A proof of the Gutzwiller Semiclassical Trace Formula using Coherent States Decomposition

TL;DR

This paper provides a straightforward proof of the Gutzwiller semiclassical trace formula by employing coherent states, simplifying previous heuristic and rigorous approaches in quantum-classical correspondence.

Contribution

It introduces a novel proof method for the Gutzwiller trace formula using coherent states, offering a simpler and more direct approach compared to prior techniques.

Findings

Proof using coherent states simplifies the derivation of the trace formula.

Provides a more accessible and direct proof compared to heuristic and Fourier integral methods.

Enhances understanding of quantum-classical correspondence in semiclassical analysis.

Abstract

The Gutzwiller semiclassical trace formula links the eigenvalues of the Scrodinger operator ^H with the closed orbits of the corresponding classical mechanical system, associated with the Hamiltonian H, when the Planck constant is small ("semiclassical regime"). Gutzwiller gave a heuristic proof, using the Feynman integral representation for the propagator of ^H. Later on mathematicians gave rigorous proofs of this trace formula, under different settings, using the theory of Fourier Integral Operators and Lagrangian manifolds. Here we want to show how the use of coherent states (or gaussian beams) allows us to give a simple and direct proof.