On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder

Krymskii, S.; Filonov, N.

doi:10.1090/spmj/1746

On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder
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by S. T. Krymskii and N. D. Filonov
Translated by: S. T. Krymskii

St. Petersburg Math. J. 34 (2023), 79-92

DOI: https://doi.org/10.1090/spmj/1746

Published electronically: December 16, 2022

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Abstract:

Consider the equation $-\Delta u + Vu = 0$ in the half-cylinder $[0, \infty ) \times (0,2\pi )^d$ with periodic boundary conditions. Assume that the potential $V$ is bounded. The possible rate of decay at infinity for a nontrivial solution is studied. It is shown that the fastest rate of decay is $e^{-cx}$ for $d=1$ or $2$ and $e^{-cx^{4/3}}$ for $d\ge 3$; here $x$ is the axial variable.

References

J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
Daniel M. Elton, Decay rates at infinity for solutions to periodic Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1113–1126. MR 4091054, DOI 10.1017/prm.2018.87
Friedrich Götze, Lattice point problems and values of quadratic forms, Invent. Math. 157 (2004), no. 1, 195–226. MR 2135188, DOI 10.1007/s00222-004-0366-3
A. L. Gusarov, Liouville theorems for elliptic equations in a cylinder, Trudy Moskov. Mat. Obshch. 42 (1981), 254–266 (Russian). MR 622005
E. M. Landis, Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables), Uspehi Mat. Nauk 18 (1963), no. 1 (109), 3–62 (Russian). MR 0150437
A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov, The Landis conjecture on exponential decay, arXiv:2007.07034.
V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Mat. Sb. 182 (1991), no. 3, 364–383 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 343–361. MR 1110071, DOI 10.1070/SM1992v072n02ABEH001414
Gordon Pall, The distribution of integers represented by binary quadratic forms, Bull. Amer. Math. Soc. 49 (1943), 447–449. MR 8084, DOI 10.1090/S0002-9904-1943-07947-5
Maxim M. Skriganov and Alexander V. Sobolev, Variation of the number of lattice points in large balls, Acta Arith. 120 (2005), no. 3, 245–267. MR 2188843, DOI 10.4064/aa120-3-3