{"title":"薛定谔特征函数衰变和阿格蒙气泡的有效边界","authors":"Stefan Steinerberger","doi":"10.1007/s11856-024-2641-x","DOIUrl":null,"url":null,"abstract":"<p>We study solutions of −Δ<i>u</i> + <i>Vu</i> = λ<i>u</i> on ℝ<sup><i>n</i></sup>. Such solutions localize in the ‘allowed’ region {<i>x</i> ∈ ℝ<sup><i>n</i></sup>: <i>V</i>(<i>x</i>) ≤ λ} and decay exponentially in the ‘forbidden’ region {<i>x</i> ∈ ℝ<sup><i>n</i></sup>: <i>V</i>(<i>x</i>) > λ}. One way of making this precise is Agmon’s inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon’s estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Effective bounds for the decay of Schrödinger eigenfunctions and Agmon bubbles\",\"authors\":\"Stefan Steinerberger\",\"doi\":\"10.1007/s11856-024-2641-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study solutions of −Δ<i>u</i> + <i>Vu</i> = λ<i>u</i> on ℝ<sup><i>n</i></sup>. Such solutions localize in the ‘allowed’ region {<i>x</i> ∈ ℝ<sup><i>n</i></sup>: <i>V</i>(<i>x</i>) ≤ λ} and decay exponentially in the ‘forbidden’ region {<i>x</i> ∈ ℝ<sup><i>n</i></sup>: <i>V</i>(<i>x</i>) > λ}. One way of making this precise is Agmon’s inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon’s estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.</p>\",\"PeriodicalId\":14661,\"journal\":{\"name\":\"Israel Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Israel Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11856-024-2641-x\",\"RegionNum\":2,\"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":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2641-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Effective bounds for the decay of Schrödinger eigenfunctions and Agmon bubbles
We study solutions of −Δu + Vu = λu on ℝn. Such solutions localize in the ‘allowed’ region {x ∈ ℝn: V(x) ≤ λ} and decay exponentially in the ‘forbidden’ region {x ∈ ℝn: V(x) > λ}. One way of making this precise is Agmon’s inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon’s estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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