) &=& 24 \\ {\\rm ch}_{0}^{(R) k=1}\n(h=\\frac{1}{4},l=0;\\tau,0) \\nonumber \\\\\n&& + {\\tilde F}(\\tau) \\ {\\rm ch}^{(R)\nk=1}(h=\\frac{1}{4},l=\\frac{1}{2};\\tau,0) \\nonumber \\\\\n&=& 24 \\ \\wp(\\tau,\\frac{1}{2})K^{2}(\\tau,\\frac{1}{2}).\n\\end{eqnarray} \nThis gives\n\\begin{eqnarray}\n{\\tilde F}(\\tau) &=& 2 \\ \\frac{\\theta_{2}(\\tau,0)^4 -\n\\theta_{4}(\\tau,0)^4}{\\prod_{n=1}^{\\infty}(1-q^n)^3} - 24 \\  \n{\\tilde h}_{3}(\\tau), \n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \n{\\tilde h}_{3}(\\tau) &=&\n\\frac{1}{\\theta_{3}(\\tau,0)}\\sum_{m=-\\infty}^{\\infty}\n\\frac{q^{\\frac{m^2}{2}}}{1+q^{m-\\frac{1}{2}}}.\n\\end{eqnarray}\n\nNow, we will turn to the case of the symmetric product.\nThe elliptic genus of $S^k K3$ can be \nalso expanded by the characters of the underlying N=4 SCA.\nIn general, it has the following expansion:\n\\begin{eqnarray}\n&&\\hspace{-2cm}Z[S^k K3](\\tau, z+\\frac{1}{2}) \n\\qquad \\left( = {\\rm Tr}(-1)^{- 2 {\\bar\nJ}_{0}^{3}}{\\bar q}^{{\\bar\nL}_{0}-\\frac{c}{24}}q^{L_{0}-\\frac{c}{24}}y^{2 J_{0}^{3}} \\right) \\nonumber \\\\\n&&\\hspace{-1cm}= \\chi(S^k K3) \\ {\\rm ch}_{0}^{(R) k}\n(h=\\frac{k}{4},l=0;\\tau,z) + \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}}F_{l}(\\tau) \\ {\\rm ch}^{(R) k}\n(h=\\frac{k}{4},l;\\tau,z), \\label{eq:charfnprod}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nF_{l}(\\tau) = \\sum_{n=0}^{\\infty}a_{n}^{(l)}q^n, \\qquad \\quad (a_{0}^{(l)}\n\\in {\\bf Z}, \\ a_{n}^{(l)} \\in {\\bf Z}_{\\geq 0} \\ (n>0)). \\label{eq:ffun}\n\\end{eqnarray}        \nAlthough the characters of the massless representations with $l \\neq\n0$ may appear in $Z[S^k K3](\\tau,z+1\/2)$, we can reduce them\nto $l=0$ by applying (\\ref{eq:relation}) recursively. $a_{n}^{(l)}$ in\n(\\ref{eq:ffun}) provides the degeneracy of the representation \n${\\cal M}_{(h,l)}^{k} \\otimes {\\tilde {\\cal M}}_{0 ({\\tilde l})}^{k}$ \nwith $h=n+k\/4$ and isospin $l$. $\\sum_{l}a_{n}^{(l)}$ provides \nthe number of the representations ${\\cal M}_{(h,l)}^{k} \n\\otimes {\\tilde {\\cal M}}_{0 ({\\tilde l})}^{k}$ with $h=n+k\/4$. \n\nBecause of eq.(\\ref{eq:generalgenus}), \nthe spectrum of the massive primary states of this $\\sigma$-model on \n$S^k K3$ also\nbecomes discrete and all states have dimension $h = {\\bf Z}_{\\geq 0}\n+k\/4$ in $Z[S^k K3](\\tau,z+1\/2)$. This corresponds to the fact \nthe extremal BTZ black holes have the discrete mass $N\/l$ in the context of\nthe Maldacena duality. \n\nThe function $F_{l}(\\tau)$ \\ (or $\\sum_{l}F_{l}(\\tau)$) in\neq.(\\ref{eq:charfnprod}) can be determined in principle by an \nanalogous way with the case of $K3$.\nBut it is a hard task to obtain the exact functional form of \n$Z[S^k K3](\\tau,z)$ such as (\\ref{eq:pfnprod}) for the case of \ngeneral $k$. \nHowever, as discussed in section 2.2, what we need for \nthe counting of the microscopic states comparable with \nthe Bekenstein-Hawking entropy is   \nthe asymptotic form of $a_{n}^{(l)}$ (or $\\sum_{l} a_{n}^{(l)}$) \nat $n \\rightarrow \\infty$, since eq.(\\ref{eq:extentropy}) is valid\nfor the region $N \\gg Q_{1}Q_{5} \\gg 1$.\nWe will consider this asymptotic form in the next section.\n\n\\section{State counting via the N=(4,4) $\\sigma$-model}   \nIn this section, we will discuss the degeneracy of  \n1\/8 BPS states of the D1-D5\nsystem in IIB superstring theory and the degeneracy of the primary states \ncorresponding to the extremal BTZ black holes.\nIn the previous section, we obtain two different expressions\nof the elliptic genus of $S^k K3$. \nWe will first use the expression in terms of a weak\nJacobi form and count the number of the 1\/8 BPS states by using a\nTauberian theorem. Then using the expression in terms of the \ncharacters of N=4 SCA, we will discuss the\ndegeneracy of the massive primary states and obtain the\nmicroscopic entropy of the corresponding extremal BTZ black holes.\n\n\\subsection{Counting the 1\/8 BPS states}\nLet us start by studying the asymptotic behavior of the elliptic genus\n$Z[S^k K3](\\tau,1\/2)$ as $\\tau \\downarrow 0$\\footnote{\n$\\tau \\downarrow 0 \\stackrel{{\\rm def}}{\\Longleftrightarrow} \\tau = iT\n\\ (T \\in {\\bf R}_{>0}),\\  {\\rm and} \\ T \\rightarrow 0.$}. \nDue to the structure theorem the elliptic genus\nhas the form (\\ref{eq:generalgenus})\n\\begin{eqnarray*}\nZ[S^k K3](\\tau,\\frac{1}{2}) = \\\n{\\cal G}_{2 k}\\left( E_{4}(\\tau),E_{6}(\\tau),\\wp(\\tau,\\frac{1}{2})\n\\right) K^{2 k}(\\tau,\\frac{1}{2}). \n\\end{eqnarray*}\nThe asymptotics can be obtained \nfrom those of the constituents in (\\ref{eq:generalgenus}).\n$\\wp(\\tau,1\/2)$, $E_{4}(\\tau)$ and\n$E_{6}(\\tau)$ behave as \n$\\wp(\\tau,1\/2) \\rightarrow (-1\/12)(-i \\tau)^{-2}, \\\nE_{4}(\\tau) \\rightarrow 1 (-i \\tau)^{-4}$, and   \n$E_{6}(\\tau) \\rightarrow (-1)(-i \\tau)^{-6}$. \nTherefore the asymptotics of ${\\cal G}_{2k}$ becomes\n\\begin{eqnarray}\n{\\cal G}_{2k}(\\tau,\\frac{1}{2}) \\rightarrow {\\tilde c}(k) (-i \\tau)^{-2 k}\n\\qquad {\\rm as} \\ \\tau \\downarrow 0,\n\\end{eqnarray}\nwhere ${\\tilde c}(k)$ is a constant which depends on $k$ and the\npolynomial form of ${\\cal G}_{2k}$.\nThe asymptotics of $K^{2}(\\tau,1\/2)$ can be read from \n$\\theta_{2}(\\tau,0) \\rightarrow 1 (-i \\tau)^{-\\frac{1}{2}}$ and \n$\\eta(\\tau) \\rightarrow 1 (-i \\tau)^{-\\frac{1}{2}} e^{-\\frac{\\pi i}{12\n\\tau}}$,\n\\begin{eqnarray}\nK^{2}(\\tau,\\frac{1}{2}) \\rightarrow (-1)(-i \\tau)^{2} e^{\\frac{\\pi i}{2\n\\tau}} \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n\\end{eqnarray}\n\nTherefore, gathering these asymptotics, we obtain\n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2}) \\rightarrow c(k)(-i \\tau)^{0}\ne^{\\frac{\\pi i k}{2 \\tau}} \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n \\quad \\left( c(k) = (-1)^k {\\tilde c}(k) \\right) \\label{eq:asymz}\n\\end{eqnarray}\n\nThe elliptic genus has the Fourier expansion of the form\n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2})=\\sum_{n=0}^{\\infty} a_{n}q^n.\n\\label{eq:expgenus}\n\\end{eqnarray}\nAgain, due to the structure theorem, the coefficients satisfy $a_{n}\n\\leq a_{n+1}$. (See Appendix A for the explicit Fourier expansions of \nvarious functions.) Each coefficient $a_{n}$ represent the number of\nthe 1\/2 BPS states of $h=n+k\/4$ in the N=(4,4) $\\sigma$-model.\nThis is the number of\nthe $1\/8$ BPS states of the mass specified by\n$h=n+k\/4$ in the D1-D5 system \\cite{Strominger-Vafa,Maldacena-Strominger}.\nWe can estimate the asymptotic form of $a_{n}$ by using the following\nTauberian theorem\\cite{Kac1,Kac2}.\n\n\\noindent \n{\\bf Theorem 3.} Let $f(\\tau)$ be a function \n\\begin{eqnarray*}\nf(\\tau) = q^{\\lambda} \\sum_{n=0}^{\\infty} a_{n} q^n \\qquad (q=e^{2 \\pi\ni \\tau})\n\\end{eqnarray*}\nwhich satisfies following conditions:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi}\n\\renewcommand{\\theenumi}{(\\roman{enumi})}\n\\item $f(\\tau)$ is a holomorphic function on ${\\bf H}^{+}$.\n\\item $a_{n} \\in {\\bf R}$ and $a_{n} \\leq a_{n+1}$ for all $n$.\n\\item There exist $c \\in {\\bf C}$, $d \\in {\\bf R}$ and $N \\in \n      {\\bf R}_{>0}$ such that \n      \\begin{eqnarray*}\n      f(\\tau) \\rightarrow c (-i \\tau)^{-d} e^{\\frac{2 \\pi i N}{\\tau}}\n      \\qquad {\\rm as} \\ \\tau \\downarrow 0.\n      \\end{eqnarray*}\n\\end{enumerate}\nThen, the behavior of $a_{n}$ at large $n$ is\n\\begin{eqnarray*}\na_{n} \\sim \\frac{c}{\\sqrt{2}}N^{-\\frac{1}{2}(d-\\frac{1}{2})}\nn^{\\frac{1}{2}(d-\\frac{3}{2})} e^{2 \\pi \\sqrt{4 N n}}  \\qquad {\\rm as} \\ \nn \\rightarrow \\infty, \n\\end{eqnarray*}\nwhere $a_{n} \\sim b_{n}$ as $n \\rightarrow \\infty$ means $\\lim_{n\n\\rightarrow \\infty} b_{n}\/a_{n} = 1$.  \n\nDue to this theorem, the asymptotic form of $a_{n}$ can be read from\nthe estimation (\\ref{eq:asymz})\\footnote{We can\nexpect $c(k)$ is not large number due to the explicit\nexample of lower $k$. (See \\cite{Kawai2}.)}\n\\begin{eqnarray}\na_{n} \\sim \\frac{c(k)}{\\sqrt{2}} \\left( \n\\frac{k}{4} \\right)^{\\frac{1}{4}} n^{-\\frac{3}{4}} e^{2 \\pi \\sqrt{k\nn}}. \\label{eq:bpsnum}\n\\end{eqnarray}  \n\n\\subsection{Counting the massive primary states}\nWe can also expand the elliptic genus by the characters of N=4 SCA\n\\begin{eqnarray}\n&&\\hspace{-2cm}Z[S^k K3](\\tau,\\frac{1}{2}) \\nonumber \\\\ \n=&&\\hspace{-0.5cm}\\chi(S^k K3) \\ {\\rm ch}_{0}^{(R) k} \n(h=\\frac{k}{4},l=0;\\tau,0) + \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}}F_{l}(\\tau) \\ {\\rm ch}^{(R) k}\n(h=\\frac{k}{4},l;\\tau,0).\n\\label{eq:charexp}\n\\end{eqnarray}\nEach coefficient of $F_{l}(\\tau) =\n\\sum_{n=0}^{\\infty}a_{n}^{(l)} q^{n}$ counts the number of the \nmassive representation ${\\cal M}_{(h{\\rm =}n+k\/4,l)}^{k}$ \nin $Z[S^k K3](\\tau,z)$\nof the N=(4,4) $\\sigma$-model. \nIn particular $\\sum_{l}a_{N}^{(l)}$ will be identified\nwith the number of the primary state (\\ref{eq:extprimary}).\nTo obtain the asymptotic form $\\sum_{l}a_{n}^{(l)}$, we may again\nutilize the Tauberian theorem. For this purpose we need to know the\nasymptotic behavior of $\\sum_{l}F_{l}(\\tau)$ as $\\tau \\downarrow 0$.\n\nSince we have obtained the asymptotics of the elliptic genus\n(\\ref{eq:asymz}), the asymptotics of $\\sum_{l}F_{l}(\\tau)$ becomes\ntractable if we can properly estimate the constituents of the\nmassless and massive characters. \nLet us remind that the character of \nmassive representation ${\\cal M}_{(h,l)}^{k}$ is\ngiven by eq.(\\ref{eq:charmass}).\nThe asymptotic behavior of\nthe character of $\\widehat{{\\rm SU}(2)}_{k}$ \nof the isospin $l$, eq.(\\ref{eq:charaffine}), is given by\\cite{Kac1,Kac2}\n\\begin{eqnarray}\n\\chi_{k}^{l}(\\tau, 0) \\rightarrow a(k,l) \\exp \\left( \n\\frac{\\pi i}{12 \\tau} c_{k} \\right)\n\\qquad {\\rm as} \\ \\tau \\downarrow 0,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na(k,l) &=& \\sqrt{\\frac{2}{k+2}}\\sin\\left(\\frac{(2 l + 1) \\pi}{k+2}\\right),\n\\nonumber \\\\\nc_{k} &=& \\frac{3 k}{k+2}.\n\\end{eqnarray}\nTherefore, combining those of $\\theta_{2}(\\tau,0)$ and $\\eta(\\tau)$,\nwe obtain\n\\begin{eqnarray}\n{\\rm ch}^{(R) k}(h=\\frac{k}{4},l;\\tau,0) \\rightarrow \na(k{\\rm -}1,l{\\rm -}\\frac{1}{2}) (-i \\tau)^{\\frac{1}{2}} \\exp \\left(\n\\frac{\\pi i}{12 \\tau} (3 + c_{k-1}) \\right) \\quad \n{\\rm as} \\ \\tau \\downarrow 0. \\label{eq:asymchar}\n\\end{eqnarray}\n \nAs for the character of massless\nrepresentation ${\\cal M}_{0 (l{\\rm =}0)}^{k}$,\nwe can obtain the upper bound of the asymptotic\nbehavior by means of eq.(\\ref{eq:relation}):\n\\begin{eqnarray}\n\\hspace{-1cm}{\\rm ch}_{0}^{(R) k}\n(h=\\frac{k}{4},l=0;\\tau,0)|_{\\tau \\downarrow 0}\n&\\leq& {\\rm ch}^{(R) k}(h=\\frac{k}{4},l=\\frac{1}{2};\\tau,0)|_{\\tau\n\\downarrow 0} \\nonumber \\\\\n\\hspace{-1cm}&=& a(k,l{\\rm =}0)(-i \\tau)^{\\frac{1}{2}}\\exp\n\\left( \\frac{\\pi i}{12 \\tau} (3 + c_{k-1}) \\right),\n\\end{eqnarray}\nwhere $f(\\tau)|_{\\tau \\downarrow 0}$ means the leading asymptotic of\n$f(\\tau)$ as $\\tau \\downarrow 0$.\nFrom this estimation, the dominant contribution of the \nasymptotic behavior of $Z[S^k K3](\\tau,1\/2)$ turns out to come\nfrom the part of the massive representations. \nWe can neglect the contribution of the massless representations \nin the asymptotics.\n\nNow we can obtain the asymptotic behavior of \n$\\sum_{l}F_{l}(\\tau)$.\nTaking the limit $\\tau \\downarrow 0$ in eq.(\\ref{eq:charexp}), \n\\begin{eqnarray}\nZ[S^k K3](\\tau,\\frac{1}{2}) \\rightarrow \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau)|_{\\tau\n\\downarrow 0} \\times (-i \\tau)^{\\frac{1}{2}}\\exp \\left( \\frac{\\pi i}{12 \n\\tau} (3+c_{k-1}) \\right) \\qquad {\\rm as} \\ \\tau\n\\downarrow 0, \\label{eq:asymf} \n\\end{eqnarray}\nwhere ${\\tilde F}_{l}(\\tau) = \\sum_{n=0}^{\\infty} {\\tilde\na}_{n}^{(l)}q^{n} = a(k{\\rm -1},l{\\rm -1\/2})F_{l}(\\tau)$.\nThe asymptotic behavior of \n$\\sum_{l}{\\tilde F}_{l}(\\tau)$\nas $\\tau \\downarrow 0$ can be read by comparing (\\ref{eq:asymf}) with\n(\\ref{eq:asymz}) \n\\begin{eqnarray}\n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau) \\rightarrow\n\\ c(k)(-i \\tau)^{- \\frac{1}{2}} \\exp \\left( \\frac{\\pi i (6 k -\n(3+c_{k-1}))}{12 \\tau} \\right) \\quad {\\rm as} \\ \\tau \\downarrow 0.\n\\end{eqnarray}\n\n$\\sum_{l}{\\tilde F}_{l}(\\tau)$ has the\nFourier expansion of the form\\footnote{These coefficients also satisfy \n$b_{n} \\leq b_{n+1}$.} \n\\begin{eqnarray}\n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde F}_{l}(\\tau) = \n\\sum_{n=0}^{\\infty} b_{n}q^n.\n\\qquad \\left( b_{n} = \\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde\na}_{n}^{(l)} \\right) \\label{eq:expf}\n\\end{eqnarray} \nDue to the Tauberian theorem, the asymptotic form of\n$b_{n}$ becomes:\n\\begin{eqnarray} \nb_{n} = \\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde a}_{n}^{(l)} \\sim\n\\frac{c(k)}{\\sqrt{2}}n^{- \\frac{1}{2}} \\exp \\left( 2 \\pi\n\\sqrt{(k-\\frac{(3+c_{k-1})}{6}) n} \\right) \\qquad {\\rm as} \\ n \\rightarrow\n\\infty. \\label{eq:numpri}\n\\end{eqnarray}\nTherefore, we conclude that the degeneracy of the massive\nprimary state of the dimension $h=n+k\/4$ \nat $n\\rightarrow\\infty$ is given by eq.(\\ref{eq:numpri}).\n\nAccording to the argument in section 3.2, the degeneracy of the \nprimary state (\\ref{eq:prisigma}) corresponds to the degeneracy \nof the microscopic states of the extremal BTZ black hole \n${\\rm BTZ}_{(N,N\/l)}$. At the limit $N\\gg k \\gg 1$\\footnote{The \ndifference between $\\sum_{l}{\\tilde a}_{n}^{(l)}$ and\n$\\sum_{l}a_{n}^{(l)}$ is \nirrelevant in this semiclassical limit.}, that is, the semiclassical limit of\nthree-dimensional gravity, the degeneracy of the state \n(\\ref{eq:extprimary}) becomes   \n\\begin{eqnarray} \n\\sum_{l=\\frac{1}{2}}^{\\frac{k}{2}} {\\tilde a}_{N}^{(l)} \\sim\n\\frac{c(k)}{\\sqrt{2}}N^{- \\frac{1}{2}} \\exp (2 \\pi\n\\sqrt{(k-1) N}) \\qquad {\\rm as} \\ N \\rightarrow \\infty, \\label{eq:microent}\n\\end{eqnarray}\nwhere $k=Q_{1}Q_{5}+1$.\nThe logarithm of eq.(\\ref{eq:microent}) can be regarded as\nthe microscopic entropy of the extremal BTZ black hole with $Ml=J=N$.\nIt becomes\n\\begin{eqnarray}\nS_{{\\rm micro}} = 2 \\pi \\sqrt{Q_{1}Q_{5}N} + {\\cal O}(\\log N, \\\n\\log c(k)).\n\\end{eqnarray} \nThis completely agrees with the entropy formula\neq.(\\ref{eq:extentropy}). \nThis provides a justification of the\nidentification of the extremal BTZ black hole states with\nthe primary states (\\ref{eq:extprimary}) of the N=(4,4) $\\sigma$-model.\n\n\\section{Discussion} \n\nUntil now, our study is limited to the case of the extremal BTZ black holes.\nThe non-extremal BTZ black holes can also appear as the near\nhorizon geometry of the non-BPS solitonic solutions in IIB supergravity.\nAccording to the duality, the non-extremal BTZ black\nholes may be also identified with the Virasoro primary states \n\\begin{eqnarray}\n   |h,l=0\\rangle \\otimes |{\\tilde h},{\\tilde l}=0\\rangle  \\ \\quad {\\rm with}\n   \\quad h, {\\tilde h} > \\frac{k}{4}, \\label{eq:nonextpri}\n\\end{eqnarray}\nin the corresponding N=(4,4) $\\sigma$-model.\n\nThese states are in the tensor product of the massive representations both in\nthe left and right moving sectors.\nSo we must consider not the elliptic genus but the full\npartition function of the N=(4,4) $\\sigma$-model for the counting of\nthe degeneracy of the states.\nIt is known that the full partition function, which depends on the moduli\nof $S^k K3$, has the contributions from the massive primary states of\n$h={\\bf Q}_{>0}+k\/4$ (${\\bf Q}$ : rational numbers)\\cite{Eguchi4}. \nTherefore, the patition\nfunction and the counterparts of $\\sum_{l}F_{l}(\\tau)$ can not have the\nforms (\\ref{eq:expgenus}) and (\\ref{eq:expf}). So the Tauberian\ntheorem can not be applied to the counting of the primary state \n(\\ref{eq:nonextpri}). \nWe need the further investigations\nfor the well-defined counting of the microscopic states of \nthe non-extremal BTZ black holes. \n\nThrough this paper, we have discussed only the case of $M_{4}=K3$. \nThe similar arguments in section 2.2 hold for        \nthe case of $M_{4}=T^4$. However the elliptic genus of the\ncorresponding N=(4,4) $\\sigma$-model vanishes identically, since\nthis $\\sigma$-model has the extra $U(1)^{4}$ symmetry other than\nN=4 superconformal symmetry. So one cannot count the degeneracy\nof the state (\\ref{eq:extprimary}) by means of the elliptic genus.\nIn \\cite{Maldacena3}, the counting of the 1\/8 BPS states has been\nargued by using another topological index called new supersymmetric\nindex. And they pointed out that the representations\nof large N=4 superconformal algebra must be considered.\nWe may expect that the similar argument in this paper can be carried out\nvia the relation between this new supersymmetric\nindex and the characters of large N=4 superconformal algebra.\n\n{\\bf Acknowledgements} \\\\\nThe authors thank H. Umetsu and D. Tomino \nfor their collaboration at the early stage of this work and \nalso for their useful discussions and comments. \nThe authors thank also T. Kawai for his useful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSuppose $X$ is a smooth\ncompact manifold and $\\varphi_t:X\\to X$ is an Anosov flow\ngenerated by a smooth vector field $ V $, $ \\varphi_t := \\exp t V $.\nCorrelation functions for a flow are defined as\n\\begin{equation}\n\\label{eq:corr}\n\\rho_{f,g} ( t ) := \\int_X f ( \\varphi_{-t} ( x ) ) g ( x ) dx , \\ \\\nf , g \\in C^\\infty ( X ) , \\ \\ t > 0 ,\n\\end{equation}\nwhere $ dx $ is a Lebesgue density on $ X $.\nThe power spectrum is defined as the (inverse) Fourier-Laplace transform\nof $ \\rho_{f,g} $:\n\\begin{equation}\n\\label{eq:powersp}\n\\widehat \\rho_{f,g} ( \\lambda ) := \\int_0^\\infty \\rho_{f,g} ( t ) e^{ i \\lambda t } dt,\n\\ \\ \\ \\Im \\lambda > 0 .\n\\end{equation}\nFaure--Sj\\\"ostrand \\cite{FaSj} proved that\n\\[   ( P - \\lambda)^{-1}  : C^\\infty ( X )\n\\to {\\mathcal D}' ( X ) , \\ \\ P := \\frac 1 i V , \\ \\ \\Im \\lambda \\gg 1 , \\]\ncontinues to a meromorphic family of operators on all of $ \\mathbb C $. Using the fact\nthat $ f ( \\varphi_{-t} ( x ) ) = [\\exp ( - i t P ) f ] ( x ) $ this easily shows\nthat $ \\widehat \\rho_{f,g} ( \\lambda ) $ has a meromorphic continuation.\nThe poles of this continuation depend only on $ P $ and their study\nwas initiated in the work of Ruelle \\cite{Rue} and Pollicott \\cite{Po}.\nThey are called {\\em Pollicott--Ruelle resonances} and their set is denoted\nby $ \\Res ( P )$. The finer properties of the correlations are then\nrelated to the distribution of these resonances. This is particularly\nclear in the work of Liverani \\cite{Liv} and Tsujii \\cite{Ts} on contact\nAnosov flows, see also Nonnenmacher--Zworski \\cite{NZ} for semiclassical generalizations.\n\nAn equivalent definition of Pollicott--Ruelle resonances was given\nby Dyatlov--Zworski \\cite{DZ2}: they are limits (with multiplicities)\nof the eigenvalues of $ P + i \\epsilon \\Delta_g $, $ \\epsilon \\to 0 + $,\nwhere $ - \\Delta_g \\geq 0 $ is a Laplacian for some { Riemannian metric $g$} on $ X $.\nBecause of a connection to Brownian motion this shows stochastic stability of these resonances.\n\nIn this note we address the basic question about the size of the set of\nresonances: is their number always infinite? Despite the long\ntradition of the subject this appeared to be unknown for arbitrary\nAnosov flows on compact manifolds.\nIn Theorem \\ref{thm2}, we show that\nin sufficiently large strips the counting function of resonances cannot\nbe bounded by $r^\\delta$, $ \\delta < 1 $.\n\n\\begin{figure}\n\\includegraphics[width=6in]{gaps}\n\\caption{The {\\em spectral gap} $ \\nu_0 $ is the supremum of $ \\nu $\nsuch that there are {\\em no} resonances with $ - \\nu < \\Im \\lambda $,\n$ \\lambda \\neq 0 $.\nFor contact Anosov flows it is known that $ \\nu_0 > 0 $ \\cite{Liv},\\cite{NZ},\\cite{Ts}.\nThe {\\em essential spectral gap}, $  \\nu_1 $, is the supremum of $ \\nu $\nsuch that there are only finitely many resonances with $ \\Im \\lambda > - \\nu $.\nOur result states that the essential spectral gap is finite for any\nAnosov flow on a compact manifold.}\n\\end{figure}\n\n{General upper bounds\non the number of resonances in strips were established by\nFaure--Sj\\\"ostrand \\cite{FaSj} (and with a sharper \nexponent in the case of contact flows by Datchev--Dyatlov--Zworski \\cite{DDZ}):\nfor any $A > 0 $ there exists $ C $ such that\n\\begin{equation}\n\\label{eq:counting}\n\\#(\\Res(P)\\cap\\{\\Im\\mu>-A, |\\Re\\mu - r |\\leq \\sqrt r\\})\n\\leq Cr^{n- \\frac12} .\n\\end{equation}}\nOn the other hand, for contact Anosov flows satisfying certain pinching conditions on Lyapunov exponents, Faure--Tsujii \\cite{FaTs} showed that\nthe resonances satisfy a precise counting law in strips, agreeing with\nthe upper bound of \\cite{DDZ}. That is a far reaching generalization of the results known in constant curvature: see Dyatlov--Faure--Guillarmou \\cite{DFG} for recent results in that case and references.\n\nThe new counting result is proved by\nestablishing a local trace formula relating resonances to periods of\nclosed trajectories and to the their Poincar\\'e maps. Hence we denote\nby  $\\mathcal{G}$ periodic orbits $\\gamma$ of the flow, by  $T_\\gamma$ the period of $\\gamma$ and by $T_\\gamma^\\#$ the primitive period. We let $\\mathcal{P}_\\gamma$ be the linearized Poincar\\'{e} map -- see \\S \\ref{pr}.\nWith this notation we can state our {\\em local trace formula}:\n\\begin{thm}\n\\label{thm1}\nFor any $A>0$ there exists\na distribution $F_A\\in\\mathcal {S}'(\\mathbb{R})$ supported in $[0,\\infty)$ such that\n\\begin{equation}\n\\label{localtrace}\n\\sum_{\\mu\\in\\Res(P), \\Im\\mu>-A}e^{-i\\mu t}+F_A(t)\n=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^{\\#}\\delta(t-T_\\gamma)}\n{|\\det(I-\\mathcal{P}_\\gamma)|},\\;\\;\\;\\; t>0\n\\end{equation}\nin the sense of distribution on $(0,\\infty)$.\nMoreover, the Fourier transform of $ F_A $ has an analytic extension\nto $\\Im\\lambda <  A$ which satisfies,\n\\begin{equation}\n\\label{error}\n|\\widehat{F}_A(\\lambda) |=\\mathcal{O}_{A,\\epsilon}( \\langle \\lambda \\rangle^{2n+1}), \\ \\\n\\Im\\lambda < A - \\epsilon, \\text{ for any $\\epsilon>0$.}\n\\end{equation}\n\\end{thm}\n\nThe trace formula \\eqref{localtrace} can be motivated as follows.\nFor the case of geodesic flows of compact Riemann surfaces,\n$X=S^*(\\Gamma\\backslash\\mathbb{H}^2)$, where $ \\Gamma $ is\nco-compact subgroup of $ {\\rm{SL}}_2 ( \\mathbb R) $,\nand $\\varphi_t$ is the geodesic flow, we have a global trace formula:\n\\begin{equation}\n\\label{globaltrace}\n\\sum_{\\mu\\in\\Res(P)}e^{-i\\mu t}=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|},\\;\\;\\; t>0.\n\\end{equation}\nHere the set of resonances is given by\n$\\Res(P)=\\{\\mu_{j,k}=\\lambda_j-i(k+\\frac{1}{2}),j,k\\in\\mathbb{N}\\}$ (up to exceptional values on the imaginary axis),\nwhere $\\lambda_j$'s are the eigenvalues\nof the Laplacian on $ \\Gamma \\backslash \\mathbb H^2 $.  This follows\nfrom the Atiyah--Bott--Guillemin trace formula and\nthe Selberg trace formula -- see \\cite{DFG} and references given there.\n{  The bound $ \\langle \\lambda \\rangle^{ 2 n +1 } $ in \n\\eqref{error} is probably not\noptimal and comes from very general estimates presented in \\S \\ref{estflt}.\nIt is possible that \\eqref{globaltrace} is valid for all Anosov flows.}\n\n\n\nMelrose's Poisson formula for\nresonances valid for Euclidean infinities \\cite{Me,SZ,Z1} and some\nhyperbolic infinities \\cite{GZ} suggests that \\eqref{globaltrace}\ncould be valid for general Anosov flows but that is unknown.\n\nIn general, the validity of \\eqref{globaltrace} follows from, but is not equivalent to, the finite order (as an entire function) of the analytic continuation of\n\\begin{equation}\n\\label{zeta}\n\\zeta_1(\\lambda) { :=} \\exp\\left(-\\sum_\\gamma\\frac{T_\\gamma^\\#e^{i\\lambda T_\\gamma}}{T_\\gamma|\\det(I-\\mathcal{P}_\\gamma)|}\\right).\n\\end{equation}\nThis finite order property is only known under certain analyticity assumptions on $X$ and $\\varphi_t$ -- see Rugh \\cite{Ru} and Fried \\cite{Fr}. {  \nThe notation $ \\zeta_1 $ is motivated by the factorization of the Ruelle\nzeta function -- see \\cite[(2.5)]{DZ}.}\n\nAs a consequence of the local trace formula \\eqref{localtrace},\nwe have the following weak lower bound on the number of resonances in a sufficiently wide strip near the real axis. It is formulated using the Hardy-Littlewood notation: $f=\\Omega(g)$ if it is\nnot true that $|f|=o(|g|)$.\n\n\\begin{thm}\n\\label{thm2}\nFor every $\\delta\\in(0,1)$ there exists a constant $A_\\delta>0$ such that if $A>A_\\delta$, then\n\\begin{equation}\n\\label{lowerbound}\n\\#(\\Res(P)\\cap\\{\\mu \\in \\mathbb C \\, : \\, |\\mu|\\leq r, \\ \\Im\\mu>-A\\})=\\Omega(r^\\delta).\n\\end{equation}\nIn particular, there are infinitely many resonances in any strip $\\Im\\mu>-A$ for $A$ sufficiently large. \n\\end{thm}\n\n\\medskip\n\\noindent{\\bf Remarks.} 1. { An explicit bound for the constant $A_\\delta$ is given by \\eqref{adelta} in the proof. This also gives an explicit bound $A_0=\\inf\\{A_\\delta:0<\\delta<1\\}$ for the essential spectral gap.} In the case of analytic semiflows (see \\cite{Naud1}) Fr\\'ederic Naud \\cite{Naud} pointed out that a better estimate of the essential spectral gap is possible: there are infinitely many resonances in any strip\n$ \\Im \\lambda > - \\frac32 P ( 2 ) - \\epsilon $, where \n{  $ P  (s ) := P ( s \\psi^u ) $ is the\ntopological pressure associated to the unstable Jacobian -- see \\eqref{eq:press} and \\eqref{eq:psiu}}.\nIn Appendix \\ref{weakmix}, Fr\\'ed\\'eric Naud shows how similar methods and Theorem \\ref{thm1} give\na narrower strip with infinitely many resonances for weakly mixing\nAnosov flows.\n\n\\noindent\n2. In the case of flows obtained by suspending Anosov maps the \ngrowth of the number of resonances in strips is linear -- see\nAppendix \\ref{suspe} by Fr\\'ederic Naud \nfor a detailed discussion of analytic perturbations of linear \nmaps. That means that the exponent $ \\delta $ close to one is\nclose to be optimal in general.\n\n\\medskip\n\n\nThe proof of Theorem \\ref{thm1} uses the microlocal approach to\nAnosov dynamics due to Faure--Sj\\\"ostrand \\cite{FaSj} and\nDyatlov--Zworski \\cite{DZ}. In particular we use the fact that\n\\begin{equation*}\n\\frac{d}{d\\lambda} \\log \\zeta_1 ( \\lambda ) = \\tr^\\flat e^{ i \\lambda t_0 } \\varphi_{-t_0}^* ( P - \\lambda)^{-1} ,\n\\end{equation*}\nand that the right hand side continues meromorphically with poles\nwith integral residues. Here the flat trace, $ \\tr^\\flat $,  is defined using a formal integration over the diagonal,  see \\S \\ref{flat},  with the justification provided by the crucial wave front set relation, see \\S \\ref{wavefront}. Some of the techniques are also related to the proof of Sj\\\"{o}strand's local trace formula for scattering resonances in the semiclassical limit \\cite{S}. It is possible that an alternative\nproof of Theorem \\ref{thm1} could be obtained using the methods of Giulietti--Liverani--Pollicott \\cite{GLP} employed in their proof of Smale's conjecture about zeta function (\\cite{DZ} provided a simple microlocal proof of that conjecture).\n\nThe proof of Theorem \\ref{thm2} is based on the proof of a similar\nresult in Guillop\\'{e}--Zworski \\cite{GZ} which in turn was inspired\nby the work of Ikawa \\cite{Ik} on existence of resonances in\nscattering by several convex bodies.\n\n\\def\\smallsection#1{\\smallskip\\noindent\\textbf{#1}.}\n\n\\smallsection{Acknowledgements}\nWe would like to thank Semyon Dyatlov for helpful discussions and in particular for suggesting the decomposition \\eqref{decomposition} which simplified the wave front arguments. We are also grateful to Fr\\'ed\\'eric Naud for sharing his\nunpublished work \\cite{Naud} with us and to the anonymous referee for\nuseful suggestions. This material is based\nupon work supported by\nthe National Science Foundation under the grant and DMS-1201417.\n\n\\smallsection{Notation} We use the following notational\nconventions: $ \\langle x \\rangle := ( 1 + |x|^2 )^{\\frac12} $,\n$ \\langle u , \\varphi \\rangle$, for the the distributional\npairing of $ u \\in \\mathcal D' ( X ) $ (distributions on a compact\nmanifold $ X $), and $ \\langle u , v \\rangle_{ H}$ for\nthe Hilbert space inner product on $ H $.\nWe write $ f =  \\mathcal O_\\ell ( g)_B $ to mean that\n$ \\|f \\|_B  \\leq C_\\ell  g $ where the norm (or any seminorm) is in the\nspace $ B$, and the constant $ C_\\ell  $ depends on $ \\ell $. When either $ \\ell $ or $ B $ are absent then the constant is universal or the estimate is scalar, respectively. When $ G = \\mathcal O_\\ell ( g )_{B_1\\to B_2 } $ then the operator $ B : H_1  \\to H_2 $ has its norm bounded by $ C_\\ell g $. By $ \\neigh_U ( \\rho ) $\nwe mean a (small) neighbourhood of $ \\rho$ in the space $ U$.\nWe refer to \\cite{DZ} and \\cite{Z2} for the notational conventions from microlocal\/semiclassical analysis as they appear in the text.\n\n\\section{Preliminaries}\n\\label{pr}\n\\subsection{Anosov flows}\nLet $X$ be a compact Riemannian manifold, $V\\in C^\\infty(X;TX)$ be\na smooth non vanishing vector field and\nand $\\varphi_t=\\exp tV:X\\to X$ the corresponding flow.\n\nThe flow is called an {\\em Anosov flow}\nif the tangent space to $X$ has a continuous decomposition\n$T_xX=E_0(x)\\oplus E_s(x)\\oplus E_u(x)$ which is invariant under the flow: $d\\varphi_t(x)E_\\bullet(X)=E_\\bullet(\\varphi_t(X))$,\n$\\bullet=s,u$, $E_0(x)=\\mathbb{R}V(x)$,\nand satisfies\n\\begin{equation*}\n\\begin{split}\n|d\\varphi_t(x)v|_{\\varphi_t(x)}\\leq Ce^{-\\theta|t|}|v|_x,&\\;\\;\\; v\\in E_u(x), \\ \\ t<0\\\\\n|d\\varphi_t(x)v|_{\\varphi_t(x)}\\leq Ce^{-\\theta|t|}|v|_x,&\\;\\;\\; v\\in E_s(x),\n\\ \\ t>0,\n\\end{split}\n\\end{equation*}\nfor some fixed $C$ and $\\theta>0$.\n\n\\subsection{Anisotropic Sobolev spaces}\nLet us put $P=-iV:C^\\infty(X)\\to C^\\infty(X)$;\nthen the principal symbol of $ P $, $p\\in S^1(T^*X)$ (see\n\\cite[\\S 18.1]{H3} or \\cite[\\S 14.2]{Z2} for this standard notation; an\noverview of semiclassical and microlocal preliminaries needed in this\npaper can be found in \\cite[\\S 2.3]{DZ}) is given by\n$p(x,\\xi)=\\xi(V(x))$ which is homogeneous of degree 1.\nThe Hamilton flow of $ p $ is\nthe symplectic lift of $ \\varphi_t $ to the cotangent bundle:\n$e^{tH_p}(x,\\xi)=(\\varphi_t(x),({}^Td\\varphi_t(x))^{-1}\\xi)$.\nWe can define the dual decomposition $T^\\ast_xX=E^\\ast_0(x)\\oplus E^\\ast_s(x)\\oplus E^\\ast_u(x)$ where $E^\\ast_0(x),E^\\ast_s(x),E^\\ast_u(x)$ are dual to $E_0(x),E_u(x),E_s(x)$, respectively. Then\n\\begin{equation*}\n\\begin{split}\n\\xi\\not\\in E^\\ast_0(x)\\oplus E^\\ast_s(x)\\Rightarrow d(\\kappa(e^{tH_p}(x,\\xi)),\\kappa(E^\\ast_u))\\to0 \\text{ as } t\\to+\\infty\\\\\n\\xi\\not\\in E^\\ast_0(x)\\oplus E^\\ast_u(x)\\Rightarrow d(\\kappa(e^{tH_p}(x,\\xi)),\\kappa(E^\\ast_s))\\to0 \\text{ as } t\\to-\\infty.\n\\end{split}\n\\end{equation*}\nHere $\\kappa:T^\\ast X\\setminus0\\to S^\\ast X := T^*X \/ \\mathbb R_+ $\nis the natural projection.\n\nA microlocal version of anisotropic Sobolev spaces of\nBlank--Keller--Liverani~\\cite{BKL}, Baladi--Tsujii~\\cite{BT}\nand other authors was provided by Faure-Sj\\\"{o}strand \\cite{FaSj}.\nHere we used a simplified version from Dyatlov-Zworski \\cite{DZ}. For that\nwe construct a function $m_G\\in C^\\infty(T^\\ast X\\setminus0;[-1,1])$ which is homogeneous of degree 0, is supported in a small neighbourhood of $E_s^\\ast\\cup E_u^\\ast$ and satisfies\n\\begin{equation*}\nm_G=1 \\text{ near } E_s^\\ast;\\;\\;\\  m_G=-1 \\text{ near } E_u^\\ast;\\;\\;\\ H_pm_G\\leq0 \\text{ everywhere. }\n\\end{equation*}\nNext, we choose a pseudodifferential operator\n$ G\\in\\Psi^{0+}(X)$, {  $ \\sigma(G)=m_G(x,\\xi)\\log\\langle\\xi\\rangle $}.\nThen for $ s > 0 $, $\\exp(\\pm sG)\\in\\Psi^{s+}(X)$ -- see \\cite[\\S 8.2]{Z2}. The anisotropic Sobolev spaces are defined as\n\\begin{equation*}\nH_{sG}:=\\exp(-sG)L^2(X), \\ \\ \\ \\|u\\|_{H_{sG}}: =\\|\\exp(sG)u\\|_{L^2}.\n\\end{equation*}\nBy the construction of $G$, we have $H^s\\subset H_{sG}\\subset H^{-s}$.\n\n\\subsection{Properties of Resolvent}\nWe quote the following results about the resolvent of $P$, see \\cite[Propositions 3.1, 3.2]{DZ}:\n\\begin{lem}\nFix a constant $C_0>0$. Then for $s>0$ large enough depending on $C_0$, $P-\\lambda:D_{sG}\\to H_{sG}$ is a Fredholm operator of index 0 in the region $\\{\\Im\\lambda>-C_0\\}$. Here the domain $D_{sG}$ of $P$ is the set of $u\\in H_{sG}$ such that $Pu$ (in the distribution sense) is in $H_{sG}$ and it is a Hilbert space with norm $\\|u\\|_{D_{sG}}^2=\\|u\\|_{H_{sG}}^2+\\|Pu\\|_{H_{sG}}^2$.\n\\end{lem}\n\n\\begin{lem}\nLet $s>0$ be fixed as above. Then there exists a constant $C_1$ depending on $s$, such that for $\\Im\\lambda>C_1$, the operator $P-\\lambda:D_{sG}\\to H_{sG}$ is invertible and\n\\begin{equation}\n\\label{resolvent}\n(P-\\lambda)^{-1}=i\\int_0^\\infty e^{i\\lambda t}\\varphi_{-t}^\\ast dt,\n\\end{equation}\nwhere $\\varphi_{-t}^\\ast=e^{-itP}$ is the pull back operator by $\\varphi_t$. The integral converges in operator norm $H^s\\to H^s$ and $H^{-s}\\to H^{-s}$.\n\\end{lem}\n\nThe analytic Fredholm theory now shows that the resolvent $ \\lambda\n\\mapsto R(\\lambda)=(P-\\lambda)^{-1}:H_{sG}\\to H_{sG}$ forms a meromorphic family of operators with poles of finite rank. In the region $\\Im\\lambda>-C_0$, the Ruelle-Pollicott resonances are defined as the poles of $R(\\lambda)$. They can be described as the meromorphic continuation of the Schwartz kernel of the operator on the right-hand side, thus are independent of the choice of $s$ and the weight $G$.\nThe mapping properties of $ ( P - \\lambda)^{-1}$ and formula \\eqref{resolvent}\nshow that the power spectrum \\eqref{eq:powersp} has a meromorphic\ncontinuation with the same poles. We note here that our definition\n\\eqref{eq:powersp} is different from the definition in \\cite{Rue} but\nthe formula there can be expressed in terms of \\eqref{eq:powersp}.\n\n{We recall the following general upper bounds on the number of resonances from \nFaure--Sj\\\"ostrand \\cite{FaSj}:\n\\begin{prop}\nLet $\\Res(P)$ be the set of Ruelle-Pollicott resonances. Then for any $C_0>0$,\n\\begin{equation}\n\\label{hupperbound}\n\\#(h\\Res(P))\\cap D(1,C_0 h^{\\frac12} )=\\mathcal{O}(h^{-n+ \\frac12}),\n\\end{equation}\nwhich is equivalent to \\eqref{eq:counting}. In particular,\n\\begin{equation}\n\\label{upperbound}\n\\#\\Res(P)\\cap\\{\\mu:|\\Re\\mu|\\leq r,\\Im\\mu>-C_0\\}=\\mathcal{O}(r^n). \n\\end{equation}\n\\end{prop}}\n\n\\subsection{Complex absorbing potentials}\nIt is convenient to introduce a semiclassical parameter $h$ and to consider the operator $hP\\in\\Psi_h^1(X)$ (for the definitions of pseudodifferential\noperators and wave front sets we\nrefer to \\cite[\\S 14.2]{Z2} and  \\cite[\\S 2.3, Appendix C]{DZ})\n with semiclassical principal symbol $p=\\sigma_h(hP)(x,\\xi)=\\xi(V_x)$.\nThen we introduce a semiclassical adaption $G(h)\\in\\Psi_h^{0+}(X)$ of the operator $G$ with\n\\begin{equation*}\n\\sigma_h(G(h))=(1-\\chi(x,\\xi))m_G(x,\\xi)\\log|\\xi|,\n\\end{equation*}\nwhere $\\chi\\in C_0^\\infty(T^\\ast X)$ is equal to 1 near the zero section. In this way, $H_{sG(h)}=H_{sG}$ but with a new norm depending on $ h $. We also\ndefine an $h$-dependent norm on the domain of $hP$,  $D_{sG(h)} = D_{s G} $:\n\\begin{equation*}\n\\|u\\|_{D_{sG(h)}} :=\\|u\\|_{H_{sG(h)}}+\\|hPu\\|_{H_{sG(h)}}.\n\\end{equation*}\n\nNow we modify $hP$ by adding a semiclassical pseudodifferential complex absorbing potential $-iQ_\\delta\\in\\Psi_h^0(X)$ which is localized to a neighbourhood of the zero section:\n\\begin{equation}\n\\label{eq:WFQd}\n\\WF_h(Q_\\delta)\\subset\\{|\\xi|<\\delta\\}, \\ \\ \\sigma_h(Q_\\delta)>0 \\text{ on } \\{|\\xi|\\leq\\delta\/2\\}, \\ \\ \\sigma_h(Q_\\delta)\\geqslant0 \\text{ everywhere}.\n\\end{equation}\n{  (For the definition of $ \\WF_h ( A ) \\subset \\overline T^* X $ and\nof the compactified cotangent bundle $ \\overline T^* X $, \nsee \\cite[\\S C.2]{DZ}.)}\nInstead of $P_h(z)=hP-z$, we consider the operator $P^\\delta_h(z)=hP-iQ_\\delta-z$ acting on $H_{sG(h)}$ which is equivalent to the conjugated operator\n\\begin{equation}\n\\label{eq:conj}\nP^{\\delta,s}_h(z)=e^{sG(h)}P_\\delta(z)e^{-sG(h)}=P^\\delta_h (z)+s[G(h),hP]+\\mathcal{O}(h^2)_{\\Psi_h^{-1+}}\n\\end{equation}\nacting on $L^2$. We recall the crucial \\cite[Proposition 3.4]{DZ}:\n\\begin{lem}\n\\label{l:3.4}\nFix a constant $C_0>0$ and $\\delta >0$. Then for $s>0$ large enough depending on $C_0$ and $h$ small enough, the operator\n\\begin{equation*}\nP^\\delta_h(z):D_{sG(h)}\\to H_{sG(h)}, \\ \\ \\ -C_0h\\leq\\Im z\\leq1, \\ \\\n |\\Re z|\\leq 2h^{1\/2},\n\\end{equation*}\nis invertible, and the inverse $R^\\delta_h(z)$, satisfies\n$\\|R^\\delta_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}$.\n\\end{lem}\n\n\\subsection{Finite rank approximation}\n\\label{fra}\nFor our application we need to make $ Q_\\delta $ a finite rank\noperator.\nIt is also convenient to make a further assumption on the\nsymbol of $ Q_\\delta$.\nAs long as \\eqref{eq:WFQd} holds,\nLemma \\ref{l:3.4} still applies.\n\nFrom now on we fix $ \\delta > 0 $ and put\n\\[  Q = Q_\\delta = f ( - h^2 \\Delta_g ), \\ \\ f \\in  {C^\\infty_{\\rm{c}}} ( ( -2 \\delta, 2 \\delta ),\n[ 0 , 1 ]) , \\ \\\nf ( s ) = 1, \\ \\ |s |\\leq \\delta . \\]\nThen (see for instance \\cite[Theorem 14.9]{Z2})\n\\begin{equation}\n\\label{finiterank}\n\\rank Q=\\mathcal O (h^{-n}),  \\ \\ Q \\geq 0 , \\ \\ \\sigma_h ( Q ) = f ( |\\xi|_g^2 ).\n\\end{equation}\nFor technical convenience only (so that we can\ncite easily available results in the proof of Proposition \\ref{flattracees}\nin the appendix)\nwe make an additional assumption on $ f$: for some $ 0 < \\alpha < \\frac12$,\n\\begin{equation}\n\\label{eq:condfk}\n| f^{(k)} ( x ) | \\leq C_k f ( x ) ^{1- \\alpha} .\n\\end{equation}\nThis can be achieved by building $ f$ from functions of the form equal to $\ne^{-1\/x} $ for $ x > 0 $ and $ 0 $ for $ x \\leq 0 $. (In that case\n\\eqref{eq:condfk} holds for all $ \\alpha > 0 $.)\n\n\nLemma \\ref{l:3.4} shows that\nfor  $-C_0h\\leq\\Im z\\leq 1\n$, $|\\Re z|\\leq 2h^{1\/2}$, \n\\begin{equation}\n\\label{eq:wideP} \n\\widetilde{P}_h(z): =hP-iQ-z, \n\\end{equation} \nis also invertible and its inverse $\\widetilde{R}_h(z)$ satisfies\n\\begin{equation}\n\\label{modifiedresolvent}\n\\|\\widetilde{R}_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}.\n\\end{equation}\nIn the upper half plane we have the following estimate on the original resolvent:\n\\begin{equation}\n\\label{eq:orres}\n \\|R_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\leq Ch^{-1}, \\ \\ \\\nC_1h\\leq\\Im z\\leq1, \\ \\ |\\Re z|\\leq 2h^{1\/2},\n\\end{equation}\nprovided that $ C_1 $ is large enough. This follows from the\nFredholm property and the estimate $ \\Im \\langle e^{ s G ( h ) } P_h ( z )\ne^{ -s G ( h ) } u , u \\rangle_{L^2 }\n\\geq h \\| u\\|_{L^2} $, $ \\Im z > C_1 h $ -- see \\eqref{eq:conj}.\n\n\\subsection{Wavefront set condition}\n\\label{wavefront}\nWe need to study the wavefront set and semiclassical wavefront set of $R_h(z)$ and $\\widetilde{R}_h(z)$. For the definitions and notations of the wavefront sets and the semiclassical wavefront sets, we refer to \\cite[Chapter VIII]{H}, \\cite[Section 8.4]{Z2} and \\cite[Appendix C]{DZ} and \\cite{A}.\n\nWe recall the following wavefront set condition and semiclassical wavefront set conditions for the resolvent $R(\\lambda)$ and $\\widetilde{R}_h(z)$ from \\cite[Proposition 3.3]{DZ}. {  (For the definition of the standard wave front set $ \\WF $\nsee \\cite[\\S C.1]{DZ} and for the definition of the twisted \nwave front set $ \\WF' $, \\cite[(C.2)]{DZ} -- the reason for the twist\nis to have $ \\WF ( I ) $ equal to the diagonal in $ T^*X \\times T^* X $.)}\n\\begin{prop}\n\\label{p:3.3}\nLet $C_0$ and $s$ be as above and assume $\\lambda$ is not a resonance with $\\Im\\lambda>-C_0$, then\n\\begin{equation}\n\\label{wavefrontset}\n\\WF'(R(\\lambda))\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast),\n\\end{equation}\nwhere $\\Delta(T^\\ast X)$ is the diagonal in $T^\\ast X$ and $\\Omega_+$ is the positive flow-out of $e^{tH_p}$ on $\\{p=0\\}$:\n\\begin{equation}\n\\label{eq:Omegapl}\n\\Omega_+=\\{(e^{tH_p}(x,\\xi),x,\\xi) \\, : \\, t\\geqslant0, \\ p(x,\\xi)=0\\}.\n\\end{equation}\nAlso, if $R_h(z)=h^{-1}R(z\/h)$, then\n\\begin{equation}\n\\label{semiwf1}\n\\WF_h'(R_h(z))\\cap T^\\ast(X\\times X)\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast),\n\\end{equation}\nand\n\\begin{equation}\n\\WF_h'(R_h(z))\\cap S^\\ast(X\\times X)\\subset\\kappa(\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}).\n\\end{equation}\n\\end{prop}\n\nNow we determine the wavefront set and the semiclassical wavefront set of $\\widetilde{R}_h(z)$. First, \n{  by inserting the resolvent formula\n$$\\widetilde{R}_h(z)=R_h(z)+iR_h(z)Q\\tilde{R}_h(z)$$\ninto another resolvent formula\n$$\\widetilde{R}_h(z)=R_h(z)+i\\tilde{R}_h(z)QR_h(z),$$}\nwe write\n$$\\widetilde{R}_h(z) = R_h ( z ) + i R_h ( z ) Q R_h ( z ) -\nR_h ( z ) Q \\widetilde{R}_h ( z ) Q R_h ( z ).$$\nThen since $Q$ is a smoothing operator, $\\WF(Q)=\\emptyset$, we have\n$$\\WF'(R_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast.$$\nSimilarly, since $Q\\widetilde{R}_h(z)Q$ is also a smoothing operator,\n$$\\WF'(R_h(z)Q\\widetilde{R}_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast.$$\nTherefore we get the same wavefront set condition as $R_h(z)$:\n\\begin{equation}\n\\WF'(\\widetilde{R}_h(z))\\subset\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast).\n\\end{equation}\n\nFor the semiclassical wavefront set, we already know from \\cite[Proposition 3.4]{DZ} that\n\\begin{equation}\n\\label{eq:WFRh} \\WF_h'(\\widetilde{R}_h(z))\\cap T^\\ast(X\\times X)\\subset\\Delta(T^\\ast X)\\cup\\Omega_+.\n\\end{equation}\nMoreover, since $\\WF_h'(Q)\\cap S^\\ast(X\\times X)=\\emptyset$, we have\n$$\\WF_h'(R_h(z)QR_h(z))\\subset E_u^\\ast\\times E_s^\\ast,$$\nand similarly, $\\WF_h'(Q\\widetilde{R}_h(z)Q)\\cap S^\\ast(X\\times X)=\\emptyset$. Therefore\n\\begin{equation}\n\\label{eq:WFhR}\n\\WF_h'(\\widetilde{R}_h(z))\\cap S^\\ast(X\\times X)\\subset\n\\kappa(\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}).\n\\end{equation}\n\n\\subsection{Flat trace}\n\\label{flat}\nConsider an operator $B:C^\\infty(X)\\to\\mathcal{D}'(X)$ with\n\\begin{equation}\\label{ftc}\n\\WF'(B)\\cap\\Delta(T^\\ast X)=\\emptyset.\n\\end{equation}\nThen we can define the flat trace of $B$ as\n\\begin{equation}\n\\label{eq:flat}\n\\tr^\\flat B=\\int_X(\\iota^\\ast K_B)(x)dx:=\\langle\\iota^\\ast K_B,1\\rangle\n\\end{equation}\nwhere $\\iota:x\\mapsto (x,x)$ is the diagonal map, $K_B$ is the Schwartz kernel of $X$ with respect to the density $dx$ on $X$. The pull back $\\iota^\\ast K_B\\in\\mathcal{D}'(X)$ is well-defined under the condition \\eqref{ftc} (see \\cite[Section 8.2]{H}).\n\n\\subsection{Dynamical zeta function and Guillemin's trace formula}\nThe zeta function $ \\zeta_1 $ defined in \\eqref{zeta}\nis closely related to the Ruelle zeta function -- see \\cite{GLP},\\cite{DZ}\nand references given there.\nThe right hand side in \\eqref{zeta} converges for $\\Im\\lambda>C_1$ and\nit continues analytically to the entire plane.\n The Pollicott-Ruelle resonances are exactly the zeros of $ \\zeta_1 $.\n  We recall the (Atiyah--Bott--)Guillemin's trace formula \\cite{Gu} (see \\cite[Appendix B]{DZ} for a proof):\n\\begin{equation}\n\\label{Guillemin}\n\\tr^\\flat e^{-itP}=\\sum_{\\gamma\\in\\mathcal{G}}\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}, \\ \\ \\   t>0.\n\\end{equation}\nTherefore we have\n\\begin{equation*}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)=\\frac{1}{i}\\sum_\\gamma\\frac{T_\\gamma^\\# e^{i\\lambda T_\\gamma}}{|\\det(I-\\mathcal{P}_\\gamma)|}\n=\\frac{1}{i}\\int_0^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt.\n\\end{equation*}\nFrom \\eqref{Guillemin}, $\\tr^\\flat e^{-itP}=0$ on $(0,t_0)$ if $t_0<\\inf\\{T_\\gamma:\\gamma\\in\\mathcal{G}\\}$. Formally, we can write (see \\cite[\\S 4]{DZ} for the\njustification)\n\\begin{equation*}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)\n=\\frac{1}{i}\\int_{t_0}^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt\n=\\tr^\\flat\\left(\\frac{1}{i}e^{-it_0(P-\\lambda)}\n\\int_0^\\infty e^{it\\lambda}e^{-itP}dt\\right).\n\\end{equation*}\nTherefore by \\eqref{resolvent} we have\n\\begin{equation}\n\\label{zetaresolvent}\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)=\\tr^\\flat(e^{-it_0(P-\\lambda)}(P-\\lambda)^{-1}).\n\\end{equation}\nThe wavefront set condition \\eqref{wavefrontset} shows that\n\\begin{equation*}\n\\begin{split}\n& \\WF'(e^{-it_0(P-\\lambda)}(P-\\lambda)^{-1})\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\}\n\\end{split}\n\\end{equation*}\nwhich does not intersect $\\Delta(T^\\ast X)$. This justifies taking the\nflat trace \\eqref{eq:flat}.\n\nTherefore $\\frac{d}{d\\lambda} \\log\\zeta_1$ has a meromorphic continuation to all of $\\mathbb{C}$ with simple poles and\npositive integral residues. That is equivalent to having a holomorphic\ncontinuation of $ \\zeta_1 $. This strategy for proving Smale's conjecture\non the meromorphy of Ruelle zeta functions is the starting point of\nour proof of the local trace formula.\n\n\\section{Estimates on flat traces}\n\\label{estflt}\n\nThe key step in the proof of the trace formula is the following estimate on the flat trace of the propagated resolvent.\n\n\\begin{prop}\n\\label{flattracees}\nLet $\\widetilde{P}_h(z)$ and $\\widetilde{R}_h(z)$ {  be given by \n\\eqref{eq:wideP} and \\eqref{modifiedresolvent},} and let $t_0\\in(0,\\inf T_\\gamma)$. Then\n\\begin{equation}\n\\label{eq:Tofz}  T(z ) := \\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)),\n\\end{equation}\nis well defined and holomorphic in $ z $ when $ - C_0 h \\leq \\Im z \\leq  1, $\n$ |\\Re z | \\leq C_1 h^{\\frac12} $. Moreover, in that range of $ z $,\n\\begin{equation}\n\\label{eq:flattr1}\n T ( z ) = \\mathcal O_{C_0, C_1 } (h^{-2n-1}).\n\\end{equation}\n\\end{prop}\n\nThe proof is based on a quantitative study of the proof of \\cite[Theorem 8.2.4]{H} and on the wave front properties established\nin \\cite[\\S 3.3,3.4]{DZ}. The general idea is the following: the wave\nfront set condition shows that the trace is well defined. The analysis\nbased on the properties of the semiclassical wave front set shows more:\nthe contribution from a microlocal neighbourhood of fiber infinity is $ \\mathcal O ( h^\\infty ) $. The contribution away from fiber infinity can be controlled\nusing the norm estimate on $ \\widetilde R_h ( z ) $. Since the weights\ndefining the $ H_{sG} $ spaces are supported near infinity, the norm\nestimates are effectively $ L^2 $ estimates.\n\nFor the proof of \\eqref{eq:flattr1} we first review\nthe construction of the flat trace under the wave front set\ncondition. Suppose that $ u \\in \\mathcal D' ( X \\times X ) $\nsatisfies the (classical) wave front condition\n\\begin{equation}\n\\label{eq:wfcond}   \\WF ( u ) \\cap N^* \\Delta ( X ) = \\emptyset, \\ \\  \\Delta ( X )\n= \\{ ( x, x ) :  x\\in X \\} \\subset X \\times X.\n\\end{equation}\nIf $ u $ is a Schwartz kernel of an operator $ T $ then\n$ \\tr^\\flat T :=  \\langle \\iota^* u , 1 \\rangle $, where $ \\iota :\n\\Delta ( X ) \\hookrightarrow X \\times X $. We will recall why \\eqref{eq:wfcond}\nallows the definition $ \\iota^* u $.\nFor any $x_0\\in X$, we can choose a neighbourhood $U$ of $x_0$ in $X$ equipped with a local coordinate patch. For simplicity, we abuse the notation and assume $x_0\\in U\\subset \\mathbb{R}^n$. Then $\\iota(x_0)=(x_0,x_0)\\in U\\times U\\subset \\mathbb{R}^n\\times\\mathbb{R}^n$. The conormal bundle to the diagonal is locally given by\n$$N_\\iota=\n\\{(x,x,\\xi,-\\xi)\\in ( U\\times U) \\times\n( \\mathbb{R}^n\\times \\mathbb{R}^n) \\}.$$\nPut $ \\Gamma := \\WF ( u ) $ and\n$ {\\Gamma}_{(x,y)}=\\{(\\xi,\\eta) : (x, y , \\xi, \\eta )\\in\\Gamma\\}$. Then\n\\[ {\\Gamma}_{(x_0,x_0)}\\cap\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}=\\emptyset.\n\\]\nSince ${\\Gamma}_{(x_0,x_0)}$ is closed, we can find a conic neighbourhood, $V$, of $ {\\Gamma}_{(x_0,x_0)}$ in $\\mathbb{R}^n\\times\\mathbb{R}^n\\setminus0$ such that\n$$V\\cap\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}=\\emptyset.$$\nWe can also find a compact neighbourhood $Y_0$ of $(x_0,x_0)$ such that $V$ is a neighbourhood of ${\\Gamma}_{(x,y)}$ for every $(x,y)\\in Y_0$.\nNext we choose a\nneighbourhood $X_0$ of $x_0$ such that $X_0\\times X_0\\Subset Y_0$. Then we have for every $x\\in X_0, (\\xi,\\eta)\\in V$,\n\\begin{equation}\n\\label{eq:nonstat} {}^t\\iota'(x)\\cdot(\\xi,\\eta)=\\xi+\\eta\\neq0.\n\\end{equation}\n\nMoreover, we can choose $ V$ so that its complement, $\\complement V $, is\na small conic neighbourhood of $\\{(\\xi,-\\xi):\\xi\\in\\mathbb{R}^n,\\xi\\neq0\\}$. In particular\nthere exists a constant $C>0$ such that in ${\\complement V}$, $C^{-1}|\\eta|\\leq|\\xi|\\leq C|\\eta|$. We can also assume that\n\\begin{equation}\n\\label{eq:CV}\n\\complement V = - \\complement V .\n\\end{equation}\n\nFinally we choose $\\psi(x)\\in C^\\infty(U)$ equal to 1 on $X_0$ such that $\\varphi(x,y)=\\psi(x)\\psi(y)\\in C_0^\\infty(Y_0)$, then for any $\\chi\\in C_0^\\infty(X_0)$, $u\\in C^\\infty(X\\times X)$, we have\n\\begin{equation}\n\\label{eq:extend} \\langle\\iota^\\ast u,\\chi\\rangle =\\langle\\iota^\\ast(\\varphi u),\\chi\\rangle\n=(2\\pi)^{-2n}\\int\\widehat{\\varphi u}(\\xi,\\eta)I_\\chi(\\xi,\\eta)d\\xi d\\eta,\n\\end{equation}\nwhere\n$$I_\\chi(\\xi,\\eta)=\\int \\chi(x)e^{i\\langle\\iota(x),(\\xi,\\eta)\\rangle}dx\n=\\int\\chi(x)e^{ix\\cdot(\\xi+\\eta)}dx.$$\nWe claim that as long as \\eqref{eq:wfcond} holds the right hand\nside of \\eqref{eq:extend} is well defined and hence the pull back\n$ \\iota^* u $ is a well defined distribution.\n\nTo see this, we first notice that if $(\\xi,\\eta)\\in V$, then\n\\eqref{eq:nonstat} shows that the phase is not stationary and hence,\n$|I_\\chi(\\xi,\\eta)|\\leq C_{N,\\chi}(1+|\\xi|+|\\eta|)^{-N} $, for all $N$.\nOn the other hand, we have\n\\begin{equation}\n\\label{eq:widehat} |\\widehat{\\varphi u}(\\xi,\\eta)|=\\left|\\int\\psi(x)\\psi(y)u(x,y)e^{-i(x\\cdot\\xi+y\\cdot\\eta)}dxdy\\right|.\\end{equation}\nThe construction of $ V $ and \\eqref{eq:wfcond} imply  that if $(\\xi,\\eta)\\not\\in V$, then\n$|\\widehat{\\varphi u}(\\xi,\\eta)|\\leq C_N(1+|\\xi|+|\\eta|)^{-N}$, for all $ N$.\nWhen $ ( \\xi, \\eta ) \\in V $ then, there exists $M>0$ such that\n$|\\widehat{\\varphi u}(\\xi,\\eta)|\\leq C_N(1+|\\xi|+|\\eta|)^M$.\nTherefore $\\langle\\iota^\\ast u,\\chi\\rangle$ is well defined. Now to define $\\langle\\iota^\\ast u,1\\rangle$, we first choose a finite partition of unity $1=\\sum\\chi_j$ where $\\chi_j$ is constructed as above for some $x_j\\in X$ (playing the role of $x_0$)\nand then choose the corresponding $\\psi_j$'s (playing the role of $ \\psi $).\nThis concludes our review of the proof\nthat\n$ \\tr^\\flat T = \\langle \\iota^* u  , 1\\rangle  $ is well defined when \\eqref{eq:wfcond}\nholds.\n\nAll of this can be applied to $ u = K$, the\nSchwartz kernel of  $e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)$,\nwith quantitative bounds in terms of $ h $.\nWe first estimate the wave front set of\n$e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)$. For that we\nneed the following\n\\begin{lem}\n\\label{l:prop}\nFor $ t \\geq 0 $,\n\\begin{equation*}\n\\begin{split}\n \\WF_h' ( e^{ - i t h^{-1} \\widetilde P_h ( z ) } )\n\\cap T^\\ast(X\\times X)&\n\\subset \\{ ( e^{ t_0 H_p } ( x, \\xi ) , ( x, \\xi ) ) : ( x ,\\xi ) \\in T^\\ast X \\},\\\\\n\\WF_h' ( e^{ - i t h^{-1} \\widetilde P_h ( z ) } )\n\\cap S^\\ast(X\\times X)&\n\\subset \\kappa(\\{ ( e^{ t_0 H_p } ( x, \\xi ) , ( x, \\xi ) ) : ( x ,\\xi ) \\in T^\\ast X\\setminus\\{0\\} \\}).\n\\end{split}\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nWe first note that the inclusion is obviously true for the\n$ \\WF_h' ( e^{ - i t P } ) $ since the operator is the pull back by\n$ \\varphi_{-t}^* $. Hence the statement above will follow from showing\nthat $ V ( t ) := e^{ i t P} e^{ - i t h^{-1} \\widetilde P_h ( z ) } $ is a pseudodifferential\noperator. If $ B \\in \\Psi^0_h $ satisfies\n\\[ \\WF_h ( B ) \\cap\n\\cup_{ 0 \\leq |t'| \\leq t }\\,  e^{t'H_p}  (\\WF_h ( Q ) ) = \\emptyset , \\]\nthen\n$ B e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) }  =  B + {\\mathcal O} ( h^\\infty )_{\n\\mathcal D' \\to {C^\\infty} } $.\n\nIn fact, we can use Egorov's theorem (a trivial case since $ e^{ it P } =\n\\varphi_t ^* $) to see that\n\\begin{equation}\n\\begin{split}\n h D_t  \\left( B e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) } \\right) & =\ni B e^{ it P } Q  e^{ - i t h^{-1} ( h P - i Q ) } \\\\\n& = i e^{ i t P } B ( t ) Q e^{ - i t h^{-1} ( h P - iQ ) } =\n\\mathcal O ( h^\\infty )_{ \\mathcal D' \\to {C^\\infty} } ,\n\\end{split}\n\\end{equation}\nwhere $ B ( t ) := e^{-  i t P } B e^{ i t P  } $ satisfies\n$ \\WF_h ( B ( t ) ) \\cap \\WF_h ( Q ) = \\emptyset $.\nBy switching the sign of $ P $ and taking adjoints we see\nthat the we also have\n$$  e^{ i t P} e^{ - i t h^{-1} ( h P - i Q ) }  B =  B + {\\mathcal O} ( h^\\infty )_{\n\\mathcal D' \\to {C^\\infty} } .$$\nHence it is enough to prove that, for $ \\alpha $ in \\eqref{eq:condfk},\n$ e^{ i tP } e^{ - i h^{-1} t ( P - i Q ) } A \\in \\Psi_\\alpha ( X ) $,\n$ \\alpha < {\\frac12} $, for $ A \\in \\Psi^{\\rm{comp}}( X ) $. But that\nis included in \\cite[Proposition { A}.3]{NZ}.\n\\end{proof}\n\n\\medskip\n\\noindent\n{\\bf Remark.} {  \nThe assumption \\eqref{eq:condfk} in the construction of $ \\widetilde P_h ( z ) $ and used in the proof of Lemma \\ref{l:prop}\nis made for convenience only as we can then cite\n\\cite[Proposition { A}.3]{NZ}. }\n\n\\medskip\n\nInclusions \\eqref{eq:WFRh} and \\eqref{eq:WFhR} and\nLemma \\ref{l:prop} show that\n\\begin{equation*}\n\\begin{split}\n& \\WF_h'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n\\cap T^\\ast(X\\times X)\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\}\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n& \\WF_h'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n\\cap S^\\ast(X\\times X)\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\kappa\\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\setminus\\{0\\}\\}.\n\\end{split}\n\\end{equation*}\nIn particular, for all $0 < h < 1 $,\n\\begin{equation*}\n\\begin{split}\n& \\WF'(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\\subset\\\\\n& \\ \\ \\ \\ \\ \\ \\{((x,\\xi),(y,\\eta)) \\; :\\;\n(e^{-t_0H_p}(x,\\xi),(y,\\eta))\\in\\Delta(T^\\ast X)\\cup\\Omega_+\\cup(E_u^\\ast\\times E_s^\\ast)\\},\n\\end{split}\n\\end{equation*}\n{  satisfying \\eqref{ftc},} that is, does not intersect with $\\Delta(T^\\ast X)$ and hence $\\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))$ is well-defined.\n\n\nUsing a microlocal partition of unity,\n$ I=\\sum_{j=1}^J B_j + \\mathcal O ( h^\\infty )_{ \\mathcal D'\n\\to {C^\\infty} } $, $ B_j\\in\\Psi^0_h(X)$ {  (see for instance \\cite[Proposition E.34]{res})}\nwe only need to prove that\n\\[\n\\begin{split}\n& {\\rm(i)}  \\WF_h(B) \\subset \\neigh_{ T^* X } ( x_0, \\xi_0 ) ,  \\\n(x_0,\\xi_0)\\in T^\\ast X  \\Rightarrow\n\\tr^\\flat  e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B\n= \\mathcal O ( h^{-{  2 } n-1} ) , \\\\\n& {\\rm(ii)}  \\WF_h(B) \\subset \\neigh_{ \\overline T^* X } ( x_0, \\xi_0 ) ,  \\\n(x_0,\\xi_0)\\in S^\\ast X  \\Rightarrow\n\\tr^\\flat e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B = \\mathcal O ( h^{\\infty} ) ,\n\\end{split} \\]\nIn case (ii), $ W :=\\neigh_{ \\overline T^* X } ( x_0, \\xi_0 ) $ is the image of\nthe closure of a conic neighbourhood of $(x_0,\\xi_0)$ in $T^\\ast X$, under\nthe map $ T ^* X \\to \\overline T^* X $.\n{  In fact, given (i) and (ii), we can use a microlocal partition of unity to write\n$$ e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)=\\sum_{j=1}^J\n e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z) B_j+O(h^\\infty)_{\\mathcal{D}'\\to {C^\\infty}}$$\nwhere each $B_j$ satisfies either (i) or (ii) and this proves \n\\eqref{eq:flattr1}.}\n\nFor each case, we repeat the construction with the Fourier transform replaced by the semiclassical Fourier transform. Let\n$u=K_h$ be the Schwartz kernel\nof $e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B$.\nThen, in the notation of \\eqref{eq:extend},\n\\begin{equation}\n\\label{eq:iota}\n\\langle\\iota^\\ast u,\\chi\\rangle =\\langle\\iota^\\ast(\\varphi u),\\chi\\rangle\n=(2\\pi h)^{-2n}\\int\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)I_{\\chi,h}(\\xi,\\eta)d\\xi d\\eta,\n\\end{equation}\nwhere now\n\\begin{equation}\n\\label{eq:Ichih} I_{\\chi,h}(\\xi,\\eta)=\\int\\chi(x)e^{i\\langle\\iota(x),(\\xi,\\eta)\\rangle\/h}dx\n=\\int\\chi(x)e^{ix\\cdot(\\xi+\\eta)\/h}dx. \n\\end{equation}\n\n If $\\WF_h(B)$ is contained in a small compact neighbourhood $W$ of $(x_0,\\xi_0)$, we can assume in the partition of unity $1=\\sum\\chi_j$ (see the\nargument following \\eqref{eq:widehat}), $\\pi(W)\\subset X_0$ for some coordinate patch $X_0$ and $\\pi(W)\\cap\\supp\\chi_j=\\emptyset$ except for the one in this coordinate patch, say $\\psi=\\psi_0$. For $j\\neq0$,\nsince\n\\begin{equation*}\n\\WF_h'(\\varphi_j u)\\subset\\WF_h' ( u ) \\cap[(\\overline{T}^\\ast X)\\times\\WF_h(B)]\n\\cap[(\\overline{T}^\\ast\\supp\\psi_j )\\times(\\overline{T}^\\ast\\supp\\psi_j)]=\\emptyset,\n\\end{equation*}\nwe have\n\\begin{equation*}\n \\mathcal{F}_h(\\varphi_j u)(\\xi,\\eta)=\\mathcal{O}(h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty}),\n\\end{equation*}\nand thus\n$ \\langle\\iota^\\ast u,\\chi_j \\rangle=\\mathcal{O}(h^\\infty)$.\nTherefore we only need to consider the coordinate patch $X_0$ centered at $x_0$ and the corresponding $\\chi,\\psi$ constructed as before.\nWe note that  $ I_{\\chi, h } (\\xi, \\eta ) = \\mathcal O (h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty})$ uniformly for $(\\xi,\\eta)\\in V$ ($ I_{\\chi, h } $ is defined in \\eqref{eq:Ichih}\nand again we use the notation introduced before \\eqref{eq:extend}).\nHence we only need to to estimate\n\\begin{equation*}\n\\left|\\int_{{\\complement V}}\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)I_{\\chi,h}(\\xi,\\eta)d\\xi d\\eta\\right|\\leq\\int_{{\\complement V}}|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|d\\xi d\\eta.\n\\end{equation*}\nHere\n\\begin{equation}\n\\begin{split}\n\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n=&\\int \\psi(x)\\psi(y)u(x,y)e^{-i(x\\cdot\\xi+y\\cdot\\eta)\/h}dxdy\\\\\n=&\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), \\psi(x)e^{- ix\\cdot\\xi\/h}\\rangle,\n\\end{split}\n\\end{equation}\nwhere $ \\langle \\bullet, \\bullet \\rangle $ denotes distributional pairing.\nWe also note that  \n\\[ \\WF_h(\\psi(x)e^{ix\\cdot\\xi\/h})=\\supp\\psi\\times\\{\\xi\\}, \\ \\ \n\\WF_h(\\psi(y)e^{-iy\\cdot\\eta\/h})=\\supp\\psi\\times\\{-\\eta\\}, \\ \\ \n ( \\xi , \\eta ) \\in \\complement V .\\]\n\nIn case (i), we assume \n\\[ \\WF_h(B)\\subset W=W_1 \\times W_2 \\text{ where $W_1=\\pi(W)\\subset X_0$\nand  $W_2 \\subset\\mathbb{R}^n$ are compact.} \\]\n We make the following observation:\nif $ \\widetilde W_2 = \\{ \\xi' : \\exists \\, \\eta' \\in W_2  \\ ( \\xi', \\eta' )\n\\in \\complement V \\} $, then either $ - \\eta \\notin W_2 $ or $ - \\xi \\in \\widetilde W_2 $.\n(Here we used the symmetry \\eqref{eq:CV}.)\nHence if $A\\in\\Psi_h^{{\\rm{comp}}} (X)$, $ \\WF_h ( I - A ) \\cap\n\\widetilde{W}_1\\times\\widetilde{W}_2 = \\emptyset$,\nwhere $\\widetilde{W}_1$ is a small neighbourhood of $\\supp\\psi$, then\n\\begin{equation*}\n\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)=\n\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), A(\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty).\n\\end{equation*}\nTherefore\n\\begin{equation*}\n\\begin{split}\n|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|\n=&\\;|\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}), A(\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle|+\\mathcal{O}(h^\\infty)\\\\\n\\leq &\\; C\\|Ae^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B\\|_{L^2\\to L^2}+\\mathcal{O}(h^\\infty)\\leq Ch^{-1}\\\\\n\\end{split}\n\\end{equation*}\nwhere we use the estimate \\eqref{modifiedresolvent} and the fact that microlocally on $\\WF_h(A)\\times\\WF_h(B)$ which is a compact set in $T^\\ast(X\\times X)$, $H_{sG(h)}$ is equivalent to $L^2$ uniformly.\nCombined with \\eqref{eq:iota} this finishes the proof for case (i).\n\nIn case (ii), we again assume that \n$ \\WF_h(B)\\subset W=W_1\\times W_2$ where $W_1=\\pi(W)\\subset X_0$ is a small compact neighbourhood of $x_0$ but now $W_2\\subset\\bar{\\mathbb{R}}^n=\\mathbb{R}^n\\cup\\partial \\, \\bar {\\mathbb{R}}^n$ is a small conic neighbourhood of $\\xi_0\\in\n\\partial \\,\\bar{ \\mathbb{R}}^n$ intersecting with $\\{|\\xi|\\geqslant C\\}$. As in case (i),\nwe put\n$ \\widetilde W =\\widetilde{W}_1\\times\\widetilde{W}_2$ such that $\\widetilde{W}_1$ is a small neighbourhood of $\\supp\\psi$ and $\\widetilde{W}_2$ is a small neighbourhood of ${\\complement V}(W_2)$, which is again a small conic neighbourhood of $\\xi_0$.\n\nWe then  choose $A\\in\\Psi_h^0(X)$ such that $ \\WF_h ( I - A) \\cap \\widetilde{W}\n= \\emptyset $,  and  $\\WF_h(A)$ is contained in a small neighbourhood of $\\widetilde{W}$.\nWe have\n\\[ (\\xi,\\eta)\\in {\\complement V} \\ \\Longrightarrow  \\ \\text{ (a)  $-\\eta\\notin W_2$\nor (b) $- \\xi\\in\\widetilde{W}_2$.}\n\\]\nIn the case (a)  we have\n\\begin{equation*}\n|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|=\\mathcal{O}(h^\\infty(1+|\\xi|+|\\eta|)^{-\\infty}).\n\\end{equation*}\nIn the case\n (b) we  need a uniform estimate for $\\langle\\xi\\rangle^N|\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)|$ where $N $ is large.\nTo do this, we use the notation from the proof of Lemma \\ref{l:prop} and write\n\\begin{equation*}\n\\begin{split}\n\\langle\\xi\\rangle^N\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n=&\\;\\langle e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h}\\rangle\\\\\n=&\\;\\langle \\varphi_{-t_0}^\\ast V (t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty)\\\\\n=&\\;\\langle V(t_0)\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle+\\mathcal{O}(h^\\infty).\n\\end{split}\n\\end{equation*}\nWe notice that $\\WF_h(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})=\\supp\\psi\\times\\{-\\xi\\}$, and\n\\begin{equation*}\n\\|\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h}\\|_{H_h^{-N}}=\\mathcal{O}(1)\n\\end{equation*}\nuniformly in $\\xi$. Since $ t_0 $ is small we can choose $W$ and $\\widetilde{W}$ small\nenough, so that $e^{-t_0 H_p}\\widetilde{W}\\cap\\widetilde{W}=\\emptyset$.\nThen we choose a microlocal partition of unity,  $A_1^2+A_2^2=I +\n\\mathcal O ( h^\\infty )_{ \\mathcal D' \\to {C^\\infty}} $,\nsuch that $e^{-tH_p}\\widetilde{W}\\subset\\el_h(A_1)$,\n$\\WF_h(A_1)$ is a small neighbourhood of $e^{-tH_p} \\widetilde W $ and\n $\\WF_h(A_2)\\cap e^{-t_0H_p}(\\WF_h(A))=\\emptyset$. We have\n\\begin{equation}\n\\label{eqa}\n\\begin{split}\n& \\langle\\xi\\rangle^N\\mathcal{F}_h(\\varphi u)(\\xi,\\eta)\n= \\langle A_1 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_1\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{-ix\\cdot\\xi\/h})\\rangle\\\\\n&  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + \\langle A_2 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_2\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{- ix\\cdot\\xi\/h})\\rangle+\n\\mathcal \\mathcal{O}(h^\\infty).\n\\end{split}\n\\end{equation}\n\nWe recall the following propagation estimate \\cite[Propositon 2.5]{DZ} which\nis essentially the clasical result of Duistermaat--H\\\"ormander:\n\\begin{prop}\n\\label{ppg}\nAssume that $P_0\\in\\Psi_h^1(X)$ with semiclassical principal symbol $p-iq\\in S^1_h(X)\/hS_h^0(X)$ where $p\\in S^1(X;\\mathbb{R})$ is independent of $h$ and $q\\geqslant0$ everywhere. Assume also that $p$ is homogeneous of degree 1 in $\\xi$ for $|\\xi|$ large enough. Let $e^{tH_p}$ be the Hamiltonian flow of $p$ on $\\overline{T}^\\ast X$ and $u(h)\\in\\mathcal{D}'(X)$, then if $A_0,B_0,B_1\\in\\Psi_h^0(X)$ and for each $(x,\\xi)\\in\\WF_h(A_0)$, there exists $T\\geqslant0$ with $e^{-TH_p}(x,\\xi)\\in\\el_h(B_0)$ and $e^{tH_p}(x,\\xi)\\in\\el_h(B_1)$ for $t\\in[-T,0]$. Then for each $m$,\n\\begin{equation}\n\\|A_0u\\|_{H^m_h(X)}\\leq C\\|B_0u\\|_{H^m_h(X)}+Ch^{-1}\\|B_1P_0u\\|_{H_h^m(X)}+\\mathcal{O}(h^\\infty).\n\\end{equation}\n\\end{prop}\n\nWe apply the proposition to $u=\\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h})$, $P_0=\\widetilde{P}_h(z)$, $A_0=A_1V(t_0)$ with $\\el_h(B_0)$ containing $e^{-TH_p}(\\WF_h(A_0))$ for $T>0$ small enough and $e^{tH_p}(x,\\xi)\\in\\el_h(B_1)$ for $t\\in [-T,0]$. Furthermore, we can choose $B_1$ so that $\\WF_h(B_1)\\cap\\WF_h(B)=\\emptyset$.\n\\[ \\begin{split} \\|A_0u\\|_{H_h^N} & \\leq C\\|B_0u\\|_{H_h^N}+Ch^{-1}\\|B_1B(\\psi(y)e^{-iy\\cdot\\eta})\\|_{H_h^N}+\\mathcal{O}(h^\\infty)\\\\\n& =C\\|B_0u\\|_{H_h^N}+\\mathcal{O}(h^\\infty). \\end{split} \\]\nHowever, the semiclassical wavefront set condition of $\\widetilde{R}_h(z)$ shows that $\\WF_h(B_0)\\cap\\WF_h(u)=\\emptyset$, thus\n$\\|A_0u\\|_{H_h^N}=\\mathcal{O}(h^\\infty)$\nand\nwe know the term corresponding to $A_1$ in the sum of \\eqref{eqa} is $\\mathcal{O}(h^\\infty)$. For the other term involving $ A_2 $, we use\n$$\\|A_2\\varphi_{t_0} ^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{ix\\cdot\\xi\/h})\\|_{H^{P}_h }\n\\leq \\mathcal{O}(h^\\infty)\\|\\varphi_{t_0} ^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{ix\\cdot\\xi\/h})\\|_{H_h^{-N}}=\\mathcal{O}(h^\\infty),$$\nfor any $ P $.\nThis is paired with the term estimated by\n$$\\|A_2 V ( t_0 ) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h})\\|_{H_h^{-P }}\\leq\nC\\|\\psi(y)e^{-iy\\cdot\\eta\/h}\\|_{H_h^P } \\leq C \\langle \\eta \\rangle^P , $$\nfor some $ P$. Hence\n\\[\n\\langle A_2 V(t_0) \\widetilde{R}_h(z)B(\\psi(y)e^{-iy\\cdot\\eta\/h}),A_2\\varphi_{t_0}^\\ast A(\\langle\\xi\\rangle^N\\psi(x)e^{- ix\\cdot\\xi\/h}) \\rangle = \\mathcal O ( h^\\infty \\langle\n\\eta \\rangle^P ). \\]\nReturning to \\eqref{eqa} we see that\n\\[ \\langle \\xi \\rangle^N \\mathcal F_h ( \\varphi u ) (\\xi, \\eta ) =\n\\mathcal O ( h^\\infty \\langle\n\\eta \\rangle^P ). \\]\nSince $ |\\xi | $ is comparable $ |\\eta | $ in $ \\complement V $, we have\n$$ \\mathcal F_h ( \\varphi u ) (\\xi, \\eta ) = \\mathcal O ( h^\\infty\n\\langle ( \\xi , \\eta ) \\rangle^{ -N + P } ) $$ \nand that concludes the\nproof of \\eqref{eq:flattr1}.\n\n\n\n\n\n\n\\section{Proof of the trace formula}\n\\subsection{Sketch of the proof}\nWe first indicate basic ideas of the proof before we go into the details -- the\nprinciple is quite simple but the implementation involves the use of\nthe results of \\cite{DZ} and of some ideas from \\cite{S}.\n\nIn general, a trace formula such as \\eqref{localtrace} follows from the finite order of the analytic continuation of $\\zeta_1(\\lambda)$ in the strip $\\Im\\lambda\\geqslant-A$, that is,\nfrom having the following estimate valid away from small neighbourhoods of resonances:\n\\begin{equation}\n\\label{finiteorder}\n\\left|\\frac{d}{d\\lambda} \\log\\zeta_1(\\lambda)\\right|=\\mathcal{O}(\\langle\\lambda\\rangle^{2n+1}).\n\\end{equation}\nTo obtain the distributional identity \\eqref{localtrace}\nwe take $\\psi\\in C_0^\\infty(0,\\infty)$ and compute the following integral in two different ways\n\\begin{equation*}\n\\int_{\\mathbb{R}}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}  \\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation*}\nOn one hand, we pass the integral contour to $\\mathbb{R}+iB$, where $B>C_1$ so that \\eqref{zeta} converges. Since there are no resonances in the upper half plane, we have\n\\begin{equation*}\n\\begin{split}\n\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\\left(\\frac{1}{i}\\int_0^\\infty e^{it\\lambda}\\tr^\\flat e^{-itP}dt\\right)d\\lambda\n=&\\frac{1}{i}\\int_0^\\infty\\left(\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)e^{it\\lambda}d\\lambda\\right)\\tr^{\\flat}e^{-itP}dt\\\\\n=&\\int_0^\\infty\\psi(t)\\tr^{\\flat}e^{-itP}dt.\n\\end{split}\n\\end{equation*}\nGuillemin's trace formula \\eqref{Guillemin} gives\n\\begin{equation}\n\\int_{\\mathbb{R}}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda=\\left\\langle \\sum_\\gamma\\frac{T_\\gamma^\\#\\delta(t-T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}, \\psi \\right\\rangle.\n\\end{equation}\nOn the other hand, we pass the integral contour to $\\mathbb{R}-iA$ and we get the contribution from the poles of $\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)$ which are exactly the Pollicott-Ruelle resonances,\n\\begin{equation}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\psi}(\\mu)=\\left\\langle\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}e^{-i\\mu t}, \\psi \\right\\rangle.\n\\end{equation}\nThe remainder is exactly\n\\begin{equation}\n\\langle F_A, \\psi\\rangle: =\\int_{\\mathbb{R}-iA}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda,\n\\end{equation}\nand we want to show that $ F_A $ can be extended to a tempered distribution\nsupported on $ [ 0, \\infty ) $ and that it satisfies \\eqref{error}. The\nestimate \\eqref{finiteorder} is crucial here.\n\nTo see \\eqref{finiteorder}, we decompose\n\\begin{equation}\n\\label{decomposition}\n\\begin{split}\ne^{-it_0(P-\\lambda)}(P-\\lambda)^{-1} & =\ne^{-it_0(P-i \\widetilde Q-\\lambda)}(P-i \\widetilde Q-\\lambda)^{-1}\n+[(P-\\lambda)^{-1}-(P-i \\widetilde Q-\\lambda)^{-1}]\\\\\n& \\ \\ \\ \\ - i\\int_0^{t_0}[e^{-it(P-\\lambda)}-e^{-it(P-i \\widetilde Q-\\lambda)}]dt,\n\\end{split}\n\\end{equation}\nwhere $ \\widetilde Q = h^{-1} Q $ for a suitably chosen $ h$ depending on the range of $ \\lambda $'s. This is valid from $ \\Im \\lambda \\gg 0 $ and then continues analytically to $ \\mathbb C $ on the level of distributional Schwartz kernels.\n\nThe first term is holomorphic in $\\lambda$ and can be estimated by Proposition \\ref{flattracees} in the semiclassical setting.\n\nThe second term on the right hand side of \\eqref{decomposition} is of trace class if $\\lambda$ is not a resonance. To see this, we use the following formula\n\\begin{equation}\n\\label{res1}\n\\begin{split}\n(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}  & =\n[(P-\\lambda)^{-1}(P-i \\widetilde Q-\\lambda)-I](P-i\\widetilde Q-\\lambda)^{-1}\\\\\n& =-(P-\\lambda)^{-1}i \\widetilde Q(P-i \\widetilde Q-\\lambda)^{-1}\\\\\n\\end{split}\n\\end{equation}\nto get\n\\begin{equation}\n\\label{res2}\n(P-\\lambda)^{-1}=(P-i \\widetilde Q-\\lambda)^{-1}[I+i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}.\n\\end{equation}\nBy using \\eqref{res2} in  \\eqref{res1} we obtain\n\\begin{equation}\n(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}=\n-(P-i\\widetilde Q-\\lambda)^{-1}[I+i \\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}\ni\\widetilde Q(P-i \\widetilde\nQ-\\lambda)^{-1}.\n\\end{equation}\nIf we denote { $F(\\lambda)=I+i\\widetilde{Q}(P-i\\widetilde Q-\\lambda)^{-1}$}, then\n\\begin{equation*}\nF'(\\lambda)=\\frac{d}{d\\lambda}F(\\lambda)=i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-2}.\n\\end{equation*}\nMoreover, $F(\\lambda) - I $ and $ F' ( \\lambda ) $ are operators\nof finite rank. By the cyclicity of the trace, we have\n\\begin{equation*}\n\\begin{split}\n\\tr[(P-\\lambda)^{-1}-(P-i\\widetilde Q-\\lambda)^{-1}]=&\n-\\tr[I+i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-1}]^{-1}i\\widetilde Q(P-i\\widetilde Q-\\lambda)^{-2}\\\\\n=&-\\tr F'(\\lambda)F(\\lambda)^{-1}=-\\frac{d}{d\\lambda}\\log\\det F(\\lambda).\n\\end{split}\n\\end{equation*}\nTherefore it can be controlled by the rank of $\\widetilde Q$ and the norm of $F(\\lambda)$.\n\nThe third term in \\eqref{decomposition} can be handled by Duhamel's principle: if $u(t): =e^{-it(P-i\\widetilde Q-\\lambda)}f$, then\n\\begin{equation*}\n\\partial_t u(t)=-i(P-i\\widetilde Q-\\lambda)u(t), \\ \\ \\ u(0)=f.\n\\end{equation*}\nRewriting the equation as\n$\\partial_t u(t)+i(P-\\lambda)u(t)=-\\widetilde Qu(t)$ ,\nwe get\n\\begin{equation*}\nu(t)=e^{-it(P-\\lambda)}f-\\int_0^t e^{-i(t-s)(P-\\lambda)}\\widetilde Qu(s)ds.\n\\end{equation*}\nTherefore\n\\begin{equation*}\ne^{-it(P-\\lambda)}-e^{-it(P-i\\widetilde Q-\\lambda)}=\\int_0^te^{-i(t-s)(P-\\lambda)}\\widetilde Qe^{-is(P-i\\widetilde Q-\\lambda)}ds.\n\\end{equation*}\nThis shows that the left hand side is also of trace class and its trace class norm is controlled by the trace class norm of $\\widetilde Q$.\n\n{  To carry out the strategy above} we need to choose correct contours and to obtain\na local version of \\eqref{finiteorder} using $ \\det F ( \\lambda ) $.\nFor that we break the infinite contour into a family of finite contours and use the semiclassical reduction to treat the zeta function on each contour separately.\nThat involves choices of $ h $ so that $ z = h \\lambda $ is in an\nappropriate range.\n\n\\subsection{The contours for integration}\nIn this section, we choose contours for integration. First, we decompose the region $\\Omega=\\{\\lambda\\in\\mathbb{C}:-A\\leq \\Im\\lambda\\leq B\\}$ into dyadic pieces: fix $ E > 0 $ and put $\\Omega=\\bigcup_{k\\in\\mathbb{Z}}\\Omega_k$, where $\\Omega_0=\\Omega\\cap\\{-E\\leq\\Re\\lambda\\leq E\\}$ and\n\\begin{gather*}\n\\Omega_k:=\\Omega\\cap\\{2^{k-1}E\\leq\\Re\\lambda\\leq 2^kE\\}, \\ \\ k>0\\\\\n\\Omega_{-k}:=\\Omega\\cap\\{-2^kE\\leq\\Re\\lambda\\leq - 2^{k-1}E\\}, \\ \\ k>0.\n\\end{gather*}\nFor each $k$, we write $\\gamma_k=\\partial\\Omega_k=\\bigcup_{j=1}^4\\gamma^j_k$ with counterclockwise orientation.\n\n\\begin{figure}[ht]\n\\includegraphics[width=6.5in]{contours}\n\\caption{Integration contours}\n\\end{figure}\n\nNext, we shall modify $\\gamma^2_k,\\gamma^3_k$ and $\\gamma^4_k$ to avoid the resonances. For simplicity, we only work for $k>0$ as the case for $k<0$ can be handled by symmetry. We choose $\\widetilde{\\gamma}^2_k,\\widetilde{\\gamma}^3_k$ and $\\widetilde{\\gamma}^4_k$ lying in\n\\begin{equation*}\n([2^{k-1}E-1,2^kE+1]+i[-A-1,B])\\setminus([2^{k-1}E+1,\n2^kE-1]+i[-A,B])\n\\end{equation*}\nso that $\\widetilde{\\gamma}^2_k\\subset[2^{k-1}E-1,2^{k-1}E+1]\n+i[-A,B]$ connects $2^{k-1}E+iB$ with a point $w_k$ which lies\non $[2^{k-1}E-1,2^{k-1}E+1]-iA$, $\\widetilde{\\gamma}^4_k=-\\widetilde{\\gamma}^2_{k+1}$; $\\widetilde{\\gamma}^3_k\\subset[2^{k-1}E-1,2^kE+1]+i[-A-1,-A]$ connects $w_k$ with $w_{k+1}$. The region bounded by  $\\widetilde{\\gamma}_k :=\\bigcup_{j=1}^4\\widetilde{\\gamma}^j_k$\n is denoted as $\\widetilde{\\Omega}_k$, (we write $\\widetilde{\\gamma}^1_k=\\gamma^1_k$). Then we have { \n\\begin{equation*}\n\\Omega\\subset\\widetilde{\\Omega}=\\bigcup_{k\\in\\mathbb{Z}}\\widetilde{\\Omega}_k\n\\subset\\{\\lambda\\in\\mathbb{C},-A-1\\leq\\Im\\lambda\\leq B\\}\n\\end{equation*}}\nand all $\\widetilde{\\Omega}_k$ have disjoint interiors.\n\nFor convenience, we turn into the semiclassical setting.\nLet $W_h=h\\widetilde{\\Omega}_k$ where $h^{-1\/2}=2^kE$, then\n\\begin{equation*}\n\\textstyle [\\frac{1}{2}h^{1\/2}+h,h^{1\/2}-h]+i[-Ah,Bh]\\subset W_h\\subset\n[\\frac{1}{2}h^{1\/2}-h,h^{1\/2}+h]+i[(-A-1)h,Bh].\n\\end{equation*}\nMoreover, $\\rho_h:=\\partial W_h=\\bigcup_{j=1}^4\\rho_h^j$ where $\\rho_h^1$ is the horizontal segment $[\\frac{1}{2}h^{1\/2},h^{1\/2}]+iBh$ with negative orientation; $\\rho_h^2\\subset[\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h]+i[-Ah,Bh]$ connects $\\frac{1}{2}h^{1\/2}+iBh$ with a point $z_h\\in[\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h]-iAh$; $\\rho_h^4\\subset[h^{1\/2}-h,h^{1\/2}+h]+i[-Ah,Bh]$ connects a point $z'_h\\in[h^{1\/2}-h,h^{1\/2}+h]-iAh$ with $h^{1\/2}+iBh$;\nand $\\rho_h^3\\subset[\\frac{1}{2}h^{1\/2}-h,h^{1\/2}+h]$ connects $z_h$ with $z'_h$.\n\nWe have the following contour integration\n\\begin{equation}\n\\label{contour}\n\\oint_{\\rho_h}\\widehat{\\psi}_h(z)\\frac{d}{dz}\\log\\zeta_h(z)dz=\n\\sum_{z_j\\in \\Res_h(P)\\cap W_h}\\psi_h(z_j).\n\\end{equation}\nHere we write $\\widehat{\\psi}_h(z)=\\widehat{\\psi}(z\/h)$, $\\zeta_h(z)=\\zeta_1(z\/h)$, $\\Res_h(P)=h\\Res(P)$.\n\nWe rewrite the decomposition \\eqref{decomposition} in this scaling:\n\\begin{equation}\n\\label{hdecom}\n\\begin{split}\n\\frac{d}{dz}\\log\\zeta_h(z) & = h \\tr^\\flat(e^{-it_0h^{-1}P_h(z)}R_h(z))\\\\\n& = \\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\n+ \\tr(R_h(z)-\\widetilde{R}_h(z))\\\\\n& -\\frac{i}{h}\\tr\\int_0^{t_0}[e^{-ith^{-1}P_h(z)}-e^{-ith^{-1}\\widetilde{P}_h(z)}]dt.\n\\end{split}\n\\end{equation}\nThen as in the discussion after \\eqref{decomposition}, in the region $-C_0h\\leq \\Im z\\leq 1, |\\Re z|\\leq 2h^{1\/2}$, we can apply Proposition\n\\ref{flattracees} to obtain\n\\begin{equation}\n\\label{hes1}\n\\left|\\tr^\\flat(e^{-it_0h^{-1}\\widetilde{P}_h(z)}\\widetilde{R}_h(z))\\right|=\\mathcal{O}(h^{-2n-1}).\n\\end{equation}\nAlso we have\n\\begin{equation}\n\\label{hes2}\n\\left \\|\\int_0^{t_0}[e^{-ith^{-1}P_h(z)}-e^{-ith^{-1}\\widetilde{P}_h(z)}]dt \\right\\|_{\\rm{tr}} =\\mathcal{O}(h^{-n-1}).\n\\end{equation}\nFor the second term, we have\n\\begin{equation*}\n\\tr(R_h(z)-\\widetilde{R}_h(z))=-\\frac{d}{dz}\\log\\det F(z),\n\\end{equation*}\nwhere\n$F(z)=I+iQ\\widetilde{R}_h(z)$ is a Fredholm operator and the poles for $F(z)^{-1}$ coincides with the resonances. Moreover, by \\eqref{finiterank}, \\eqref{modifiedresolvent} and Weyl's inequality, we have\n\\begin{equation}\n\\label{detes1}\n|\\det F(z)|\\leq (Ch^{-1})^{Ch^{-n}}\\leq Ce^{Ch^{-n-1}}.\n\\end{equation}\n\nMoreover, when $\\Im z\\geqslant C_1h$, we have $F(z)=I+iQ\\widetilde{R}_h(z)=P_h(z)\\widetilde{R}_h(z)$, so $F(z)$ is invertible and $F(z)^{-1}=\\widetilde{P}_h(z)R_h(z)$. Therefore\n\\begin{equation*}\n\\begin{split}\n\\|F(z)^{-1}\\|_{H_{sG(h)}\\to H_{sG(h)}}\n\\leq\\|\\widetilde{P}_h(z)\\|_{D_{sG(h)}\\to H_{sG(h)}}&\\|R_h(z)\\|_{H_{sG(h)}\\to D_{sG(h)}}\\\\\n\\leq\\|\\widetilde{P}_h(z)\\|_{D_{sG(h)}\\to H_{sG(h)}}&(\\|R_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}}\\\\\n&+\\|hPR_h(z)\\|_{H_{sG(h)}\\to H_{sG(h)}})\n\\leq Ch^{-1}.\n\\end{split}\n\\end{equation*}\nWe can also write $F(z)^{-1}=I-iQR_h(z)$ which gives the estimate\n\\begin{equation}\n\\label{detes2}\n|\\det F(z)^{-1}|\\leq (Ch^{-1})^{Ch^{-n}}\\leq Ce^{Ch^{-n-1}}.\n\\end{equation}\n\nWe recall a lower modulus theorem due to H. Cartan { (see \\cite[\\S 11.3, Theorem 4]{Le})} : Suppose that $g$ is holomorphic in $D(z_0,2eR)$ and $g(z_0)=1$. Then for any $\\eta>0$,\n\\begin{equation}\n\\label{lowermod}\n\\log|g(z)|\\geqslant-\\log(15e^3\/\\eta)\\log\\max_{|z-z_0|<2eR}|g(z)|,\\;\\;\nz\\in D(z_0,R)\\setminus\\mathcal{D},\n\\end{equation}\nwhere $\\mathcal{D}$ is a union of discs with the sum of radii less than $\\eta R$. With the help of this lower modulus theorem, we can make a suitable choice of integration contour.\n\n\\begin{lem}\nWe can choose $\\widetilde{\\gamma}_k$ suitably such that in addition to the assumptions above, we have\n\\begin{equation}\n\\label{detbound}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation}\nwhen $z\\in\\rho_h$.\n\\end{lem}\n\\begin{proof}\nWe shall apply the lower modulus theorem with $z_0=\\frac{1}{2}h^{1\/2}+iBh\/2$ and $R=C_0'h$ where $C_0'$ is large enough, so that\n\\begin{equation*}\n\\begin{split}\n[\\textstyle{\\frac{1}{2}h^{1\/2}-h,\\frac{1}{2}h^{1\/2}+h}]+i[(-A-1)h,Bh]   & \\subset D(z_0,R) \\subset D(z_0,2eR) \\\\\n & \\subset[-2h^{1\/2},2h^{1\/2}]+i[-C_0h,1].\n\\end{split}\n\\end{equation*}\nIn addition, we let $\\eta$ be small enough, so that $\\eta R<h$.\nSince $\\widetilde{R}_h(z)$ is holomorphic in $D(z_0,2eR)$, so is $\\det F(z)$. Moreover, from \\eqref{detes1}, we see\n\\begin{equation*}\n\\log\\max_{|z-z_0|<2eR}|\\det F(z)|\\leq Ch^{-n-1}.\n\\end{equation*}\n\nOn the other hand, at $z=z_0$, by \\eqref{detes2}\n\\begin{equation*}\n|\\det F(z_0)^{-1}|\\leq e^{-Ch^{-n-1}}.\n\\end{equation*}\nApplying the lower modulus theorem with $g(z)=\\det F(z)\\det F(z_0)^{-1}$, we can choose $\\widetilde{\\gamma}^2_k$ so that\n\\begin{equation*}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation*}\nwhen $z\\in h\\widetilde{\\gamma}^2_k$. Taking $\\widetilde{\\gamma}^4_k=-\\widetilde{\\gamma}^2_{k+1}$ as required, it is clear that we also have\n\\begin{equation*}\n|\\log\\det F(z)|=\\mathcal{O}(h^{-n-1})\n\\end{equation*}\nwhen $z\\in h\\widetilde{\\gamma}^4_k$.\n\n{  Finally, we can apply the lower modulus theorem to a sequence of balls $D(z_{0j},R)$ which are translations of $D(z_0,R)$ as above by $jh$, $0\\leqslant j\\leqslant \\frac{1}{2}h^{-1\/2}$. In particular, $z_{0j}=z_0+jh$ and we have\n\\begin{equation*}\n\\begin{split}\n[\\textstyle{\\frac{1}{2}h^{1\/2}+jh-\\frac{1}{2}h,\\frac{1}{2}h^{1\/2}+jh+\\frac{1}{2}h}]+i[(-A-1)h,-Ah]   & \\subset D(z_0,R) \\subset D(z_0,2eR) \\\\\n & \\subset[-2h^{1\/2},2h^{1\/2}]+i[-C_0h,1].\n\\end{split}\n\\end{equation*}\nWe can now repeat the argument above and notice that all the estimates hold uniformly in $j$ to conclude that we can choose a curve in $[\\frac{1}{2}h^{1\/2}+jh-\\frac{1}{2}h,\\frac{1}{2}h^{1\/2}+jh+\\frac{1}{2}h]$ for each $j$ such that $|\\log\\det F(z)|=O(h^{-n-1})$ uniformly in $j$ on these curves and these curves form the curve $h\\tilde{\\gamma}_3$ connecting $z_h$ and $z_h'$.}\n\\end{proof}\n\nNow by the upper bound on the number of resonances, we have\n\\begin{equation}\n\\label{resw}\n\\#\\Res_h(P)\\cap W_h=\\mathcal{O}(h^{-\\frac{n}{2}-1}).\n\\end{equation}\nFrom the proof of the lemma above, we can actually construct $\\rho_h$ so that \\eqref{detbound} holds in a neighborhood of $\\rho_h$ of size $\\sim h^{\\frac{n}{2}+2}$. By Cauchy's inequality, we see that for $z\\in\\rho_h$,\n\\begin{equation}\n\\label{hes3}\n\\left|\\frac{d}{dz}\\log\\det F(z)\\right|=\\mathcal{O}(h^{-\\frac{3n}{2}-3}).\n\\end{equation}\nNow combining \\eqref{hes1}, \\eqref{hes2} and \\eqref{hes3}, we have the estimate for $z\\in\\rho_h$,\n\\begin{equation}\n\\label{zetabound}\n\\left|\\frac{d}{dz}\\log\\zeta_h(z)\\right|=\\mathcal{O}(h^{-2n-1}),\n\\end{equation}\nwhich is equivalent to \\eqref{finiteorder}.\n\n\\subsection{End of the proof}\nNow we finish the proof of the local trace formula by summing the contour integrals \\eqref{contour}.\n\nFirst, since $\\widehat{\\psi}(\\lambda)=\\mathcal{O}(\\langle\\lambda\\rangle^{-\\infty})$ as $\\Re\\lambda\\to\\pm\\infty$ as well as all of its derivatives when $\\Im\\lambda$ is bounded, we have $\\widehat{\\psi}_h(z)=\\mathcal{O}(h^\\infty)$ as well as all its derivatives. Therefore by \\eqref{zetabound},\n\\begin{equation*}\n\\int_{\\widetilde{\\gamma}_k^2}\\widehat{\\psi}(\\lambda)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda) d\\lambda =\\int_{h\\widetilde{\\gamma}_k^2}\n\\widehat{\\psi}_h(z)\\frac{d}{dz}\\log\\zeta_h(z)dz=\\mathcal{O}(h^\\infty).\n\\end{equation*}\nMoreover, both of the sums\n\\begin{equation*}\n\\sum_{k\\in\\mathbb{Z}}\\int_{\\widetilde{\\gamma}_k^1}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n=-\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n\\end{equation*}\nand\n\\begin{equation*}\n\\sum_{k\\in\\mathbb{Z}}\\int_{\\widetilde{\\gamma}_k^3}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda=\\int_{\\Gamma}\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n\\end{equation*}\nconverges absolutely. Here $\\Gamma=\\bigcup_{k=-\\infty}^\\infty\\widetilde{\\gamma}_k^3$.\nOn the other hand, by the upper bound of the resonance \\eqref{resw}, the sum\n\\begin{equation*}\n\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega}}\\widehat{\\psi}(\\mu_j)\n=\\sum_{k\\in\\mathbb{Z}}\\sum_{z_j\\in\\Res_h(P)\\cap W_k}\\widehat{\\psi}_h(z_j)\n\\end{equation*}\nalso converges absolutely. Hence we have the following identity\n\\begin{equation*}\n\\int_{\\mathbb{R}+iB}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda\n=\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega}}\\widehat{\\psi}(\\mu_j)+\n\\int_{\\Gamma}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation*}\nIn other words,\n\\begin{equation*}\n\\sum_{\\mu\\in\\Res(P), \\Im\\mu>-A}\\widehat{\\psi}(\\mu)\n=\\sum_{\\gamma}\\frac{T_\\gamma^{\\#}\\delta(t-T_\\gamma)}\n{|\\det(I-\\mathcal{P}_\\gamma)|}+\\langle\\psi,F_A\\rangle\n\\end{equation*}\nwhere\n\\begin{equation}\n\\label{errorform}\n\\langle\\psi, F_A\\rangle=-\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega},\\Im\\mu_j\\leq-A}\\widehat{\\psi}(\\mu_j)+\\int_{\\Gamma}\\widehat{\\psi}(\\lambda)\n\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation}\nThis proves \\eqref{localtrace}.\n\nSo far, the distribution $F_A$ is only defined in $\\mathcal{D}'(0,\\infty)$. However, the right-hand side in \\eqref{localtrace} has an obvious extension to $\\mathbb{R}$ by zero on the negative half line as it is supported away from 0. By the polynomial upper bounds \\eqref{upperbound} on the number of resonances in the strip $\\Im\\mu>-A$, the sum\n\\begin{equation*}\nu_A(t)=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}e^{-i\\mu t}\n\\end{equation*}\nalso has an extension to $\\mathbb{R}$ which has support in $[0,\\infty)$. We only need to show that $u_A$ is of finite order: For any $\\varphi\\in C_0^\\infty(0,\\infty)$,  $k\\geqslant0$, we have\n$\\widehat{\\varphi^{(k)}}(\\lambda)=(i\\lambda)^k\\widehat{\\varphi}(\\lambda)$. Therefore we can write\n\\begin{equation*}\n\\langle u_A,\\varphi\\rangle=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}(\\mu)=\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}(i\\mu)^{-k}\\widehat{\\varphi^{(k)}}(\\mu)\n\\end{equation*}\nWhen $k$ is large, the sum\n\\begin{equation*}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}|\\mu|^{-k}\n\\end{equation*}\nconverges absolutely. Therefore we have the finite order property of $u_A$. Moreover, any two such extensions of $u_A$ are only differed by a distribution $v$ supported at $\\{0\\}$, that is,  a linear combination of delta function and its derivatives.\n\nNow we can certainly extend $F_A$ to a distribution on $\\mathbb{R}$ with support in $[0,\\infty)$. Since $\\check{v}$ is a polynomial in the whole complex plane. Therefore choice of the extension of $u_A$ does not affect the estimate on $\\widehat{F}_A$.\n\nFinally, we give the {  desired} estimate on $\\widehat{F}_A$. This follows from the fact $e^{\\eta t}F_A\\in\\mathcal{S}'$ for any $\\eta<A$ and \\cite[Theorem 7.4.2]{H}. To see this, we only need to show that we can extend \\eqref{errorform} to $\\psi\\in C^\\infty(\\mathbb{R})$ with support in $(0,\\infty)$ such that $\\varphi=e^{-\\eta t}\\psi\\in\\mathcal{S}$. If $\\psi$ has compact support, then\n\\begin{equation*}\n\\widehat{\\psi}(\\lambda)=\\int\\psi(t)e^{-it\\lambda}dt\n=\\int\\varphi(t)e^{-it(\\lambda+i\\eta)}dt=\\widehat{\\varphi}(\\lambda+i\\eta).\n\\end{equation*}\nTherefore \\eqref{errorform} can be rewritten as\n\\begin{equation}\n\\label{deffa}\n\\langle\\psi, F_A\\rangle=-\\sum_{\\mu_j\\in\\Res(P)\\cap\\widetilde{\\Omega},\\Im\\mu_j\\leq-A}\\widehat{\\varphi}(\\mu_j+i\\eta)+\\int_{\\Gamma}\\widehat{\\varphi}(\\lambda+i\\eta)\\frac{d}{d\\lambda}\\log\\zeta_1(\\lambda)d\\lambda.\n\\end{equation}\nAgain, by the estimate \\eqref{finiteorder} and the upper bound on the resonances \\eqref{upperbound}, this converges as long as $\\supp\\varphi\\subset(0,\\infty)$ which gives the definition of $e^{\\eta t}F_A$ in $\\mathcal{S}'$. The order of $\\widehat{F}_A(\\lambda)$ comes from the formula \\eqref{deffa} and \\eqref{finiteorder}, \\eqref{upperbound}.\n\n\\section{Proof of the weak lower bound on the number of resonances}\nNow we prove the weak lower bound on the number of resonances. The strategy is similar to the proof in \\cite{GZ}  and we proceed by contradiction. Let\n\\begin{equation*}\nN_A(r)=\\#(\\Res(P)\\cap\\{|\\mu|\\leq r,\\Im\\mu>-A\\})\n\\end{equation*}\nand assume that\n\\begin{equation}\n\\label{contra}\nN_A(r)\\leq P(\\delta,A)r^\\delta.\n\\end{equation}\n\nWe fix a test function $\\varphi\\in C_0^\\infty(\\mathbb{R})$ with the following properties:\n\\begin{equation*}\n\\varphi\\geqslant0, \\ \\ \\varphi(0)>0, \\ \\ \\supp\\varphi\\subset[-1,1].\n\\end{equation*}\nNext we set $\\varphi_{l,d}(t)=\\varphi(l^{-1}(t-d))$ where $d>1$ and $l<1$, so that $\\varphi_{l,d}\\in C_0^\\infty(0,\\infty)$. Therefore we can apply the local trace formula to get\n\\begin{equation}\n\\label{tr1}\n\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}_{l,d}(\\mu)\n+\\langle F_A,\\varphi_{l,d}\\rangle=\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}.\n\\end{equation}\n\nFirst, we note that by Paley-Wiener theorem,\n\\begin{equation}\n\\label{PW}\n|\\widehat{\\varphi}_{l,d}(\\zeta)|=|l\\widehat{\\varphi}(l\\zeta)e^{-id\\zeta}|\\leq\nC_N l e^{(d-l)\\Im\\zeta}(1+|l\\zeta|)^{-N},\n\\end{equation}\nfor $\\Im\\zeta\\leq0$ and any $ N \\geq 0 $.\n\nBy the assumption, we have the following estimate on the sum on the left-hand side of \\eqref{tr1},\n\\begin{equation}\n\\label{es1}\n\\begin{split}\n\\left|\\sum_{\\mu\\in\\Res(P),\\Im\\mu>-A}\\widehat{\\varphi}_{l,d}(\\mu)\\right|\\leq&\\; Cl\\int_0^\\infty(1+l r)^{-N}dN_A(r)\\\\\n\\leq&\\; Cl\\int_0^\\infty\\frac{d}{dr}[(1+l r)^{-N}]N_A(r)dr\\\\\n\\leq&\\; CP(\\delta,A)l\\int_0^\\infty\\frac{d}{dr}[(1+l r)^{-N}]r^\\delta dr\\leq Cl^{1-\\delta}.\n\\end{split}\n\\end{equation}\n\nThe remainder term $\\langle F_A,\\varphi_{l,d}\\rangle$ on the left-hand side of \\eqref{tr1} can be rewritten as\n\\begin{equation*}\n\\langle\\check{F}_A,\\widehat{\\varphi}_{l,d}\\rangle\n=\\int_{\\mathbb{R}}\\widehat{F}_A(-\\zeta)\\widehat{\\varphi}_{l,d}(\\zeta)d\\zeta.\n\\end{equation*}\nBy \\eqref{error}, we can pass the contour to $\\mathbb{R}+i(\\epsilon-A)$ to get\n\\begin{equation}\n\\label{es2}\n\\begin{split}\n|\\langle F_A,\\varphi_{l,d}\\rangle|\n\\leq &\n\\int_{\\mathbb{R}+i(\\epsilon-A)}|\\widehat{F}_A(-\\zeta)|\n|\\widehat{\\varphi}_{l,d}(\\zeta)|d\\zeta\\\\\n\\leq &\\;\nCl e^{(d-l)(\\epsilon-A)}\n\\int_{\\mathbb{R}+i(\\epsilon-A)}\\langle\\zeta\\rangle^{2n+1}(1+l|\\zeta|)^{-2n-3}d\\zeta\\\\\n\\leq &\\; Cl^{-2n-1}e^{(d-l)(\\epsilon-A)}\n\\end{split}\n\\end{equation}\nwhere we use \\eqref{PW} with $ N =2n+3$.\n\n{ \nOn the other hand, to get a lower bound of the right-hand side of \\eqref{tr1}, we fix one primitive periodic orbit $\\gamma_0$ and let $d=kT_{\\gamma_0}$, $k\\in\\mathbb{N}$. Since every term there is nonnegative, we ignore all but the term corresponding to $\\gamma_d$ which is the $k$-times iterate of $\\gamma_0$ and get\n\\begin{equation*}\n\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}\n\\geq\\frac{T_{\\gamma_d}^\\#\\varphi(0)}{|\\det(I-\\mathcal{P}_{\\gamma_d})|}=\\frac{T_{\\gamma_0}\\varphi(0)}{|\\det(I-\\mathcal{P}_{\\gamma_0}^k)|}.\n\\end{equation*}\nLet $\\lambda_1,\\ldots,\\lambda_{n-1}$ be the eigenvalues of $\\mathcal{P}_{\\gamma_0}$, then for some $\\alpha$ depending only on $\\lambda_j$'s,\n\\begin{equation*}\n|\\det(I-\\mathcal{P}_{\\gamma_0}^k)|=|(1-\\lambda_1^k)\\cdots(1-\\lambda_{n-1}^k)|\\leq Ce^{k\\alpha}=Ce^{\\theta_0d},\n\\end{equation*}\nif $\\theta_0=\\alpha\/T_{\\gamma_0}$.} \nThis gives the lower bound\n\\begin{equation}\n\\label{es3}\n\\sum_{\\gamma}\\frac{T_\\gamma^\\#\\varphi_{l,d}(T_\\gamma)}{|\\det(I-\\mathcal{P}_\\gamma)|}\\geq Ce^{-\\theta_0d}.\n\\end{equation}\n\n\n\nCombining \\eqref{es1},\\eqref{es2},\\eqref{es3}, we have the following inequality\n\\begin{equation*}\nCl^{1-\\delta}+Cl^{-2n-1}e^{(d-l)(\\epsilon-A)}\\geq Ce^{-\\theta d}.\n\\end{equation*}\nWe first choose $l=e^{-\\beta d}$, then we have\n\\begin{equation*}\nCe^{-\\beta d(1-\\delta)}+Ce^{(d-l)(\\epsilon-A)+(2n+1)\\beta d}\\geq Ce^{-\\theta_0d}.\n\\end{equation*}\nNotice that the constants $C$'s may depend on $A$, but not on $d$. If we choose $\\beta$ and $A$ large while $\\epsilon$ small so that $\\beta(1-\\delta)>\\theta_0$ and $A-\\epsilon-(2n+1)\\beta>\\theta_0$, then we get a contradiction as $d\\to\\infty$. This can be achieved when $A>A_\\delta$ where\n\\begin{equation}\n\\label{adelta}\nA_\\delta=\\theta_0(1+(2n+1)(1-\\delta)^{-1}).\n\\end{equation}\nThis finishes the proof of Theorem \\ref{thm2}.\n\n\\medskip\n\n\\noindent\n{\\bf Remark.} { \nFrom the proof, we see that the essential gap is bounded by $A_0=\\theta_0(2n+2)$, where $\\theta_0$ given above only depends on the Poincar\\'{e} map associated to a primitive periodic orbit $\\gamma_0$. More explicitly, \n$$\\theta_0=\\frac{1}{T_{\\gamma_0}}\\sum_{\\lambda\\in\\sigma(\\mathcal{P}_{\\gamma_0}):|\\lambda|>1}\\log|\\lambda|.$$\nA weaker bound not depending on the specific orbit is given by\n$\\theta_0\\leq\\theta d_u$ where $d_u=\\dim E_u$ is the dimension of the unstable fiber and $\\theta$ is the Lyapunov constant of the flow given in \\S 2.1.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}