Subdeterminant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs

Waltner, Daniel; Gnutzmann, Sven; Tanner, Gregor; Richter, Klaus

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.

Citation

Waltner, D., Gnutzmann, S., Tanner, G., & Richter, K. (2013). Subdeterminant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs. Physical Review E, 87(5), Article 052919. https://doi.org/10.1103/PhysRevE.87.052919

• Journal Article Type: Article
• Acceptance Date: Jan 30, 2013
• Online Publication Date: May 29, 2013
• Publication Date: May 29, 2013
• Deposit Date: Dec 10, 2020
• Publicly Available Date: Dec 18, 2020
• Journal: Physical Review E
• Electronic ISSN: 1550-2376
• Publisher: American Physical Society
• Peer Reviewed: Peer Reviewed
• Volume: 87
• Issue: 5
• Article Number: 052919
• DOI: https://doi.org/10.1103/PhysRevE.87.052919
• Keywords: Chaotic Dynamics; Mesoscale and Nanoscale Physics
• Public URL: https://nottingham-repository.worktribe.com/output/2460174
• Publisher URL: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.87.052919