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

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.