Asymptotics of the Whispering Gallery-Type in the Eigenproblem for the Laplacian in a Domain of Revolution Diffeomorphic To a Solid Torus

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI:10.1134/S1061920823040131

D.S. Minenkov, S.A. Sergeev

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Abstract

We consider the eigenproblem for the Laplacian inside a three-dimensional domain of revolution diffeomorphic to a solid torus, and construct asymptotic eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type. The whispering gallery-type asymptotics are localized near the boundary of the domain, and an explicit analytic representation in terms of Airy functions is constructed for such asymptotics. There are several different scales in the problem, which makes it possible to apply the procedure of adiabatic approximation in the form of operator separation of variables to reduce the initial problem to one-dimensional problems up to a small correction. We also discuss the relationship between the constructed whispering gallery-type asymptotics and classical billiards in the corresponding domain, in particular, such asymptotics correspond to almost integrable billiards with proper degeneracy. We illustrate the results in the case when a domain of revolution is obtained by the rotation of a triangle with rounded wedges.

DOI 10.1134/S1061920823040131

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实心圆环差分革命域中拉普拉斯函数特征问题中的 "悄悄话画廊 "渐近论

摘要我们考虑了拉普拉斯函数在与实体环相差形的三维旋转域内的特征问题，并构造了耳语画廊型渐近特征值和特征函数（准节点）。耳语画廊型渐近线定位在域边界附近，并为这种渐近线构建了明确的艾里函数解析表示。问题中有几个不同的尺度，这使得应用算子变量分离形式的绝热近似程序，将初始问题简化为一维问题成为可能。我们还讨论了所构建的耳语画廊型渐近与相应域中经典台球之间的关系，特别是这种渐近对应于具有适当退化性的几乎可积分台球。我们以带圆角楔的三角形旋转得到的旋转域为例说明了这一结果。 doi 10.1134/s1061920823040131

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

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.