{"title":"连续统随机Schrödinger算子的水平间距和泊松统计量","authors":"Adrian Dietlein, A. Elgart","doi":"10.4171/jems/1033","DOIUrl":null,"url":null,"abstract":"For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"163 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2020-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Level spacing and Poisson statistics for continuum random Schrödinger operators\",\"authors\":\"Adrian Dietlein, A. Elgart\",\"doi\":\"10.4171/jems/1033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.\",\"PeriodicalId\":50003,\"journal\":{\"name\":\"Journal of the European Mathematical Society\",\"volume\":\"163 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2020-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the European Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/jems/1033\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the European Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jems/1033","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Level spacing and Poisson statistics for continuum random Schrödinger operators
For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(logL) for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E.
期刊介绍:
The Journal of the European Mathematical Society (JEMS) is the official journal of the EMS.
The Society, founded in 1990, works at promoting joint scientific efforts between the many different structures that characterize European mathematics. JEMS will publish research articles in all active areas of pure and applied mathematics. These will be selected by a distinguished, international board of editors for their outstanding quality and interest, according to the highest international standards.
Occasionally, substantial survey papers on topics of exceptional interest will also be published. Starting in 1999, the Journal was published by Springer-Verlag until the end of 2003. Since 2004 it is published by the EMS Publishing House. The first Editor-in-Chief of the Journal was J. Jost, succeeded by H. Brezis in 2004.
The Journal of the European Mathematical Society is covered in:
Mathematical Reviews (MR), Current Mathematical Publications (CMP), MathSciNet, Zentralblatt für Mathematik, Zentralblatt MATH Database, Science Citation Index (SCI), Science Citation Index Expanded (SCIE), CompuMath Citation Index (CMCI), Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), ISI Alerting Services, Journal Citation Reports/Science Edition, Web of Science.