{"title":"Real Semiclassical Approximation for the Asymptotics of Jacobi Polynomials Given by a Difference Equation","authors":"A.V. Tsvetkova","doi":"10.1134/S1061920824040162","DOIUrl":null,"url":null,"abstract":"<p> The paper is devoted to constructing the global asymptotics of Jacobi polynomials by the method of “real semiclassics for problems with complex phases,тАЩтАЩ which is based on the study of recurrence relations. The method is based on the semiclassical approximation and the study of the geometry and types of singularities of the arising Lagrangian manifolds. While manifolds with a turning point in whose neighborhood the asymptotics is determined by the Airy function are well studied, the methods for the case in which the asymptotics is determined by the Bessel functions are not so well developed. In this paper, we demonstrate the application of the above-mentioned method in both situations, in particular, we describe the Lagrangian manifold that arises in the second case. </p><p> <b> DOI</b> 10.1134/S1061920824040162 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 4","pages":"774 - 784"},"PeriodicalIF":1.7000,"publicationDate":"2025-02-09","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/S1061920824040162","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
The paper is devoted to constructing the global asymptotics of Jacobi polynomials by the method of “real semiclassics for problems with complex phases,тАЩтАЩ which is based on the study of recurrence relations. The method is based on the semiclassical approximation and the study of the geometry and types of singularities of the arising Lagrangian manifolds. While manifolds with a turning point in whose neighborhood the asymptotics is determined by the Airy function are well studied, the methods for the case in which the asymptotics is determined by the Bessel functions are not so well developed. In this paper, we demonstrate the application of the above-mentioned method in both situations, in particular, we describe the Lagrangian manifold that arises in the second case.
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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