Classical and quantum resonances for hyperbolic surfaces

TL;DR

This paper establishes a detailed link between classical Ruelle resonances and quantum resonances on hyperbolic surfaces, revealing a nuanced correspondence that varies at negative integers.

Contribution

It provides an explicit correspondence between classical and quantum resonant states on hyperbolic surfaces, including special cases at negative integers involving holomorphic sections.

Findings

Classical and quantum resonant states are explicitly connected on hyperbolic surfaces.

The correspondence involves holomorphic sections at negative integers.

The results apply to both compact and convex co-compact hyperbolic surfaces.

Abstract

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.