Singular spectral shift function for relatively trace class perturbations

Author: Tom Daniels

Daniels, Tom, 2018 Singular spectral shift function for relatively trace class perturbations, Flinders University, College of Science and Engineering

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Recently discovered is a natural decomposition of the Lifshitz-Krein spectral shift function (SSF) a la Lebesgue into the sum of absolutely continuous and singular SSF's. The latter part represents the flow of singular spectrum and takes integer values even within the essential spectrum. The singular SSF may be alternatively characterised as the either of the so-called total resonance index or singular mu-invariant. The first of these measures the total number of poles of the sandwiched resolvent, considered as a function of the coupling parameter, which split from the unit interval as the spectral parameter is perturbed off of the real axis, counting the poles that move into the upper half-plane with a positive sign and those that move to the lower half-plane with a negative sign. The second measures the sum of winding numbers of the eigenvalues of the scattering matrix as it is continuously deformed to the identity in two different ways: by shrinking the coupling parameter to 0 and by sending the imaginary part of the spectral parameter to 1. This document is in part a review of these facts, which were first established by N. Azamov under the assumption of a trace class perturbation, and also generalises their proofs to the case of relatively trace class perturbations, thereby making them applicable for instance to Schroedinger operators with bounded potentials undergoing integrable perturbations.

Keywords: Spectral shift function, singular spectral shift function, Schrodinger operators, resonance index, singular mu-invariant, stationary scattering theory

Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2018
School: College of Science and Engineering
Supervisor: Nurulla Azamov