{"title":"On Perturbation of Thresholds in Essential Spectrum under Coexistence of Virtual Level and Spectral Singularity","authors":"D.I. Borisov, D.A. Zezyulin","doi":"10.1134/s106192084010059","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the perturbation of the Schrödinger operator on the plane with a bounded potential of the form <span>\\(V_1(x)+V_2(y),\\)</span> where <span>\\(V_1\\)</span> is a real function and <span>\\(V_2\\)</span> is a compactly supported function. It is assumed that the one-dimensional Schrödinger operator <span>\\( \\mathcal{H} _1\\)</span> with the potential <span>\\(V_1\\)</span> has two real isolated eigenvalues <span>\\( \\Lambda _0,\\)</span> <span>\\( \\Lambda _1\\)</span> in the lower part of its spectrum, and the one-dimensional Schrödinger operator <span>\\( \\mathcal{H} _2\\)</span> with the potential <span>\\(V_2\\)</span> has a virtual level at the boundary of its essential spectrum, i.e., at <span>\\(\\lambda=0\\)</span>, and a spectral singularity at the inner point of the essential spectrum <span>\\(\\lambda=\\mu>0\\)</span>. In addition, the eigenvalues and the spectral singularity overlap in the sense of the equality <span>\\( \\lambda _0:= \\Lambda _0+\\mu= \\Lambda _1.\\)</span> We show that a perturbation by an abstract localized operator leads to a bifurcation of the internal threshold <span>\\( \\lambda _0\\)</span> into four spectral objects which are resonances and/or eigenvalues. These objects correspond to the poles of the local meromorphic continuations of the resolvent. The spectral singularity of the operator <span>\\( \\mathcal{H} _2\\)</span> qualitatively changes the structure of these poles as compared to the previously studied case where no spectral singularity was present. This effect is examined in detail, and the asymptotic behavior of the emerging poles and corresponding spectral objects of the perturbed Schrödinger operator is described. </p><p> <b> DOI</b> 10.1134/S106192084010059 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-03-19","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://doi.org/10.1134/s106192084010059","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}
引用次数: 0
Abstract
We study the perturbation of the Schrödinger operator on the plane with a bounded potential of the form \(V_1(x)+V_2(y),\) where \(V_1\) is a real function and \(V_2\) is a compactly supported function. It is assumed that the one-dimensional Schrödinger operator \( \mathcal{H} _1\) with the potential \(V_1\) has two real isolated eigenvalues \( \Lambda _0,\)\( \Lambda _1\) in the lower part of its spectrum, and the one-dimensional Schrödinger operator \( \mathcal{H} _2\) with the potential \(V_2\) has a virtual level at the boundary of its essential spectrum, i.e., at \(\lambda=0\), and a spectral singularity at the inner point of the essential spectrum \(\lambda=\mu>0\). In addition, the eigenvalues and the spectral singularity overlap in the sense of the equality \( \lambda _0:= \Lambda _0+\mu= \Lambda _1.\) We show that a perturbation by an abstract localized operator leads to a bifurcation of the internal threshold \( \lambda _0\) into four spectral objects which are resonances and/or eigenvalues. These objects correspond to the poles of the local meromorphic continuations of the resolvent. The spectral singularity of the operator \( \mathcal{H} _2\) qualitatively changes the structure of these poles as compared to the previously studied case where no spectral singularity was present. This effect is examined in detail, and the asymptotic behavior of the emerging poles and corresponding spectral objects of the perturbed Schrödinger operator is described.
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.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
