{"title":"Method of Potential Operators for Interaction Problems on Unbounded Hypersurfaces in \\(\\mathbb{R}^{n}\\) for Dirac Operators","authors":"V. S. Rabinovich","doi":"10.1134/S1061920823040167","DOIUrl":null,"url":null,"abstract":"<p> We consider the <span>\\(L_{p}\\)</span>-theory of interaction problems associated with Dirac operators with singular potentials of the form <span>\\(D=\\mathfrak{D}_{m,\\Phi }+\\Gamma\\delta_{\\Sigma}\\)</span> where </p><p> is a Dirac operator on <span>\\(\\mathbb{R}^{n}\\)</span>, <span>\\(\\alpha_{1},\\alpha_{2},\\dots,\\alpha _{n},\\alpha_{n+1}\\)</span> are Dirac matrices, <span>\\(m\\)</span> is a variable mass, <span>\\(\\Phi \\mathbb{I}_{N}\\)</span> electrostatic potential, <span>\\(\\Gamma\\delta_{\\Sigma}\\)</span> is a singular potential with support on smooth hypersurfaces <span>\\(\\Sigma \\subset\\mathbb{R}^{n}.\\)</span> </p><p> We associate with the formal Dirac operator <span>\\(D\\)</span> the interaction (transmission) problem on <span>\\(\\mathbb{R}^{n}\\diagdown\\Sigma\\)</span> with the interaction conditions on <span>\\(\\Sigma\\)</span>. Applying the method of potential operators we reduce the interaction problem to a pseudodifferential equation on <span>\\(\\Sigma.\\)</span> The main aim of the paper is the study of Fredholm property of these pseudodifferential operators on unbounded hypersurfaces <span>\\(\\Sigma\\)</span> and applications to the study of Fredholmness of interaction problems on unbounded smooth hypersurfaces in Sobolev and Besov spaces. </p><p> <b> DOI</b> 10.1134/S1061920823040167 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"674 - 690"},"PeriodicalIF":1.7000,"publicationDate":"2023-12-25","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/S1061920823040167","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 \(L_{p}\)-theory of interaction problems associated with Dirac operators with singular potentials of the form \(D=\mathfrak{D}_{m,\Phi }+\Gamma\delta_{\Sigma}\) where
is a Dirac operator on \(\mathbb{R}^{n}\), \(\alpha_{1},\alpha_{2},\dots,\alpha _{n},\alpha_{n+1}\) are Dirac matrices, \(m\) is a variable mass, \(\Phi \mathbb{I}_{N}\) electrostatic potential, \(\Gamma\delta_{\Sigma}\) is a singular potential with support on smooth hypersurfaces \(\Sigma \subset\mathbb{R}^{n}.\)
We associate with the formal Dirac operator \(D\) the interaction (transmission) problem on \(\mathbb{R}^{n}\diagdown\Sigma\) with the interaction conditions on \(\Sigma\). Applying the method of potential operators we reduce the interaction problem to a pseudodifferential equation on \(\Sigma.\) The main aim of the paper is the study of Fredholm property of these pseudodifferential operators on unbounded hypersurfaces \(\Sigma\) and applications to the study of Fredholmness of interaction problems on unbounded smooth hypersurfaces in Sobolev and Besov spaces.
Abstract We consider the \(L_{p}\)Theory of interaction problems associated with Dirac operators with singular potentials of form （D=\mathfrak{D}_{m、\其中 $$mathfrak{D}_{m,\Phi}=\sum_{j=1}^{n}\alpha_{j}(-i\partial_{x_{j}})+m\alpha_{n+1}+\Phi\mathbb{I}_{N}$$ 是 \(\mathbb{R}^{n}\)上的狄拉克算子、\(\alpha_{1},\alpha_{2},\dots,\alpha _{n},\alpha_{n+1}}\)是狄拉克矩阵，\(m\)是可变质量，\(\Phi \mathbb{I}_{N}\)是静电势、\(((Gamma\delta_{\Sigma}\)是一个奇异势，在光滑超曲面上有支持。\我们把\(\mathbb{R}^{n}\diagdown\Sigma\)上的相互作用（传输）问题和\(\Sigma\)上的相互作用条件与形式上的狄拉克算子\(D\)联系起来。）本文的主要目的是研究这些伪微分算子在无界超曲面 \(\Sigma\) 上的弗里德霍姆性质，并将其应用于研究索波列夫和贝索夫空间中无界光滑超曲面上相互作用问题的弗里德霍姆性。 doi 10.1134/s1061920823040167
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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