{"title":"On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder","authors":"S. Krymskii, N. Filonov","doi":"10.1090/spmj/1746","DOIUrl":null,"url":null,"abstract":"<p>Consider the equation <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus normal upper Delta u plus upper V u equals 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>V</mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">-\\Delta u + Vu = 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the half-cylinder <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma normal infinity right-parenthesis times left-parenthesis 0 comma 2 pi right-parenthesis Superscript d\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>×<!-- × --></mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mi>π<!-- π --></mml:mi>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>d</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">[0, \\infty ) \\times (0,2\\pi )^d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with periodic boundary conditions. Assume that the potential <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper V\">\n <mml:semantics>\n <mml:mi>V</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">V</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> 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 <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e Superscript minus c x\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>e</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>c</mml:mi>\n <mml:mi>x</mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">e^{-cx}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>d</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">d=1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e Superscript minus c x Super Superscript 4 slash 3\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>e</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>c</mml:mi>\n <mml:msup>\n <mml:mi>x</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>4</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n </mml:msup>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">e^{-cx^{4/3}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d greater-than-or-equal-to 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>d</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">d\\ge 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>; here <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x\">\n <mml:semantics>\n <mml:mi>x</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">x</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the axial variable.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1746","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
Consider the equation −Δu+Vu=0-\Delta u + Vu = 0 in the half-cylinder [0,∞)×(0,2π)d[0, \infty ) \times (0,2\pi )^d with periodic boundary conditions. Assume that the potential VV 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−cxe^{-cx} for d=1d=1 or 22 and e−cx4/3e^{-cx^{4/3}} for d≥3d\ge 3; here xx is the axial variable.
考虑半圆柱体[0，∞）×（0，2π）d[0，\infty）\times（0,2\pi）中的方程−Δu+V u=0-\Δu+Vu=0^d具有周期性边界条件。假设电势V V是有界的。研究了一个非平凡解在无穷远处的可能衰变率。结果表明，当d＝1 d＝1或2 2时，最快的衰变率是e−c x e ^｛-cx｝；当d≥3时，最慢的衰变率为e−c×第3页；这里x x是轴向变量。
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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