# Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula

## TL;DR

This paper develops a geometric quantization framework for Hamiltonian flows, deriving a Gutzwiller trace formula and semi-classical estimates, with potential applications in contact topology.

## Contribution

It introduces a novel approach to quantize Hamiltonian flows using Berezin-Toeplitz operators and derives an explicit Gutzwiller trace formula within this setting.

## Key findings

- Established a Gutzwiller trace formula for Kostant-Souriau operators.
- Computed the leading term explicitly in the trace formula.
- Provided semi-classical estimates for quantum Hamiltonian dynamics.

## Abstract

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via parallel transport in the quantum spaces over the induced path of almost complex structures, and we establish various semi-classical estimates. In particular, we establish a Gutzwiller trace formula for the Kostant-Souriau operator and compute explicitly the leading term. We then describe a potential application to contact topology.