Quantum-classical correspondence on associated vector bundles over locally symmetric spaces

TL;DR

This paper establishes a detailed correspondence between classical resonant states and quantum eigenvalues on certain vector bundles over rank-one locally symmetric spaces, using representation theory and Lie theory.

Contribution

It provides a precise description of classical resonant states vanishing in unstable directions and relates them to quantum eigenvalues, revealing an exact spectral band structure.

Findings

Classical resonant states correspond to generalized eigenspaces of invariant differential operators.

An explicit isomorphism maps resonant states to quantum eigenfunctions.

The Pollicott-Ruelle spectrum exhibits an exact band structure.

Abstract

For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states on $\mathbf V_\tau$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^\ast\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,\sigma)$ on compatible associated vector bundles $\mathbf W_\sigma$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott-Ruelle spectrum. Further, under some mild assumptions on the representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_\tau$ and $\mathbf W_\sigma$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott-Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_\sigma$. Our methods of proof are based on representation theory and Lie theory.