Reduced Gutzwiller formula with symmetry: case of a Lie group

2005/09/09 by Roch Cassanas, Cassanas, Roch
Mathematics · Physics and Astronomy · #58J70 #81Q50 #81R30 #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP #msc:58J70 #msc:81Q50 #msc:81R30

paper · pdf · doi:10.48550/arxiv.math-ph/0509014

23 pages

arxiv created 2005/09/09 · arxiv updated 2009/12/01

Abstract

We consider a classical Hamiltonian H on ℝ^2d, invariant by a Lie group of symmetry G, whose Weyl quantization H is a selfadjoint operator on L²(ℝ^d). If χ is an irreducible character of G, we investigate the spectrum of its restriction H_χ to the symmetry subspace L²_χ(ℝ^d) of L²(ℝ^d) coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of H_χ in an interval of ℝ, and interpret it geometrically in terms of dynamics in the reduced space ℝ^2d/G. Besides, oscillations of the spectral density of H_χ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of ℝ^2d.

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