Singular spectral shift and Pushnitski $\mu$-invariant

TL;DR

This paper demonstrates that for trace class perturbations, the singular part of Pushnitski's μ-invariant is angle-independent, providing an alternative proof of the spectral shift function's integer-valuedness and deriving the Birman-Krein formula.

Contribution

It shows the angle-independence of the singular μ-invariant part and offers a new proof of the spectral shift function's integer-valuedness for trace class perturbations.

Findings

Singular part of Pushnitski μ-invariant is angle-independent.

Provides an alternative proof of the integer-valuedness of the spectral shift function.

Derives the Birman-Krein formula for trace class perturbations.

Abstract

In this paper it is shown that in case of trace class perturbations the singular part of Pushnitski $\mu$-invariant does not depend on the angle variable. This gives an alternative proof of integer-valuedness of the singular part of the spectral shift function. As a consequence, the Birman-Krein formula for trace class perturbations follows.