Closed orbits and spatial density oscillations in the circular billiard

TL;DR

This paper analyzes the semiclassical oscillations in particle and energy densities within a circular billiard, classifying all orbits and deriving analytical properties to improve understanding of quantum-classical correspondence.

Contribution

It provides a complete classification of all closed orbits in the circular billiard and derives analytical expressions for their properties, advancing semiclassical methods.

Findings

Successful semiclassical calculations of density oscillations

Complete classification of all orbits including bifurcations

Demonstrated convergence of the closed-orbit sum

Abstract

We present a case study for the semiclassical calculation of the oscillations in the particle and kinetic-energy densities for the two-dimensional circular billiard. For this system, we can give a complete classification of all closed periodic and non-periodic orbits. We discuss their bifurcations under variation of the starting point r and derive analytical expressions for their properties such as actions, stability determinants, momentum mismatches and Morse indices. We present semiclassical calculations of the spatial density oscillations using a recently developed closed-orbit theory [Roccia J and Brack M 2008 Phys. Rev. Lett. 100 200408], employing standard uniform approximations from perturbation and bifurcation theory, and test the convergence of the closed-orbit sum.