Reduced Gutzwiller formula with symmetry: case of a Lie group

Abstract

We consider a classical Hamiltonian $H$ on ${\mathbb{R}}^{2d}$, invariant by a Lie 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 semi-classical Weyl asymptotics for the eigenvalues counting function of $\hat{H}\_\chi$ in an interval of $\mathbb{R}$, and interpret it geometrically in terms of dynamics in the reduced space ${\mathbb{R}}^{2d}/G$. Besides, oscillations of the spectral density of $\hat{H}\_\chi$ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of ${\mathbb{R}}^{2d}$.

23 pages