A proof of the Gutzwiller Semiclassical Trace Formula using Coherent States Decomposition
M. Combescure, J. Ralston, D. Robert
TL;DR
This paper delivers a concise, direct derivation of the semiclassical Gutzwiller trace formula by leveraging coherent-state decomposition (gaussian beams) and the stationary-phase method on a complex phase. By expanding around Gaussian wave packets and tracking the propagation via the linearized flow and metaplectic representation, it elucidates how classical actions, monodromy, and Maslov indices determine the quantum spectral density in the semiclassical limit. The approach yields explicit leading-order contributions from isolated nondegenerate periodic orbits and clarifies the role of the energy shell, Liouville measure, and observable averages along orbits, all under a clean-flow hypothesis. Overall, it provides a transparent alternative to Fourier integral operator methods, linking quantum traces directly to classical closed trajectories with explicit $\hbar$-dependent corrections.
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.