A Gutzwiller trace formula for large hermitian matrices

TL;DR

This paper develops a semiclassical approximation for quantum dynamics in finite-dimensional systems with toroidal phase space, deriving a Gutzwiller trace formula for continuous-time evolution.

Contribution

It introduces a novel semiclassical Fourier integral operator framework for quantum maps with continuous time, leading to a Gutzwiller trace formula in this setting.

Findings

Derived a Gutzwiller trace formula for finite-dimensional quantum systems.

Constructed semiclassical Fourier integral operators approximating quantum evolution.

Discussed semiclassical quantisation conditions for eigenvalues.

Abstract

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schr\"odinger equation. In the framework of Weyl quantisation, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantisation condition for eigenvalues as well as some simple examples.