Semi-classical trace formula, isochronous case. Application to conservative systems

TL;DR

This paper derives a semi-classical trace formula for conservative systems with discrete energy surface periods, extending known formulas to include supplementary terms related to linearized flow, and applies it to quadratic Hamiltonians and integrable systems.

Contribution

It introduces a semi-classical trace formula for systems with clean flow, incorporating new terms from the linearized flow, and provides practical conditions for conservative systems with symmetries.

Findings

Derived a leading term in the semi-classical trace formula for discrete period energy surfaces.

Extended the trace formula to include supplementary terms involving the linearized flow.

Applied the formula to quadratic Hamiltonians and obtained the Berry-Tabor formula for integrable systems.

Abstract

Under conditions of clean flow we compute the leading term in the STF when the set of periods of the energy surface is discrete. Comparing to the case of non-degenerate periodic orbits, we obtain a supplementary term which is given in terms of the linearized flow. As particular cases, we give a STF for quadratic Hamiltonians and we obtain the Berry-Tabor formula for integrable systems. For conservative systems (i.e. systems with several first integrals), we give practical conditions to get a clean flow and interpret the leading term of the STF for a compact symmetry. We give several examples to illustrate our computation.