Trace formulas for general Hermitian matrices: unitary scattering approach and periodic orbits on an associated graph

Gnutzmann, Sven; Smilansky, Uzy

Abstract

Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix H to a unitary analogue. In the first type the unitary matrix is is the spectral parameter. The new feature is that the spectral parameter appears in the final form as an argument of Eulerian polynomials - thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin's theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

Citation

Gnutzmann, S., & Smilansky, U. (2019). Trace formulas for general Hermitian matrices: unitary scattering approach and periodic orbits on an associated graph. Journal of Physics A: Mathematical and Theoretical, 53(3), Article 035201. https://doi.org/10.1088/1751-8121/ab5986

• Journal Article Type: Article
• Acceptance Date: Nov 20, 2019
• Online Publication Date: Dec 23, 2019
• Publication Date: Dec 23, 2019
• Deposit Date: Jan 15, 2020
• Publicly Available Date: Apr 28, 2025
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 3
• Article Number: 035201
• DOI: https://doi.org/10.1088/1751-8121/ab5986
• Keywords: Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/3610716
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/ab5986