{"title":"半圆柱体中Schrödinger方程解在无穷远处的衰减速率","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":"{\"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}","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
摘要
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。
On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder
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.
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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