Semiclassical wave functions in billiards built on classical trajectories. Energy quantization, scars and periodic orbits

TL;DR

This paper introduces a semiclassical wave function construction based on classical trajectories and skeletons, applicable to chaotic systems, and demonstrates its effectiveness in calculating spectra and analyzing scars in billiards.

Contribution

It presents a new skeleton-based semiclassical method for wave functions that handles chaotic dynamics and caustic singularities, extending traditional approaches.

Findings

Successfully applied to circular, rectangular, and stadium billiards.

Provides a new algorithm for semiclassical approximation.

Offers insights into scar phenomena in quantum chaos.

Abstract

A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schr\"odinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars