{"title":"Interaction of Relativistic Particles with Singular Potentials Supported by a Periodic Graph","authors":"V.S. Rabinovich","doi":"10.1134/S1061920825600163","DOIUrl":null,"url":null,"abstract":"<p> We consider the interaction of relativistic particles described by two-dimensional Dirac operators with delta-type singular potentials supported by periodic graphs <span>\\(\\Gamma\\subset\\mathbb{R}^{2}\\)</span>. This problem can be regarded as a relativistic analog of the Kronig–Penney model of electron propagation in solid state physics. We associate with this problem an unbounded operator in the Hilbert space <span>\\(L^{2}(\\mathbb{R}^{2},\\mathbb{C}^{2})\\)</span>. The study of spectral properties of these operators is reduced to the study of the Fredholmness of singular integral operators on the graph <span>\\(\\Gamma\\)</span>. We obtain necessary and sufficient conditions for the Fredholmness of these operators as ellipticity conditions on the edges, matrix conditions at the vertices, and conditions of invertibility of limit operators which are periodic operators on the graph <span>\\(\\Gamma\\)</span>. We apply the Bloch–Floquet theory to the study of invertibility of limit operators. </p><p> <b> DOI</b> 10.1134/S1061920825600163 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"32 2","pages":"365 - 378"},"PeriodicalIF":1.5000,"publicationDate":"2025-07-29","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/S1061920825600163","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 consider the interaction of relativistic particles described by two-dimensional Dirac operators with delta-type singular potentials supported by periodic graphs \(\Gamma\subset\mathbb{R}^{2}\). This problem can be regarded as a relativistic analog of the Kronig–Penney model of electron propagation in solid state physics. We associate with this problem an unbounded operator in the Hilbert space \(L^{2}(\mathbb{R}^{2},\mathbb{C}^{2})\). The study of spectral properties of these operators is reduced to the study of the Fredholmness of singular integral operators on the graph \(\Gamma\). We obtain necessary and sufficient conditions for the Fredholmness of these operators as ellipticity conditions on the edges, matrix conditions at the vertices, and conditions of invertibility of limit operators which are periodic operators on the graph \(\Gamma\). We apply the Bloch–Floquet theory to the study of invertibility of limit operators.
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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