{"title":"Uniform Spectral Asymptotics for a Schrödinger Operator on a Segment with Delta-Interaction","authors":"D.I. Borisov, D.M. Polyakov","doi":"10.1134/S1061920824020018","DOIUrl":null,"url":null,"abstract":"<p> We consider a Schrödinger operator on the segment <span>\\((0,1)\\)</span> subject to the Dirichlet condition and perturb it by a delta-potential concentrated at the point <span>\\(x= \\varepsilon \\)</span>, where <span>\\( \\varepsilon \\)</span> is a small positive parameter. We show that the perturbed operator converges to the unperturbed one in the norm resolvent sense and this also implies the convergence of the spectrum. However, the latter convergence is true only inside each compact set on the complex plane and it does not characterize the behavior of the total ensemble of the eigenvalues under the perturbation. Our main result is the spectral asymptotics for the eigenvalues of the perturbed operator with an estimate for the error term uniform in the small parameter. This asymptotics involves an additional nonstandard term, which allows us to describe a global behavior of the total ensemble of the eigenvalues under the perturbation. </p><p> <b> DOI</b> 10.1134/S1061920824020018 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 2","pages":"149 - 161"},"PeriodicalIF":1.7000,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920824020018","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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Abstract
We consider a Schrödinger operator on the segment \((0,1)\) subject to the Dirichlet condition and perturb it by a delta-potential concentrated at the point \(x= \varepsilon \), where \( \varepsilon \) is a small positive parameter. We show that the perturbed operator converges to the unperturbed one in the norm resolvent sense and this also implies the convergence of the spectrum. However, the latter convergence is true only inside each compact set on the complex plane and it does not characterize the behavior of the total ensemble of the eigenvalues under the perturbation. Our main result is the spectral asymptotics for the eigenvalues of the perturbed operator with an estimate for the error term uniform in the small parameter. This asymptotics involves an additional nonstandard term, which allows us to describe a global behavior of the total ensemble of the eigenvalues under the perturbation.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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