Reduced Gutzwiller formula with symmetry: case of a Lie group

TL;DR

This paper extends the Gutzwiller trace formula to systems with Lie group symmetries, providing semi-classical spectral asymptotics for symmetry-restricted quantum Hamiltonians and linking spectral oscillations to periodic orbits in reduced phase space.

Contribution

It introduces a reduced Gutzwiller trace formula for quantum systems with Lie group symmetries, connecting spectral properties to classical dynamics in the symmetry-reduced space.

Findings

Derived semi-classical Weyl asymptotics for eigenvalue counting functions.

Provided a geometric interpretation of spectral asymptotics in reduced phase space.

Formulated a Gutzwiller trace formula involving periodic orbits of the reduced dynamics.

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 $\mathbb{R}^{2d}$.