{"title":"On the Birman Problem in the Theory of Nonnegative Symmetric Operators with Compact Inverse","authors":"M. M. Malamud","doi":"10.1134/s0016266323020090","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Large classes of nonnegative Schrödinger operators on <span>\\(\\Bbb R^2\\)</span> and <span>\\(\\Bbb R^3\\)</span> with the following properties are described: </p><p> 1. The restriction of each of these operators to an appropriate unbounded set of measure zero in <span>\\(\\Bbb R^2\\)</span> (in <span>\\(\\Bbb R^3\\)</span>) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent; </p><p> 2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis. </p><p> The obtained results give a solution of a problem by M. S. Birman. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0016266323020090","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Large classes of nonnegative Schrödinger operators on \(\Bbb R^2\) and \(\Bbb R^3\) with the following properties are described:
1. The restriction of each of these operators to an appropriate unbounded set of measure zero in \(\Bbb R^2\) (in \(\Bbb R^3\)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;
2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.
The obtained results give a solution of a problem by M. S. Birman.
期刊介绍：
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
