Reduced Gutzwiller formula with symmetry: case of a finite group

TL;DR

This paper develops a reduced Gutzwiller trace formula for quantum systems with finite group symmetries, providing semi-classical spectral asymptotics that incorporate symmetry reduction and periodic orbit contributions.

Contribution

It introduces a novel semi-classical analysis of spectral densities for symmetric Hamiltonians, deriving a reduced Gutzwiller trace formula that accounts for symmetry effects.

Findings

Derived reduced semi-classical spectral asymptotics near non-critical energies.

Established a Gutzwiller trace formula incorporating symmetry reduction.

Demonstrated the appearance of periodic orbits in the reduced space.

Abstract

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_\chi$ to the symmetry subspace $L^2\_\chi(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give reduced semi-classical asymptotics of a regularised spectral density describing the spectrum of $\hat{H}\_\chi$ near a non critical energy $E\in\mathbb{R}$. If $\Sigma\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $\Sigma\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $\Sigma\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.