Reduced Gutzwiller formula with symmetry: case of a finite group
Résumé
We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\widehat{H}$ 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 $\widehat{H}_\chi$ 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 $\widehat{H}_\chi$ 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.
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