Reduced Gutzwiller Formula with Symmetry: Case of a Lie Group

We consider a classical Hamiltonian H on R, invariant by a Lie group of symmetry G, whose Weyl quantization Ĥ is a selfadjoint operator on L(R). If χ is an irreducible character of G, we investigate the spectrum of its restriction Ĥχ to the symmetry subspace L2χ(R ) of L(R) coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of Ĥχ in an interval of R, and interpret it geometrically in terms of dynamics in the reduced space R/G. Besides, oscillations of the spectral density of Ĥχ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R.