Semiclassical Asymptotics on Stratified Manifolds

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL Russian Journal of Mathematical Physics Pub Date : 2024-06-28 DOI:10.1134/s1061920824020110

V.E. Nazaikinskii

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Abstract

We study the problem on semiclassical asymptotics for (pseudo)differential equations with singularities on a stratified manifold of a special form—the orbit space \(X\) of a smooth action of a compact Lie group \(G\) on a smooth manifold \(M\). The operators under consideration are obtained as the restriction of \(G\)-invariant operators with smooth coefficients on \(M\) to the subspace of \(G\)-invariant functions, naturally identified with functions on \(X\), and have singularities on strata of positive codimension. The asymptotics are associated with Lagrangian manifolds in the phase space defined by the Marsden–Weinstein symplectic reduction of the cotangent bundle \(T^*M\) under the action of the group \(G\); rapidly oscillating integrals defining the Maslov canonical operator on such manifolds contain exponentials as well as special functions related to representations of the group \(G\). For the simplest stratified manifold—a manifold with boundary obtained as the orbit space of a semi-free action of the group \( \mathbb{S} ^1\) on a closed manifold—the corresponding construction of semiclassical asymptotics was realized earlier. Note that, in this case, the class of equations under consideration on manifolds with boundary includes the linearized shallow water equations in a basin with a sloping beach. The present paper deals with the general case.

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分层流形上的半经典渐近论

摘要我们研究了在特殊形式的分层流形上具有奇点的（伪）微分方程的半经典渐近问题--在光滑流形\(M\)上紧凑李群\(G\)的光滑作用的轨道空间\(X\)。所考虑的算子是作为在\(M\)上具有平滑系数的\(G\)不变算子对\(G\)不变函数子空间的限制而得到的，自然地与\(X\)上的函数相一致，并且在正标度层上具有奇点。渐近线与相空间中的拉格朗日流形有关，相空间是由\(G\)组作用下的余切束\(T^*M\)的马斯登-温斯坦交映还原所定义的；定义了这些流形上的马斯洛夫典范算子的快速振荡积分包含指数以及与\(G\)组的表示相关的特殊函数。对于最简单的分层流形--作为封闭流形上的群\( \mathbb{S} ^1\)的半自由作用的轨道空间而得到的具有边界的流形--半经典渐近的相应构造早先已经实现。需要注意的是，在这种情况下，所考虑的有边界流形上的方程类别包括具有倾斜海滩的盆地中的线性化浅水方程。本文讨论的是一般情况。

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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.