Abstract

We describe the semiclassical asymptotics of the solution to the Cauchy problem for the Schrödinger equation with a delta potential localized on a codimension \(1\) surface. The initial condition represents a rapidly oscillating wave packet. We show that the asymptotics is expressed in terms of the Maslov canonical operator on a pair of Lagrangian manifolds in the extended phase space; the form of the delta potential defines a mapping between these manifolds that describes the reflection and scattering of the wave packet.

中文翻译：

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Delta势定于Codimension 1曲面上的Schrödinger方程Cauchy问题解的半经典渐近性

摘要

我们描述了δ势位于余维\（1 \）面上的Schrödinger方程Cauchy问题解的半经典渐近性。初始条件表示一个快速振荡的波包。我们证明渐近性是在扩展相空间中的一对拉格朗日流形上用Maslov正则算子表示的。三角电位的形式定义了这些歧管之间的映射，该映射描述了波包的反射和散射。