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Reduced Gutzwiller formula with symmetry: case of a Lie group

Roch Cassanas
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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 $χ$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_χ$ to the symmetry subspace $L^2\_χ(\mathbb{R}^d)$ 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 $\hat{H}\_χ$ 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 $\hat{H}\_χ$ are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of $\mathbb{R}^{2d}$.

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Roch Cassanas. 2005-09-09. Reduced Gutzwiller formula with symmetry: case of a Lie group. https://arxiv.org/abs/math-ph/0509014

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