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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 thespectral shift function of any pair of operators from this space is aninteger-valued function. The proof uses a natural decomposition of Pushnitski's$\mu$-invariant into "absolutely continuous" and "singular" parts. As acorollary, the Birman-Krein formula follows.

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