Semiclassical catastrophe theory of simple bifurcations

TL;DR

This paper extends catastrophe theory to analyze bifurcations in Hamiltonian systems using an improved stationary phase method, providing accurate semiclassical trace formulas that match quantum results near bifurcations.

Contribution

It introduces an enhanced ISPM approach for bifurcation analysis, accounting for orbit contributions and improving semiclassical trace formulas in systems with symmetries.

Findings

Enhanced trace formulas accurately describe bifurcations.

Good agreement with quantum results at bifurcation points.

Method applicable to systems with continuous symmetries.

Abstract

The Fedoriuk-Maslov catastrophe theory of caustics and turning points is extended to solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second- and third-order expansion of the classical action near the stationary point. A considerable enhancement of contributions of the two orbits (pair of consisting of the parent and newborn orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries, where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold. This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points.