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

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.

25 pages, one figure

References in corpus (6)

• Semiclassical Foundation of Universality in Quantum Chaos
• Periodic-Orbit Theory of Level Correlations
• Resummation and the semiclassical theory of spectral statistics
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Quantum ergodicity on graphs
• Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations

Cited by in corpus (4)

• Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity
• Trace formulas for general Hermitian matrices: Unitary scattering approach and periodic orbits on an associated graph
• New approach to periodic orbit theory of spectral correlations
• Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales