A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs

TL;DR

This paper introduces a sub-determinant approach to pseudo-orbit expansions in quantum maps and graphs, revealing how unitarity influences spectral determinants and enabling more detailed relations between pseudo-orbits.

Contribution

It proposes grouping pseudo-orbits into sub-determinants to incorporate unitarity and derive new identities, improving understanding of spectral properties in quantum systems.

Findings

Explicitly shows cancellation of long orbits via sub-determinants

Reformulates Newton identities and spectral density using sub-determinants

Analyzes impact on spectral correlation functions and form factors

Abstract

We study implications of unitarity for pseudo-orbit expansions of the spectral determinants of quantum maps and quantum graphs. In particular, we advocate to group pseudo-orbits into sub-determinants. We show explicitly that the cancellation of long orbits is elegantly described on this level and that unitarity can be built in using a simple sub-determinant identity which has a non-trivial interpretation in terms of pseudo-orbits. This identity yields much more detailed relations between pseudo orbits of different length than known previously. We reformulate Newton identities and the spectral density in terms of sub-determinant expansions and point out the implications of the sub-determinant identity for these expressions. We analyse furthermore the effect of the identity on spectral correlation functions such as the auto-correlation and parametric cross correlation functions of the spectral determinant and the spectral form factor.