Reduced Gutzwiller formula with symmetry: case of a finite group

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.

20 pages

References in corpus (2)

• A proof of the Gutzwiller Semiclassical Trace Formula using Coherent States Decomposition
• Toeplitz operators and Hamiltonian torus action

Cited by in corpus (2)

• Equivariant spectral asymptotics for h-pseudodifferential operators
• Semi-classical trace formula, isochronous case. Application to conservative systems