{"title":"Paraxial Diffraction on a Delta Potential","authors":"E.A. Zlobina, A.P. Kiselev","doi":"10.1134/S1061920825600679","DOIUrl":null,"url":null,"abstract":"<p> A special Cauchy problem for the Schrödinger equation with a delta potential localized on a half-line is addressed. From the viewpoint of high-frequency parabolic-equation heuristics, the problem could be viewed as an approximation to a paraxial (i.e., nearly tangential) diffraction of a plane wave incident on a screen. Explicit solution of the problem, found with the help of integral transformations, is subjected to exhaustive asymptotic investigation for all values of the complex coefficient of the potential. The asymptotic findings are qualitatively interpreted using diffraction terminology and quantitatively compared with the results of diffraction theory. Some effects that have no analogs in the related diffraction problems are noted. The solution is shown to be, in a certain range of parameters, an asymptotic solution of the Helmholtz equation with a delta potential. </p><p> <b> DOI</b> 10.1134/S1061920825600679 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"32 3","pages":"597 - 613"},"PeriodicalIF":1.5000,"publicationDate":"2025-09-29","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/S1061920825600679","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
A special Cauchy problem for the Schrödinger equation with a delta potential localized on a half-line is addressed. From the viewpoint of high-frequency parabolic-equation heuristics, the problem could be viewed as an approximation to a paraxial (i.e., nearly tangential) diffraction of a plane wave incident on a screen. Explicit solution of the problem, found with the help of integral transformations, is subjected to exhaustive asymptotic investigation for all values of the complex coefficient of the potential. The asymptotic findings are qualitatively interpreted using diffraction terminology and quantitatively compared with the results of diffraction theory. Some effects that have no analogs in the related diffraction problems are noted. The solution is shown to be, in a certain range of parameters, an asymptotic solution of the Helmholtz equation with a delta potential.
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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