Abstract

We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla \) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.

中文翻译：

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在半刚性壁为界的域中，算子$$ \ nabla D（x）\ nabla $$的弹跳球特征函数的渐近性

摘要

我们考虑在算子\（\ nabla D（x）\ nabla \）的半经典谱上的问题，在二维域（半刚性壁）的边界上具有Bessel型退化。众所周知，与拉格朗日流形相关的渐近本征函数可以使用对Maslov规范算子的修改来构造。我们获得了与相应汉密尔顿系统的最简单周期轨迹相关联的渐近特征函数，并在域边界上进行了反射。