diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzplth" "b/data_all_eng_slimpj/shuffled/split2/finalzzplth" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzplth" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\nThe Hill-Valley Evolutionary Algorithm (HillVallEA) \\cite{Maree18, Maree18b} is a real-valued multi-modal evolutionary algorithm, that automatically detects niches in the search space, based on the Hill-Valley test. This test states that two solutions belong to the same niche (valley) when there is no hill in between. To do so, a number of intermediate solutions are sampled and evaluated. Hill-Valley Clustering (HVC) is an iterative approach to efficiently cluster an entire population of solutions into niches. The resulting clusters are used to initialze a population-based core search algorithm, in this case, AMaLGaM-Univariate \\cite{bosman08} is used.\n\n\\section{Adaptations to the HillVallEA19}\nSome small adaptations have been made to HillVallEA18 \\cite{Maree18b} to further enhance its performance. Source code of HillVallEA is available at \\url{github.com\/scmaree\/HillVallEA}.\n\n\\subsection{Adaptive initial population sampling}\nIn HillVallEA, after all local optimizers have been terminated (and there is still budget remaining), a new initial population is sampled. When this population is sampled uniformly random, previously explored basins get re-explored every time a new population is initialized. To reduce this computational overhead, we store for each solution of the previous initial population to which cluster it belonged. Then, new solutions are sampled based on rejection sampling. A sample is rejected with probability $P = 0.9$ if its nearest $d+1$ solutions of the previous initial population belonged to the same cluster. In that case, it is very likely that this solution will end up exploring the same basin as the cluster of the previous generation. \n\nAdditionally, better spreading the initially sampled population has been shown to improve performance of evolutionary algorithms \\cite{wessing15}. Minimax sampling method or Latin hypercube sampling \\cite{wessing15} grow slow very quickly as the problem dimensionality or the sample size increase. We therefore use a greedy scattered subset selection method \\cite{rodrigues14}. To construct a population of $N$ solutions, we sample $2N$ using the strategy above, and use greedy scattered subset selection to reduce this to $N$ solutions.\n\n\n\\subsection{Force accept of low-fitness solutions in Hill-Valley test}\nPreviously in hill-valley clustering, all nearest-better solutions were tested with the hill-valley test. Especially for problems with a large number of low-fitness local optima, such as the Shubert function (problems 6 and 8 in the benchmark), this results in many resources spent on obtaining accurate low-fitness clusters that are later discarded. Therefore, during hill-valley clustering, the hill-valley test is only performed on solutions that belong to the fittest half of the selection, or on solutions-pairs that are more than the expected edge length (EEL) \\cite{Maree18} apart (i.e., $N_t > 1$).\n\n\n\\subsection{Recalibration of the recursion scheme}\nThe population sizing scheme within HillVallEA is parameterized as $\\xi = (N,N^{\\mbox{inc}}, N_C,N_C^{\\mbox{inc}}, )$, with initial population size $N$, population size increment $N^{\\mbox{inc}}$, cluster size $N_C$, and cluster size increment $N_C^{\\mbox{inc}}$. Previously, these parameters were set to $\\xi = (2^8d, 2, 1, 1.2)$, where $d$ is the problem dimensionality. As the initial population sampling has been adapted, setting $\\xi = (2^6, 2, 0.8, 1.1)$ was found to enhance performance of the HillVallEA. That is, using both a smaller population and smaller clusters initially, and increasing cluster size at a slower pace.\n\n\\section{Experiment Setup}\nWe evaluate the performance of HillVallEA on the test problems in the CEC2013 niching benchmark suite \\cite{CEC2013NichingCompetition}. The benchmark consists of 20 problems, as shown in Table~\\ref{tab:2dbenchmarks}, to be solved within a predefined budget in terms of function evaluations. For each of the benchmark problems, the location of the optima and the corresponding fitness values are known, however, these are only used for to measure performance, and not used during optimization. All benchmark functions are defined on a bounded domain. All experiments are repeated 50 times, and resulting performance measures are averaged over all repetitions. Note that no problem-specific parameter tuning has been performed.\n\n\\subsection{Performance Metrics}\n\\label{sec:measures}\nTwo performance measures are used, from which three scoring scenarios are computed, according to the competition guidelines. Let $\\mathcal{O}$ be the set of presumed optima obtained by an algorithm, and let $g$ be the number of distinct global optima within $\\mathcal{O}$. Finally, let $G_p$ be the number of global optima for problem $p$. Then, we define the peak ratio (PR) as $\\mbox{PR} = g \/ G_p$ and the success rate (SR) as $\\mbox{SR} = g \/ |\\mathcal{O}|$. Both measures should be maximized, with maximum $1$. From these two measures, three scoring scenarios are constructed. \n\n\\begin{enumerate}\n\\item[S1] The first scenario is simply the PR. \n\\item[S2] The second scenario is known as the static $F_1$ measure, defined as $F_1 = \\frac{2 \\cdot PR \\cdot SR}{PR + SR}$.\n\\item[S3] The third and final scenario is the dynamic $F_1$ (dyn$F_1$), which is the area under the curve of the $F_1$ over time (in number of function evaluations). For this, sort the solutions in $\\mathcal{O}$ based on the number of function evaluations $f_i$ before a solution $o_i\\in\\mathcal{O}$ was considered a global optimum, with the first-obtained solution first. Let $\\mathcal{O}_{[1:t]}$ with $t\\in[1,|\\mathcal{O}|]$ be the subset of $\\mathcal{O}$ containing the first $t$ solutions and let $B_p$ be the function evaluation budget for problem $p$. Then we can write the dyn$F_1$ as,\n$$\\mbox{dyn}F_1 = \\left(\\frac{B_p - f_{|\\mathcal{O}|}}{B_p}\\right) F_1(\\mathcal{O}) + \\sum_{i = 2}^{|\\mathcal{O}|} \\left( \\frac{f_i - f_{i-1} }{B_p}\\right) F_1(\\mathcal{O}_{[1:i-1]}).$$\n\\end{enumerate}\n\nAccording to the competition guidelines, a solution is marked as a distinct global optima for five different accuracy levels $\\varepsilon = \\{10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, 10^{-5}\\}$. Then, for each of the accuracy levels, for each problem, the scenario score is the average of the scores over the five accuracy levels.\n\n\\subsection{Algorithms}\n\\label{sec:algorithms}\nWe compare performance of HillVallEA-2019 to all algorithms that previously participated in the niching competitions held at the GECCO and CEC conferences in 2016, 2017 and 2018. The raw solution sets are used. The obtained solutions are re-evaluated and the scores under the different scenarios are computed given the definitions stated above. Note that the algorithms are not re-run. The included algorithms are,\nNEA2+ \\cite{preuss12}, RLSIS \\cite{wessing15}, RS-CMSA \\cite{ahrari17} , HillVallEA18 \\cite{Maree18b}, SDE-Ga (No known reference, developed by Jun-ichi Kushida) and finally the method that we discus in this paper, HillVallEA19. \n\n\n\\section{Results and discussion}\nTables~\\ref{tab:s1}, \\ref{tab:s2} and \\ref{tab:s3} shows the score per problem and per algorithm under respectively scenario S1, S2 and S3. Table~\\ref{tab:ranks} shows the overall score and the corresponding ranks. \n\nThe SR of HillVallEA is 1 in all cases, and almost always for RS-CMSA, which shows that the similar post-processing step that both algorithms perform is successful at removing duplicates and local optima. Problems 1-5 and 10 are fully solved by all methods in all runs for all accuracy levels (except for two runs of NEA2+), which suggesting that these problems are too simple. \n\nNo method can obtain the final two Weierstrass peaks of Composition Function 4 fully in any dimension (problem 15, 17, 19 and 20), which may indicate that this is a needle in a haystack problem. Similarly to problem 18, for which the obtained peak ratio was 0.667 for all methods (except NEA2+). Especially, SDE-Ga obtains for problems 13, 14, 16, and 18 a maximum peak ratio of 0.667, being unable to solve the Weierstrass function. \n\nTo conclude, HillVallEA19 was shown to be an improvement over HillVallEA18 under all scenarios. RS-CMSA comes directly after in all three scenarios. SDE-Ga performs well under S1 and S2, especially for problems 8 and 9, but performance deteriorates for the higher-dimensional problems. SDE-Ga obtains solutions very late in the convergence process, resulting in a very low S3 score. Overall, HillVallEA19 performs best under all scenarios.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Niching benchmark suite from the CEC2013 special session on multi-modal optimization \\cite{CEC2013NichingCompetition}. For each problem the function name, problem dimensionality $d$, number of global optima $\\#gopt$, and local optima $\\#lopt$ and budget in terms of function evaluations are given.}\n\\label{tab:2dbenchmarks}\n\\small\n\\begin{tabular}{cccccc}\n\\toprule\n\\# & Function name & $d$ & \\#gopt & \\#lopt & budget \\\\\n\\toprule\n1 & Five-Uneven-Peak Trap & 1 & 2 & 3 & 50K \\\\\n2 & Equal Maxima & 1 & 5 & 0 & 50K \\\\\n3 & Uneven Decreasing Maxima & 1 & 1 & 4 & 50K \\\\\n4 & Himmelblau & 2 & 4 & 0 & 50K\\\\\n5 & Six-Hump Camel Back & 2 & 2 & 5 & 50K \\\\\n6 & Shubert & 2 & 18 & many & 200K \\\\\n7 & Vincent & 2 & 36 & 0 & 200K \\\\\n8 & Shubert & 3 & 81 & many & 400K \\\\\n9 & Vincent & 3 & 216 & 0 & 400K \\\\\n10 & Modified Rastrigin & 2 & 12 & 0 & 200K \\\\\n11 & Composition Function 1 & 2 & 6 & many &200K \\\\\n12 & Composition Function 2 & 2 & 8 &many & 200K \\\\\n13 & Composition Function 3 & 2 & 6 &many & 200K \\\\\n14 & Composition Function 3 & 3 & 6 &many & 400K \\\\\n15 & Composition Function 4 & 3 & 8 &many & 400K \\\\\n16 & Composition Function 3 & 5 & 6 &many & 400K \\\\\n17 & Composition Function 4 & 5 & 8 &many & 400K \\\\\n18 & Composition Function 3 & 10 & 6 & many &400K \\\\\n19 & Composition Function 4 & 10 & 8 &many & 400K \\\\\n20 & Composition Function 4 & 20 & 8 &many & 400K \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S1 (peak ratio) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s1}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & & & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n p & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n 1 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 2 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 3 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 4 & 0.998 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 5 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 6 & 0.997 & 0.872 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n 7 & 0.840 & 0.920 & 0.997 & 1.000 & 1.000 & 1.000 \\\\\n 8 & 0.568 & 0.189 & 0.871 & 1.000 & 0.920 & 0.975 \\\\\n 9 & 0.552 & 0.584 & 0.730 & 0.992 & 0.945 & 0.972 \\\\\n 10 & 0.997 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 11 & 0.955 & 1.000 & 0.997 & 0.733 & 1.000 & 1.000 \\\\\n 12 & 0.796 & 0.950 & 0.948 & 0.800 & 1.000 & 1.000 \\\\\n 13 & 0.947 & 0.938 & 0.997 & 0.667 & 1.000 & 1.000 \\\\\n 14 & 0.813 & 0.799 & 0.810 & 0.667 & 0.917 & 0.923 \\\\\n 15 & 0.721 & 0.720 & 0.748 & 0.750 & 0.750 & 0.750 \\\\\n 16 & 0.683 & 0.675 & 0.667 & 0.667 & 0.687 & 0.723 \\\\\n 17 & 0.723 & 0.738 & 0.703 & 0.703 & 0.750 & 0.750 \\\\\n 18 & 0.650 & 0.667 & 0.667 & 0.667 & 0.667 & 0.667 \\\\\n 19 & 0.505 & 0.515 & 0.502 & 0.555 & 0.585 & 0.593 \\\\\n 20 & 0.398 & 0.422 & 0.482 & 0.460 & 0.482 & 0.480 \\\\\n\n\\bottomrule\n avg & 0.807 & 0.800 & 0.856 & 0.833 & 0.885 & 0.892 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S2 (static $F_1$) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s2}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & & & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n \\# & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n 1 & 1.000 & 0.993 & 0.996 & 1.000 & 1.000 & 1.000 \\\\\n 2 & 1.000 & 0.993 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 3 & 1.000 & 0.993 & 0.987 & 1.000 & 1.000 & 1.000 \\\\\n 4 & 0.960 & 0.978 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 5 & 0.947 & 0.949 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 6 & 0.997 & 0.924 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n 7 & 0.614 & 0.947 & 0.999 & 1.000 & 1.000 & 1.000 \\\\\n 8 & 0.723 & 0.315 & 0.931 & 1.000 & 0.958 & 0.987 \\\\\n 9 & 0.646 & 0.733 & 0.844 & 0.996 & 0.972 & 0.986 \\\\\n 10 & 0.997 & 0.988 & 1.000 & 1.000 & 1.000 & 1.000 \\\\\n 11 & 0.971 & 0.992 & 0.998 & 0.733 & 1.000 & 1.000 \\\\\n 12 & 0.881 & 0.967 & 0.972 & 0.800 & 1.000 & 1.000 \\\\\n 13 & 0.966 & 0.941 & 0.998 & 0.723 & 1.000 & 1.000 \\\\\n 14 & 0.894 & 0.865 & 0.893 & 0.799 & 0.953 & 0.958 \\\\\n 15 & 0.835 & 0.831 & 0.855 & 0.857 & 0.857 & 0.857 \\\\\n 16 & 0.811 & 0.795 & 0.800 & 0.800 & 0.813 & 0.837 \\\\\n 17 & 0.838 & 0.843 & 0.823 & 0.824 & 0.857 & 0.857 \\\\\n 18 & 0.787 & 0.794 & 0.800 & 0.800 & 0.800 & 0.800 \\\\\n 19 & 0.668 & 0.676 & 0.668 & 0.712 & 0.735 & 0.741 \\\\\n 20 & 0.563 & 0.590 & 0.650 & 0.627 & 0.650 & 0.647 \\\\\n\\bottomrule\n avg & 0.855 & 0.855 & 0.911 & 0.884 & 0.930 & 0.934 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Scores obtained under Scenario S3 (dynamic $F_1$) for each of the algorithms per problem $p$. Higher is better, 1 is the maximum score. Scores are averaged over 50 runs and five accuracy levels. Average (avg.) score computed over all 20 problems. }\n\\label{tab:s3}\n\\smaller\n\\begin{tabular}{c|cccc|cc}\n\\toprule\n& & & & & \\multicolumn{2}{|c}{HillVallEA} \\\\\n \\# & NEA2+ & RLSIS & RS-CMSA & SDE-Ga & HillVallEA18 & HillVallEA19 \\\\ \n\\toprule\n 1 & 0.982 & 0.987 & 0.911 & 0.572 & 0.992 & 0.995 \\\\\n 2 & 0.917 & 0.983 & 0.959 & 0.626 & 0.987 & 0.989 \\\\\n 3 & 0.895 & 0.982 & 0.949 & 0.456 & 0.992 & 0.994 \\\\\n 4 & 0.959 & 0.975 & 0.932 & 0.528 & 0.973 & 0.977 \\\\\n 5 & 0.971 & 0.979 & 0.944 & 0.416 & 0.982 & 0.983 \\\\\n 6 & 0.917 & 0.699 & 0.933 & 0.525 & 0.951 & 0.966 \\\\\n 7 & 0.659 & 0.855 & 0.928 & 0.413 & 0.960 & 0.966 \\\\\n 8 & 0.464 & 0.209 & 0.715 & 0.324 & 0.750 & 0.805 \\\\\n 9 & 0.550 & 0.573 & 0.654 & 0.157 & 0.791 & 0.818 \\\\\n 10 & 0.988 & 0.983 & 0.984 & 0.539 & 0.979 & 0.982 \\\\\n 11 & 0.961 & 0.980 & 0.967 & 0.254 & 0.983 & 0.983 \\\\\n 12 & 0.829 & 0.859 & 0.909 & 0.303 & 0.958 & 0.963 \\\\\n 13 & 0.932 & 0.897 & 0.923 & 0.372 & 0.957 & 0.964 \\\\\n 14 & 0.862 & 0.834 & 0.832 & 0.434 & 0.867 & 0.882 \\\\\n 15 & 0.806 & 0.782 & 0.785 & 0.544 & 0.824 & 0.836 \\\\\n 16 & 0.799 & 0.788 & 0.777 & 0.449 & 0.776 & 0.793 \\\\\n 17 & 0.803 & 0.779 & 0.688 & 0.419 & 0.787 & 0.816 \\\\\n 18 & 0.719 & 0.748 & 0.730 & 0.160 & 0.721 & 0.763 \\\\\n 19 & 0.624 & 0.627 & 0.560 & 0.035 & 0.634 & 0.656 \\\\\n 20 & 0.491 & 0.496 & 0.502 & 0.003 & 0.514 & 0.524 \\\\\n\\bottomrule\n avg & 0.806 & 0.801 & 0.829 & 0.376 & 0.869 & 0.883 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\caption{Algorithm ranks based on the three scenarios. }\n\\label{tab:ranks}\n\\smaller\n\\begin{tabular}{c|cc|cc|cc|c}\n\\toprule\nAlgorithm & S1 & rank & S2 & rank & S3 & rank & average rank \\\\\n\\toprule\n RLSIS & 0.800 & 6& 0.855 &5 & 0.801 & 5& 5.3\\\\\n NEA2+ & 0.807 & 5& 0.855 & 6 & 0.806 &4 & 5\\\\ \n SDE-Ga & 0.833 & 4& 0.884 & 4 & 0.376 & 6& 4.7\\\\ \n RS-CMSA & 0.856 &3 & 0.911 & 3& 0.829 & 3& 3\\\\ \n HillVallEA18 & 0.885 &2 & 0.930 & 2 & 0.869 &2 & 2\\\\ \n HillVallEA19 & 0.892& 1& 0.934 & 1 & 0.883 & 1& 1\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Pure outputs}\\label{s:uno}\n\nProving that proper quantum mover $\\mbox{PQM}(p)$\ncannot be implemented on a physical system S is equivalent to showing the following:\n\n{\\em Given $p \\in [0,1[$ there is no deterministic physical process \n$\\cal E$ which transforms an unknown pure vector $|\\psi\\ran$ of a generic \nHilbert space ${\\cal H}_S$ according to the mapping\n\\begin{eqnarray}\n\t|\\psi\\ran \\; \\stackrel{\\cal E}{\\longrightarrow} \\; |\\psi_p\\ran \\label{THETRA} \\;,\n\\end{eqnarray}\nwhere $|\\psi_p \\ran$ is a pure state of ${\\cal H}_S$ depending on $|\\psi\\rangle$ \nwhich satisfies the identity\n\\begin{eqnarray}\n\t|\\bra{\\psi} \\psi_p\\ran |^2= p \\;.\n\\label{unnot}\n\\end{eqnarray}}\nThe adjective ``deterministic'' in the hypothesis plays an important role.\nOn one hand it allows us to exclude from the analysis those processes which \nimplement~(\\ref{THETRA}) probabilistically, i.e. allowing some probability of failure.\nAs in the case of the cloning and the U-NOT machines~\\cite{DEMA,PASQ,BUZEK,BUZEK1}\nsuch transformations are not prohibited by quantum mechanics and the above no-go theorem \ndoes not apply to them. In addition the term ``deterministic'' indicates that here \nwe are considering scenarios where the output state $|\\psi_p\\ran$ associated with \n$|\\psi\\ran$ is uniquely determined: i.e. if we run the transformation many times \non various copies of a certain input $|\\psi\\ran$, the output state will be always \nthe same vector $|\\psi_p\\ran$. This point is important since the vectors $|\\psi_p\\ran$ \nwhich satisfy the property~(\\ref{unnot}) form a dense subset of ${\\cal H}_S$. A transformation \n(\\ref{THETRA}) which randomly picks up one of these $|\\psi_p\\ran$ is described by \na $\\mbox{GQM}(p)$ map whose output state is not pure: these operations will be discussed in the \nnext section where we will see that the possibility of implementing them depends \nexplicitly on the dimension of ${\\cal H}_S$.\n\nThe most general deterministic physical transformations~(\\ref{THETRA}) \nof the system S are the linear, completely positive,\ntrace preserving (LCPT) maps $\\cal E$ which for all $|\\psi\\ran \\in {\\cal H}_S$ \ngive\n\\begin{eqnarray}\n\t{\\cal E}(|\\psi\\rangle \\langle \\psi|) \n\t&=&| \\psi_p \\rangle \\langle \\psi_p | \n\\label{cpt}\\;,\n\\end{eqnarray}\nwith $|\\psi_p\\ran$ unit vector of ${\\cal H}_S$ such that Eq.~(\\ref{unnot}) \napplies~\\cite{NOTA0}.\nIn the qubit case ($d=2$) and for $p<1\/3$ \nthe impossibility of implementing the transformation~(\\ref{cpt}) \ncan be established by considering the maximal fidelity achievable in performing \nan approximated U-NOT~\\cite{BUZEK,PASQ}. In fact, by expressing the trace of \n${\\cal E}(|\\psi\\rangle \\langle \\psi|)$ in the orthonormal basis $|\\psi\\rangle$, \n$|\\psi_\\perp\\rangle$ of ${\\cal H}_S$ we get \n\\begin{eqnarray}\n\t\\langle \\psi_\\perp | {\\cal E}(|\\psi\\rangle\n\t\\langle \\psi|) |\\psi_\\perp\\rangle = 1 - p\\;.\n\\label{ULTIMA}\n\\end{eqnarray}\nThe thesis then follows by observing that the left-hand side term of this \nexpression is the fidelity of the output state of $\\cal E$ with respect to \nthe orthogonal vector $|\\psi_\\perp\\rangle$ which according to~\\cite{BUZEK,PASQ} is\nupper bounded by $2\/3$. \n\nThe above argument does not apply if $d > 2$ and for generic $p$. One can still \nprove that no transformations~(\\ref{cpt}) exist by considering a unitary \nrepresentation~\\cite{CHUANG} of $\\cale$, i.e.\n \\begin{eqnarray}\n{\\cal E}(| \\psi\\rangle \\langle \\psi|) = \n\\mbox{Tr}_A \\big[ \\; U \\; \\left( \n\\ket{\\psi}\\bra{\\psi} \\otimes \\ket{0}_A\\bra{0}\\right) \\; U^\\dag \\;\\big]\\;, \\label{cpt1}\n\\end{eqnarray}\nwhere $U$ is a unitary operator acting on S and on an ancillary system A, \nwhile $|0\\ran_A$ is a fiduciary state of A. According to Eq.~(\\ref{cpt}) the pure states \nof ${\\cal H}_S$ are mapped into pure states of the same space, this implies that\nthe state $U \\; \\ket{\\psi} \\otimes \\ket{0}_A$ cannot be entangled, i.e.\n\\begin{eqnarray}\nU \\; \\ket{\\psi} \\otimes \\ket{0}_A = \\ket{\\psi_p} \\otimes \\ket{\\phi(\\psi)}_A \\label{cpt2}\n\\end{eqnarray}\nwith $\\ket{\\phi(\\psi)}_A$ a pure state of A which, in principle, depends on the input \nstate $|\\psi\\ran$ of S. By applying this relation to the unit vector\n\\begin{eqnarray}\n|\\psi\\ran = \\alpha \\; |k\\ran + \\beta \\; |k^\\prime\\ran\\label{VETTORI}\\;,\n\\end{eqnarray}\n(with $|k\\ran$, $|k^\\prime\\ran$ orthonormal states of ${\\cal H}_S$ and $\\alpha$, $\\beta$ complex\nsuch that $|\\beta|^2=1-|\\alpha|^2$) and using the linearity of $U$ one gets\n\\begin{eqnarray}\n\\ket{\\psi_p} \\otimes \\ket{\\phi(\\psi)}_A = \n\\alpha\\; |k_p\\ran \\otimes \\ket{\\phi(k)}_A + \\beta \\; |k_p^\\prime\\ran \\otimes\n\\ket{\\phi(k^\\prime)}_A \\nonumber \\\\\n\\label{cpt3}\n\\end{eqnarray}\nwith $|k_p\\ran$, $\\ket{\\phi(k)}_A$ and $|k_p^\\prime\\ran$, $\\ket{\\phi(k^\\prime)}_A$\ndefined as in Eq.~(\\ref{cpt2}). In particular $|k_p\\ran$ and $|k_p^\\prime\\ran$ are, \nrespectively, the {\\em unit vectors} of ${\\cal H}_S$ which describes the \noutputs of the map $\\cale$ associated with $\\ket{k}$, $\\ket{k^\\prime}$: by hypothesis \nthey satisfy the relations $\\bra{k_p}k\\ran = \\sqrt{p} e^{i \\phi}$, \n$\\bra{k^\\prime_p}k^\\prime \\ran = \\sqrt{p} e^{i \\phi^\\prime}$ for some \nphases $\\phi$ and $\\phi^\\prime$.\nBy projecting the left hand side of Eq.~(\\ref{cpt3}) on $|\\psi\\ran$ we derive the following \nidentity\n\\begin{eqnarray}\np&=& \\| \\big( |\\alpha|^2 \\sqrt{p} \\; e^{i \\phi} +\n \\alpha \\beta^*\\; \\bra{k^\\prime} k_p\\ran \\; \\big) |\\phi(k)\\ran_A \\nonumber \\\\\n&&+ \\big(|\\beta|^2 \\sqrt{p} \\; e^{i \\phi^\\prime}\n+ \\alpha^*\\beta\\; \\bra{k} k^\\prime_p\\ran \\; \\big) |\\phi(k^\\prime)\\ran_A \\|^2 \\;,\\label{cpt10}\n\\end{eqnarray}\nwhich must hold for all the complex amplitudes $\\alpha$ and $\\beta$.\nOne can verify that \nthis condition is satisfied if and only if\n\\begin{eqnarray}\n \\bra{k^\\prime} k_p\\ran = \\bra{k} k^\\prime_p\\ran = 0 \n\\label{cpt11}\\;,\n\\end{eqnarray}\n(see Appendix~\\ref{appendixA} for details).\nThe impossibility of the transformation $\\cal E$\nfinally follows by observing that Eq.~(\\ref{cpt11}) is in contradiction\nwith $|k_p\\ran$ being a \nunit vector of ${\\cal H}_S$. In fact since Eq.~(\\ref{cpt11})\napplies for all $|k\\ran$ and\n$|k^\\prime\\ran$ orthonormal we have\n\\begin{eqnarray}\n\\| \\ket{k_p} \\|^2 = |\\bra{k} k_p\\ran|^2 + \\sum_{k^\\prime} \n|\\bra{k^\\prime} k_p\\ran|^2 = p \\;, \\label{NUOVA}\n\\end{eqnarray}\nwhere the sum is performed over the set of vectors~$\\{|k^\\prime\\ran\\}_{k^\\prime}$ which\ncomplete $|k\\ran$ to an orthonormal basis of ${\\cal H}_S$. \nIt should be stressed that the above derivation holds for all value of $p<1$ and for \nHilbert space ${\\cal H}_S$ (also infinite-dimensional Hilbert spaces).\n\n\\section{Mixed outputs}\\label{s:due}\n\nConsider now the case of generalized \nquantum movers $\\mbox{GQM}(p)$ with not necessarily pure\noutput states $\\rho_b$.\nThese mappings include all those transformations which implement a proper quantum mover\ntransformation~(\\ref{THETRA}) \nprobabilistically, i.e. where the output state is randomly selected among the set\nof the vectors $|\\psi_p\\ran$ which satisfies the condition of Eq.~(\\ref{unnot}). \nDifferently from the previous case, these generalized quantum \nmovers are \nallowed at least for some restricted values of $p$. \n What is more interesting \nis that the generalized quantum movers $\\mbox{GQM}(p)$ can be implemented by means of \nLCPT maps only if $p$ exceeds a threshold $p_{\\mbox{\\small t}}(d)$ which explicitly \ndepends on $d$. \nHere we provide both upper and \nlower bounds for this threshold, i.e.\n\\begin{eqnarray}\n 1\/(2 d-1)\\; \\leqslant \\; p_{\\mbox{\\small t}}(d) \\;\\leqslant \\; 1\/(d +1) \\label{UPLOW}\\;.\n\\end{eqnarray}\nSuch bounds coincide for $d=2$ so that $p_{\\mbox{\\small t}}(2)=1\/3$\nand the threshold of the qubit case is completely determined by the above inequalities.\nAs in the case of Eq.~(\\ref{ULTIMA}) this is \na direct consequence of the $2\/3$ bound on the optimal fidelity associated with \nan approximate realization of a U-NOT~\\cite{BUZEK,PASQ}.\n\nTo derive the upper bound of Eq.~(\\ref{UPLOW}) it is sufficient to provide a LCPT \ntransformation that satisfies Eq.~(\\ref{movers1}) for $p\\geqslant 1\/(d+1)$. \nFor instance let us analyze the linear mapping which takes any trace-class \noperator $\\Lambda$ of ${\\cal H}_S$ to\n\\begin{eqnarray}\n\t{\\cal N}(\\Lambda) \\equiv \\frac{dp -1}{d-1} \\Lambda + \n\t\\frac{1-p}{d-1} \\mbox{Tr}[ \\Lambda ] \\; \\openone \\;,\n\\label{esempio}\n\\end{eqnarray}\nwhere $\\openone$ is the identity operator of ${\\cal H}_S$. The transformation ${\\cal N}$ \nbelongs to the set of universal inverter super-operators that have been introduced in \nRefs.~\\cite{BUZEK1,HORO} in the context of entanglement criteria for bipartite systems.\nMoreover, for $p=1\/(d+1)$ Eq.~(\\ref{esempio}) gives the approximated U-NOT \nmapping~\\protect\\cite{BUZEK,PASQ,DEMA} which provides optimal output fidelity for $d=2$ \nand, in the case $d>2$, provides optimal fidelity in the \nclass of LCPT maps which are covariant \nwith respect to arbitrary unitary transformations of ${\\cal H}_S$~\\cite{BUZEK1}.\nOne easily verifies that the transformation (\\ref{esempio}) \nsatisfies the condition~(\\ref{movers1}) \nfor all~$|\\psi\\ran$. However, it turns out~\\cite{BUZEK1} that the map ${\\cal N}$ \nis completely positive only for \n$p\\geqslant \\; 1\/(d +1)$ and this fixes the upper bound for the threshold \n$p_{\\mbox{\\small t}}(d)$.\nIt should be stressed that in agreement with the no-go theorem \non the deterministic quantum movers, the map ${\\cal N}$ \nof Eq.~(\\ref{esempio}) transforms pure states of ${\\cal H}_S$ in mixed outputs.\nIn fact, the purity of the output state is\n$$\\mbox{Tr} \\left[ \\; \n{\\cal N}(|\\psi\\ran\\lan\\psi|)^2 \\; \\right] = \\left(\\frac{1-p}{d-1}\\right)^2 +\np^2, $$\nwhich is strictly less than $1$ for $p\\geqslant \\; 1\/(d +1)$ and $d\\geqslant 2$. \n\n\nTo derive the lower bound of Eq.~(\\ref{UPLOW})\n assume that a LCPT map $\\cale$ satisfying Eq.~(\\ref{movers1}) for all \ninput states $|\\psi\\ran$ exists.\nBy applying this relation to the vector of Eq.~(\\ref{VETTORI}) we get the\nfollowing equation\n\\begin{eqnarray}\np&=&|\\alpha|^2 \\bra{k}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k\\ran\n+ |\\beta|^2 \\bra{k^\\prime}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k^\\prime\\ran \\nonumber \\\\\n&&+\\; [\\; \\alpha \\beta^* \\bra{k^\\prime}\\: \\cale (|\\psi\\ran\\lan\\psi|)\\; |k\\ran + \\mbox{c.c.} \\; ]\\;,\n\\label{equazione}\n\\end{eqnarray} \nwith\n\\begin{eqnarray} \n\\cale (|\\psi\\ran\\lan\\psi|) &\\equiv& |\\alpha|^2 \\cale (|k\\ran\\lan k|) \n+ |\\beta|^2 \\cale (|k^\\prime \\ran\\lan k^\\prime |) \\nonumber\\\\\n&&+\\; \n [ \\; \\alpha \\beta^* \\cale (|k\\ran\\lan k^\\prime|) + \\mbox{h.c.} \\; ] \\;.\\label{equazione1}\n\\end{eqnarray}\nAgain by requiring Eq.~(\\ref{equazione}) to hold for all $\\alpha$ and $\\beta$ we get the following \nconstraints on the map $\\cale$ which apply for all $|k\\ran$ and $|k^\\prime\\ran$\northonormal \n\\begin{eqnarray}\n\\left\\{ \n\\begin{array}{l}\n2p = \\lan k k^\\prime k^\\prime k\\ran\n+ \\lan k^\\prime k k k^\\prime \\ran\n+ (\\; \\lan k k k^\\prime k^\\prime \\ran + c.c \\;)\\\\\n\\lan k k^\\prime k k^\\prime \\ran= 0 \\\\\n\\lan k k k k^\\prime \\ran +\\lan k k^\\prime k k\\ran \n =0\n \\;,\n\\end{array}\\right. \\label{equo}\n\\end{eqnarray}\nwhere we defined the quantities\n$\\lan j k \\ell m \\ran \\equiv \\bra{j} \\cale (|k \\ran\\lan \\ell|) |m\\ran \n= \\lan m \\ell k j \\ran^*$ (see Appendix~\\ref{appendixB}).\nThe previous equations can be supplemented by another constraint which can be obtained \nby applying $(\\cale_S\\otimes{\\cal I}_A)$ \nto the entangled state \n$$|\\Psi\\ran_{SA} \\equiv \\frac{1}{\\sqrt{d}}\\sum_{k=1}^d |k\\ran_S\\otimes |k\\ran_A,$$\nand requiring its trace \nto be properly normalized to 1\n[In the above expression $\\{|k\\ran_{S,A}\\}_k$ are orthonormal bases of \n${\\cal H}_S$ and ${\\cal H}_A$,\nrespectively, and ${\\cal I}_A$ is the identity mapping on the ancillary\nspace ${\\cal H}_A$].\n This yields\n\\begin{equation}\n1= \\frac{1}{d} \\sum_{k=1}^{d}\\sum_{k^\\prime=1}^d \\lan k^\\prime kk k^\\prime \\ran \n = p + \\sum_{k \\neq k^\\prime} \n\\lan k^\\prime kk k^\\prime \\ran \/d \\;,\n\\label{traccia} \n\\end{equation} \nwhere we used the hypothesis~(\\ref{movers1}) to write \n$\\lan k kk k \\ran = p$ for all $|k\\ran$.\nEquation~(\\ref{traccia}) can be further simplified by replacing the first relation\nof the system~(\\ref{equo}). This gives \n$$ \\sum_{k \\neq k^\\prime} \n\\lan k k k^\\prime k^\\prime \\ran = d(d p -1).$$\nThe lower bound of Eq.~(\\ref{UPLOW}) finally follows by comparing this\nrelation with the inequality\n$\n|\\lan k k k^\\prime k^\\prime\\ran | \\leqslant p\n$, obtained by introducing a Kraus decomposition\nof the LCPT map $\\cale$ \nand applying the Cauchy-Schwarz inequality to the left hand side term~\\cite{FOOTNOTE4}. \n\nWe finally comment on the relations of $\\mbox{GQM}(p)$ with the U-NOT. One can easily verify \nthat the set of the linear transformations which implement $\\mbox{GQM}(p)$\nis closed under convex combination, e.g. given ${\\cal E}_{p_i}$ LCPT implementations \nof $\\mbox{GQM}(p_i)$ on S and given $\\{\\lambda_i\\}_i$ a set of probabilities,\nthe transformation $\\sum_i \\lambda_i {\\cal E}_{p_i}$ \nis a LCPT map which\nimplements the generalized mover $\\mbox{GQM}(p)$ on S \nwith $p=\\sum_i \\lambda_i p_i$ being the\naverage of the $p_i$'s.\nThis observation allows us to establish that if the U-NOT could be implemented through \na LCPT transformation, then convex combinations of such a map and the identity map will \nimplement the $\\mbox{GQM}(p)$ for $p<1$.\n\n\n\\section{Quantum Movers and Entanglement}\\label{s:tre}\nIn this last section we discuss an application of quantum movers in the\ncontext of entanglement witness. \nA central issue in quantum information \nis to determine whether or not a given density matrix $R$ of a \ncomposite system S+A is entangled. The identification of classes of impossible \ntransformations \n can become a \npowerful tool in this respect~\\cite{HORO3}. In our case, \none can construct a one parameter set of positive \n{\\em but} not completely positive linear maps $\\{{\\cal M}^{(p)}\\}_{p}$ which \nimplement the generalized quantum mover transformation $\\mbox{GQM}(p)$ on the system S.\nThe presence of entanglement in $R$ will then be detected by \nlooking for the presence of negative eigenvalues in ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R)$ \nfor some values of $p$. \nMoreover, one can use $p$ as an effective ``measure'' of the entanglement in $R$ as one \nexpects intuitively only highly entangled states to feel the non complete positivity of \nthe transformation ${\\cal M}^{(p)}$ with $p\\sim 1$. A suitable choice for the operators \n$\\{{\\cal M}^{(p)}\\}_{p}$ is obtained by adding to the transformations~(\\ref{esempio}) \na linear term which gives \nnull contribution to Eq.~(\\ref{movers1}), e.g.\n\\begin{eqnarray}\n{\\cal M}^{(p)} (\\Lambda) \\equiv \\frac{dp -1}{d-1} \\Lambda + \n\\frac{1-p}{d-1} \\mbox{Tr}[ \\Lambda ]\\; \\openone + i [ \\Lambda, \\Theta ]\n\\label{NEWesempio1}\n\\end{eqnarray}\nwith $\\Theta$ being a Hermitian operator on ${\\cal H}_S$. \nFor the sake of simplicity let us consider the $d=2$ case where we can select \n$\\Theta = \\sqrt{p(1-p)} \\; \\sigma_{\\hat{n}}$ with $\\sigma_{\\hat{n}}$ being the Pauli \noperator along the $\\hat{n}$ direction. With this choice Eq.~(\\ref{NEWesempio1}) yields a \nmapping which, for all $p\\in[0,1[$ is positive but not completely positive (this can be \nverified for instance by a direct inspection of the corresponding Choi \nmatrix~\\cite{CHOI}). As a test consider the set of Werner states~\\cite{WER}\n\\begin{eqnarray}\nR_q \\equiv \\frac{1-q}{4} \\, \\openone_S \\otimes \\openone_A \n\\, + \\, q \\, |\\Psi_-\\rangle_{SA} \\langle \\Psi_-| \\; \n\\label{werner},\n\\end{eqnarray}\nwhich have concurrence~\\cite{CONC} $C= \\max \\{ 0, (3q-1)\/2\\}$ and are hence \nseparable if and only if $q\\leqslant 1\/3$ \n[in the above expression $|\\Psi_-\\rangle = (|01\\rangle_{SA} - |10\\rangle_{SA})\/\\sqrt{2}$\nis the singlet state].\nThe eigenvalues of ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R_q)$ can\nbe easily computed: three of them are always positive and the fourth\nis equal to\n\\begin{eqnarray}\n\\lambda(q,p)\\equiv q(p-(3q-1)\/2q)\/2\\;. \n\\end{eqnarray}\nFor $R_q$ separable (i.e. $q\\leqslant 1\/3$) this quantity is positive for all $p$. \nHowever for $R_q$ entangled (i.e. $q> 1\/3$) $\\lambda(q,p)$ is negative for all \n$p$ smaller than the critical value $(3 q -1)\/2q=(3\/2) [ 1-1\/(2 C+1)]$ which\nmonotonically increases with $C$. As intuitively expected the more entanglement is in $R_q$\nthe more transformations ${\\cal M}_S^{(p)}\\otimes {\\cal I}_A(R_q)$ \nare able to detect it. \n\n\\section{Conclusions}\\label{s:conc}\nWe have introduced a new class of impossible\nquantum operations, the quantum movers which include, as a \nspecial case, the U-NOT transformation of Refs.~\\cite{BUZEK,PASQ}.\nWe proved that it is not possible to transform an unknown state $|\\psi\\ran$\nof a system S under the requirement that the final output configuration $|\\psi_p\\ran$\nhas a fixed (non unitary) component along the direction of the initial state.\nBy relaxing the requirement on the purity of the final configuration of the system,\nthere exists a threshold $p_{\\mbox{\\small t}}(d)$ for the value of the input-output fidelity \nabove which mixed quantum movers can be implemented. We related the movers to the U-NOT\nand we showed an application of these transformation in the context of separability\ncriteria.\nThe approach introduced in this work can be further extended. One can, for example, \nanalyze the case in which the output fidelity of a quantum mover is bounded \nin a given interval, e.g. $F\\in [p_1,p_2]$. Moreover, as in the case of a U-NOT\ntransformation, \nit may be interesting to analyze under which conditions the $\\mbox{GQM}(p)$ with $p$ \nbelow the threshold value $p_t(d)$ can be realized approximatively.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nUltra high energy (UHE) cosmic neutrinos play a key role in the\ndetermination of the composition of the ultra high energy cosmic\nrays (UHECRs) and their origin. UHE neutrinos are expected to be\nproduced in astrophysical sources in the decays of charged pions\ncreated in the interactions of UHECRs with matter or radiation. They\ncan be also produced via interaction of the UHECRs with the cosmic\nmicrowave background during propagation to the Earth (cosmogenic\nneutrinos). UHE cosmic neutrinos are not deviated by magnetic fields\nand could point back to their sources.\n\nRecently, three neutrinos of energy 1-2 PeV, as well as tens of\nneutrinos above 10 TeV were detected with the IceCube experiment\n\\cite{IceCube:2014}. The cosmic neutrinos with energies near 1 EeV\nare detectable with the Surface Detector (SD) of the Pierre Auger\nObservatory (PAO) \\cite{PAO}. In order to isolate neutrino-induced\nevents at the SD of the PAO, it is necessary to look for deeply\npenetrating quasi-horizontal (inclined) air showers\n\\cite{Berezinsky:1969}-\\cite{Zas:2005}. The PAO can efficiently\nsearch for two types of neutrino-induced inclined air showers (see\nfig.~\\ref{nu_events}):\n\\begin{enumerate}\n \\item\nDownward-going (DG) neutrino-induced showers. They are initiated by\nneutrinos moving with large zenith angle $\\theta$ which interact in\nthe atmosphere close to the SD. At the PAO the search is restricted\nto showers with $\\theta > 60^\\circ$ \\cite{Auger:2015}. Note that the\nbackground from hadronic showers above $10^{17}$ eV is\n$\\mathrm{O}(1)$ in 20 years, and it is negligible above $10^{19}$ eV\n\\cite{Anchordoqui:2010}.\n \\item\nEarth-skimming (ES) showers induced by upward tau neutrinos. They\ncan interact in the Earth's crust producing tau leptons. The latter\nare efficiently produced at zenith angles $90^\\circ < \\theta <\n95^\\circ$ \\cite{Auger:2015}. The tau leptons escape the Earth and\ndecay in the atmosphere close to the SD\n\\cite{Bertou:2002}-\\cite{Feng:2002}.\n\\end{enumerate}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm,clip]{nu_events}\n\\caption{Different types of showers induced by DG and ES neutrinos\n(fig.~1 in \\cite{Auger:2011}).}\n\\label{nu_events}\n\\end{figure}\n\nRecently, the Auger Collaboration reported on searches for DG\nneutrinos in the zenith angle bins $60^\\circ - 75^\\circ$ and\n$75^\\circ - 90^\\circ$, as well as for ES neutrinos. The data were\ncollected by the SD of the PAO from 1 January 2004 until 20 June\n2013 \\cite{Auger:2015}\n\\footnote{This search period is equivalent of 6.4 years of a\ncomplete Auger SD working continuously \\cite{Auger:2015}.}\nNo neutrino candidates were found. Assuming the diffuse flux of UHE\nneutrinos to be $dN\/dE_\\nu = k E_\\nu^{-2}$ in the energy range\n$1.0\\times10^{17}$ eV - $2.5\\times10^{19}$ eV, the stringent limit\nwas obtained:\n\\begin{equation}\\label{Auger_bound}\nk < 6.4 \\times 10^{-9} \\mathrm{\\ GeV \\ cm^2 \\ s^{-1} \\ sr^{-1}} \\;.\n\\end{equation}\nThis Auger limit is a factor 3.64 below the Waxman-Bachall bound on\nneutrino production in optically thin astrophysical sources\n\\cite{WB:2001}:\n\\begin{equation}\\label{WM_bound}\nE_\\nu^2 \\frac{dN}{dE_\\nu} = 2.33 \\times 10^{-8} \\mathrm{\\ GeV \\ cm^2\n\\ s^{-1} \\ sr^{-1}} \\;.\n\\end{equation}\n\nIn the Standard Model (SM) neutrino-nucleon cross sections are\nexpected to be very small in comparison with hadronic cross sections\neven at UHEs \\cite{Sarkar:2008}. That is why, the UHE cosmic\nneutrinos can be regarded as unique probes of new interactions. In\nthe present paper a theory with an extra dimension (ED) is\nconsidered to be a ``new physics'' theory. We will see that effects\ncoming from the ED can be significant or even dominant in the $\\nu\nN$-scattering at UHEs.\n\n\n\\section{General solution for the Randall-Sundrum metric}\n\\label{sec:RS}\n\nIn ref.~\\cite{Randall:1999} the 5-dimensional space-time with\nnon-factorizable geometry was suggested as an alternative to the\nmodels with flat extra dimensions. The Randall-Sundrum (RS) model\n\\cite{Randall:1999} predicts an existence of heavy Kaluza-Klein (KK)\nexcitations (massive gravitons). These massive KK resonances are\nintensively searched for at the LHC.\n\nThe RS scenario is described by the following background warped\nmetric\n\\begin{equation}\\label{RS_background_metric}\n\\quad ds^2 = e^{-2 \\sigma (y)} \\, \\eta_{\\mu \\nu} \\, dx^{\\mu} \\,\ndx^{\\nu} - dy^2 \\;,\n\\end{equation}\nwhere $\\eta_{\\mu\\nu}$ is the Minkowski tensor with the signature\n$(+,-,-,-)$, and $y$ is an extra coordinate. The periodicity\ncondition $y=y + 2\\pi r_c$ is imposed, and the points $(x_\\mu,y)$\nand $(x_\\mu,-y)$ are identified. Thus, one has a model of gravity in\nthe AdS$_5$ space-time compactified to the orbifold $S^1\\!\/Z_2$. The\norbifold has two fixed points, $y=0$ and $y=\\pi r_c$. It is assumed\nthat there are two branes located at these points. All the SM fields\nlive on one of these branes.\n\nThe classical action of the RS scenario is given by\n\\cite{Randall:1999}\n\\begin{align}\\label{action}\nS &= \\int \\!\\! d^4x \\!\\! \\int_{-\\pi r_c}^{\\pi r_c} \\!\\! dy \\,\n\\sqrt{G} \\, (2 \\bar{M}_5^3 \\mathcal{R}\n- \\Lambda) \\nonumber \\\\\n&+ \\int \\!\\! d^4x \\sqrt{|g^{(1)}|} \\, (\\mathcal{L}_1 - \\Lambda_1) +\n\\int \\!\\! d^4x \\sqrt{|g^{(2)}|} \\, (\\mathcal{L}_2 - \\Lambda_2) \\;,\n\\end{align}\nwhere $G_{MN}(x,y)$ is the 5-dimensional metric, with $M,N =\n0,1,2,3,4$, $\\mu = 0,1,2,3$. The quantities\n\\begin{equation}\ng^{(1)}_{\\mu\\nu}(x) = G_{\\mu\\nu}(x, y=0) \\;, \\quad\ng^{(2)}_{\\mu\\nu}(x) = G_{\\mu\\nu}(x, y=\\pi r_c)\n\\end{equation}\nare induced metrics on the branes, $\\mathcal{L}_1$ and\n$\\mathcal{L}_2$ are brane Lagrangians, $G = \\det(G_{MN})$, $g^{(i)}\n= \\det(g^{(i)}_{\\mu\\nu})$. $\\bar{M}_5$ is the \\emph{reduced}\n5-dimensional Planck scale. The quantity $\\Lambda$ is a\n5-dimensional cosmological constant, while $\\Lambda_{1,2}$ are brane\ntensions.\n\nFor the first time, the function $\\sigma(y)$ was obtained in\n\\cite{Randall:1999} (see also the expression at the bottom of\nfig.~\\ref{two_solutions}):\n\\begin{equation}\\label{sigma_1}\n\\sigma_0(y) = \\kappa |y| - C_0 \\;,\n\\end{equation}\nwhere $\\kappa$ is a parameter with a dimension of mass, which\ndefines the curvature of the 5-dimensional space-time.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=8cm,clip]{two_solutions}\n\\caption{Two equivalent solutions of Einstein-Hilbert's equations\nfor the function $\\sigma(y)$ related to the different branes located\nat $y=0$ and $y=\\pi r_c$.}\n\\label{two_solutions}\n\\end{figure}\nThis solution is consistent with the orbifold symmetry $y\n\\rightarrow - y$. However, it is attributed to the brane $y=0$ (see\nfig.~\\ref{two_solutions}), and it is\nnot symmetric with respect to the branes\n\\footnote{Note that in contrast to \\eqref{sigma_1}, the RS1 solution\n\\cite{Randall:1999} has no constant term.}\n\nInstead of \\eqref{sigma_1}, one can use another \\emph{equivalent}\nsolution related to the brane $y=\\pi r_c$ (see the expression at the\ntop of fig.~\\ref{two_solutions}):\n\\begin{equation}\\label{sigma_2}\n\\sigma_\\pi (y) = -\\kappa |y - \\pi r_c| + |\\kappa| \\pi r_c - C_\\pi\n\\;,\n\\end{equation}\nIn order to get an expression symmetric with respect to both branes,\nwe take half the sum of expressions \\eqref{sigma_1} and\n\\eqref{sigma_2}. As a result, we come to the solution:\n\\begin{equation}\\label{sigma_symmetric}\n\\sigma (y) = \\frac{\\kappa}{2} ( |y| - |y - \\pi r_c| ) +\n\\frac{|\\kappa| \\pi r_c}{2} - C \\;,\n\\end{equation}\nwhere $C = (C_0 + C_\\pi)\/2$. The constant terms in\n\\eqref{sigma_symmetric} are chosen in such a way that one has\n$\\sigma (y) = \\kappa y - C$ within the interval $0 < y < \\pi r_c$.\n\nNote that neither original RS1 solution nor generalized RS-like\nsolution \\eqref{sigma_symmetric} obeys periodicity in $y$\n\\emph{explicitly}. One has to keep the periodicity condition in\n``mind''. Fortunately, a new solution was recently proposed\n\\cite{Kisselev:2015} which is both symmetric with respect to the\nbranes and \\emph{periodic} function of $y$:\n\\begin{equation}\\label{sigma}\n\\sigma(y) = \\frac{\\kappa r_c }{2} \\left[ \\left| \\mathrm{Arccos}\n\\left(\\cos \\frac{y}{r_c} \\right) \\right| - \\left| \\pi -\n\\mathrm{Arccos} \\left(\\cos \\frac{y}{r_c} \\right)\\right| \\right] +\n\\frac{\\pi \\,|\\kappa| r_c }{2} - C \\;,\n\\end{equation}\nwith the fine tuning relations:\n\\begin{equation}\\label{fine_tuning}\n\\Lambda = -24 \\bar{M}_5^3\\kappa^2 \\;, \\quad \\Lambda_1 = -\n\\,\\Lambda_2 = 24 \\bar{M}_5^3 \\kappa \\;.\n\\end{equation}\nHere $\\mathrm{Arccos(z)}$ is a \\emph{principal value} of the\nmultivalued inverse trigonometric function $\\arccos(z)$. It is\ndefine by the inequality (see, for instance, \\cite{Arccos}):\n\\begin{equation}\\label{Arccos}\n0 \\leqslant \\mathrm{Arccos}(z) \\leqslant \\pi \\;, \\quad -1 \\leqslant\nz \\leqslant 1 \\;.\n\\end{equation}\nIt means that \\cite{Arccos}\n\\begin{equation}\\label{arccos}\n\\mathrm{Arccos} (\\cos x) = \\left\\{\n\\begin{array}{ll}\n x - 2n \\pi , & 2n \\pi \\leqslant x \\leqslant (2n +1)\\pi \\;, \\\\\n - x + 2(n +1)\\pi , & (2n +1)\\pi \\leqslant x \\leqslant 2(n +1)\\pi\n \\;,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $n=0,1, \\ldots$ It follows from \\eqref{sigma}, (\\ref{arccos})\nthat $\\sigma(y) + C = \\kappa y$ for $0 < y < \\pi r_c$, $\\sigma(y) +\nC = \\kappa (2\\pi r_c - y)$ for $\\pi r_c < y < 2\\pi r_c$. and so on\n(see fig.~\\ref{two_solutions}).\n\nOur solution \\eqref{sigma} \\\\\n- is symmetric with respect to the\nbranes: $\\sigma(y)$ remains unchanged if\n$y \\rightarrow \\pi r_c - y$, $k \\rightarrow -k$\n\\footnote{Under the replacement $y \\rightarrow \\pi r_c - y$ the\npositions of the branes are interchanged ($y=0 \\rightleftarrows\ny=\\pi r_c$), while under the replacement $k \\rightarrow -k$ their\ntensions are interchanged ($\\Lambda_1 \\rightleftarrows \\Lambda_2$).}\n\\\\\n- obeys the orbifold $Z_2$-symmetry: $\\sigma(y) = \\sigma(-y)$;\n\\\\\n- reproduces the jumps of $\\sigma'(y)$ on both branes: $\\sigma''(y)\n= 2\\kappa [\\delta (y) - \\delta (y - \\pi r_c)]$, $0 \\leqslant y\n\\leqslant \\pi r_c$; \\\\\n- is the periodic function of the extra coordinate: $\\sigma(y + 2\\pi\nr_c n) = \\sigma(y)$, $n=\\pm 1, \\pm 2, \\ldots$ Note that the warped\nfunction $\\sigma(y)$ depends on the constant $C$.\n\n\n\\section{RS-like scenario with the small curvature}\n\\label{sec:RSSC}\n\nBy taking different values of $C$ in eq.~\\eqref{sigma}, we come to\nquite \\emph{diverse physical scenarios} \\cite{Kisselev:2015},\n\\cite{Kisselev:2016}. One of them ($C = 0$) is in fact the RS1 model\n\\cite{Randall:1999}. Another scheme ($C = \\kappa \\pi r_c$) describes\na geometry with a small curvature of five-dimensional space-time\n(RSSC model) \\cite{Giudice:2005}-\\cite{Kisselev:2006}. In the RSSC\nmodel the hierarchy relation looks like (we assume that $\\kappa \\pi\nr_c \\gg 1$)\n\\begin{equation}\\label{hierarchy_relation_RSSC}\n\\bar{M}_{\\mathrm{Pl}}^2 = \\frac{\\bar{M}_5^3}{\\kappa} \\, e^{2\\kappa\n\\pi r_c} \\;.\n\\end{equation}\nSo, one can take, for instance, $\\kappa \\sim 1$ GeV, $M_5 \\sim 1$\nTeV. Let us underline that the original RS scenario does not admit\nthe parameters to lie in this region, since $\\kappa \\sim \\bar{M}_5\n\\sim M_{\\mathrm{Pl}}$ TeV in it \\cite{Randall:1999}.\n\nThe masses of the KK gravitons are proportional to the curvature\nparameter $\\kappa$:\n\\begin{equation}\\label{m_n_2}\nm_n = x_n \\kappa \\;,\n\\end{equation}\nwhere $x_n$ are zeros of the Bessel function $J_1(x)$. The coupling\nof the massive gravitons to the SM fields is given by the constant:\n\\begin{equation}\\label{Lambda_pi_2}\n\\Lambda_\\pi = \\bar{M}_5 \\left( \\frac{\\bar{M}_5}{\\kappa}\n\\right)^{1\/2} .\n\\end{equation}\nThus, the RSSC scenario predicts a spectrum of the KK gravitons\nsimilar to that of the ADD model\n\\cite{Arkani-Hamed:1998}-\\cite{Antoniadis:1998}. For the LHC\nphenomenology of the RSSC model, see \\cite{Kisselev:2008},\n\\cite{Kisselev:2013}.\n\n\n\\section{Neutrino-nucleon amplitude in the RSSC model}\n\\label{sec:nu_N}\n\nFor the UHE cosmic neutrinos, the scattering takes place in the\ntrans-Planckian kinematical region:\n\\begin{equation}\\label{kinem_region}\n\\sqrt{s} \\gtrsim \\bar{M}_5 \\gg -t \\;,\n\\end{equation}\nwith $\\sqrt{s}$ being the colliding energy and $t = - q_{\\bot}^2$\nfour-dimensional momentum transfer. Remember that the inequality\n$\\kappa \\ll \\bar{M}_5$ is also satisfied. In the eikonal\napproximation, which is valid in the kinematical region\n\\eqref{kinem_region}, the neutrino-proton amplitude is of the form:\n\\begin{equation}\\label{ampl}\nA_{\\nu \\rm p}(s, t) = 4 \\pi i \\, s \\!\\! \\int\\limits_0^{\\infty} \\! db\nb J_0(b \\, q_{\\bot}) \\left\\{ 1 - \\exp [i \\chi(s, b)] \\right\\} \\;.\n\\end{equation}\nIn its turn, the eikonal is given by the Fourier-Bessel\ntransformation of the eikonal:\n\\begin{equation}\\label{eikonal}\n\\chi(s, b) = \\frac{1}{4 \\pi s} \\! \\int\\limits_0^{\\infty} \\!\ndq_{\\bot} q_{\\bot} J_0(q_{\\bot} b) \\, A_{\\nu \\rm p}^{\\rm B} (s, t)\n\\;.\n\\end{equation}\n\nThe Born amplitude is given by the sum of gravi-Reggeons, i.e.\nreggeized gravitons in the $t$-channel. Because of a presence of the\nED, the Regge trajectory of the graviton is splitting into an\ninfinite sequence of trajectories enumerated by the KK number\n$n$~\\cite{Kisselev:2004}:\n\\begin{equation}\\label{trajectories}\n\\alpha_n(t) = 2 + \\alpha_g' t - \\alpha_g' \\, m_n^2, \\quad n = 0, 1,\n\\ldots.\n\\end{equation}\nIn string theories, the slope of the gravi-Reggeons is universal,\nand $\\alpha_g' = M_s^{-2}$, where $M_s$ is the string scale. As a\nresult, in the RSSC model the gravity Born amplitude for the\nneutrino scattering off \\emph{a point-like} particle looks like\n\\cite{Kisselev:2005}, \\cite{Kisselev:2009}:\n\\begin{equation}\\label{Born_part_ampl}\nA_{\\rm grav}^{\\mathrm{B}}(s, t) = \\frac{\\pi \\alpha'_g s^2}{2\n\\Lambda_{\\pi}^2} \\sum_{n \\neq 0} \\left[ i - \\cot\\frac{\\pi\n\\alpha_n(t)}{2} \\right] \\left( \\frac{s}{\\bar{M}_5}\n\\right)^{\\alpha_n(t) -2} \\;.\n\\end{equation}\n\nThe \\emph{hadronic} Born amplitude in \\eqref{eikonal} is defined by\na convolution of the gravity amplitude~\\eqref{Born_part_ampl} and\nskewed ($t$-dependent) PDFs $f_i(x,\\mu^2; t)$:\n\\begin{equation}\\label{Born_had_ampl}\nA_{\\nu \\rm p}^{\\rm B}(s, t) = \\sum_{i=q,\\bar{q},g} \\int\\limits_0^1\ndx A_{\\rm grav}^{\\mathrm{B}}(xs, t) \\, f_i(x,\\mu^2; t) \\;.\n\\end{equation}\nWe take the $t$-dependent PDFs in a factorized form:\n\\begin{equation}\\label{skewed dist}\nf_i(x,\\mu^2; t) = f_i(x, \\mu^2) D(t) \\;.\n\\end{equation}\nFor the PDFs $f_i(x, \\mu^2)$ we use the CT14 parametrization\n\\cite{CT14} and put $\\mu^2 = |t|$. As for a suppression factor\n$D(t)$, at hadron scales it was evaluated to be $D(t) =\n\\exp(tr_0^2)$, with $r_0^2 = 0.62 \\mathrm{\\ GeV}^{-2}$\n\\cite{Petrov:2002}. For larger $|t|$, we assume $D(t)$ to have a\npower-like behavior. As a result, we come to the parametrization:\n\\begin{equation}\\label{damping_factor}\nD(t) =\n\\left\\{\n \\begin{array}{ll}\n \\exp(t r_0^2) \\;, & |t| \\leqslant q_0^2 \\;, \\\\\n \\left(\\mathrm{e} |t|\/q_0^2 \\right)^{-q_0^2 r_0^2} \\;, & |t| > q_0^2 \\;,\n \\end{array}\n\\right.\n\\end{equation}\nwhere $q_0 = m_\\rho$ is the $\\rho$-meson mass, and $\\ln \\mathrm{e} =\n1$.\n\nThe total cross sections with the account of the contribution from\nthe KK gravitons are shown in fig.~\\ref{cross_sections}. We adopted\nthe SM $\\nu N$ cross sections from \\cite{Sarkar:2008}. One can see,\nthe total cross section noticeably exceeds the SM one at energies\nhigher than $10^{18}$ or $10^{19}$ eV, depending on the reduced\n5-dimensional Planck scale $\\bar{M}_5$.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{cs_mm}\n\\caption{Solid lines (from above): the neutrino total cross sections\nfor $\\bar{M}_5 = 1.5$, 2.0 and 3.0 TeV. Dashed line: neutrino CC\ntotal cross section.}\n\\label{cross_sections}\n\\end{figure}\n\n\n\\section{Expected number of neutrino events at the PAO}\n\\label{sec:ev_num}\n\nIn \\cite{Anchordoqui:2010} the following functional dependence of\nthe DG event rate on the new physics cross section\n$\\sigma_{\\mathrm{NP}}$ was proposed for a bin in neutrino energy\n($10^{18.5} \\mathrm{\\ eV} < E_\\nu < 10^{19.5} \\mathrm{\\ eV}$):\n\\begin{equation}\\label{DG:BSM_vs_SM_prel}\nN_{\\mathrm{BSM}}^{\\mathrm{DG}} = N_{\\mathrm{SM}}^{\\mathrm{DG}} \\,\n\\frac{\\sigma_{\\mathrm{CC}} +\n\\sigma_{\\mathrm{NP}}}{\\sigma_{\\mathrm{CC}}} \\;,\n\\end{equation}\nwhere $N_{\\mathrm{SM}}^{\\mathrm{DG}}$ is the number of DG events in\nthe absence of new physics, $\\sigma_{\\mathrm{CC}}$ is a charged\ncurrent (CC) total cross section. $N_{\\mathrm{BSM}}^{\\mathrm{DG}}$\nis the number of DG events with the account of interactions beyond\nthe SM. We propose a more general formula:\n\\begin{equation}\\label{DG:BSM_vs_SM}\n\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{DG}} (E_\\nu) =\n\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{DG}} (E_\\nu)\\,\n\\frac{\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}(E_\\nu) +\n\\sigma_{\\mathrm{NP}}(E_\\nu)}{\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}(E_\\nu)}\n\\;,\n\\end{equation}\nwhere $\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{DG}}$\n($\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{DG}}$) is the \\emph{exposure}\nof the SD of the PAO with (without) account of the new interaction.\nIn addition, instead of $\\sigma_{\\mathrm{CC}}$, an effective SM\ncross section $\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}}$ is introduce in\n\\eqref{DG:BSM_vs_SM}:\n\\begin{equation}\\label{sigma_eff}\n\\sigma_{\\mathrm{SM}}^{\\mathrm{eff}} = \\sigma_{\\mathrm{CC}}\n\\!\\!\\sum_{i=e,\\mu,\\tau} \\! \\, m_{\\mathrm{CC}}^i +\n3\\sigma_{\\mathrm{NC}} \\, m_{\\mathrm{NC}} + \\sigma_{\\mathrm{CC}}\\,\nm_{\\mathrm{mount}} \\;.\n\\end{equation}\nHere $m_{\\mathrm{CC}}^i$ and $m_{\\mathrm{NC}}$ are relative mass\napertures for charged current and neutral current (NC) interactions\nof the DG neutrinos at the PAO. The mass aperture\n$m_{\\mathrm{mount}}$ corresponds to the CC interaction of a $\\tau$\nneutrino within the mountains around the PAO (see\nfig.~\\ref{nu_events}). The relative mass apertures as functions of\nthe neutrino energy where calculated using data from Table~I of\nref.~\\cite{Auger:2011}\n\\footnote{In particular, $\\sum_{i=e,\\mu,\\tau} m_{\\mathrm{CC}}^i$ is\nequal to 0.69 (0.65) at $10^{18}$ ($10^{19}$) eV.}\nNote that $\\sum_{i=e,\\mu,\\tau} m_{\\mathrm{CC}}^i + 3 m_{\\mathrm{NC}}\n+ m_{\\mathrm{mount}} = 1$.\n\nIn contrast to the DG neutrino exposure, the exposure of the ES\nneutrinos \\emph{decreases} with the rise of the neutrino total\ncross section\n\\footnote{Is is a generalization of the formula for the rate for ES\nshowers in the range $10^{18.5} \\mathrm{\\ eV} < E_\\nu < 10^{19.5}\n\\mathrm{\\ eV}$ \\cite{Anchordoqui:2010}.}\n\\begin{equation}\\label{ES:BSM_vs_SM}\n\\mathcal{E}_{\\mathrm{BSM}}^{\\mathrm{ES}}(E_\\nu) =\n\\mathcal{E}_{\\mathrm{SM}}^{\\mathrm{ES}}(E_\\nu) \\,\n\\frac{\\sigma_{\\mathrm{CC}}^2(E_\\nu)}{[\\sigma_{\\mathrm{CC}}(E_\\nu) +\n\\sigma_{\\mathrm{NP}}(E_\\nu)]^2} \\;.\n\\end{equation}\n\nThe formulas \\eqref{DG:BSM_vs_SM} and \\eqref{ES:BSM_vs_SM} allowed\nus to calculate expected exposures of the SD of the PAO for the\nperiod 1 January 2004 - 20 June 2013. The Auger data on exposures\nfor the SM neutrino interactions in the region from\n$\\log(E_\\nu\/\\mathrm{eV}) = 17$ to 20.5 in steps of 0.5 were used\n(see also fig.~3 from \\cite{Auger:2015}). The results of our\ncalculations are presented in fig.~\\ref{Auger_exposures}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{exp_DG_BSM}\n\\includegraphics[width=7cm,clip]{exp_ES_BSM}\n\\caption{Left panel: expected Auger exposures for the DG neutrinos\nwith zenith angle $75^\\circ < \\theta < 90^\\circ$. Red, green and\nblue line corresponds to $\\bar{M}_5 = 1.5$, 2.0 and 3.0 TeV,\nrespectively. Dashed line: SM exposure taken from \\cite{Auger:2015}.\nRight panel: the same but for the ES neutrinos.}\n\\label{Auger_exposures}\n\\end{figure}\n\nThe number of neutrino events is given by the formula:\n\\begin{equation}\\label{numbers_events}\nN_{\\mathrm{ev}} = \\int \\frac{dN_\\nu}{dE_\\nu}(E_\\nu) \\,\n\\mathcal{E}(E_\\nu) \\, dE_\\nu \\;.\n\\end{equation}\nWe assume differential neutrino flux $dN\/dE_\\nu \\sim E_\\nu^{-2}$, as\nwell as a ratio $\\nu_e : \\nu_\\mu : \\nu_\\tau = 1:1:1$. In\nfig.~\\ref{BSM_vs_SM} we compare the number of neutrino events\npredicted in the RSSC model with the expected number of neutrino\nevents in the absence of a new physics. Both the DG events with\nzenith angles $60^\\circ < \\theta < 90^\\circ$ and ES events was taken\ninto account in the full energy region of the sensitivity of the SD\n($1.0\\times10^{17} - 2.5\\times10^{19}$ eV).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{ev_BSM-SM_2_2_m}\n\\caption{The ratio of the neutrino events with and without effects\nfrom EDs as a function of the gravity scale $\\bar{M}_5$, expected at\nthe SD of the PAO. The event numbers correspond to the energy range\n$1.0\\times10^{17} - 2.5\\times10^{19}$ eV.}\n\\label{BSM_vs_SM}\n\\end{figure}\nAs was already mentioned above, no neutrino events were seen at the\nSD of the PAO. Thus, from fig.~\\ref{BSM_vs_SM} we obtain the lower\nbound on 5-dimensional \\emph{reduced} Planck scale:\n\\begin{equation}\\label{M5_bound}\n\\bar{M}_5 > 2.75 \\mathrm{\\ TeV} \\;.\n\\end{equation}\nCorrespondingly, for the 5-dimensional Planck scale we find $M_5 >\n2.75 \\times (2\\pi)^{1\/3} = 5.07$ TeV.\n\nWe have also calculated an expected ratio of the ES neutrino showers\nto the DG neutrino showers with the zenith angles $75^\\circ < \\theta\n< 90^\\circ$ (see fig.~\\ref{ES_vs_DG}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=7cm,clip]{ev_DG-ES_2_2}\n\\caption{The expected ratio of the ES neutrinos to the DG neutrinos\n(with zenith angle $75^\\circ < \\theta < 90^\\circ$) at the SD of the\nPAO as a function of the gravity scale $M_5$.}\n\\label{ES_vs_DG}\n\\end{figure}\nWe predict $N_{\\mathrm{ES}}\/N_{\\mathrm{DG}}$ to be 2.8 (5.7) for\n$\\bar{M}_5 = 2.8$ (4.0) TeV. For the SM interactions, this ratio was\nestimated by the Auger Collaboration to be equal to 6.0\n\\cite{Auger:2015}.\n\n\n\\section{Conclusions}\n\nIn the present paper we have studied the neutrino-induced inclined\n(quasi-horizontal) events at the Surface Detector of the Pierre\nAuger Observatory in the Randall-Sundrum scenario with the extra\ndimension and warped metric. We have presented the general solution\nfor the metric \\eqref{sigma} which is symmetric with respect to both\nbranes and explicitly periodic in extra variable. In the framework\nof the RS-like model with the small curvature of the 5-dimensional\nspace-time, the exposures of the SD of the PAO for the downward\ngoing and Earth-skimming neutrinos are estimated\n(fig.~\\ref{Auger_exposures}). The lower bound on the fundamental\ngravity scale $\\bar{M}_5$ is obtained \\eqref{M5_bound}. The ratio of\nthe number of the ES air showers to the number of the DG showers,\n$N_{\\mathrm{ES}}\/N_{\\mathrm{DG}}$, is calculated as a function of\n$\\bar{M}_5$ (fig.~\\ref{ES_vs_DG}).\n\n\n\\section*{Acknowledgements}\n\nThe authors are indebted to J. Alvarez-Mu\\~{n}iz for sending\nnumerical data on the Auger exposures shown in fig.~3 from\nref.~\\cite{Auger:2015}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{Section1}\nConsider the time-dependent Ginzburg-Landau system of equations\n\\begin{subequations}\n\\label{eq:1}\n\\begin{alignat}{2}\n \\frac{\\partial\\psi}{\\partial t}- \\nabla_{\\kappa A}^2\\psi +i\\kappa\\phi \\psi=\\kappa^2(1-|\\psi|^2)\\psi & \\quad \\text{ in }(0,+\\infty)\\times \\Omega \\,,\\\\\n \\frac{1}{c}\\Big(\\frac{\\partial A}{\\partial t}+\\nabla\\phi\\Big)+ \\text{\\rm curl\\,}^2A\n =\\frac{1}{\\kappa}\\Im(\\bar\\psi\\nabla_{\\kappa A}\\psi) &\n \\quad \\text{ in } (0,+\\infty)\\times \\Omega\\,, \\\\\n \\psi=0 &\\quad \\text{ on } (0,+\\infty)\\times \\partial\\Omega_c\\,, \\\\\n \\nabla_{\\kappa A}\\psi\\cdot\\nu=0 & \\quad \\text{ on } (0,+\\infty)\\times \\partial\\Omega_i\\,, \\\\\n \\frac{\\partial\\phi}{\\partial\\nu} = - c\\kappa J(x) &\\quad \\text{ on } (0,+\\infty)\\times \\partial\\Omega_c\\,, \\\\\n \\frac{\\partial\\phi}{\\partial\\nu}=0 &\\quad \\text{ on } (0,+\\infty)\\times \\partial\\Omega_i \\,, \\\\[1.2ex]\n \\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A(t,x) \\, ds = \\kappa h_{ex} \\,,&\\quad {\\text{ on } (0,+\\infty)}\\\\\n \\psi(0,x)=\\psi_0(x) & \\quad \\text{ in } \\Omega\\,, \\\\\nA(0,x)=A_0(x) & \\quad \\text{ in } \\Omega \\,.\n\\end{alignat}\n\\end{subequations}\nIn the above $\\psi$ denotes the order parameter, $A$ the\nmagnetic potential, $\\phi$ the electric potential, $\\kappa$\nthe Ginzburg-Landau parameter, which is a material property, and\n$$c=\\kappa^2\/\\sigma$$ where $\\sigma$ is the normal conductivity of the\nsample. Finally, $h_{ex}$, or the average magnetic field on $\\partial\\Omega$\ndivided by $\\kappa$, is constant in time. Length has been scaled with\nrespect to the penetration depth (see \\cite{alhe14}). \\\\\n Unless otherwise stated we\nshall assume in the sequel that $$\\kappa\\geq1\\,.$$ We further assume that\n$(\\psi_0,A_0)\\in H^1(\\Omega,{\\mathbb{C}})\\times H^1(\\Omega,{\\mathbb{R}}^2)$ and that\n\\begin{equation}\\label{condinit}\n \\|\\psi_0\\|_\\infty\\leq 1 \\,.\n\\end{equation}\nWe use the notation $\\nabla_A=\\nabla-iA$, $\\Delta_A=\\nabla_A^2$ and $ds$ for the induced\nmeasure on $\\pa \\Omega$. We have also used above the standard notation\n\\begin{displaymath}\n \\Xint-_{\\partial \\Omega} = \\frac{1}{\\ell(\\partial \\Omega)}\\int \\,.\n\\end{displaymath}\nThe domain $\\Omega\\subset\\subset{\\mathbb{R}}^2$ has the same characteristics as in \\cite{alhe14},\nin particular its boundary $\\pa \\Omega$ contains a smooth interface,\ndenoted by $\\partial\\Omega_c\\,$, with a conducting metal which is at the normal\nstate. Thus, we require that $\\psi$ vanishes on $\\partial\\Omega_c$ in (\\ref{eq:1}c).\n\nWe make the following assumptions on the current $J$\n\\begin{equation}\n\\label{eq:2}\n(J1)\\quad J\\in C^2(\\overline{\\partial\\Omega_c},{\\mathbb{R}})\\,,\n\\end{equation}\n\\begin{equation}\n \\label{eq:3}\n(J2) \\quad \\int_{\\partial\\Omega_c}J \\,ds=0 \\,,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:75aa}\n(J3)\\quad \\mbox{the sign of } J \\mbox{ is constant on each\nconnected component of } \\partial\\Omega_c\\,.\n\\end{equation}\nWe allow for $J\\neq0$ at the corners despite the fact that no current is\nallowed to enter the sample through the insulator. The rest of\n$\\partial\\Omega$, denoted by $\\partial\\Omega_i$ is adjacent to an insulator. By\nconvention, we extend $J$ as equal to $0$ on $\\partial \\Omega_i$.\n\nTo simplify some of our regularity arguments we introduce the\nfollowing geometrical assumption (for further discussion we refer the\nreader to Appendix A in \\cite{alhe14}) on $\\partial\\Omega$:\n\\begin{equation}\\label{propertyR}\n(R1)\\,\\left\\{\n\\begin{array}{l}\n(a) \\; \\pa \\Omega_i \\mbox{ and } \\pa \\Omega_c \\mbox{ are of class } C^3\\,;\\\\\n(b) \\mbox{ Near each edge, } \\pa\n\\Omega_i \\mbox{ and } \\pa \\Omega_c\n \\mbox{ are} \\\\ \\quad \\mbox{ flat and meet with an angle of } \\frac \\pi 2\\,.\n\\end{array}\\right.\n\\end{equation}\n We also require that\n\\begin{equation}\\label{hyptopolo}\n(R2) \\quad\\quad \\mbox{Both } \\partial\\Omega_c \\mbox{ and } \\partial\\Omega_i \\mbox{ have two components}.\n\\end{equation}\n\\begin{figure}\n \\centering\n\\setlength{\\unitlength}{0.0005in}\n\\begingroup\\makeatletter\\ifx\\SetFigFont\\undefined%\n\\gdef\\SetFigFont#1#2#3#4#5{%\n 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\\caption{Typical superconducting sample. The arrows denote the direction of the current flow ($J_{in}$ for the inlet, and $J_{out}$ for the outlet).}\n \\label{fig:1}\n\\end{figure}\n\\nopagebreak\nFigure 1 presents a typical sample with properties (R1) and\n(R2), where the current flows into the sample from one connected\ncomponent of $\\partial\\Omega_c$, and exits from another part, disconnected from\nthe first one. Most wires would fall into the above class of domains.\n\nThe system \\eqref{eq:1} is invariant to the gauge transformation\n\\begin{equation}\n\\label{eq:4}\n A'= A + \\nabla\\omega\\,, \\; \\phi'= \\phi-\n \\frac{\\partial\\omega}{\\partial t}\\,, \\; \\psi' = \\psi e^{i\\omega} \\,.\n\\end{equation}\nWe thus choose, as in \\cite{alhe14}, the Coulomb gauge, i.e., we assume\n\\begin{equation}\n \\label{eq:5}\n \\begin{cases}\n \\text{\\rm div\\,} A =0 & \\text{ in } (0,+\\infty)\\times \\Omega\\,, \\\\\n A\\cdot\\nu =0 & \\text{ on }(0,+\\infty)\\times \\partial\\Omega\\,,\n \\end{cases}\n\\end{equation}\nwhere the divergence is computed with respect to the spatial coordinates only.\n\n\nFrom (\\ref{eq:25}b), (\\ref{eq:25}d), and (\\ref{eq:25}f), we know that $\\text{\\rm curl\\,}\nA$ is constant on each connected component of $\\partial \\Omega_i$. Let then\n$\\{\\partial\\Omega_{i,j}\\}_{j=1}^{2}$ denote the set of connected components of\n$\\partial\\Omega_i$. We can write, for $j=1,2\\,$,\n\\begin{equation}\n\\label{eq:6}\n\\text{\\rm curl\\,} A|_{\\partial\\Omega_{i,j}} =\nh_j \\, \\kappa\\,,\n\\end{equation}\nwhere $h_1$ and $h_2$ are constants.\n\nNote that $h_1$ and $h_2$ can be determined from\n$J$ and $h_{ex}$ via the formula \\cite{alhe14}\n\\begin{equation}\n\\label{eq:7}\n h_j= h_{ex} - \\Xint-_{\\pa \\Omega} \\, |\\Gamma(\\tilde x, x_j)| \\,\n J(\\tilde x) ds (\\tilde x)\\, \\mbox{ for any } x_j \\in \\pa\n \\Omega_{i,j}\\,, \\quad j=1,2\\,,\n\\end{equation}\nwhere $\\Gamma(x,x_0)$ is the portion of $\\partial\\Omega$ connecting $x_0$ and $x$ in\nthe positive trigonometric direction. We assume that $h_{ex}$ and $J$ are such\nthat\n\\begin{equation}\n \\label{eq:8}\nh_1h_2 <0 \\,,\n\\end{equation}\nand without any loss of generality we can assume $h_2>0$. Let\n\\begin{equation}\n \\label{eq:9}\nh=\\text{\\rm max}(|h_1|,|h_2|)\\,.\n\\end{equation}\nWe assume that\n\\begin{equation}\n\\label{hyph}\nh>1\n\\end{equation}\nand distinguish below between two\n different cases\n\\begin{subequations}\n\\label{eq:10}\n \\begin{equation}\n \\label{eq:5a}\n 11\\,.\n \\end{equation}\n\n We focus attention in this work on the exponential decay of $\\psi$ in\n regions where $|B_n|>1$. For steady-state solutions of \\eqref{eq:1}\n in the absence of electric current ($J=0$) we may set $\\phi\\equiv0$ and the\n magnetic field is then constant on the boundary. The exponential\n decay of $\\psi$ away from the boundary has been termed ``surface\n superconductivity'' and has extensively been studied (cf.\n \\cite{fohe09} and the references within). More recently, the case of\n a non-constant magnetic field has been studied as well\n \\cite{attar2014ground,heka15}. In these works $\\phi$ still identically\n vanishes but nevertheless $\\nabla B_n\\neq0$ in view of the presence of a\n current source term $\\text{\\rm curl\\,} h_{ex}$ in (\\ref{eq:1}b). In particular\n in \\cite{heka15} it has been established, in the large $\\kappa$ limit for\n the case $1\\ll h\\ll\\kappa$ that $\\psi$ is exponentially small away from\n $B_n^{-1}(0)$.\n\n In the absence of electric current the time-dependent case is of\n lesser interest, since every solution of \\eqref{eq:1} converges to a\n steady-state solution \\cite{lid97, feta01}. This result has been obtained in \\cite{feta01} by\n using the fact that the Ginzburg-Landau energy functional is a\n Lyapunov function in this case. In contrast, when $J\\neq0$ this\n property of the energy functional is lost, and convergence to a\n steady-state is no-longer guaranteed. In \\cite{alhe14} the global\n stability of the normal state has been established for $h=\\mathcal O(\\kappa)$\n in the large $\\kappa$ limit. In the present contribution, for the same\n limit, we explore the behavior of the solution for $11$. We do that for both steady-state solutions (whose\n existence we need to assume) and time-dependent ones. We also study\n the large-domain limit, where we obtain weaker results for\n steady-state solutions only.\n\n\nLet, for $j=1,2$,\n \\begin{equation}\n\\label{eq:13}\n \\omega_{j} = \\{x\\in \\Omega \\, : \\, (-1)^jB_n(x)>1 \\} \\,.\n \\end{equation}\nOur first theorem concerns steady state solutions and their\nexponential decay, in certain subdomains of $\\Omega$, in the large $\\kappa$ limit.\n\\begin{theorem}\n\\label{mainsteadystate}\n Let for $\\kappa \\geq 1$, $(\\psi_\\kappa,A_\\kappa, \\phi_\\kappa)$ be a time-independent solution of \\eqref{eq:1}.\n Suppose that for some $j\\in\\{1,2\\}$ we have\n \\begin{equation}\n\\label{eq:14}\n 1<|h_j| \\,.\n \\end{equation}\n Then, for any compact set $K\\subset \\omega_j \\cup \\partial \\Omega_c $, there\n exist $C>0$, $\\alpha >0$, and\n $\\kappa_0\\geq 1 $, such that for any $\\kappa\\geq \\kappa_0$ we have\n\\begin{equation}\n\\label{eq:15}\n\\int_{K} |\\psi_\\kappa(x)|^2 \\,dx \\leq C e^{- \\alpha \\kappa} \\,.\n\\end{equation}\nIf, in addition,\n\\begin{equation}\n \\frac{1}{\\Theta_0}<|h_j| \\,,\n \\end{equation}\n then (\\ref{eq:15}) is satisfied for any compact subset $K\\subset\\overline{\\omega_j}$.\n\\end{theorem}\nIn addition to the above exponential decay, which is limited to the\nregion where the normal magnetic field is large, we establish a weaker\ndecay of $\\psi_\\kappa$ in the entire domain.\n\\begin{proposition}\n\\label{entire}\n Under the assumptions \\eqref{eq:2}-\\eqref{eq:10} there exists\n $C(J,\\Omega)>0$ such that, for $\\kappa \\geq 1$,\n \\begin{equation}\n\\label{eq:16}\n\\| \\psi_\\kappa \\|_2 \\leq C(J,\\Omega) \\, (1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6} \\,.\n \\end{equation}\n\\end{proposition}\nTheorem \\ref{mainsteadystate} is extended to the time dependent case\nin the following way.\n\\begin{theorem}\n \\label{main}\n Let $(\\psi_\\kappa,A_\\kappa, \\phi_\\kappa)$ denote a time-dependent solution of\n \\eqref{eq:1}. Assuming $c=1$, under the conditions of Theorem\n \\ref{mainsteadystate} and \\eqref{condinit}, for any compact set $K\\subset\n \\omega_j\\cup \\partial \\Omega_c$ there exist $C>0$ , $\\alpha >0$, and $\\kappa_0\\geq 1$, such\n that for any $\\kappa\\geq \\kappa_0$ we have\n\\begin{equation}\n\\label{eq:17}\n\\limsup_{t\\to\\infty}\\int_{K} |\\psi_\\kappa(t,x)|^2 \\,dx \\leq C e^{- \\alpha \\kappa} \\,.\n\\end{equation}\n\\end{theorem}\n\n\nFinally, we consider steady-state solutions of (\\ref{eq:1}) in the\nlarge domain limit, i.e., we set $\\kappa=c=1$ and stretch $\\Omega$ by a factor\nof $R \\gg 1 $. Let $\\Omega^R$ be the image of $\\Omega$ under the map $x\\to Rx$.\nWe consider again steady-state solutions of \\eqref{eq:1}.\n\\begin{subequations}\n\\label{eq:18}\n\\begin{alignat}{2}\n& \\Delta_A\\psi + \\psi \\left( 1 - |\\psi|^{2} \\right)-i\\phi\\psi =0 & \\quad \\text{ in } \\Omega^R\\, ,\\\\\n & \\text{\\rm curl\\,}^2A + \\nabla\\phi = \\Im(\\bar\\psi\\, \\nabla_A\\psi) & \\quad \\text{ in } \\Omega^R\\,,\\\\\n &\\psi=0 &\\quad \\text{ on } \\partial\\Omega_c^R\\,, \\\\\n &\\nabla_A\\psi\\cdot\\nu=0 & \\quad \\text{ on } \\partial\\Omega_i^R\\,,\\\\\n & \\frac{\\partial\\phi}{\\partial\\nu}= \\frac{F(R)}{R}J &\\quad \\text{ on } \\partial\\Omega_c^R\\,,\\\\\n& \\frac{\\partial\\phi}{\\partial\\nu}= 0 &\\quad \\text{ on } \\partial\\Omega_i^R\\,,\\\\[1.2ex]\n&\\Xint-_{\\partial\\Omega^R}\\text{\\rm curl\\,} A(x)\\,ds = F(R)h_{ex}\\,.\n\\end{alignat}\n\\end{subequations}\nIn the above $F(R)=R^{\\gamma}$ for some $0<\\gamma<1$. We study the above\nproblem in the limit $R\\to\\infty$. Assuming again (\\ref{eq:2})-(\\ref{eq:10})\nwe establish the following result.\n\\begin{proposition}\n\\label{largedomain}\n Let $(\\psi,A,\\phi)$ denote a solution of \\eqref{eq:18}.\n Then, there exists a compact set $K\\subset\\Omega$, $C>0$, $R_0>0$, and $\\alpha >0$,\nsuch that for any $R>R_0\\,$ we have\n\\begin{equation}\n\\label{eq:126a}\n\\int_{K_R} |\\psi (x)|^2 \\,dx \\leq C e^{-\\alpha R} \\,,\n\\end{equation}\nwhere $K_R$ is the image of $K$ under the map $x\\to Rx$.\n\\end{proposition}\nNote that $h_{ex}$ must be of $\\mathcal O(J)$, otherwise $(0,A_n,\\phi_n)$ would\nbe the unique solution.\n\n\nPhysically \\eqref{eq:126a} demonstrates that there is a\nsignificant portion of the superconducting sample which remains,\npractically, at the normal state, for current densities which may be\nvery small. This result stands in\ncontrast with what one finds in standard physics handbooks\n\\cite{poetal99} where the critical current density, for which the\nfully superconducting state looses its stability, is tabulated a\nmaterial property. However, our results suggest that the critical\ncurrent depends also on the geometry of the superconducting sample. In\nfact, according to Proposition \\ref{largedomain}, this current density\nmust decay in the large domain limit. In two-dimensions, our result\nsuggests that one should search for a critical current (and not\ncurrent density), whereas in three-dimensions a density with respect\nto cross-section circumference (instead of area) should be obtained.\n\nWe note that Proposition \\ref{largedomain} is certainly not\noptimal. In fact, we expect the following conjecture to be true.\n\\begin{conjecture}\n \\label{conj}\nUnder the conditions of Proposition \\ref{largedomain}, for any\ncompact set $K\\subset\\Omega\\setminus B_n^{-1}(0)$, there exist $R_0>0$, $C>0$, and $\\alpha >0$,\nsuch that for any $R>R_0$ \\eqref{eq:126a} is satisfied.\n\\end{conjecture}\n\n\nThe rest of this contribution is organized as follows. In the next\nsection we establish some preliminary results related to the\neigenvalues of the magnetic Laplacian in the presence of\nDirichlet-Neumann corners. We use these results in Section 3 where we\nestablish Theorem \\ref{mainsteadystate} and Proposition~\\ref{entire}.\nIn Section 4 we consider the time-dependent problem and establish, in\nparticular, Theorem~\\ref{main}. Finally, in the last section, we\nobtain some weaker results for steady-state solutions of (\\ref{eq:1})\nin the large domain limit.\n\n\\section{Magnetic Laplacian Ground States}\n\\label{sec:2}\nIn this section, we analyze the spectral properties of the Schr\\\"odinger\noperator with constant magnetic field in a sector. The Neumann problem\nhas been addressed by V. Bonnaillie-No\\\"el in \\cite{Bon}. In the sequel\nwe shall need, however, a lower bound for the ground state energy of\nthe above operator on a Dirichlet-Neumann sector, i.e., a Dirichlet\ncondition is prescribed on one side of the sector and the magnetic\nNeumann condition on the other side. We begin by the following\nauxiliary lemma whose main idea has been introduced to us by M.~Dauge\n\\cite{da14}. Hereafter the norms in the Lebesgue spaces $L^p(\\Omega)$,\n$L^p(\\Omega,\\mathbb R^2)$ and $L^p(\\Omega,\\mathbb C)$ will be denoted by\n$\\|\\cdot\\|_{L^p(\\Omega)}$ or $\\|\\cdot\\|_p$, and the norms in the Sobolev spaces\n$W^{k,p}(\\Omega,\\mathbb R)$, $W^{k,p}(\\Omega,\\mathbb R^2)$ and $W^{k,p}(\\Omega,\\mathbb C)$\nwill be denoted by $\\|\\cdot\\|_{W^{k,p}(\\Omega)}$ or $\\|\\cdot\\|_{k,p}$.\n \\begin{lemma}\n\\label{lem:Dirichlet-Neumann}\n Let $S_\\alpha$ denote an infinite sector of angle $\\alpha \\in (0,\\pi]$, i.e.,\n \\begin{displaymath}\n S_\\alpha = \\{ (x,y)\\in{\\mathbb{R}}^2 \\,: \\, 0<\\arg(x+iy)<\\alpha \\}\\,.\n \\end{displaymath}\nLet further\n \\begin{displaymath}\n {\\mathcal H}_\\alpha = \\{ u\\in H^1(S_\\alpha) \\,: \\, u(r\\cos\\alpha,r\\sin\\alpha)=0\\,, \\; \\forall r>0\\}\\,,\n \\end{displaymath}\nand\n\\begin{displaymath}\n \\Theta^{DN}_\\alpha = \\inf_{\\begin{subarray}{c}\n u\\in {\\mathcal H}_\\alpha \\\\\n \\|u\\|_2 =1\n \\end{subarray}}\n\\int_{S_\\alpha} |(\\nabla-iF) u|^2 \\,dx \\,,\n\\end{displaymath}\nwhere $F$ is a magnetic potential satisfying $\\text{\\rm curl\\,} F = 1$ in\n$S_\\alpha$. Then,\n \\begin{equation}\n \\label{eq:19}\n \\Theta^{DN}_\\alpha =\\Theta_0\\,,\n \\end{equation}\nwhere\n \\begin{equation}\n\\label{eq:20}\n \\Theta_0 = \\inf_{\\begin{subarray}{c}\n u\\in H^1(S_\\pi) \\\\\n \\|u\\|_2 =1\n \\end{subarray}}\n\\int_{S_\\pi} |(\\nabla-iF)u|^2 \\,dx \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n Let $0<\\alpha_1<\\alpha_2\\leq\\pi$ and $u\\in{\\mathcal H}_{\\alpha_1}$. Let further\n $\\tilde{u}\\equiv u$ in $S_{\\alpha_1}$ and $\\tilde{u}\\equiv0$ in\n $S_{\\alpha_2}\\setminus S_{\\alpha_1}$. Clearly $\\tilde{u}\\in{\\mathcal H}_{\\alpha_2}$, and hence it\n follows that $\\Theta^{DN}_{\\alpha_1}\\geq\\Theta^{DN}_{\\alpha_2}$. Consequently,\n \\begin{displaymath}\n \\Theta^{DN}_\\alpha \\geq \\Theta^{DN}_\\pi\\,, \\quad \\forall\\alpha\\leq\\pi.\n \\end{displaymath}\n From the definition of $\\Theta_0$ it follows, however, that\n $\\Theta^{DN}_\\pi\\geq\\Theta_0$, and hence, $ \\Theta^{DN}_\\alpha \\geq \\Theta_0$ for all\n $\\alpha\\in(0,\\pi]$. The proof of \\eqref{eq:19} now easily follows \n from the proof of Persson's Theorem \\cite[Appendix B]{fohe09}\n providing the upper bound $\\Theta_0$ for the essential spectrum of the\n magnetic Dirichlet-Neumann Laplacian in $S_\\alpha$.\n\\end{proof}\n\nLet ${\\mathcal D}\\subset\\Omega$ have a smooth boundary, except at the corners of $\\partial\\Omega$. As\nin \\cite{alhe14} we let\n\\begin{equation}\n \\label{eq:21}\n \\mu_\\epsilon(\\mathcal A,{\\mathcal D}) = \\inf_{\n \\begin{subarray}{c}\n u\\in {\\mathcal H}({\\mathcal D}) \\\\\n \\|u\\|_2 =1\n \\end{subarray}} \\int_{\\mathcal D} |\\epsilon\\nabla-i\\mathcal A u|^2 \\,dx \\,,\n\\end{equation}\nwherein\n\\begin{displaymath}\n {\\mathcal H}({\\mathcal D}) = \\{ u\\in H^1({\\mathcal D})\\,:\\, u=0 \\text{ on } \\partial{\\mathcal D}\\setminus (\\partial{\\mathcal D}\\cap \\partial\\Omega_i)\\,\\} \\,.\n\\end{displaymath}\nLet\n\\begin{displaymath}\n {\\mathcal S} = \\overline{\\partial\\Omega_c}\\cap\\overline{\\partial\\Omega_i}\\cap\\overline{\\partial{\\mathcal D}}\n\\end{displaymath}\ndenote the corners of $\\Omega$ belonging to $\\partial{\\mathcal D}$. Following \\cite{boda06}\nwe set for a given magnetic potential $\\mathcal A \\in C^1(\\overline{\\Omega},\\mathbb R^2)$,\n\\begin{displaymath}\n b=\\inf_{x\\in{\\mathcal D}}|\\text{\\rm curl\\,} \\mathcal A| \\quad ; \\quad b^\\prime = \\inf_{x\\in\\partial{\\mathcal D}\\cap\\partial\\Omega_i}|\\text{\\rm curl\\,} \\mathcal A|\\,.\n\\end{displaymath}\nThe following proposition is similar to a result in \\cite{boda06}\nobtained for a Neumann boundary condition. Here we treat a\nDirichlet-Neumann boundary condition and allow, in addition, some\ndependence of the magnetic potential on the semi-classical parameter.\n\\begin{proposition}\n Let $a\\in W^{1,\\infty}({\\mathcal D},\\mathbb R^2)$. There exist $C>0$ and $\\epsilon_0>0$ such that for all\n$0<\\epsilon<\\epsilon_0$ we have\n\\begin{equation}\n \\label{eq:22}\n \\mu_\\epsilon(\\mathcal A+\\epsilon^{1\/2}a,{\\mathcal D}) \\geq \\epsilon \\min(b,\\Theta_0b^\\prime) [1-C(1+\\|\\nabla a\\|_\\infty^2)\\epsilon^{1\/3}]\\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\n The case $b=0$ being trivial, we assume that $b>0$. We begin by\n introducing for any $\\epsilon >0$ a partition of unity (cf. also\n \\cite{hemo96}), i.e., families $\\{\\eta_i\\}_{i=1}^K\\subset C^\\infty(\\Omega)$, and\n $\\{x_i\\}_{i=1}^K\\subset{\\mathcal D}$ satisfying\n\\begin{displaymath}\n \\sum_{i=1}^K\\eta_i^2=1 \\quad ; \\quad {\\rm supp}\\,\\eta_i \\subset B(x_i,\\epsilon^{1\/3}) \\quad\n ; \\quad \\sum_{i=1}^K|\\nabla\\eta_i|^2 \\leq \\frac{C}{\\epsilon^{2\/3}} \\,,\n\\end{displaymath}\nwhere $C>0$ is independent of $\\epsilon$.\n\nIt can be easily verified that for any $u\\in H^1(\\Omega,{\\mathbb{C}})$\n\\begin{equation}\n\\label{eq:23}\n \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])u\\|_2^2=\\sum_{i=1}^K\\big[\n \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])(\\eta_iu)\\|_2^2 - \\epsilon^2\\|u\\nabla\\eta_i\\|_2^2 \\big]\\,.\n\\end{equation}\nWe now set\n\\begin{displaymath}\n v_i=\\eta_iu\\exp(-i\\epsilon^{-1\/2}a(x_i)\\cdot x) \\,,\n\\end{displaymath}\nto obtain that\n$$\\aligned\n& \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])(\\eta_iu)\\|_2^2\\\\\n=&\\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}(a-a(x_i))])v_i\\|_2^2\\\\\n \\geq&\n(1-\\epsilon^{1\/3})\\|(\\epsilon\\nabla-i\\mathcal A)v_i\\|_2^2 - \\epsilon^{2\/3}\\|a-a(x_i))v_i\\|_2^2\\\\\n \\geq&\n(1-\\epsilon^{1\/3})\\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\|v_i\\|_2^2 - C\\epsilon^{4\/3}\\|v_i\\|_2^2\\,.\n\\endaligned\n$$\nSubstituting the above into \\eqref{eq:23} yields\n\\begin{equation}\n\\label{eq:24}\n \\|(\\epsilon\\nabla-i[\\mathcal A+\\epsilon^{1\/2}a])u\\|_2^2 \\geq (1-\\epsilon^{1\/3})\\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\|u\\|_2^2 - C\\epsilon^{4\/3}(1+\\|\\nabla a\\|_\\infty^2)\\|u\\|_2^2\\,.\n\\end{equation}\nBy following the same steps of the proof of Theorem 7.1 in \\cite{boda06}\nwe can establish that\n\\begin{equation}\\label{eq:74a}\n \\mu_\\epsilon(\\mathcal A,{\\mathcal D}) \\geq \\epsilon \\min(b,\\Theta_0b^\\prime,\\Theta^{DN}_\\alpha\\inf_{x\\in{\\mathcal S}}|\\text{\\rm curl\\,} \\mathcal A| )\n (1-C\\epsilon^{1\/2})\\,.\n\\end{equation}\nThe lemma now follows from \\eqref{eq:19}, \\eqref{eq:24}, and\n\\eqref{eq:74a}.\n\\end{proof}\n\n\n\n\n\\section{Steady-State Solutions}\\label{sec:3}\n\nWe begin by considering steady-state solutions of \\eqref{eq:1}\n$(\\psi_k,A_\\kappa,\\phi_\\kappa)\\in H^1(\\Omega,{\\mathbb{C}}\\times{\\mathbb{R}}^2\\times{\\mathbb{R}})$ in\nthe limit $\\kappa\\to + \\infty$. Hence, we look at the\nsystem \\cite[Section 5]{alhe14}\n\\begin{subequations}\n\\label{eq:25}\n\\begin{alignat}{2}\n - & \\nabla_{\\kappa A_\\kappa}^2\\psi_\\kappa+i\\kappa\\phi_\\kappa \\psi_\\kappa =\\kappa^2(1-|\\psi_\\kappa|^2)\\psi_\\kappa & \\quad \\text{ in } \\Omega \\,,\\\\\n - &\\text{\\rm curl\\,}^2A_\\kappa+ \\frac{1}{c}\\nabla\\phi_\\kappa =\\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) &\n \\quad \\text{ in } \\Omega\\,, \\\\\n &\\psi_\\kappa =0 &\\quad \\text{ on } \\partial\\Omega_c\\,, \\\\\n &\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\cdot\\nu=0 & \\quad \\text{ on } \\partial\\Omega_i\\,, \\\\\n & \\frac{\\partial\\phi_\\kappa}{\\partial\\nu} = - c\\kappa J(x) &\\quad \\text{ on } \\partial\\Omega_c\\,, \\\\\n &\\frac{\\partial\\phi_\\kappa}{\\partial\\nu}=0 &\\quad \\text{ on } \\partial\\Omega_i \\,, \\\\[1.2ex]\n &\\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_\\kappa \\, ds = \\kappa h_{ex} \\,,&\n\\end{alignat}\n\\end{subequations}\nwith the additional gauge restriction \\eqref{eq:5}. In the above\n$(\\psi_k,A_\\kappa,\\phi_\\kappa)$ is the same as $(\\psi,A,\\phi)$ in \\eqref{eq:1}. The\nsubscript $\\kappa$ has been added to emphasize the limit we consider\nhere. We assume in addition (\\ref{eq:2})-(\\ref{eq:10}). By the strong\nmaximum principle we easily obtain that\n\\begin{equation}\n\\label{eq:26}\n \\|\\psi_\\kappa\\|_\\infty <1 \\,.\n\\end{equation}\n\n\n\nLet $h$ be given by\n\\eqref{eq:9}. It has been demonstrated in \\cite{alhe14} that for\nsome $h_c>0$, when $h0$ and $\\kappa\\geq 1$,\n\\begin{displaymath}\n D_\\delta(\\kappa) = \\{ x\\in\\Omega \\,: \\,|B_\\kappa(x)|>(1+\\delta)\\kappa \\} \\,,\n\\end{displaymath}\nand\n\\begin{equation}\n\\label{eq:35}\n S_\\delta = \\{ x\\in\\Omega \\,: \\,|B_n(x)|>(1+\\delta)\\}\\,.\n\\end{equation}\nBy either (\\ref{eq:10}a) or (\\ref{eq:10}b), it follows that for $0<\\delta <\nh-1$, $S_\\delta\\neq\\emptyset\\,$. Below we show that the same is true for $ D_\\delta(\\kappa)$.\nNote that \\eqref{eq:8} implies that $S_\\delta$ consists of two disjoint\nsets:\n\\begin{equation}\\label{sdj}\nS_{\\delta} = S_{\\delta,1} \\cup S_{\\delta,2}\\,,\n\\end{equation}\none near $\\partial\\Omega_{i,1}$ denoted by $S_{\\delta,1}$, and one\nnear $\\partial\\Omega_{i,2}$ denoted by $S_{\\delta,2}\\,$. \\\\\nWe then let\n\\begin{equation}\n \\label{eq:36}\n\\mathcal C_{\\delta,j}=\\partial S_{\\delta,j}\\setminus (\\partial\\Omega\\cap\\partial S_{\\delta,j})\\,,\\quad j=1,2\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{Cdi}\n \\mathcal C_\\delta=\\mathcal C_{\\delta,1}\\cup \\mathcal C_{\\delta,2} \\,.\n\\end{equation}\n\nWe can now state and prove\n\\begin{lemma}\\label{lemma3.1}\n For any $0<\\alpha<1$ there exists $ \\kappa_0=\\kappa_0(\\Omega,J,\\alpha)>0$ such that for\n all $\\kappa \\geq \\kappa_0$ and $0<\\delta0$\n\\begin{equation}\n\\label{eq:43}\n\\|A_\\kappa\\|_\\infty \\leq C\\, \\kappa \\,.\n\\end{equation}}\n\nTaking the divergence of (\\ref{eq:31}a) yields, with the aid of (\\ref{eq:31}b),\n\\begin{equation}\n\\label{eq:38}\n \\begin{cases}\n - \\Delta\\phi_{1,\\kappa} = - \\frac{c}{\\kappa}\\, \\text{\\rm div\\,} \\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa\n }\\psi_\\kappa) & \\text{ in } \\Omega\\,, \\\\\n \\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega \\,.\n \\end{cases}\n\\end{equation}\nMultiplying the above equation by $\\phi_{1,\\kappa }$ and integrating by parts\nthen yields, with the aid of \\eqref{eq:26} and (\\ref{eq:25}c,d),\n\\begin{equation}\n\\label{eq:39}\n \\|\\nabla \\phi_{1,\\kappa} \\|_2 \\leq \\frac{c}{\\kappa}\\, \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2 \\,.\n \\end{equation}\nTaking the inner product of (\\ref{eq:25}a) with $\\psi_\\kappa$ yields, after\nintegration by parts\n\\begin{equation}\n\\label{eq:40}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2^2 = \\kappa^2\\, \\|\\psi_\\kappa\\|_2^2 \\,.\n\\end{equation}\nBy \\eqref{eq:39} we then obtain that\n\\begin{displaymath}\n\\|\\nabla\\phi_{1,\\kappa}\\|_2 \\leq Cc\\,.\n\\end{displaymath}\nSince $\\text{\\rm curl\\,} B_{1,\\kappa}=\\nabla_\\perp B_{1,\\kappa}\\,$, the boundedness of $ \\|\\nabla\nB_{1,\\kappa}\\|_2$ then easily follows from the above and\n\\eqref{eq:31}. Consequently,\n \\begin{equation}\n\\label{eq:41}\n \\frac{1}{c}\\|\\nabla \\phi_{1,\\kappa} \\|_2+ \\|\\nabla B_{1,\\kappa}\\|_2 \\leq C\\,.\n\\end{equation}\nNote that $\\nabla\\phi_{1,\\kappa}$ and $\\nabla_\\perp B_{1,\\kappa}$ are respectively the $L^2$ projections\nof $\\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa)$ on\n\\begin{displaymath}\n H^0_0(\\text{\\rm curl\\,}, \\Omega)=\\{\\widehat V \\in L^2(\\Omega,\\mathbb R^2)\\,:\\, \\text{\\rm curl\\,} \\widehat V=0\\}\\,,\n\\end{displaymath}\nand\n\\begin{displaymath}\n \\mathcal H^0_d:= \\{\\widehat W \\in L^2(\\Omega,\\mathbb R^2)\\,:\\, \\text{\\rm div\\,} \\widehat\n W=0\\mbox{ and } \\widehat W \\cdot \\nu =0\\mbox{ on } \\pa \\Omega \\}\\,.\n\\end{displaymath}\n\nNext, we attempt to estimate $\\|\\nabla \\phi_{1,\\kappa} \\|_p$ and $\\|\\nabla B_{1,\\kappa}\\|_p$ for\nany \\linebreak $p>2$. Since $\\Omega$ is simply-connected, we may conclude\nfrom \\eqref{eq:5}, (\\ref{eq:29}a) and Remark B.2 in \\cite{alhe14} that\nthere exists for any $p>2$ a constant \\linebreak $C(p,\\Omega)>0$ such that\n\\begin{displaymath}\n \\|A_\\kappa\\|_{1,p} \\leq C\\, \\|B_\\kappa\\|_p \\,,\n\\end{displaymath}\nfor all $\\kappa\\geq 1$. Sobolev embeddings then imply\n\\begin{equation}\n \\label{eq:42}\n\\|A_\\kappa\\|_\\infty \\leq C\\, \\|B_\\kappa\\|_p\\,.\n\\end{equation}\nSince $ \\|\\nabla B_{1,\\kappa}\\|_2$ is uniformly bounded for all $\\kappa \\geq 1$, we\nobtain from \\eqref{eq:32}, the Poincar\\'e inequality, and Sobolev embeddings\nthat, for any $p>2$ there exists a constant $C(p,\\Omega)>0$ such that we have\n\\begin{equation}\\label{estB1p}\n \\|B_{1,\\kappa}\\|_p \\leq C(p,\\Omega)\\,.\n\\end{equation}\n\nHence, recalling from \\eqref{reguphi} that $B_n\\in L^p$ and independent\nof $c$ and $\\kappa$, as $J$ is independent of $\\kappa$, there\nexists a constant $C>0$ such that\n\\begin{equation}\\label{estBp}\n \\|B_\\kappa\\|_p=\\|B_{1,\\kappa}+\\kappa B_n\\|_p\\leq C\\, \\kappa\\,.\n\\end{equation}\nCombining the above computations with \\eqref{eq:42} then yields\n\\eqref{eq:43}.\n\\vspace{1ex}\n\n\n\n{\\it Step 2: Prove \\eqref{eq:37}.}\n\\vspace{1ex}\n\nWe first rewrite (\\ref{eq:25}a,c,d) in the following form\n\\begin{displaymath}\n \\begin{cases}\n\\Delta\\psi_\\kappa = 2i\\kappa A_\\kappa\\cdot\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa + |\\kappa A_\\kappa|^2\\psi_\\kappa- \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)+i\\kappa\\phi_\\kappa\\psi_\\kappa & \\text{ in } \\Omega\\, ,\\\\\n \\psi_\\kappa =0 & \\text{on } \\partial\\Omega_c \\,, \\\\\n\\frac{\\partial\\psi_\\kappa}{\\partial\\nu}=i\\kappa (A_\\kappa \\cdot\\nu) \\psi_\\kappa=0\\, & \\text{ on } \\partial\\Omega_i \\,,\\\\\n \\end{cases}\n\\end{displaymath}\nwhere the last equality follows from (\\ref{eq:5}). By \\eqref{eq:43},\n\\eqref{eq:34}, the fact $\\|\\psi_\\kappa\\|_\\infty\\leq 1$, Proposition~A.3 and\nRemark~A.4 in \\cite{alhe14} (note that $\\psi_\\kappa$ vanishes at the\ncorners) we obtain that for some $C(\\Omega,p,J)$\n\\begin{displaymath}\n \\|\\psi_\\kappa\\|_{2,p} \\leq C\\big[\\kappa^4 + \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\big]\\,, \\quad \\forall p>2\\,.\n\\end{displaymath}\nSobolev embedding and \\eqref{eq:43} then yield\n\\begin{equation}\n\\label{eq:44}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_\\infty \\leq C\\big[\\kappa^4 +\n \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\big] \\,.\n\\end{equation}\nWe now use a standard interpolation theorem to obtain that\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_2^{2\/p} \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_\\infty^{1-2\/p}\\,.\n\\end{displaymath}\n\n\n\n\nSubstituting \\eqref{eq:44} in conjunction with \\eqref{eq:40} into the\nabove inequality then yields\n\\begin{equation}\n\\label{eq:45}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq C\\Big[\\kappa^4 + \\kappa^2\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\Big]^{1-2\/p}\\kappa^{2\/p}\\,.\n\\end{equation}\nSuppose first that\n\\begin{displaymath}\n \\kappa^2< \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\,.\n\\end{displaymath}\nThen, we have\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\leq C\\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p^{1-2\/p}\\kappa^{2(1-1\/p)}\\,.\n\\end{displaymath}\nHence,\n\\begin{equation}\n\\label{eq:46}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p \\leq C\\kappa^{p-1} \\,.\n\\end{equation}\nNext, assume that\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq \\kappa^2 \\,,\n\\end{displaymath}\nto obtain that\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\leq C\\kappa^{4-6\/p} \\,.\n\\end{displaymath}\nFrom the above, together with \\eqref{eq:46} we easily conclude that,\nfor any $20$,\nwe have\n\\begin{equation}\\label{reglp}\n \\|\\nabla\\phi_{1,\\kappa}\\|_p \\leq C \\frac{c}{\\kappa} \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_p\\,, \\quad \\forall\\kappa \\geq 1\\,,\\; 21\\,,\\; 2(1+\\delta+\\kappa^{-\\alpha})\\,.\n\\end{displaymath}\nFrom \\eqref{B1k}, for some $C(p,\\Omega,J)>0$ we have that\n\\begin{displaymath}\n \\aligned\n |B_\\kappa(x)|>&\\kappa|B_n(x)| - |B_{1,\\kappa}(x)| \\geq(1+\\delta+\\kappa^{-\\alpha})\\kappa - C \\kappa^{3(p-2)\/p}\\\\\n =&(1+\\delta)\\kappa+[\\kappa^{1-\\alpha}- C \\kappa^{3(p-2)\/p}].\n \\endaligned\n\\end{displaymath}\nBy choosing\n\\begin{displaymath}\n 20$ for sufficiently large $\\kappa\\,$. Thus\n$$|B_\\kappa(x)|>(1+\\delta)\\kappa\\,,\n$$\nand hence $x\\in D_\\delta(\\kappa)$. Consequently, $S_{\\delta+\\kappa^{-\\alpha}}\\subset D_\\delta(\\kappa)\\,$.\n\\end{proof}\n\nAs a byproduct of the proof, we also obtain\n\\begin{proposition}\n For any $20$ such that\n \\begin{equation}\n \\label{eq:49}\n\\|A_{1,\\kappa}\\|_{2,p} \\leq C \\kappa^{3(p-2)\/p}\\,,\\quad \\forall \\kappa \\geq \\kappa_0\\,.\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\n The proof follows immediately from \\eqref{eq:31}, \\eqref{eq:48}, and\n Proposition B.3 in \\cite{alhe14}.\n\\end{proof}\n\nWe can now prove the following semi-classical Agmon estimate for\n$\\psi_\\kappa$, establishing that it must be exponentially small in $S_\\delta$.\n\\begin{proposition}\nSuppose that $h$ satisfies (\\ref{eq:10}b). Let then $j\\in\\{1,2\\}$ be such that\n$h_j>1\/\\Theta_0$. There exist $C>0$ and\n$\\delta_0>0$, such that, for any $0<\\delta\\leq \\delta_0\\,$, some $\\kappa_0(\\delta)$ can be found,\nfor which\n \\begin{equation}\n \\label{eq:50}\n\\kappa \\geq \\kappa_0(\\delta) \\Rightarrow\\int_{S_{\\delta,j}} \\exp\\Big(\\delta^{1\/2}\\kappa \\,\n d(x,\\mathcal C_{\\delta,j})\\Big) |\\psi_\\kappa|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}} \\,,\n \\end{equation}\nwhere $S_{\\delta,j}$ is introduced in \\eqref{sdj} and $\\mathcal C_{\\delta,j}$ in \\eqref{eq:36}.\n\\end{proposition}\n\\begin{proof}\nFor $\\delta >0$, let $\\eta\\in C^\\infty(\\Omega,[0,1])$ satisfy\n\\begin{equation}\n\\label{eq:51}\n \\eta(x)=\n \\begin{cases}\n 1 & x\\in S_{\\delta,j}\\,, \\\\\n 0 & x\\in\\Omega\\setminus S_{\\delta\/2,j} \\,.\n \\end{cases}\n \\end{equation}\n By \\eqref{eq:12} and \\eqref{reguphi}, it follows that $\\nabla B_n$ is\n bounded and independent of both $\\delta$ and $\\kappa$. Consequently,\nthere exists a constant $C_1 >0$ such that\n \\begin{displaymath}\n d(\\mathcal C_{\\delta,j},\\mathcal C_{\\delta\/2,j}) \\geq \\frac{\\delta}{C_1}\\,,\n \\end{displaymath}\nand hence, for some $C(\\Omega,J)$ and all $0<\\delta<\\delta_0$ we can choose $\\eta$ such that\n \\begin{displaymath}\n |\\nabla\\eta| \\leq \\frac{C}{\\delta} \\,.\n \\end{displaymath}\nLet further $$\\zeta=\\chi\\, \\eta$$ where\n\\begin{displaymath}\n \\chi=\n \\begin{cases}\n \\exp(\\alpha_\\delta\\kappa d(x,\\mathcal C_{\\delta,j})) &\\text{if } x\\in S_{\\delta,j}\\,, \\\\\n 1 &\\text{if } x\\in\\Omega\\setminus S_{\\delta,j}\\,.\n \\end{cases}\n\\end{displaymath}\nWe leave the determination of $\\alpha_\\delta$ to a later stage. We further\ndefine, for any $r\\in (0,r_0)$, $\\eta_r\\in C^\\infty(\\Omega,[0,1])$ and $\\tilde \\eta_r\\in C^\\infty(\\Omega,[0,1])$ such that\n\\begin{equation}\n\\label{eq:52}\n \\eta_r(x)=\n \\begin{cases}\n 1 & d(x,\\partial\\Omega_i)>r \\\\\n 0 & d(x,\\partial\\Omega_i)1+\\delta \\;;\\;d(x,\\partial\\Omega_i) > \\delta\\,\\} \\,,\\end{equation}\nand\n\\begin{equation} \\label{Gamma}\n \\Gamma_{\\delta,j}= \\partial\\omega_{\\delta,j} \\setminus\\partial\\Omega_c\\cap\\partial\\omega_{\\delta,j} \\,.\n\\end{equation}\nWe can now state\n\\begin{proposition}\n Suppose that for some $j\\in\\{1,2\\}$ we have\n \\begin{equation}\n\\label{eq:57}\n 1<|h_j| \\,.\n \\end{equation}\nThen, there exist $C>0$, $\\delta_0>0$, such that for any $0<\\delta<\\delta_0$,\nsome $\\kappa_0 (\\delta)>0$ can be found, for which\n \\begin{equation} \\label{eq:58}\n\\kappa\\geq \\kappa_0(\\delta)\\Rightarrow \\int_{\\omega_{\\delta,j}} \\exp\\Big(\\delta^{1\/2}\\kappa d(x,\n\\Gamma_{\\delta,j})\\Big) |\\psi_\\kappa|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}} \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWithout loss of generality we may assume $h_j>0$; otherwise we apply\n to \\eqref{eq:25} the transformation\n $(\\psi_\\kappa,A_\\kappa,\\phi_\\kappa)\\to(\\bar{\\psi_\\kappa},-A_\\kappa,-\\phi_\\kappa)$. Let\n \\begin{equation}\\label{check1}\n { \\check{\\chi}=}\n \\begin{cases}\n \\exp\\Big(\\frac 12 \\delta^{1\/2}\\kappa d(x,\\Gamma_{\\delta,j})\\Big) &\\text{if } x\\in \\omega_{\\delta,j}\\,, \\\\\n 1 &\\text{if } x\\in\\Omega\\setminus \\omega_{\\delta,j}\\,.\n \\end{cases}\n\\end{equation}\nLet further $\\eta$ and $\\eta_r$ be given by \\eqref{eq:51} and\n\\eqref{eq:52} respectively. Then set\n\\begin{equation}\\label{check2}\n\\check{\\zeta}= \\eta_\\delta\\, \\eta\\, \\check \\chi \\,.\n\\end{equation}\nThe proof proceeds in the same manner as in the previous proposition\nwith $\\zeta$ replaced by $\\check{\\zeta}$ with the difference that now\n$\\check{\\zeta} \\psi_\\kappa(x)$ vanishes for all $ x\\in\\partial \\Omega_i$. Consequently,\n(\\ref{eq:10}a) is no longer necessary (see \\eqref{eq:55}). We use (\\ref{eq:10}b)\nto establish that $\\omega_{\\delta,j}$ is not empty.\n\\end{proof}\n\nWe conclude this section by showing that for $\\mathcal O(\\kappa)$ currents\n(i.e. when $J$ is independent of $\\kappa$) $\\|\\psi_\\kappa\\|_2$ must be small.\nTo this end we define $\\Phi_n$ as the solution of (\\ref{eq:11b}a,b), and\n\\begin{equation}\n\\label{eq:59}\n \\int_\\Omega |\\psi_\\kappa|^2\\Phi_n \\,dx =0\\, .\n\\end{equation}\nThe above condition is a natural choice as by \\eqref{eq:33} we have\nthat\n\\begin{displaymath}\n \\int_\\Omega |\\psi_\\kappa|^2\\phi_\\kappa \\,dx =0\\, .\n\\end{displaymath}\nIt can be easily verified from \\eqref{eq:20} that\n\\begin{equation}\\label{calconst}\n \\Phi_n=\\phi_n+ C(\\kappa,c)\\,,\n\\end{equation}\nwhere $\\phi_n$ denotes the solution of \\eqref{eq:11b}.\nThe constant can be extracted from \\eqref{eq:59}:\n$$\nC (\\kappa,c) = - \\frac{\\int_\\Omega \\phi_n |\\psi_\\kappa|^2\\, dx }{\\int_\\Omega |\\psi_\\kappa|^2\\, dx}\\,,\n$$\nfrom which we get the following upper bound (independent of $\\kappa$ and $c$)\n\\begin{equation}\n\\label{eq:60}\n|C (\\kappa,c) | \\leq \\|\\phi_n\\|_\\infty < +\\infty\\,.\n\\end{equation}\n\n\\begin{proposition}\n Under Assumptions \\eqref{eq:2}-\\eqref{eq:10} there exists \\linebreak\n $C(J,\\Omega)>0$ and $\\kappa_0>0$ such that for any $\\kappa\\geq \\kappa_0$,\n \\begin{equation}\n \\label{eq:61}\n\\| \\psi_\\kappa \\|_2 \\leq C (J,\\Omega) (1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6} \\,.\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $\\Phi_{1,\\kappa}=\\phi_k-c\\kappa\\Phi_n$.\n An immediate consequence of \\eqref{eq:33} is that\n\\begin{equation}\n\\label{eq:62}\n \\begin{cases}\n -\\Delta\\Phi_{1,\\kappa} + c|\\psi|^2_\\kappa \\Phi_{1,\\kappa} = - c^2|\\psi|^2_\\kappa \\kappa\\Phi_n & \\text{ in } \\Omega\\,, \\\\\n \\frac{\\partial\\Phi_{1,\\kappa}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega\\,.\n \\end{cases}\n\\end{equation}\nTaking the inner product with $\\Phi_n$ yields, with the aid of (\\ref{eq:59}),\n\\begin{equation}\n\\label{eq:63}\n \\aligned\n& \\kappa\\| \\psi_\\kappa \\Phi_n\\|_2^2 = - \\frac{1}{c^2}\\langle\\nabla\\Phi_n,\\nabla\\Phi_{1,\\kappa}\\rangle +\\frac{1}{c}\\langle \\langle|\\psi_\\kappa|\n \\Phi_n,|\\psi_\\kappa| (\\Phi_{1,\\kappa}-(\\Phi_{1,\\kappa})_\\Omega \\rangle\\\\\n\\leq& \\frac{1}{c^2}\\|\\nabla\\Phi_n\\|_2\\|\\nabla\\Phi_{1,\\kappa}\\|_2 + \\frac{1}{c}\\| \\psi_\\kappa\n \\Phi_n\\|_2 \\| \\psi_\\kappa(\\Phi_{1,\\kappa}-(\\Phi_{1,\\kappa})_\\Omega )\\|_2\\,,\n\\endaligned\n\\end{equation}\nwhere $(\\Phi_{1,\\kappa})_\\Omega $ is the average of $\\Phi_{1,\\kappa}$ in $\\Omega$.\n\nWith the aid of (\\ref{eq:41}) (note that $\\nabla \\phi_{1,\\kappa}=\\nabla \\Phi_{1,\\kappa}$),\nthe fact that \\break $|\\psi_\\kappa|\\leq1$, and the Poincar\\'e inequality we then\nobtain\n\\begin{equation}\n \\label{eq:64}\n \\| \\psi_\\kappa \\Phi_n\\|_2 \\leq C\\kappa^{-1\/2}(1+ c^{-1\/2}) \\,.\n\\end{equation}\nWe now set\n\\begin{displaymath}\n {\\mathcal U}_\\kappa = \\{ x\\in\\Omega \\,:\\, |\\Phi_n(x) |< (1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\,\\} \\,.\n\\end{displaymath}\nBy \\eqref{eq:59} the level set $\\Phi_n^{-1}(0)$ lies inside\n$\\Omega$.\nLet $x_0\\in\\Phi_n^{-1}(\\tau)$ for some $\\tau\\neq0$, and set\n$$\\Gamma_\\perp=B_n^{-1}(B_n(x_0)).\n$$\nBy (\\ref{eq:11}a) $B_n$ is the conjugate harmonic function of $\\Phi_n$,\nand hence $\\Gamma_\\perp$ must be perpendicular to $\\Phi_n^{-1}(\\tau)$ at $x_0$. Note\nthat in \\cite[(2.3)]{alhe14} we showed that $B_n^{-1}(\\mu)$ is a simple\nsmooth curve connecting the two connected components of $\\partial\\Omega_c$ for any\n$h_1<\\mu0$ or increases if $\\tau<0$, and terminating either on $\\Phi_n^{-1}(0)$ or\non the boundary. Clearly,\n\\begin{displaymath}\n |\\tilde{\\Gamma}_\\perp|\\inf_{x\\in\\Omega}|\\nabla\\Phi_n|\\leq \\Big|\\int_{\\tilde{\\Gamma}_\\perp} \\nabla\\Phi_n ds\\Big| \\leq |\\tau|\\,,\n\\end{displaymath}\nwhere $\\int_\\Gamma \\vec{V}$ denotes the circulation of $\\vec{V}$ along the path $\\Gamma$.\nIn \\cite[\\S2.3]{alhe14} we have\nestablished that $|\\nabla\\Phi_n|=|\\nabla B_n|>0$ in $\\bar{\\Omega}$. It follows that\n\\begin{equation}\n\\label{eq:65}\n d(x_0,\\partial\\Omega\\cup\\Phi_n^{-1}(0)) \\leq |\\tilde{\\Gamma}_\\perp| \\leq C|\\tau|\\,.\n\\end{equation}\nLet\n\\begin{displaymath}\n \\tilde{{\\mathcal U}}_\\kappa(r)= \\{x\\in\\Omega \\,| \\, d(x,\\partial\\Omega\\cup\\Phi_n^{-1}(0)) \\leq\n r(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\}\\,.\n\\end{displaymath}\nBy \\eqref{eq:65} we obtain that for sufficiently large $r$ there\nexists $\\kappa_0(r)$ such that for all $\\kappa>\\kappa_0$ and\n$c\\in{\\mathbb{R}}$ we have ${\\mathcal U}_\\kappa \\subseteq \\tilde{{\\mathcal U}}_\\kappa(r)$. Consequently,\n\\begin{equation}\n\\label{eq:66}\n | {\\mathcal U}_\\kappa| \\leq C(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}(|\\Phi_n^{-1}(0)|+|\\partial\\Omega|)\\leq C(1+c^{-1\/2})^{2\/3}\\kappa^{-1\/3}\\,.\n\\end{equation}\n\nBy \\eqref{eq:64} we have that\n\\begin{displaymath}\n \\| \\psi_\\kappa\\|_{L^2(\\Omega\\setminus{\\mathcal U}_\\kappa)} \\leq C(1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6} \\,,\n\\end{displaymath}\nwhereas from \\eqref{eq:66} and \\eqref{eq:26} we learn that\n\\begin{displaymath}\n \\| \\psi_\\kappa\\|_{L^2({\\mathcal U}_\\kappa)} \\leq C(1+c^{-1\/2})^{1\/3}\\kappa^{-1\/6} \\,.\n\\end{displaymath}\nThe proposition can now be readily verified.\n\\end{proof}\nAn immediate conclusion is that whenever $c\\kappa\\gg1$, $|\\psi_\\kappa|$ is small. If\n$c=\\mathcal O(\\kappa^{-1})$, $|\\psi_\\kappa|$ may not tend to $0$ as $\\kappa\\to\\infty$. Further research\nis necessary to establish this point.\n\\begin{remark}\n If, for some $0<\\alpha<1$, we assume that $J=J(\\cdot, \\kappa)$ satisfies\n$$\n\\|J\\|\\leq C\\kappa^{-\\alpha}\\,,\n$$\nthen (\\ref{eq:63}) and\n (\\ref{eq:41}) remain valid. Assuming $c=1$, and using this time (\\ref{eq:48}), we obtain\n instead of (\\ref{eq:64}), for any $0<\\beta$, that\n \\begin{displaymath}\n \\| \\psi_\\kappa \\Phi_n\\|_2 \\leq C_\\beta \\kappa^{-1\/2}(\\kappa^{\\alpha\/2}+\\kappa^\\beta\\|\\psi_\\kappa\\|_2^{1\/2}) \\,.\n \\end{displaymath}\nUsing the above, with sufficiently small $\\beta$, we obtain, similarly to the derivation of (\\ref{eq:61})\n\\begin{displaymath}\n \\| \\psi_\\kappa\\|_2\\leq C\\kappa^{-(1-\\alpha)\/6}\\,,\n\\end{displaymath}\nwhich implies\n$$\\|\\psi_\\kappa\\|_2\\xrightarrow[\\kappa\\to + \\infty]{}0\\,.\n$$\nThis result stands in sharp contrast with the behavior obtained\nin the absence of electric potential \\cite{sase03,attar2014ground}.\n\\end{remark}\n\n\\section{Time-Dependent Analysis}\n\\label{sec:4}\nIn this section we return to the time-dependent problem as introduced\nin \\eqref{eq:1}. For convenience we set here $$c=1\\,.$$\n\\begin{subequations}\n\\label{eq:67}\n\\begin{alignat}{2}\n & \\frac{\\partial\\psi_\\kappa}{\\partial t}- \\nabla_{\\kappa A_\\kappa}^2\\psi_\\kappa+i\\kappa\\phi_\\kappa \\psi_\\kappa =\\kappa^2(1-|\\psi_\\kappa|^2)\\psi_\\kappa & \\text{ in } (0,+\\infty)\\times \\Omega& \\,,\\\\\n &\\frac{\\partial A_\\kappa}{\\partial t}+\\nabla\\phi_\\kappa +\\text{\\rm curl\\,}^2A_\\kappa\n =\\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) &\n \\text{ in } (0,+\\infty)\\times \\Omega&\\,, \\\\\n &\\psi =0 &\\text{ on } (0,+\\infty)\\times \\partial\\Omega_c&\\,, \\\\\n& \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\cdot\\nu=0 & \\text{ on } (0,+\\infty)\\times \\partial\\Omega_i&\\,, \\\\\n& \\frac{\\partial\\phi_\\kappa}{\\partial\\nu} = - \\kappa J(x) & \\text{ on }(0,+\\infty)\\times \\partial\\Omega_c&\\,, \\\\\n & \\frac{\\partial\\phi_\\kappa}{\\partial\\nu}=0 & \\text{ on } (0,+\\infty)\\times \\partial\\Omega_i &\\,, \\\\[1.2ex]\n& \\Xint-_{\\partial\\Omega}\\text{\\rm curl\\,} A_\\kappa \\, ds = \\kappa h_{ex} &\\quad \\text{ on } (0,+\\infty)&\\,, \\\\\n& \\psi(0,x)=\\psi_0(x) & \\text{ in } \\Omega&\\,, \\\\\n&A(0,x)=A_0(x) & \\text{ in } \\Omega & \\,.\n\\end{alignat}\n\\end{subequations}\nWe assume again \\eqref{condinit}-(\\ref{eq:6}), \\eqref{eq:8}, and\n\\eqref{eq:10}. Since in the time dependent case $\\phi_\\kappa$ is determined\nup to a constant in view of \\eqref{eq:4}\n and \\eqref{eq:5}, we can\nfurther impose\n\\begin{equation}\\label{eq:68}\n \\int_\\Omega\\phi_\\kappa (t,x)\\,dx =0\\,, \\quad \\forall t>0 \\,.\n\\end{equation}\nIt follows from \\eqref{condinit} by the maximum\nprinciple (see \\cite[Theorem 2.6] {alhe14}) that\n\\begin{equation}\n\\label{eq:69}\n \\|\\psi_\\kappa(t, \\cdot)\\|_\\infty\\leq 1 \\,, \\quad \\forall t\\geq0\\,.\n\\end{equation}\nWe recall from \\cite [Subsection 2.4] {alhe14} the following spectral entity\n\\begin{subequations}\n\\label{eq:70}\n \\begin{equation}\n\\lambda= \\inf_{\n \\begin{subarray}{2}\n V\\in{\\mathcal H}_d \\\\\n \\|V\\|_2 =1\n \\end{subarray}} \\| \\text{\\rm curl\\,} V\\|_2^2 \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n {\\mathcal H}_d = \\big\\{ V\\in H^1(\\Omega,{\\mathbb{R}}^2)\\,:\\, \\text{\\rm div\\,} V=0 \\,, V\\big|_{\\partial\\Omega}\\cdot\\nu=0 \\big\\} \\,.\n\\end{equation}\n\\end{subequations}\nWe further recall from \\cite [Proposition 2.5] {alhe14} that, under condition $(R_1)$ on $\\partial \\Omega$,\n\\begin{displaymath}\n \\lambda=\\lambda^D:=\\inf_{\n \\begin{subarray}{2}\n u\\in H^1_0(\\Omega) \\\\\n \\|u\\|_2 =1\n \\end{subarray}} \\| \\nabla u\\|_2^2>0 \\,.\n\\end{displaymath}\n\nWe retain our definition of the normal fields $(A_n,\\phi_n)$ via \\eqref{eq:11}. For\nthe solution $(A_\\kappa,\\phi_\\kappa)$ of \\eqref{eq:67} we set\n\\begin{equation}\\label{time-dep-AB} \\aligned\n&A_{1,\\kappa}(t,x)=A_\\kappa(t,x)-\\kappa A_n(x),\\\\\n&\\phi_{1,\\kappa}(t,x)=\\phi_\\kappa(t,x)-\\kappa\\phi_n(x),\\\\\n&B_\\kappa(t,x)=\\text{\\rm curl\\,} A_\\kappa(t,x),\\\\\n&B_{1,\\kappa}(t,x)=\\text{\\rm curl\\,} A_{1,\\kappa}(t,x).\n\\endaligned\n\\end{equation}\nClearly,\n\\begin{subequations}\n\\label{eq:71}\n \\begin{alignat}{2}\n \\frac{\\partial A_{1,\\kappa}}{\\partial t}+\\nabla\\phi_{1,\\kappa} + \\text{\\rm curl\\,} B_{1,\\kappa}\n & = \\frac{1}{\\kappa}\\Im(\\bar\\psi_\\kappa\\, \\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) & \\text{ in }\n (0,+\\infty)\\times \\Omega\\,,\\\\\n \\frac{\\partial\\phi_{1,\\kappa}}{\\partial\\nu}&= 0 & \\text{ on } (0,+\\infty)\\times \\partial\\Omega\\,, \\\\\n\\Xint-_{\\partial\\Omega}B_{1,\\kappa}(t,x)\\,ds& = 0 & \\text{ in } (0,+\\infty) \\,.\n\\end{alignat}\n\\end{subequations}\n\nWe begin by the following auxiliary estimate. We recall that \\break $\\|A(t,\\cdot)\\|_{1,2}=\\|A(t,\\cdot)\\|_{H^1(\\Omega,{\\mathbb{R}}^2)}$.\n\\begin{lemma}\n\\label{lem:parabolic-A1}\nLet $A_{1,\\kappa}$ and $B_{1,\\kappa}$ be defined by \\eqref{time-dep-AB}.\nSuppose that \\linebreak $\\|A_{1,\\kappa}(\\cdot,0)\\|_2\\leq M$ (where $M$ may depend on\n$\\kappa$). Then, under the above assumptions, there exists $ t^*(M)$ and a\nconstant $C=C(\\Omega,t^*)>0$ such that for all $t>t^*$ and $\\kappa\\geq 1$ we\nhave\n\\begin{equation}\n \\label{eq:72}\n\\|A_{1,\\kappa}(t,\\cdot)\\|_{1,2} + \\|A_{1,\\kappa}\\|_{L^2(t,t+1,H^2(\\Omega))} \\leq C.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n By \\cite[Lemma 5.3]{alhe14} there exists a constant $C=C(\\Omega)>0$ such that for\n sufficiently large\n $\\kappa$\n \\begin{displaymath}\n \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq \\Big(\\|A_{1,\\kappa}(0,\\cdot)\\|_2^2+\\frac{C}{\\kappa^2}\\Big)\ne^{-\\lambda t} + C \\int_0^te^{-\\lambda(t-\\tau)}\\|\\psi_\\kappa(\\tau,\\cdot)\\|_2^2 \\,d\\tau \\,.\n \\end{displaymath}\nFrom \\eqref{eq:69}, we then get\n \\begin{displaymath}\n \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq \\Big(\\|A_{1,\\kappa}(0,\\cdot)\\|_2^2+\\frac{C}{\\kappa^2}\\Big)\\,\ne^{-\\lambda t} + \\frac{C}{\\lambda}\n \\end{displaymath}\nWe thus have\n \\begin{equation}\\label{anytime}\n \\|A_{1,\\kappa}(t,\\cdot)\\|_2^2 \\leq (M+C) e^{-\\lambda t} + \\frac{C}{\\lambda}\n \\end{equation}\nHence, there exists $t_0^*(M)$, such that for $t \\geq t_0^*(M)$, we have\n\\begin{equation}\n\\label{eq:73}\n \\|A_{1,\\kappa}(t,\\cdot)\\|_2 \\leq \\frac{2C}{\\lambda} \\,.\n\\end{equation}\nNext, we apply \\cite[Theorem C.1 (Formula C.4)]{alhe14} to the operator\n$\\mathcal L^{(1)}$ (as introduced there in Example (4) above this theorem) to\nobtain that\n\\begin{equation}\\label{consc1}\n \\aligned\n& \\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+\n \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))}\\\\& \\qquad\n\\leq \\frac{C}{\\kappa}\\|\\Im \\{\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n A_\\kappa}\\psi_\\kappa\\}\\|_{L^2(t_0,t_0+1,L^2(\\Omega))} +\n C \\|A_{1,\\kappa}(t_0,\\cdot)\\|_{1,2} \\,.\n\\endaligned\n\\end{equation}\nwith a constant $C$ independent of $t_0$.\nSince from (\\ref{eq:67}a) (cf. \\cite{alhe14}) we can easily get\nthat\n\\begin{equation}\n\\label{eq:74}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa (t,\\cdot)\\|_2^2 \\leq \\kappa^2\\|\\psi_\\kappa(t,\\cdot)\\|_2^2-\\frac{1}{2}\n \\frac{d\\|\\psi_\\kappa (t,\\cdot)\\|_2^2}{dt} \\,,\n\\end{equation}\nwe obtain by integrating over $(t_0,t_0+1)$\n\\begin{equation}\\label{eq:70a}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa \\|^2_{L^2(t_0,t_0+1,L^2(\\Omega))} \\leq \\kappa^2\\|\\psi_\\kappa \\|_{L^2(t_0,t_0+1,L^2(\\Omega))}^2 + \\frac 12 \\|\\psi_\\kappa(t_0, \\cdot)\\|_2^2\\,,\n \\end{equation}\nand note for later reference that it implies\n \\begin{equation}\\label{eq:70aa}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa \\|_{L^2(t_0,t_0+1,L^2(\\Omega))} \\leq C(\\Omega)\\, \\kappa\\,.\n \\end{equation}\nImplementing the upper bound \\eqref{eq:70a} in \\eqref{consc1}, yields\n$$\\aligned\n& \\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+ \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\\\\n\\leq& C\\Big[1+\n \\|\\psi_\\kappa\\|_{L^2(t_0,t_0+1,L^2(\\Omega))} +\n \\frac{1}{\\kappa}\\|\\psi_\\kappa(t_0, \\cdot)\\|_2+\\|A_{1,\\kappa}(t_0,\\cdot)\\|_{1,2}\\Big] \\,.\n\\endaligned\n$$\nWe next apply \\cite[Theorem C.1 (Formula C.2)]{alhe14} to obtain in\nprecisely the same manner\n$$\n\\begin{array}{l}\n\\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}\\\\\n\\qquad \\leq C\\Big[1+\n \\|\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} +\n \\frac{1}{\\kappa}\\|\\psi_\\kappa(t_0, \\cdot)\\|_2 + \\|A_{1,\\kappa}(t_0-1,\\cdot)\\|_2 \\Big] \\,.\n \\end{array}\n$$\nThe above together with \\eqref{eq:69} and \\eqref{eq:73} yields, for\n $t_0\\geq t_0^* +1$,\n\\begin{equation}\n\\label{est-of-B}\n\\|A_{1,\\kappa}\\|_{L^\\infty(t_0,t_0+1,H^1(\\Omega))}+ \\|A_{1,\\kappa}\\|_{L^2(t_0,t_0+1,H^2(\\Omega))}\n\\leq C\\,,\n\\end{equation}\nwhich implies \\eqref{eq:72}, with $t^* = t_0^* (M) +1$.\n\\end{proof}\n\\begin{remark}\\label{usefulrem}\n Since our interest is in the limit as $t \\to +\\infty$, Lemma~\\ref{lem:parabolic-A1} allows us to assume in the sequel, without\n any loss of generality, that (\\ref{eq:72}) is satisfied for all\n $t\\geq0$. We have just to make a translation $t \\mapsto t -t^*$ and to\n observe that $\\psi_\\kappa(t^*,\\cdot)$ has the same properties as $\\psi_0$.\n \\end{remark}\n\\begin{proposition}\n \\label{lem:asymp-exp}\n Let $\\omega_{\\delta,j}$ ($j\\in\\{1,2\\}$) be defined in\n \\eqref{eq:56}.\nSuppose that\n for some $j\\in\\{1,2\\}$ we have that\n \\begin{equation}\n\\label{eq:75}\n 1<|h_j| \\,.\n \\end{equation}\n Then, there exist $C>0$ and $\\delta_0>0$, and, for any $0<\\delta<\\delta_0$, $\\kappa_0(\\delta)\\geq 1$ such that, for $\\kappa\\geq \\kappa_0(\\delta)$,\n\\begin{equation}\n\\label{eq:76}\n\\limsup_{t\\to\\infty}\\int_{\\omega_{\\delta,j} }|\\psi_\\kappa|^2(t,x) \\,dx \\leq\n \\frac{C_\\delta }{\\kappa^2} \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \n Without loss of generality we may assume $h_j>0\\,$; otherwise we apply\nto \\eqref{eq:67} the transformation\n$(\\psi_\\kappa,A_\\kappa,\\phi_\\kappa)\\to(\\bar{\\psi_\\kappa},-A_\\kappa,-\\phi_\\kappa)$.\n\\vspace{1ex}\n\n{\\it Step 1: Let, for $n\\geq1$,\n\\begin{displaymath}\n a_n = \\|{\\widehat \\zeta}\\psi_\\kappa\\|_{L^\\infty(n-1, n,L^2(\\Omega))} \\,.\n\\end{displaymath}\nProve that for all $\\delta \\in (0,1)$ and $\\kappa \\geq \\kappa_0(\\delta)$,\n\\begin{equation}\\label{inega}\n a_n^2 \\leq C \\delta^{-3} \\Big(\\kappa^{-2} + \\kappa^{-1} (a_{n} + a_{n-1})\\Big)\\,.\n\\end{equation}}\n\\vspace{2ex}\n\nLet $\\eta$ and $\\eta_r$ be given by \\eqref{eq:51} and \\eqref{eq:52}\nrespectively. Then, set\n\\begin{equation}\n\\label{eq:77}\n {\\widehat \\zeta}=\\eta \\, \\eta_\\delta\\, .\n\\end{equation}\nMultiplying (\\ref{eq:67}a) by ${\\widehat \\zeta}^{\\,2} \\bar{\\psi_\\kappa}$ and integrating by\n parts yields\n\\begin{displaymath}\n \\frac{1}{2}\\frac{d}{dt} \\left( \\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\right) \\, + \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa (t,\\cdot))\\|_2^2 \\leq\n \\kappa^2\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2 + \\|\\psi_\\kappa (t,\\cdot)\\, \\nabla{\\widehat \\zeta}\\|_2^2 \\,.\n\\end{displaymath}\nBy \\cite[Theorem 2.9] {AHS},\nwe have\n$$\\aligned\n \\|\\nabla_{\\kappa A_\\kappa}({\\widehat \\zeta}\\psi_\\kappa (t,\\cdot))\\|_2^2 \\geq& \\langle\\kappa B_\\kappa (t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\\\ =&\n \\kappa^2\\langle B_n{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot) \\rangle + \\langle\\kappa B_{1,\\kappa}(t,\\cdot)\\,{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\\\\n \\geq&\n \\kappa^2\\Big(1+\\frac{\\delta}{2}\\Big)\\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2+ \\langle\\kappa B_{1,\\kappa}{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\,,\\,{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n\\endaligned\n$$\nWe can thus write\n\\begin{displaymath}\n\\begin{array}{l}\n \\frac{1}{2}\\frac{d}{dt} \\left( \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot) \\|_2^2\\right) + \\frac{\\kappa^2\\delta}{2}\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot) \\|_2^2\\\\ \\quad \\leq\n \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\psi_\\kappa(t,\\cdot) \\nabla\\eta_\\delta\\|_2^2- \\langle\\kappa B_{1,\\kappa}(t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot),{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n \\end{array}\n\\end{displaymath}\nSince\n\\begin{displaymath}\n \\|\\psi_\\kappa (t,\\cdot)\\nabla\\eta\\|_2^2 + \\|\\psi_\\kappa(t,\\cdot) \\nabla\\eta_\\delta\\|_2^2 \\leq \\frac{C}{\\delta^2} \\,,\n\\end{displaymath}\nwe obtain that\n\\begin{equation}\n \\label{eq:78}\n \\begin{array}{l}\n \\frac{1}{2}\\frac{d}{dt} \\left( \\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2 \\right)\n + \\frac{\\kappa^2\\delta}{2}\\|{\\widehat \\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\\\ \\qquad\n \\leq \\frac{C}{\\delta^2} - \\langle \\kappa B_{1,\\kappa} (t,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\,,\\, {\\widehat \\zeta}\\psi_\\kappa (t,\\cdot) \\rangle \\,.\n\\end{array} \\end{equation}\nFrom \\eqref{eq:78} we can conclude that\n\\begin{equation}\\label{eq:79}\n\\aligned\n \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\leq& \\|{\\widehat \\zeta}\\psi_0\\|_2^2\\, e^{-\\delta\\kappa^2t} +\n \\frac{C}{\\delta^3\\kappa^2}\\\\\n &+2 \\int_0^t\n e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa(\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau,\\cdot) \\rangle \\big|\\,d\\tau \\,.\n\\endaligned\n\\end{equation}\n \nTo estimate the last term on the right-hand-side of \\eqref{eq:79}, we start from\n $$ \\aligned\n& \\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa (\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot) \\rangle\\big|\\,d\\tau \\\\\n\\leq & \\kappa\\, \\Big[\\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\n { \\|B_{1,\\kappa}(\\tau, \\cdot)\\|_2^2 \\,d\\tau} \\cdot \\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}{ \\psi_\\kappa(\\tau, \\cdot )\\|_4^4}\\,d\\tau\\Big]^{1\/2}\\,. \\\\\n \\endaligned\n $$\nWith Remark \\ref{usefulrem} in mind, we use \\eqref{eq:69} to obtain\n$$\n \\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|B_{1,\\kappa}(\\tau, \\cdot)\\|_2^2 \\,d\\tau \\leq \\frac{C}{\\delta \\kappa^2} \\,.\n$$\nImplementing the above estimate, we obtain\n$$\n\\begin{array}{l}\n\\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n B_{1,\\kappa}(\\tau,\\cdot) {\\widehat \\zeta}\\psi_\\kappa (\\tau,\\cdot) ,{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot) \\rangle\\big|\\,d\\tau\\\\\n \\qquad\n \\leq C \\delta^{-\\frac 12} \\Big[\\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}{\\psi_\\kappa(\\tau, \\cdot )\\|_4^4}\\,d\\tau\\Big]^{1\/2}\\,.\n \\end{array}\n$$\nTo control of the right hand side we now write for $t\\geq 1$\n$$\\aligned\n& \\Big[\\int_0^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2} \\\\\n& \\leq \\Big[\\int_0^{t-1} e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2} + \\Big[\\int_{t-1} ^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_4^4\\,d\\tau\\Big]^{1\/2}\\\\\n & \\leq C \\Big[\\int_{t-1} ^t e^{-\\delta\\kappa^2(t-\\tau)}\n \\|{\\widehat \\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_2^2\\,d\\tau\\Big]^{1\/2} + C \\delta^{-\\frac 12} \\kappa^{-1} e^{- \\frac{\\delta }{2} \\kappa^2}\\\\\n& \\leq C \\delta^{-\\frac 12} \\kappa^{-1} \\|\\widehat \\zeta\n \\psi_\\kappa\\|_{L^\\infty(t-1,t,L^2(\\Omega)) } + C \\delta^{-\\frac 12} \\kappa^{-1} e^{-\n \\frac{\\delta }{2} \\kappa^2} \\,.\n\\endaligned\n$$\n\nSubstituting the above into \\eqref{eq:79} yields, with a new constant\n$C$, for $\\kappa$ large enough, and for $t\\geq1\\,$,\n$$\n\\begin{array}{l}\n \\|{\\widehat \\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\\\ \\quad \\leq C \\delta^{-3} \\kappa^{-2} + C \\delta^{-1} \\kappa^{-1} e^{ - \\frac{\\delta}{2} \\kappa^2}\n + C \\delta^{-1} \\kappa^{-1} \\|\\widehat \\zeta \\psi_\\kappa\\|_{L^\\infty(t-1,t,L^2(\\Omega)) } \\,.\n \\end{array}\n$$\nFrom which we easily obtain \\eqref{inega}.\n\\vspace{1ex}\n\n\n{\\it Step 2: Prove \\eqref{eq:76}.}\n\\vspace{1ex}\n\nBy (\\ref{eq:69}) we have \n$$\n0 < a_n \\leq C \\,,\n$$\nwhich readily yields\n$$\na_n \\leq C \\delta^{-\\frac 32}\\kappa^{-\\frac 12}\\,.\n$$\nWe improve the above estimate by reimplementing \\eqref{inega}.\nTo this end we set\n$$\\widehat C := C \\delta^{- \\frac 32 }\\,,$$ and then let\n\\begin{displaymath}\n a_n= \\frac{\\widehat C}{\\kappa}\\alpha_n \\,.\n\\end{displaymath}\nSubstituting into (\\ref{inega}) yields\n\\begin{equation}\\label{inegaga}\n \\alpha_n^2 \\leq 1 + \\alpha_{n-1} +\\alpha_n \\,.\n\\end{equation}\nSuppose that for some $N\\geq0$, we have $\\alpha_N\\leq 1 + \\sqrt{2} \\,$, then\\break $\\alpha_{N+1}\\leq 1 +\\sqrt{2}$ and\nhence $\\alpha_n\\leq 1 + \\sqrt{2}$ for all $n\\geq N$.\n\nIf $\\alpha_{n-1} > 1+ \\sqrt{2}$ for any $n$, we have, with $\\hat \\alpha_n = \\alpha_n - \\frac 12\\,$,\n\\begin{displaymath}\n \\hat \\alpha_n^2\\leq \\frac 74 + \\hat \\alpha_{n-1} < \\hat \\alpha_{n-1}^2 \\,.\n\\end{displaymath}\nHence, $\\hat \\alpha_n<\\hat \\alpha_{n-1}$ which means that $\\hat \\alpha_n$\nconverges as a positive decreasing sequence, and necessarily to a\nlimit smaller than $1\/2 + \\sqrt{2}$. We thus conclude that\n\\begin{equation}\n \\label{eq:80}\n\\limsup \\alpha_n \\leq 1 + \\sqrt{2} \\,,\n\\end{equation}\nand hence\n\\begin{equation}\n\\label{ineq-1} \\limsup a_n \\leq \\frac{C (1 +\\sqrt{2})}{\\delta^{3\/2}\\kappa},\n\\end{equation}\nfrom which \\eqref{eq:76} can easily be deduced.\n\\end{proof}\n\nWe next obtain the following improvement over (\\ref{eq:70aa}) for\n$\\nabla_{\\kappa A_\\kappa}(\\widehat \\zeta \\psi_\\kappa)$.\n\n\\begin{proposition}\n Let $p \\geq 2$. For any $\\delta >0$, there exists $\\kappa_0 (\\delta)$ and $C(\\delta)$\n such that for $\\kappa \\geq \\kappa (\\delta)$ we have, with $u={\\widehat \\zeta}\\psi_\\kappa\\,,\n $ the following estimate:\n\\begin{equation}\\label{eq:89a}\n \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}\\leq C (\\delta) \\kappa^{6(1-2\/p)}\\,.\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof} {\\it Step 1: Prove that for some $C(\\delta)>0$ we have, for\n sufficiently large $\\kappa$ that\n\\begin{equation}\n \\label{eq:87}\n\\|u\\|_{L^2(t_0-1,t_0+1,H^2(\\Omega))} \\leq C\\kappa^3\\,.\n\\end{equation}}\n\\vspace{1ex}\n\nWe rewrite (\\ref{eq:67}a, c, d) in the form\n\\begin{displaymath}\n \\begin{cases}\n\\frac{\\partial\\psi_\\kappa}{\\partial t}-\\Delta\\psi_\\kappa = -2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa - |\\kappa A_\\kappa|^2\\psi_\\kappa+ \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)\\\\\n\\qquad\\qq\\qquad \\qquad - i\\kappa\\phi_\\kappa\\psi_\\kappa&\\text{ in } (0,+\\infty)\\times\\Omega,\\\\\n \\psi_\\kappa =0 &\\text{ on } (0,+\\infty)\\times \\partial\\Omega_c \\,, \\\\\n\\frac{\\partial\\psi_\\kappa}{\\partial\\nu}=i\\kappa A_\\kappa\\psi_\\kappa\\cdot\\nu=0 &\\text{ on }(0,+\\infty)\\times \\partial\\Omega_i \\,.\\\\\n \\end{cases}\n\\end{displaymath}\nClearly, by our choice of $\\widehat \\zeta$,\n\\begin{displaymath}\n \\begin{cases}\n\\frac{\\partial u}{\\partial t}-\\Delta u\n= {\\widehat \\zeta}\\big[-2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa - |\\kappa\nA_\\kappa|^2\\psi_\\kappa + \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)& \\\\\n\\qquad\\qquad\\quad - i\\kappa\\phi_\\kappa\\psi_\\kappa\\big] + 2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa\n + \\psi_\\kappa \\Delta{\\widehat \\zeta} & \\text{ in } (0,+\\infty)\\times \\Omega\\, ,\\\\\n u =0 & \\text{on } (0,+\\infty)\\times \\partial\\Omega \\,.\n\\end{cases}\n\\end{displaymath}\nBy \\cite[Theorem C.1]{alhe14} (this time applied to the Dirichlet\nLaplacian in $\\Omega$) in the interval $(t_0-1,t_0+1)$\n\\begin{equation}\\label{eq:81}\n\\aligned\n&\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\\\\n\\leq &\\big\\|{\\widehat \\zeta}\\big[-2i\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa - |\\kappa\nA_\\kappa|^2\\psi_\\kappa + \\kappa^2\\psi_\\kappa\\big( 1 - |\\psi_\\kappa|^{2}\n\\big)- i\\kappa\\phi_\\kappa\\psi_\\kappa\\big]\\big\\| _{L^2(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\\\\n&+ \\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa\n + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} + C\\|u(t_0-1,\\cdot)\\|_{2} \\,.\n\\endaligned\n\\end{equation}\n\nBy \\eqref{eq:69} we have that\n\\begin{equation}\n\\label{eq:82}\n \\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\n\\leq C(1+\\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}) \\,.\n\\end{equation}\nAs\n\\begin{displaymath}\\\n \\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}+\\|\\kappa A_\\kappa\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\,,\n\\end{displaymath}\nwe obtain in view of (\\ref{eq:70a}) and (\\ref{eq:72}) that\n\\begin{equation}\n\\label{eq:83}\n \\|\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq C\\kappa^2 \\,.\n\\end{equation}\nSubstituting the above into (\\ref{eq:82}) yields\n\\begin{equation}\n \\label{eq:84}\n\\|2\\nabla{\\widehat \\zeta}\\cdot\\nabla\\psi_\\kappa + \\psi_\\kappa \\Delta{\\widehat \\zeta}\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq C\\kappa^2\\,.\n\\end{equation}\n\nWe next observe that\n\\begin{displaymath}\n \\|{\\widehat \\zeta} |\\kappa A_\\kappa|^2\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))} \\leq\n \\kappa^2\\|A_\\kappa\\|_{L^4(t_0-1,t_0+1,L^\\infty(\\Omega,{\\mathbb{R}}^2))}^2 \\|{\\widehat \\zeta}\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\,.\n\\end{displaymath}\nSince $\\widehat \\zeta$ is supported in the set $\\omega_{\\delta,j}$, we may use\n\\eqref{eq:76}, which together with Agmon's inequality \\cite[Lemma\n13.2]{ag65}, (\\ref{time-dep-AB}), and \\eqref{eq:72}, yield, for $t_0$ large enough,\n\\begin{equation}\\label{eq:85} \\aligned\n& \\|{\\widehat \\zeta} |\\kappa A_\\kappa|^2\\psi_\\kappa \\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\\\\n\\leq& C\\kappa\\|A_\\kappa\\|_{L^\\infty(t_0-1,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))}\n\\|A_\\kappa\\|_{L^2(t_0-1,t_0+1,H^2(\\Omega,{\\mathbb{R}}^2))}\\\\\n\\leq& C\\kappa^3\\,.\n\\endaligned\n\\end{equation}\nSimilarly,\n\\begin{equation}\\label{eq:86}\n\\aligned\n& \\|{\\widehat \\zeta}\\kappa A_\\kappa\\cdot\\nabla\\psi_\\kappa\\|_{L^2(t_0-1,t_0+1,L^2(\\Omega))}\\\\\n\\leq& \\kappa\\|A_\\kappa\\|_{L^4(t_0,t_0+1,L^\\infty(\\Omega,{\\mathbb{R}}^2))}\n \\big[\\|u\\|_{L^4(t_0-1,t_0+1,H^1(\\Omega))}\n +\\|\\psi_\\kappa\\nabla{\\widehat \\zeta}\\|_{L^4(t_0-1,t_0+1,L^2(\\Omega))}\\big]\\\\\n\\leq& C\\kappa^2 +\n C\\kappa^2\\|u\\|^{1\/2}_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\|u\\|^\\frac 12 _{L^\\infty(t_0-1,t_0+1,L^2(\\Omega))} \\\\\n\\leq & C\\Big[\\kappa^2 + \\kappa^{3\/2}\\|u\\|^\\frac 12_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\Big]\\,.\n\\endaligned\n\\end{equation}\nSubstituting \\eqref{eq:86} together with \\eqref{eq:84},\nand\n\\eqref{eq:85} into \\eqref{eq:81} yields with the aid of \\eqref{eq:69}\n\\begin{equation}\\label{eq:80a}\n\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega))} \\leq C\\Big[\\kappa^3 + \\kappa^{3\/2}\\|u\\|^{\\frac 12}_{L^2(t_0-1,t_0+1,H^2(\\Omega))}\\Big]\\,\n\\end{equation}\n\n\n\nProceeding as in the proof of Proposition \\ref{lem:asymp-exp}, we can\nassume $C\\geq 1$ in \\eqref{eq:80a} and set\n$$\n\\alpha_n = C^{-1} \\kappa^{-\\frac 32} \\|u\\|_{L^2(n,n+1,H^2(\\Omega))} ^\\frac 12\\,.\n$$\nWe now can rewrite \\eqref{eq:80a} in the form\n$$\n\\alpha_n^2 \\leq ( 1 +\\alpha_{n-1} + \\alpha_n)\\,,\n$$\nwhich is precisely \\eqref{inegaga}.\nWe can thus conclude (\\ref{eq:80}), and hence, for a new value of $C$,\n\\eqref{eq:87} easily follows.\n\\vspace{1ex}\n\n{\\it Step 2: Prove that\n\\begin{equation}\n \\label{eq:89}\n \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1,H^1(\\Omega,{\\mathbb{R}}^2))} \\leq C \\kappa^3 \\,.\n\\end{equation}}\n\\vspace{2ex}\n\nIt can be easily verified that\n\\begin{equation}\n\\label{eq:88}\n \\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_{1,2}\\leq \\|u(t,\\cdot)\\|_{2,2} + \\kappa\\|\\,|A_\\kappa|\\nabla u(t,\\cdot)\\|_2 +\n \\kappa\\|u\\nabla A_{\\kappa}(t,\\cdot)\\|_2 \\,.\n\\end{equation}\nFurthermore, in the same manner we have obtained \\eqref{eq:86} we\nobtain, with the aid of \\eqref{eq:87}\n\\begin{displaymath}\n \\kappa \\|\\,|A_\\kappa|\\nabla u\\|_{L^2(t_0,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\leq C\n \\kappa^{3\/2}\\|u\\|_{L^2(t_0,t_0+1,H^2(\\Omega,{\\mathbb{R}}^2))}^{1\/2}\\leq C\\kappa^3 \\,.\n\\end{displaymath}\nBy \\eqref{eq:69} and \\eqref{eq:72} we have that\n\\begin{displaymath}\n \\kappa\\|u\\nabla A_{\\kappa}\\|_{L^2(t_0,t_0+1,L^2(\\Omega,{\\mathbb{R}}^2))} \\leq C\\kappa^2 \\,.\n\\end{displaymath}\nWe can now conclude \\eqref{eq:89} from \\eqref{eq:88}. \n\\vspace{1ex}\n\n\n{\\it Step 3: Prove \\eqref{eq:89a}.}\n\\vspace{1ex}\n\n In \\cite[(5.35)]{alhe14} it was shown that\n \\begin{equation}\n\\label{eq:90}\n \\|\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}\\leq C\\kappa^3\\,.\n \\end{equation}\n (Note that while the setting in \\cite{alhe14} is different then - in\n particular, we assume there $J\\sim\\mathcal O(\\kappa)$ - the estimate is\n still valid in the present case because $c=1$.)\nHence, we get\n\\begin{equation}\n\\label{eq:91}\n \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}\\leq C\\kappa^3\\,.\n\\end{equation}\nWe now use Gagliardo-Nirenberg interpolation inequality (see \\cite{ni59}) to\nobtain\n$$\n\\aligned\n & \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}^p\\qquad \\\\\n& \\leq C \\int_{t_0}^{t_0+1}\n \\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_{1,2}^{p-2}\\|\\nabla_{\\kappa A_\\kappa}u(t,\\cdot)\\|_2^2 \\,dt\\\\\n& \\leq C \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; H^1(\\Omega))}^{p-2} \\|\\nabla_{\\kappa\n A_\\kappa}u\\|_{L^{\\frac{4}{4-p}}(t_0,t_0+1; L^2(\\Omega))}^2 \\,.\n\\endaligned\n$$\nConsequently,\n\\begin{equation}\\label{eq:92}\n\\aligned\n&\\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^p(t_0,t_0+1; L^p(\\Omega))}^p \\\\\n\\leq &C \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; H^1(\\Omega))}^{p-2} \\|\\nabla_{\\kappa\n A_\\kappa}u\\|_{L^2(t_0,t_0+1; L^2(\\Omega))}^{4-p} \\|\\nabla_{\\kappa\n A_\\kappa}u\\|_{L^\\infty(t_0,t_0+1; L^2(\\Omega))}^{p-2} \\,.\n\\endaligned\n\\end{equation}\n Multiplying (\\ref{eq:67}a) by ${\\widehat \\zeta}^2\\bar{\\psi}_\\kappa$ and integrating over\n$\\Omega$ we obtain for the real part that\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa(t,\\cdot)}u(t,\\cdot)\\|_2^2 \\leq \\kappa^2\\|u (t,\\cdot)\\|_2^2-\\frac{1}{2}\n \\frac{d\\|u (t,\\cdot)\\|_2^2}{dt}+ \\|\\psi_\\kappa (t,\\cdot) \\nabla{\\widehat \\zeta}\\|_2^2 \\,,\n\\end{displaymath}\nIntegrating over $(t_0,t_0+1)$ and using \\eqref{eq:76} we then obtain, for sufficiently large $t_0$,\n\\begin{displaymath}\n \\|\\nabla_{\\kappa A_\\kappa}u\\|_{L^2(t_0,t_0+1; L^2(\\Omega))}^2 \\leq C \\,.\n\\end{displaymath}\nSubstituting the above, \\eqref{eq:89}, and \\eqref{eq:91}, into\n\\eqref{eq:92} we then obtain \\eqref{eq:89a}.\n\\end{proof}\n\n\nWe can now obtain the following improved regularity for $B_{1,\\kappa}$\n\\begin{proposition}\nLet $20$ such that for all $t_0>1$\n and $\\kappa>\\kappa_0(\\delta)$ we have\n\\begin{equation}\n \\label{eq:93}\n\\|B_{1,\\kappa}\\|_{L^p(t_0,t_0+1,W^{1,p}(\\omega_{\\delta,j}))} \\leq C \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \nTaking the curl of \\eqref{eq:71} yields that $B_{1,\\kappa}$ is a weak\nsolution of\n\\begin{equation}\n\\label{eq:94}\n \\begin{cases}\n \\frac{\\partial B_{1,\\kappa}}{\\partial t} - \\Delta B_{1,\\kappa} = \\frac{1}{\\kappa}\\text{\\rm curl\\,}\n \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa A_\\kappa}\\psi_\\kappa) & \\text{ in } (0,+\\infty)\\times \\Omega \\\\\n B_{1,\\kappa} =0 & \\text{ on } (0,+\\infty)\\times \\partial\\Omega \\,.\n \\end{cases}\n\\end{equation}\n Let\n\\begin{displaymath}\n \\mathcal B_{\\widehat \\zeta} = {\\widehat \\zeta}B_{1,\\kappa} \\,,\n\\end{displaymath}\nwhere the cutoff function $\\widehat{\\zeta}$ is defined by (\\ref{eq:77}).\nLet further $\\hat{\\Omega}(\\delta)\\subset\\Omega$ be smooth and satisfy\n$$\n{\\rm supp}\\,\\widehat{\\zeta}\\subset \\hat{\\Omega}(\\delta)\\,.$$\nAs for any $V\\in H^1(\\Omega,{\\mathbb{R}}^2)$ we have that\n\\begin{displaymath}\n \\text{\\rm curl\\,} V = \\text{\\rm div\\,} V_\\perp\\,,\n\\end{displaymath}\nit can be easily verified\nfrom (\\ref{eq:94}) that\n\\begin{displaymath}\n \\frac{\\partial \\mathcal B_{\\widehat \\zeta}}{\\partial t} - \\, \\Delta \\mathcal B_{\\widehat \\zeta} =\n \\frac{1}{\\kappa}{\\widehat \\zeta}\\text{\\rm div\\,}\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n A_\\kappa}\\psi_\\kappa)_\\perp\\big) -2\\text{\\rm div\\,}(B_{1,\\kappa}\\nabla{\\widehat \\zeta)}) +\n B_{1,\\kappa}\\Delta\\widehat \\zeta\\,.\n\\end{displaymath}\nConsequently,\n\\begin{equation}\n\\label{eq:95}\n\\left\\{\n \\begin{array}{rll}\n \\frac{\\partial \\mathcal B_{\\widehat \\zeta}}{\\partial t} - \\, \\Delta \\mathcal B_{\\widehat \\zeta} &=\n \\frac{1}{\\kappa}{\\widehat \\zeta}\\, \\text{\\rm div\\,}\\,\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n A_\\kappa}\\psi_\\kappa)_\\perp\\big)& \\\\\n &\\quad -\\text{\\rm div\\,}(2 B_{1,\\kappa}\\nabla{\\widehat \\zeta} +\n \\nabla \\Delta_D^{-1}(B_{1,\\kappa}\\Delta{\\widehat \\zeta}) ) & \\text{in } (t_0-1,t_0+1)\\times\\hat{\\Omega} \\\\\n \\mathcal B_{\\widehat \\zeta} &=0 & \\text{on } (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\\\\n\\mathcal B_{\\widehat \\zeta} (t_0-1,\\cdot) & = \\widehat \\zeta\\, \\mathcal B_{1,\\kappa} (t_0-1,\\cdot) &\\text{in } \\hat{\\Omega} \\,.\n \\end{array}\n \\right.\n\\end{equation}\nIn the above $\\Delta_D^{-1}$ denotes the inverse Dirichlet Laplacian in $\n\\hat{\\Omega}$.\n\nIn order to apply\n \\cite[Theorem 1.6]{by07} which is devoted to the case of parabolic\n operators written in divergence form and with zero initial condition\n we first decompose the solution of \\eqref{eq:95} into two\n Cauchy-Dirichlet problems.\n The first of them is:\n \\begin{equation} \\label{eq:117a}\n\\left\\{\n \\begin{array}{rll}\n \\frac{\\partial U_1}{\\partial t} - \\, \\Delta U_1 &= \\text{\\rm div\\,} f_1\n & \\text{ in } (t_0-1,t_0+1)\\times\\hat{\\Omega}\\,, \\\\\n U_1 &=0 & \\text{ on } (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\,,\\\\\nU_1 (t_0-1,\\cdot)& = 0 &\\text{ in } \\hat{\\Omega} \\,,\n \\end{array}\n \\right.\n\\end{equation}\nin which\n\\begin{equation}\\label{eq:117aa}\nf_1 = \\frac{1}{\\kappa}{\\widehat \\zeta}\\, \\,\\big( \\Im(\\bar{\\psi}_\\kappa\\nabla_{\\kappa\n A_\\kappa}\\psi_\\kappa)_\\perp\\big) - 2 B_{1,\\kappa}\\nabla{\\widehat \\zeta} -\n \\nabla \\Delta_D^{-1}(B_{1,\\kappa}\\Delta{\\widehat \\zeta}) \\,.\n \\end{equation}\nThe second one is:\n\\begin{equation}\n\\label{eq:117b}\n\\left\\{\n \\begin{array}{rll}\n \\frac{\\partial U_2}{\\partial t} - \\, \\Delta U_2 &= F_2\n & \\text{in } (t_0-1,t_0+1)\\times\\hat{\\Omega} \\,,\\\\\n U_2 &=0 & \\text{on } (t_0-1,t_0+1)\\times\\partial\\hat{\\Omega} \\,,\\\\\nU_2(t_0-1,\\cdot )& = \\widehat \\zeta\\, \\mathcal B_{1,\\kappa} (t_0-1,\\cdot) &\\text{in } \\hat{\\Omega} \\,,\n \\end{array}\n \\right.\n\\end{equation}\nwhere\n \\begin{equation} \\label{eq:117ba}\n F_2:= - \\frac{1}{\\kappa} \\,\\big( \\Im(\\bar{\\psi}_\\kappa \\nabla {\\widehat \\zeta}\\cdot (\\nabla_{\\kappa\n A_\\kappa}\\psi_\\kappa)_\\perp) \\big)\\,.\n \\end{equation}\nBy uniqueness of the weak solution, we have\n\\begin{equation} \\label{eq:117c}\n \\mathcal B_{\\widehat \\zeta} = U_1 + U_2 \\text{ in } (t_0-1,t_0+1)\\times\\hat{\\Omega}\\,.\n \\end{equation}\nWe now separately estimate $U_1$ and $U_2$ in $L^p (t_0,t_0+1, W^{1,p}( \\hat \\Omega))\\,$.\n\\vskip0.05in\n\n{\\it Estimate of $U_1$}\n\\vspace{1ex}\n\nWe apply \\cite[Theorem 1.6]{by07} to obtain\n\\begin{equation} \\label{eq:96}\n\\| U_1\\|_{L^p(t_0,t_0+1,W^{1,p}(\\Omega))}\\\\\n\\leq C \\,\\|f_1\\|_{L^p(t_0,t_0+1,L^p(\\Omega,{\\mathbb{R}}^2)}\\,.\n\\end{equation}\nIt can be easily verified, by the Gagliardo-Nirenberg interpolation\ninequality \\cite{ni59} that, for\nall $20\\,,$ constants $C(\\delta)$ and $\\kappa(\\delta)$ such that for any $\\kappa\n\\geq \\kappa_0(\\delta)$ and any $t_0 >1$ we have\n \\begin{equation}\n\\label{eq:89c}\n \\| U_2\\|_{ L^p(t_0,t_0+1,W^{1,p})} \\leq C (\\delta)\\,.\n \\end{equation}\n The combination of \\eqref{eq:79a} and \\eqref{eq:89c} together with\n \\eqref{eq:117c} completes the proof of the proposition.\n\\end{proof}\n\nWe can now establish the exponential decay of $\\psi_\\kappa$.\n\\begin{proposition}\n \\label{asymp-exp-2}\n Let $\\omega_{\\delta,j}$ ($j\\in\\{1,2\\}$) be given by \\eqref{eq:56}. Suppose\n that for some $k\\in\\{1,2\\}$ \\eqref{eq:75} is satisfied. Then, there\n exist $C>0$ and $\\delta_0>0$, and, for any $0<\\delta<\\delta_0$,\n $\\kappa_0(\\delta)$, such that for any $\\kappa\\geq \\kappa_0(\\delta)$ we have\n\\begin{equation}\n\\label{eq:99}\n\\limsup_{t\\to\\infty}\\int_{\\omega_{\\delta,j}} \\exp \\Big(\\delta^{1\/2}\n \\kappa d(x,\n\\Gamma_{\\delta,j})\\Big) |\\psi_\\kappa|^2(t,x) \\,dx \\leq C \\,.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} \n Without loss of generality we may, as before, assume $h_j>0$.\nLet $\\check{\\chi}$ and $\\check{\\zeta}$ be defined by \\eqref{check1} and\n\\eqref{check2}.\n\\vspace{1ex}\n \n{\\it Step 1: Prove that\n\\begin{equation}\n\\label{eq:101}\n \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 \\leq \\|\\check{\\zeta}\\psi_0\\|_2^2e^{-2\\gamma\\kappa^2t} +\n \\frac{C(\\delta)}{\\kappa^4}+ \\int_0^t\n e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle(\\tau)\\big|\\,d\\tau \\,.\n\\end{equation}}\n\\vspace{1ex}\n \nMultiplying\n(\\ref{eq:67}a) by $\\check{\\zeta}^2 \\bar{\\psi}$ and integrating by parts yields\n\\begin{displaymath}\n \\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot) \\|_2^2 \\right) + \\|\\nabla_{\\kappa A_\\kappa(t,\\cdot)}(\\check{\\zeta}\\psi_\\kappa(t,\\cdot))\\|_2^2 \\leq\n \\kappa^2\\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2 + \\|\\psi_\\kappa (t,\\cdot) \\nabla\\check{\\zeta}\\|_2^2 \\,.\n\\end{displaymath}\nBy Theorem 2.9 in \\cite{AHS} we have\n$$\\aligned\n \\|\\nabla_{\\kappa A_\\kappa}(\\check{\\zeta}\\psi_\\kappa)\\|_2^2\n& \\geq \\langle\\kappa B_\\kappa(t,\\cdot) \\check{\\zeta}\\psi_\\kappa(t,\\cdot),\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\rangle\\\\ & =\n \\kappa^2\\langle B_n\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle + \\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle \\\\\n&\\geq\n \\kappa^2\\Big(1+\\frac{\\delta}{2}\\Big)\\|\\check{\\zeta}\\psi_\\kappa\\|_2^2+ \\langle\\kappa B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle \\,.\n\\endaligned\n$$\nWe can thus write\n\\begin{equation}\\label{eq3.32}\n\\aligned\n&\\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2\\right) +\n\\kappa^2\\Big( \\frac{\\delta}{2}- \\frac{\\delta}{4}\n\\Big)\\|\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\|_2^2\\\\\n\\leq&\n \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\tilde \\chi\\,\\psi_\\kappa (t,\\cdot)\\nabla\\eta_\\delta\\|_2^2- \\langle\\kappa B_{1,\\kappa}(t,\\cdot)\\check{\\zeta}\\psi_\\kappa(t,\\cdot),\\check{\\zeta}\\psi_\\kappa(t,\\cdot)\\rangle \\,.\n\\endaligned\n\\end{equation}\nBy \\eqref{eq:76}, for every $0<\\delta\\leq\\delta_0$, we have\n\\begin{displaymath}\n \\|\\psi_\\kappa (t,\\cdot) \\nabla\\eta\\|_2^2 + \\|\\tilde \\chi\\, \\psi_\\kappa (t,\\cdot) \\,\\nabla\\eta_\\delta\\|_2^2 \\leq \\frac{C}{\\kappa^2} \\,,\n\\end{displaymath}\nwhich when substituted into \\eqref{eq3.32} yields\n\\begin{equation}\n \\label{eq:100}\n \\frac{1}{2}\\frac{d}{dt}\\left( \\|\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\|_2^2\\right) + \\kappa^2\\gamma\\|\\check{\\zeta}\\psi_\\kappa (t,\\cdot) \\|_2^2\n \\leq \\frac{C}{\\kappa^2} - \\langle\\kappa B_{1,\\kappa}(t,\\cdot) \\check{\\zeta}\\psi_\\kappa(t,\\cdot) ,\\check{\\zeta}\\psi_\\kappa (t,\\cdot)\\rangle \\,,\n\\end{equation}\nwhere $\\gamma=\\delta\/4$. We now get \\eqref{eq:101} from \\eqref{eq:100}. \n\\vspace{1ex}\n\n\n{\\it Step 2: Prove that, for all $n\\geq2$,\n\\begin{equation}\n\\label{eq3.33}\n\\aligned\n&\\|\\check{\\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_{L^\\infty(t^*+n-1,t^*+n,L^2(\\Omega))}^2 \\\\\n\\leq& C \\kappa^{-(p-2)\/p}\\|\\check{\\zeta}\\psi_\\kappa(\\tau, \\cdot )\\|_{L^\\infty(t^*+n-2,t^*+n,L^2(\\Omega))}^2 +{C\\over\\kappa^4} \\,,\n\\endaligned\n\\end{equation}\nwhere $C$ is independent of $\\kappa\\,$.}\n\\vspace{1ex}\n\nTo prove \\eqref{eq3.33} we need to estimate the last term on the\nright-hand-side of \\eqref{eq:101}. To this end we first write\n\\begin{multline*}\n \\int_0^t e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle (\\tau) \\big|\\,d\\tau \\\\\n\\leq \\kappa \\left(\\int_{t-1}^t e^{-2\\gamma\\kappa^2(t-\\tau)}\n{ \\|B_{1,\\kappa}(\\tau, \\cdot\n )\\|_{L^\\infty(\\omega_{\\delta\/2,j})}}\\,d\\tau\\right)\\,\\cdot\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(t-1,t;L^2(\\Omega))}\\\\\n+ \\kappa\\int_0^{t-1}\ne^{-2\\gamma\\kappa^2(t-\\tau)}\\|B_{1,\\kappa}(\\tau,\\cdot)\\|_1\\,d\\tau\n\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(0,t-1;L^2(\\Omega))}\\,.\n\\end{multline*}\nFor the last term on the right-hand-side we have in view of Remark~\\ref{usefulrem}, \\eqref{check1}, \\eqref{check2}, and \\eqref{eq:69} that\n\\begin{displaymath}\n \\kappa\\int_0^{t-1}\ne^{-2\\gamma\\kappa^2(t-\\tau)}\\|B_{1,\\kappa}(\\tau,\\cdot)\\|_1\\,d\\tau\n\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(0,t-1;L^2(\\Omega))}\\leq Ce^{-2\\gamma\\kappa^2}e^{C\\kappa}\\,.\n\\end{displaymath}\nHence for sufficiently large $\\kappa$ we have\n\\begin{multline}\n\\label{eq:102}\n \\int_0^t e^{-2\\gamma\\kappa^2(t-\\tau)}\\big|\\langle\\kappa\n B_{1,\\kappa}\\check{\\zeta}\\psi_\\kappa,\\check{\\zeta}\\psi_\\kappa\\rangle (\\tau) \\big|\\,d\\tau \\leq \\\\\n\\kappa \\left(\\int_{t-1}^t e^{-2\\gamma\\kappa^2(t-\\tau)} \\|B_{1,\\kappa}(\\tau, \\cdot\n )\\|_{L^\\infty(\\omega_{\\delta\/2,j})}\\,d\\tau\\right)\\,\\cdot\\|\\check{\\zeta}\\psi_\\kappa\\|_{L^\\infty(t-1,t;L^2(\\Omega))}+ Ce^{-\\gamma\\kappa^2}\\,.\n\\end{multline}\nSince by Sobolev embeddings\n\\begin{displaymath}\n \\|B_{1,\\kappa}\\|_{L^p(t-1,t,L^\\infty(\\omega_{\\delta\/2,j}))} \\leq C\\|B_{1,\\kappa}\\|_{L^p(t-1,t,W^{1,p}(\\omega_{\\delta\/2,j}))} \\,,\n\\end{displaymath}\nwe can use \\eqref{eq:93} to obtain, for\nsufficiently large $t$ and $\\kappa$, and for any $20$ we have\n \\begin{displaymath}\n \\epsilon^{-\\gamma}0$, and $\\alpha >0$,\nsuch that for any $0\\,<\\,\\epsilon\\,<\\epsilon_0\\,$ we have\n\\begin{equation}\n\\label{eq:105}\n\\int_{K} |\\psi_\\epsilon(x)|^2 \\,dx \\leq C e^{-\\alpha\/\\epsilon} \\,.\n\\end{equation}\n\\end{proposition}\n\nWe split the proof of Proposition \\ref{semiclassical} into several\nsteps, to each of them we dedicate a separate lemma. We begin by observing,\nas in Section \\ref{sec:3}, that\n\\begin{equation}\n \\label{eq:106}\n\\|\\psi_\\epsilon\\|_\\infty\\leq 1\\,.\n\\end{equation}\nLet\n\\begin{equation}\n\\label{eq:107}\n A_{1,\\epsilon}=A_\\epsilon-\\epsilon^{-\\alpha}A_n,\\quad\n\\phi_{1,\\epsilon}=\\phi_\\epsilon-\\epsilon^{-\\alpha}\\phi_n,\n\\end{equation}\nSet further\n\\begin{equation}\n\\label{eq:108}\n B_{1,\\epsilon}= \\text{\\rm curl\\,} A_{1,\\epsilon}\\,; \\quad B_\\epsilon= \\text{\\rm curl\\,} A_\\epsilon \\,.\n\\end{equation}\nBy (\\ref{eq:104}b) and (\\ref{eq:11}a), we then have\n\\begin{subequations}\n\\label{eq:109}\n\\begin{empheq}[left={\\empheqlbrace}]{alignat=2}\n &\\nabla_\\perp B_{1,\\epsilon} + \\nabla\\phi_{1,\\epsilon} = \\Im(\\bar\\psi_\\epsilon\\, \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon) & \\text{ in }\n \\Omega\\,,\\\\\n &\\frac{\\partial\\phi_{1,\\epsilon}}{\\partial\\nu}= 0 & \\text{ on } \\partial\\Omega \\,, \\\\\n&\\Xint-_{\\partial\\Omega}B_{1,\\epsilon}(x)\\,ds = 0 \\,. &\n\\end{empheq}\n\\end{subequations}\nNote that since $\\partial B_{1,\\epsilon}\/\\partial\\tau=\\partial\\phi_{1,\\epsilon}\/\\partial\\nu=0$ on $\\partial\\Omega$\nwe must have by (\\ref{eq:109}c) that\n\\begin{equation}\n\\label{eq:110}\n B_{1,\\epsilon}|_{\\partial\\Omega}\\equiv 0\\,.\n\\end{equation}\n\n\n We begin with the following auxiliary estimate.\n\\begin{lemma}\n \\label{lem:magnetic-bound}\nLet $w_\\epsilon$ denote the solution of\n\\begin{equation}\\label{eq:111}\n \\left\\{\\aligned\n& \\Delta w_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2w_\\epsilon = 0\\quad & \\text{\\rm in }\\Omega\\;, \\\\\n& w_\\epsilon = B_\\epsilon - 1 \\quad& \\text{\\rm on } \\partial\\Omega.\n\\endaligned\n\\right.\n\\end{equation}\nUnder the assumptions on $J$ and $\\Omega$ in (\\ref{eq:3})-(\\ref{hyptopolo}) we have\n\\begin{equation}\n \\label{eq:112}\n\\|B_\\epsilon-1-w_\\epsilon\\|_\\infty \\leq \\frac{1}{2} \\,.\n\\end{equation}\nFurthermore, we have\n\\begin{equation}\n\\label{eq:113}\n\\|\\nabla\\phi_\\epsilon\\|_2+ \\|\\phi_\\epsilon\\|_\\infty \\leq C\\epsilon^{-\\alpha} \\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n As can be easily verified from (\\ref{eq:104}a) we have (see in\n \\cite{al02} (formula (2.4) in the case when $\\phi_\\epsilon=0$),\n \\begin{equation}\n \\label{eq:114}\n \\frac 12 \\Delta|\\psi_\\epsilon|^2 = -\\frac{|\\psi_\\epsilon|^2}{\\epsilon^2}(1-|\\psi_\\epsilon|^2) +\n|\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2 \\,.\n \\end{equation}\nFurthermore, taking the curl of (\\ref{eq:104}b) yields\n\\begin{displaymath}\n \\Delta B_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon =\n -\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar\\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon) \\,.\n\\end{displaymath}\nNote that\n$$\\aligned\n&\\text{\\rm curl\\,} \\Im(\\bar\\psi_\\epsilon\\, \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon) \\\\\n=&\\Im(\\nabla\\bar\\psi_\\epsilon\\times \\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon\n -i\\epsilon^{-2}\\bar\\psi_\\epsilon A_\\epsilon\\times\\nabla\\psi_\\epsilon) - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon\\\\\n=&\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon) -\n \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2B_\\epsilon \\,.\n\\endaligned\n$$\nLet \n$$u_\\epsilon=B_\\epsilon-1 + {|\\psi_\\epsilon|^2\\over 2}-w_\\epsilon\\,.\n$$\nCombining the above and \\eqref{eq:114}\nyields that (cf. also \\cite{al02})\n\\begin{displaymath}\n \\begin{cases}\n \\Delta u_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2u_\\epsilon = |\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2 -\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon)\n + \\frac{1}{2\\epsilon^2} |\\psi_\\epsilon|^4\\geq 0 &\n\\text{in } \\Omega, \\\\\nu_\\epsilon = \\frac{|\\psi_\\epsilon|^2}{2} & \\text{on } \\partial\\Omega.\n \\end{cases}\n\\end{displaymath}\nBy the weak maximum principle (cf. for instance \\cite[Theorem\n8.1]{GT}) and \\eqref{eq:106} we obtain that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n u_\\epsilon (x)\\leq \\frac{1}{2}\\,.\n\\end{displaymath}\nThe lower bound in \\eqref{eq:112} follows easily by setting\n$$\\tilde{u}_\\epsilon = -B_\\epsilon-1 + {|\\psi_\\epsilon|^2\\over 2}+w_\\epsilon\n$$ \nto obtain\n\\begin{displaymath}\n \\begin{cases}\n \\Delta \\tilde{u}_\\epsilon - \\frac{1}{\\epsilon^2}|\\psi_\\epsilon|^2\\tilde{u}_\\epsilon = |\\nabla_{\\epsilon^{-2}A_\\epsilon}\\psi_\\epsilon|^2 +\n\\Im (\\nabla_{\\epsilon^{-2}A_\\epsilon}\\bar \\psi_\\epsilon\\times\\nabla_{\\epsilon^{-2}A_\\epsilon} \\psi_\\epsilon) + \\frac{1}{2\\epsilon^2} |\\psi_\\epsilon|^4 \\geq 0 &\n\\text{in } \\Omega, \\\\\n\\tilde{u}_\\epsilon = \\frac{|\\psi_\\epsilon|^2}{2} & \\text{on } \\partial\\Omega,\n \\end{cases}\n\\end{displaymath}\nupon which we use again the weak maximum principle.\n\nTo prove \\eqref{eq:113} we first obtain for $\\phi_\\epsilon$, in the same manner\nused to derive \\eqref{eq:33}, the following problem:\n\\begin{equation}\\label{eq:115}\n\\left\\{\n\\aligned\n-&\\Delta\\phi_\\epsilon + \\frac{\\rho^2_\\epsilon}{\\epsilon^2}\\phi_\\epsilon =0 \\quad& \\text{\\rm in } \\Omega\\,, \\\\\n& \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= \\epsilon^{-\\alpha}J \\quad& \\text{\\rm on } \\partial\\Omega_c\\,,\\\\\n& \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}= 0 \\quad& \\text{\\rm on } \\partial\\Omega_i\\,.\n\\endaligned\\right.\n\\end{equation}\nThen, we follow the same steps as in the proof of \\eqref{eq:34} to\nobtain the bound $\\|\\phi_\\epsilon\\|_\\infty\\leq C \\epsilon^{-\\alpha}$. Multiplying \\eqref{eq:115} by $\\phi_\\epsilon$ and\nintegrating by parts yields, using the preceding $L^\\infty$ bound \\eqref{eq:112}, we find that\n\\begin{displaymath}\n \\|\\nabla\\phi_\\epsilon\\|_2^2 \\leq \\int_{\\partial\\Omega} \\phi_\\epsilon \\frac{\\partial\\phi_\\epsilon}{\\partial\\nu}\\,ds \\leq C\\epsilon^{-2\\alpha}\\,.\n\\end{displaymath}\n\\end{proof}\nAs a corollary we get:\n\\begin{corollary}\n \\begin{equation}\n\\label{eq:116}\n\\|B_\\epsilon\\|_\\infty \\leq \\text{\\rm max}(|b_1|,b_2)\\epsilon^{-\\alpha} + \\frac{1}{2} \\,.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\n By \\eqref{eq:112} and the maximum principle we have that\n \\begin{displaymath}\n \\|B_\\epsilon-1\\|_\\infty\\leq \\|w_\\epsilon\\|_\\infty +\\frac{1}{2} \\leq \\text{\\rm max}(|b_1|,b_2)\\epsilon^{-\\alpha} - \\frac{1}{2}\\,,\n \\end{displaymath}\nwhich readily yields \\eqref{eq:116}.\n\\end{proof}\n\nWe continue with the following auxiliary estimate:\n\\begin{lemma}\\label{Lem3}\n Let $\\Omega$ and $J$ satisfy (\\ref{eq:3})-(\\ref{hyptopolo}), and $w_\\epsilon$ be a solution of \\eqref{eq:111}.\nThere exist positive constants $C$ and $\\epsilon_0$, such that, for all $0<\\epsilon<\\epsilon_0$,\n \\begin{equation}\n\\label{eq:117}\n\\|\\nabla w_\\epsilon \\|_\\infty \\leq\n\\frac{C}{\\epsilon^{1+\\alpha}} \\,.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n For convenience of notation we drop the subscript $\\epsilon$ in the proof\n and bring only its main steps, as it rather standard. We first apply\n the inverse transformation of \\eqref{eq:103} to \\eqref{eq:111} to\n obtain\n\\begin{displaymath}\n \\begin{cases}\n \\Delta w - |\\psi|^2w = 0 & \\text{in }\\Omega_R\\,, \\\\\n w = B - 1 & \\text{on } \\partial\\Omega_R \\,,\n \\end{cases}\n\\end{displaymath}\nwhere $B=\\text{\\rm curl\\,} A$.\\\\\nWe distinguish in the following between interior estimates and\nboundary estimates. Let $x_0\\in \\partial \\Omega_R$ and\n$D_r=D_r(x_0)=B(x_0,r)\\cap\\Omega_R\\,.\n$\\\\\nBy the standard elliptic estimates we then have, in view of\n\\eqref{eq:106},\n\\begin{equation}\n\\label{eq:124}\n \\|w\\|_{H^2(D_r)} \\leq C(\\|w\\|_{L^2(D_{2r})}+ \\| \\epsilon^{-\\alpha}B_n-1\\|_{H^2(D_{2r})}) \\,.\n\\end{equation}\nTo obtain the above we first observe that $B = \\epsilon^{-\\alpha} B_n$ on the\nboundary, and then use the fact that the trace of $B_n$ in $H^\\frac\n32 (\\partial\\Omega)$ and is therefore bounded from above by a proper $H^2$ norm.\n\nSimilarly,\n\\begin{equation}\n\\label{eq:125}\n \\|w\\|_{H^3(D_r)} \\leq C\\Big(\\|w\\|_{H^1(D_{2r})} + \\|\\epsilon^{-\\alpha}B_n-1\\|_{H^3(D_{2r})}\n +\\|w\\|_{L^\\infty(D_{2r})} \\||\\psi|^2\\|_{H^1(D_{2r})}\\Big) \\,.\n\\end{equation}\nUsing Kato's inequality and (\\ref{eq:18}a) yields\n\\begin{displaymath}\n \\|\\nabla|\\psi|\\,\\|_{L^2(D_r)} \\leq \\|\\nabla_A\\psi\\|_{L^2(D_r)} \\leq C\\|\\psi\\|_{L^2(D_{2r})}\\,,\n\\end{displaymath}\nthen we obtain by \\eqref{eq:112} that\n\\begin{equation}\\label{112aa}\n \\|w\\|_{H^3(D_r)} \\leq C\\epsilon^{-\\alpha}\\,.\n\\end{equation}\n\nAn interior estimate is even easier. Consider $x_0\\in \\Omega_R$ such that\n$D(x_0,2r) \\subset \\Omega_R$. There is no need in this case to include a\nboundary term in \\eqref{eq:124} and \\eqref{eq:125}. We then obtain\n\\eqref{112aa} in this case in a similar manner. \n\n\nThe proof of \\eqref{eq:117} now follows from Sobolev embeddings and\n\\eqref{eq:103}.\n\\end{proof}\n\n\nWe next define the following subdomain of $\\Omega$:\n \\begin{displaymath}\n{\\mathcal D}_\\delta(\\epsilon) = \\{ x\\in\\Omega \\,:\\, |B_\\epsilon(x)|<\\delta\\epsilon^{-\\alpha} \\} \\,.\n \\end{displaymath}\n Let further\n \\begin{displaymath}\n d_{\\delta,j}(\\epsilon) = d( {\\mathcal D}_\\delta(\\epsilon) ,\\partial\\Omega_{i,j}),\\quad j=1,2\\,,\n \\end{displaymath}\nwhere, as in the introduction, $\\{\\partial\\Omega_{i,j}\\}_{j=1}^{2}$ denotes the set of connected components of\n$\\partial\\Omega_i$. We now obtain a lower bound of\n\\begin{equation}\\label{d-delta}\nd_\\delta(\\epsilon)=\\text{\\rm max}_{j\\in\\{1,2\\}}d_{\\delta,j}(\\epsilon) .\n\\end{equation}\n\\begin{lemma}\n Let\n \\begin{equation}\n \\label{eq:118}\n\\delta_0=\\min(|b_1|,|b_2|)\\,.\n \\end{equation}\n Under the conditions of Lemma \\ref{Lem3}, there exists, for any $0<\\delta<\\delta_0\\,$, a\n positive $C_\\delta$ such that for sufficiently small $\\epsilon$ we have\n \\begin{equation}\n\\label{eq:119}\nd_\\delta(\\epsilon) \\geq C_\\delta\\, \\epsilon \\,.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Let $x\\in\\partial\\Omega_i$ and $y\\in{\\mathcal D}_\\delta(\\epsilon)$. By \\eqref{eq:117} we have\n\\begin{equation}\n\\label{eq:120}\n |w_\\epsilon(x)-w_\\epsilon(y)| \\leq \\frac{C}{\\epsilon^{1+\\alpha}} |x-y|\\,.\n\\end{equation}\nBy \\eqref{eq:112} we have\n\\begin{displaymath}\n |w_\\epsilon(x)-w_\\epsilon(y)| \\geq |B_\\epsilon(x)-B_\\epsilon(y)| -\\frac{1}{2}\\geq (\\delta_0-\\delta)\\epsilon^{-\\alpha}-\\frac{1}{2}\\,.\n\\end{displaymath}\nCombining the above with \\eqref{eq:120} yields\n\\begin{displaymath}\n |x-y| \\geq (\\delta_0-\\delta)\\epsilon-\\frac{1}{2}\\epsilon^{1+\\alpha} \\,,\n\\end{displaymath}\nwhich readily yields \\eqref{eq:119}.\n\\end{proof}\n\n\nNext we show\n\\begin{lemma} Under the conditions of Lemma \\ref{Lem3}, there exist $C>0$, $\\epsilon_0>0$ and $\\delta_0>0$ such that, for\n $0<\\epsilon\\leq \\epsilon_0 $ and $0< \\delta \\leq \\delta_0\\,$,\n \\begin{equation}\n\\label{eq:121}\n\\int_{\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon)} \\exp \\left( [2\\delta \\Theta_0\\epsilon^{-\\alpha}]^{1\/2}(4\\epsilon)^{-1}d(x,{\\mathcal D}_\\delta(\\epsilon))\\right) |\\psi_\\epsilon|^2 \\,dx \\leq \\frac{C}{\\delta^{3\/2}} \\,.\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\n Let $\\eta\\in C^\\infty(\\Omega,[0,1])$ satisfy\n \\begin{displaymath}\n \\eta(x)=\n \\begin{cases}\n 1 & x\\in \\Omega\\setminus {\\mathcal D}_\\delta(\\epsilon)\\,, \\\\\n 0 & x\\in {\\mathcal D}_{\\delta\/2}(\\epsilon) \\,.\n \\end{cases}\n \\end{displaymath}\n With the aid of \\eqref{eq:117} and \\eqref{eq:112}, it can be easily\n verified, for some $C=C(p,J,\\Omega)>0$ which is independent of both $\\delta$\n and $\\epsilon$, that for all $0<\\delta<\\delta_0$ we can construct $\\eta$ with the\n additional property\n \\begin{displaymath}\n |\\nabla\\eta| \\leq \\frac{C}{\\delta\\epsilon} \\,.\n \\end{displaymath}\nLet further\n$$\\zeta=\\chi\\eta\\,,\n$$\nwhere\n\\begin{displaymath}\n \\chi=\n \\begin{cases}\n \\exp\\big(\\alpha_\\delta\\epsilon^{-1-\\alpha\/2}d(x,{\\mathcal D}_\\delta(\\epsilon))\\big) &\\text{if } x\\in\\Omega\\setminus {\\mathcal D}_\\delta(\\epsilon)\\,, \\\\\n 1 &\\text{if } x\\in {\\mathcal D}_\\delta(\\epsilon)\\,.\n \\end{cases}\n\\end{displaymath}\nWe leave the determination of\n$\\alpha_\\delta$ to a later stage.\n\nMultiplying (\\ref{eq:25}a) by $\\zeta^2\\bar{\\psi_\\epsilon}$ and integrating by parts yields\n\\begin{displaymath}\n \\|\\nabla_{\\epsilon^{-2}A_\\epsilon}(\\zeta\\psi_\\epsilon)\\|_2^2 =\\frac{1}{\\epsilon^2}\\big[ \\|\\zeta\\psi_\\epsilon\\|_2^2 -\n \\|\\zeta^{1\/2}\\psi_\\epsilon\\|_4^4\\big] + \\|\\psi_\\epsilon\\nabla\\zeta\\|_2^2 \\,.\n\\end{displaymath}\nBy \\eqref{eq:22} we have that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n \\|\\nabla_{\\epsilon^{-2}A_\\epsilon}(\\zeta\\psi_\\epsilon)\\|_2^2 \\geq\\frac{\\Theta_0\\delta}{2\\epsilon^2}\\epsilon^{-\\alpha}\n \\|\\zeta\\psi_\\epsilon\\|_2^2 \\,,\n\\end{displaymath}\nwhere $\\Theta_0$ is defined in \\eqref{eq:20}. Consequently,\n\\begin{equation}\n\\label{eq:122}\n \\frac{\\Theta_0\\delta \\epsilon^{-\\alpha}-2}{2\\epsilon^2}\\|\\zeta\\psi_\\epsilon\\|_2^2 \\leq \\|\\psi_\\epsilon\\nabla\\zeta\\|_2^2 \\,.\n\\end{equation}\nFrom \\eqref{eq:122} we learn that\n\\begin{displaymath}\n \\|\\zeta\\psi_\\epsilon\\|_2 \\leq \\Big[\\frac{2\\epsilon^2}{\\Theta_0\\delta \\epsilon^{-\\alpha}-2}\\Big]^{1\/2} \\Big(\\alpha_\\delta\n \\epsilon^{-(1+\\alpha\/2)}\\|\\zeta\\psi_\\epsilon\\|_2 +\n \\|\\psi_\\epsilon\\nabla\\eta\\|_2\\Big) \\,,\n\\end{displaymath}\nwhere we have used the fact that $|\\nabla d(x,{\\mathcal D}_\\delta(\\epsilon))|\\leq1$ a.e.\n\nChoosing\n\\begin{displaymath}\n \\alpha_\\delta=\\frac{1}{2}\\Big[\\frac{\\Theta_0\\delta}{2}\\Big]^{1\/2}\n\\end{displaymath}\nwe obtain that for sufficiently small $\\epsilon$\n\\begin{displaymath}\n \\|\\zeta\\psi_\\epsilon\\|_2 \\leq 4\\Big[\\frac{2}{\\delta}\\Big]^{1\/2} \\epsilon^{1+\\alpha\/2}\\|\\psi_\\epsilon\\nabla\\eta\\|_2 \\leq\n \\frac{C\\epsilon^{\\alpha\/2}}{\\delta^{3\/2}}\\|\\psi_\\epsilon\\|_{L^2({\\mathcal D}_\\delta(\\epsilon)\\setminus {\\mathcal D}_{\\delta\/2}(\\epsilon))}\\,.\n\\end{displaymath}\nThe proof of \\eqref{eq:121} now follows from \\eqref{eq:106}.\n\\end{proof}\n\nWe now obtain an improved lower bound.\n\\begin{proposition}\n Let $\\delta_0$ be given by \\eqref{eq:118}. Under the conditions of Lemma\n \\ref{Lem3}, for any $0<\\delta<\\delta_0$, there exist positive constants $C_\\delta$ and $\\epsilon_\\delta$ such that for\n $0< \\epsilon \\leq \\epsilon_\\delta$ we have\n \\begin{equation}\n\\label{eq:123}\nd_\\delta(\\epsilon) \\geq C_\\delta \\,.\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\n We begin by noticing that by (\\ref{eq:104}b), \\eqref{eq:113}, and\n \\eqref{eq:121} we have\n\\begin{displaymath}\n \\|\\nabla B_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq\n \\|\\nabla\\phi_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))}+Ce^{-1\/\\epsilon} \\leq \\widehat C \\epsilon^{-\\alpha} \\,.\n\\end{displaymath}\nThen, we write\n\\begin{displaymath}\n (\\delta_0-\\delta)\\epsilon^{-\\alpha} \\leq \\|\\nabla B_\\epsilon\\|_{L^1(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq Cd_\\delta^{1\/2}\n \\|\\nabla B_\\epsilon\\|_{L^2(\\Omega\\setminus{\\mathcal D}_\\delta(\\epsilon))} \\leq Cd_\\delta^{1\/2}\\epsilon^{-\\alpha}\\,,\n\\end{displaymath}\nfrom which \\eqref{eq:123} readily follows.\n\\end{proof}\n\\begin{remark}\n\\label{rem:size}\n Note that by \\eqref{eq:121} and the above arguments, \\eqref{eq:123}\n holds even if we use $d_\\delta$ to measure the distance of ${\\mathcal D}_\\delta(\\epsilon)$ from\n any point on $\\partial\\Omega$ where $B_\\epsilon>\\delta_1\\epsilon^{-\\alpha}$ with $\\delta_1>\\delta$.\n\\end{remark}\n\nProposition \\ref{semiclassical} now follows from \\eqref{eq:121} and \\eqref{eq:123}.\n\n\n\n\n\\paragraph{\\bf Acknowledgements}\nThe authors gratefully acknowledge Phuc Nguyen for introducing to them\nByun's result \\cite{by07} and Monique Dauge for the idea behind the\nproof of Lemma \\ref{lem:Dirichlet-Neumann}. Y. Almog was partially\nsupported by NSF grant DMS-1109030 and by US-Israel BSF grant\nno.~2010194. B. Helffer was partially supported by the ANR programme\nNOSEVOL and Simons foundation visiting fellow at the Newton Institute\nin Cambridge during the completion of this article. X.B. Pan was\npartially supported by the National Natural Science Foundation of\nChina grant no. 11171111 and no. 11431005. \n\\vspace{2ex}\n\nThe authors declare that\nthey have no conflict of interest. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Conclusions}\n\\label{sec:conclusions}\nWe have studied the utilization of BERT for the task of passage re-ranking in light of its strict limitation on passage length. Using three BERT-based LTR methods, we conducted experiments on three datasets, with different passage lengths and segmentation configurations.\n\nOur findings are that using BERT representations for mid-sized (SeqLen=256 tokens) of (query, passage) pairs give the best results. We have further shown that, by breaking passages to smaller chunks and aggregating their (query, chunk) representations, we can re-rank larger passages, with only a moderate degradation in quality. Finally, we found out that point-wise and pair-wise LTR achieved similar results on the three datasets. \n\n\n\n\\section{Experimentation Analysis}\nThe primary goal of our analysis is to evaluate the behavior of the three proposed BERT-based LTR methods under various passage-length and segmentation settings.\nOur implementation of the various LTR models is based on the TensorFlow\\footnote{https:\/\/www.tensorflow.org\/} version of BERT\\footnote{ https:\/\/github.com\/google-research\/bert}. \nIn all the experiments we used the pre-trained BERT-Base, Uncased model (12-layer, 768-hidden, 12-heads, 110M parameters).\n \nWe start with a short description of our datasets and setup, followed by the empirical results of the analysis.\n\\label{sec:experiments}\n\n\\subsection{Datasets}\n\\label{sec:datasets}\nWe used three different datasets that were previously used for the non-factoid QA task~\\cite{hybrid:2018,wikiPassageQA:2018}. These datasets are next shortly described while their statistics are summarized in Table~\\ref{tab:datasets}.\n\n\\noindent \\textbf{nfL6}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/nfL6}~\\cite{hybrid:2018} consists of 87,361 Yahoo's non-factoid questions. Each query has a single correct passage.\n\n\\noindent \\textbf{WebAP}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/WebAP\/}~\\cite{hybrid:2018} consists of 82\\footnote{Two queries have no relevant answers so are omitted from our experiments.} queries with labeled passages from the TREC GOV2 web collection. Queries in this dataset are more open ended than the former dataset, and can have a variety of relevant passages.\nLabeling was done by first retrieving the top-50 web documents for each query and then marking contiguous sequences by a five graded-relevance scores (4 - Perfect, 3 - Excel, 2- Good, 1 - fair, 0 - None). Similar to~\\cite{hybrid:2018}, we took the relevant sequences (i.e., relevance $>0$) as positive passages and divided the irrelevant sequences to negative ones. Since there was not enough details in~\\cite{hybrid:2018} on how the latter was done, we assumed passage lengths are normally distributed with mean and standard deviation estimated using the relevant passages.\n\n\\noindent \\textbf{WikiPassageQA}\\footnote{https:\/\/ciir.cs.umass.edu\/downloads\/wikipassageqa\/}~\\cite{wikiPassageQA:2018} contains 4,165 non-factoid queries with passages extracted from Wikipedia articles. Here, it is assumed that for each query, there is a single Wikipedia article that contains relevant passages, which are manually marked as contiguous sequences. Unlike WebAP, each article is divided to fixed passages of $6$ sentences each, where a passage is labeled relevant if it has at least $15\\%$ overlap with a marked relevant sequence.\n\n\\begin{table}[tbh]\n\t\\center\n \\small\n\t\t\\caption{Datasets}\n \\setlength\\tabcolsep{2.0pt}\n\t\t\\begin{tabular}{l|c|c|c|c|c|}\n\t\t \\cline{2-6}\n\t\t\t & \\textbf{queries} &\\textbf{psgs} & \\textbf{min} &\n\t\t\t \\textbf{max} & \\textbf{avg} \\\\\n\t\t\t\\hline\n\t\t\t\\multicolumn{1}{|l|}{nfL6} &\n\t\t\t87,362 & 87,362 & 2 & 809 & 42.4 \\\\\n\t\t\n\t\t\t\\hline\n\t\t\t\\multicolumn{1}{|l|}{WebAP} &\n\t\t\t80 & 489,042 & 0 & 10,851 & 74.5 \\\\\n\t\t\n \\hline\n\t\t\t\\multicolumn{1}{|l|}{WikiPassageQA} &\n\t\t\t4,165 & 50,612 & 10 & 1,332 & 134.2\\\\\n\t\t\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t \\label{tab:datasets}\n\\end{table}\n\n\n\\subsection{Setup}\n\\label{sec:train}\nSimilar to~\\cite{hybrid:2018}, for each query, we\nretrieved the top-$k$ BM25 results in the nfL6 and WebAP datasets ($k=10$ for nfL6 and $k=100$ for WebAP). To this end, we indexed each passage as a document in an ElasticSearch\\footnote{https:\/\/www.elastic.co} index. For WikiPassageQA, we simply took all passages from the query's Wikipedia article.\n\nFor fine-tuning the three\nBERT models, we used the retrieved top-$k$ passages of each query $q$, and created their corresponding triplets set as follows. \n\nGiven a query $q$ with $m$ positive examples and $n$ negative examples (usually $m << n$), each positive passage participates in different $n \/ m$ triplets.\nTo cope with a reasonable number of triplets, we down-sampled nfL6 (which has a large number of queries) to have at most two negative examples for each positive one. For WikiPassageQA we down-sampled to five negative examples per each positive one. Since WebAP has a small number of queries, we used all the negative examples.\n\nWikiPassageQA is already divided into $3,332$ train, $417$ development and $416$ test queries. For the other two datasets, we used a 5-fold cross validation. Similar to~\\cite{BERT:2018}, we used a fixed number of three train epochs, hence we omitted the development set.\n\nTo evaluate the performance of the various ranking methods, we re-ranked the top-$k$ passages returned by the BM25 baseline ranker. Similar to~\\cite{hybrid:2018}, for the nfL6 and WebAP datasets, we added a correct passage at position $k$ ($k=10$ for nfL6 and $k=100$ for WebAP) if it was not retrieved within the BM25 top-$k$ passages.\n\nWe used standard Information Retrieval (IR) metrics: Precision@1 (P@1), Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR).\n\nTraining was done using the default BERT setup~\\cite{BERT:2018}, with Adam optimizer, learning rate $2e$-$5$ over three epochs.\nAll experiments were run on a Tesla K40m GPU, with a memory clock-rate of 0.745Ghz and 12Gb memory.\n\n\\subsection{Results}\n\\label{sec:results}\n\nWe compared the three studied BERT-based LTR methods (Section~\\ref{sec:triplets}) to two baselines, namely BM25 and the previously best known deep-learning (DL) method. Similar to the reported DL methods, we use P@1 for the first two datasets (nfL6 and webAP)~\\cite{hybrid:2018} and MAP for the third one (WikiPassageQA)\n~\\cite{,wikiPassageQA:2018}. Figure~\\ref{fig:all-map} depicts the results obtained by each ranker. The three BERT methods use 256 tokens for each pair (SeqLen in the BERT jargon), which was found to be the optimal size, as we discuss in Section~\\ref{sec:block_size} below.\n\nWe can observe that the three BERT methods significantly outperform the two baselines. The improvements range from $31\\%$ improvement on MAP in WikiPassageQA to $120\\%$ on P@1 in nfL6 over the best DL baseline.\nAmong the three BERT methods, the point-wise\n(\\texttt{BERT[PW]}) performed better on webAP (with P@1=$0.55$, compared to $0.53$ by the pair-wise (\\texttt{BERTlets})), while \\texttt{BERTlet} performed better on WikiPassageQA (with MAP=$0.74$ compared to $0.71$ by \\texttt{BERT[PW]}). \nAmong the two pair-wise BERT variants, the \\texttt{BERTlets} method performs better.\nThe experiments in the subsequent sections are based on \\texttt{BERTlets}.\n\n\\input{figs\/all_map}\n\n\\subsubsection{Effect of Passage Length}\n\\label{sec:block_size}\n\nAs was mentioned above, BERT imposes a limit of maximum $512$ token\n\\footnote{BERT ignores input tokens after the first SeqLen tokens} (SeqLen) in each pair when using a GPU. Figure~\\ref{fig:passage_size_map}\ndepicts the \n\\texttt{BERTlets} results for varying SeqLen. We observe that, the best P@1 and MAP are achieved for $256$ tokens. Lower values ($64$ and $128$) actually hurt the performance while increasing to $384$ does not improve and even hurts it mainly on WikiPassageQA.\n\nThe effect of the small SeqLen is mainly manifested on the WikiPassageQA which has the largest average passage-length as shown in Table~\\ref{tab:datasets}. This is actually expected, since short passages may lack enough content, while too long ones may include additional superfluous content (``noise'') to allow to fully capture such passages semantics through the BERT representation.\n\n\n\\input{figs\/passage_size_map}\n\n\\subsubsection{Effect of passage segmentation}\n\\label{sec:sentences}\n\nTo overcome the limitations of BERT on passage-length, we further segmented each passage into equal number of chunks. Given a query $q$ and passage $p$, we derived $\\texttt{BERT}(q,c)$, for chunks $c \\in p$, and combined their representation through an attention mechanism~\\cite{Attention:2016}. We tried to break to two and three chunks and pass them with SeqLen$=128$, using an attention size of $192$. The results are summarized in Table~\\ref{tab:sentences}.\n\n\\begin{table}[h]\n\t\\center\n \\small\n\t\t\\setlength\\tabcolsep{1.8pt}\n\t\t\\caption{Effect of breaking paragraphs to chunks of SeqLen$=128$ (with attention)}\n\t\t\\begin{tabular}{l|c|c||c|c||c|c|}\n\t\t\t\\cline{2-7}\n\t\t\t\\multirow{2}{*}{} &\n\t\t\t\t\\multicolumn{2}{c||}{nfL6} & \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\multicolumn{2}{c||}{WebAP} &\n\t\t\t\t\\multicolumn{2}{c|}{WikiPassageQA} \\\\\n\t\t\t\\cline{2-7}\n \t\t\t\t\t& P@1 & MRR & P@1 & MRR & MAP & MRR \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (2*128)} & .744 & .835 & .513 & .662 & .715 & .778 \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (3*128)} & .748 & .839 & .488 & .634 & .715 & .791 \\\\ \\hline\n\t\t\t\t\t\\multicolumn{1}{|l|}{BERTlets (256)} & .776 & .857 & .525 & .683 & .736 & .810 \\\\ \\hline\n\t\t\\end{tabular}\n \\label{tab:sentences}\n\\end{table}\n\nAs we can observe, using passage segmentation results in a slightly lower performance.\nHowever, given the BERT constraints on passage length, this segmentation can be used to handle large passages above the BERT limitation of $512$ tokens. We further tried to combine the chunks representations using \\textit{Max-Pooling} but it gave similar results to the attention method.\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\n\nWithin a question-answering (QA) setting, the passage-retrieval task, retrieves a (relatively) short text fragment (passage)\nthat provides a focused answer to a given question.\nAs opposed to the factoid-QA task, which requires a succeeding step of answer extraction, the non-factoid QA task aims at retrieving passages as the answers.\n\nIn this work, we focus on the passage retrieval task and more specifically, on the passage re-ranking task, i.e., given an initial ranked-list of passages, retrieved by some basic passage-retrieval method (e.g., BM25~\\cite{psgbm25}), our goal is to re-rank the passages in the list so as to position the most relevant ones higher. Similar to the latest works on this task, our approach is based on Deep-Learning (DL).\nExisting DL methods commonly train their networks directly on the labeled datasets~\\cite{hybrid:2018, wikiPassageQA:2018,mitra2019duet,tan2015lstmbased}. While achieving better results than more ``traditional'' methods, such methods still suffer from the relatively small labeled data available for training.\n\nTrying to overcome such data limitations, more recent works have utilized the pre-trained Bidirectional Encoder Representations from Transformers (BERT) language representation models~\\cite{BERT:2018}.\nEach BERT model was trained on large corpora of Wikipedia and news data,\nconsisting of a large number of layers (12\/24 in the base\/large model) based on the Transformer~\\cite{Transformer:2017} architecture.\nWhen trained on pairs of sentences, BERT allows to capture high-order interactions between a given pair of sentences. Such pre-trained models can be then fine-tuned for a specific downstream task with a relatively small amount of labeled data. Previously, such end-to-end networks have been shown to be highly effective in several NLP tasks, and very recently also for passage-retrieval~\\cite{multiTaskBert:2019, passageRerankingBert:2019}.\n\nWhile utilizing BERT for passage re-ranking is promising, there are still several technical limitations that are imposed by BERT's basic architecture. First, due to the large number of layers of BERT, it should be run on a GPU, enforcing a maximum of 512 input tokens \nper pair.\nAmong these tokens are the special \\textsf{[CLS]} token and two \\textsf{[SEP]} tokens that BERT adds to each pair. Moreover, BERT's tokenizer breaks tokens not in its dictionary into sub-tokens, which further limits the total number of tokens that can be passed for each pair.\n\nThe primary goal of our work is to study how BERT can be best utilized for the passage re-ranking task, within the non-factoid QA setting, in spite of its input length limitations. To this end, we report on an extensive empirical analysis with different passage lengths. We show that, too short passages may suffer from a lack of content, while too large passages may generate noisy representations. We further report on experiments that try to breakdown passages into smaller chunks and combine their representations into a passage-level representation.\n\n\\section{Related work}\n\\label{sec:related}\n\nFollowing the success in applying BERT~\\cite{BERT:2018} for multiple NLP tasks, some very recent works have applied it to\nthe factoid QA~\\cite{BERTserini} task, as well as to the non-factoid QA task~\\cite{multiTaskBert:2019,passageRerankingBert:2019,qiao2019understanding}.\nIn this work we focus on the non-factoid QA task.\n\nSimilar to the MRPC paraphrase task in the original BERT work, the authors of~\\cite{passageRerankingBert:2019} have fine-tuned BERT for passage re-ranking by casting it as a binary classification problem. To this end, correct passages were used as positive examples and the rest as negative examples. At run time, the classification score of passages was used for ranking. In~\\cite{multiTaskBert:2019}, the authors further fine-tuned BERT with a variation of the triplet-loss for ranking passages within a factoid QA task.\n\nIn this work we study the behaviour of BERT with respect to passage length properties.\nA couple of recent works have also investigated BERT's behaviour for retrieval tasks. In~\\cite{qiao2019understanding} an analysis of BERT's attention allocation between query-document tokens in its Transformer layers was performed, further examining its difference from soft match patterns learned by a traditional learning-to-rank (LTR) neural model. The work by~\\cite{padigela2019investigating} has compared BERT's performance to that obtained by a BM25 model with respect to various query properties (e.g., query-length and term frequencies).\n\n\n\n\n\\section{Passage Re-ranking with BERT\n\\label{sec:triplets}\n\nHere, we shortly discuss several options for fine-tuning BERT for the passage re-ranking task.\nTo this end, we exploit a given pre-trained BERT model\\footnote{ https:\/\/github.com\/google-research\/bert} for assigning an effective representation $\\texttt{BERT}(s_1,s_2)$ for any given pair of sentences $(s_1,s_2)$. As the representation itself, we consider it to be that of the \\textsf{[CLS]} token~\\cite{BERT:2018} which serves as the aggregated pair representation.\n\nWe now assume the availability of some labeled examples of queries, where each query ($q$) is potentially associated with both positive ($p^{+}$) and negative ($p^{-}$) passages.\nAs a first step, we create training examples, consisting of (positive and negative) query-passage pairs for fine-tuning BERT.\nIn our study we explore two main learning-to-rank~\\cite{Li2011ASI} (LTR) strategies with BERT, namely \\textbf{point-wise} and \\textbf{pair-wise}.\n\n\\paragraph{Point-wise}\nWithin a point-wise LTR approach, each passage is scored independently of other passages~\\cite{Li2011ASI}. To this end, we use the MRPC (Microsoft Research Paraphrase Corpus) classifier as described in~\\cite{BERT:2018}. The classifier has two labels, $\\textsf{0}$ - for negative examples (i.e., ($q,p^{-}$) pairs) and $\\textsf{1}$ - for positive examples (i.e., ($q,p^{+}$) pairs). Training the LTR model is simply implemented by minimizing the cross-entropy loss. Once trained, the classifier's confidence is used as the score. We refer to this ranking method as \\texttt{BERT[PW]}.\n\n\\paragraph{Pair-wise}\nCompared to the point-wise LTR approach, a pair-wise LTR approach is trained to learn a preference (ordering) among pairs of passages~\\cite{Li2011ASI}.\nIn this work we implement two such methods. The first method, termed \\texttt{BERTlets}, uses a \\textit{Triplet-Network}~\\cite{Triplet:2018} as follows.\nLet $v$ be a vector, having the same dimension as the obtained BERT representation\\footnote{$768$\/$1024$ for the base\/large BERT model.}. We define the score of a passage $p$ for query $q$ as:\n\\vspace{-0.1in}\n\\begin{equation}\\label{eq:score}\n score(q;p) = \\texttt{BERT}(q,p) \\cdot v,\n\\end{equation}\n\nwhere \\texttt{BERT}(q,p) is the BERT pooled representation of the CLS token of the last layer.\n\nGiven a triplet $t = (q,p^{+},p^{-})$,\nlet $s^{+} = score(q;p^{+})$ and $s^{-} = score(q;p^{-}))$ be the scores assigned to the positive and the negative passage samples for a given query $q$, respectively.\nOur goal is to tune $v$, such that \n$(q,p^{+}) \\prec (q,p^{-})$ (or $s^{+} > s^{-}$). To achieve that, we train the network with a Hinge loss~\\cite{hinge}:\n\\begin{equation}\\label{eq:loss}\n loss_t = \\max(m - (\\hat{s}^{+} -\\hat{s}^{-}), 0),\n\\end{equation}\n\nwhere $\\hat{s}^{+} = \\frac {\\exp{s^{+}}}{\\exp{s^{+}} + \\exp{s^{-}}}$, $\\hat{s}^{-} = \\frac {\\exp{s^{-}}}{\\exp{s^{+}} + \\exp{s^{-}}}$ and $m$ is the margin hyperparameter (e.g., 0.2).\n\nThe second pair-wise method, termed \\texttt{BERT[CE]}, is inspired by~\\cite{multiTaskBert:2019} and serves as an alternative to \\texttt{BERTlets}. This method uses a similar technique as \\texttt{BERTlets}, but instead of triplets, it is trained by comparing a given positive pair $(q,p^{+})$ with any given negative pair $(q,p^{-})$. It also learns a vector $v$ and assigns a similar $score(q,p)$ as in Eq.~\\ref{eq:score}. Yet, instead of the Hinge loss, its loss is the negative log-likelihood of the positive example (i.e., $-log (\\hat{s}^{+})$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}