Pushnitski's $\mu$-invariant and Schr\"odinger operators with embedded eigenvalues

TL;DR

This paper demonstrates that under specific conditions, the singular part of the spectral shift function for operator pairs with embedded eigenvalues is integer-valued, utilizing Pushnitski's $ ext{-} ext{invariant}$ decomposition.

Contribution

It establishes the integer-valued nature of the singular spectral shift function for operators with embedded eigenvalues, extending the understanding of spectral invariants.

Findings

Singular spectral shift function is integer-valued under certain assumptions.

Decomposition of Pushnitski's $ ext{-} ext{invariant}$ into absolutely continuous and singular parts.

Derivation of the Birman-Krein formula as a corollary.

Abstract

In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitski's $\mu$-invariant into "absolutely continuous" and "singular" parts. As a corollary, the Birman-Krein formula follows.