Non-trivial singular spectral shift functions exist

TL;DR

This paper proves the existence of operator pairs with non-zero singular spectral shift functions on the absolutely continuous spectrum, revealing new insights into spectral theory.

Contribution

It demonstrates the existence of irreducible operator pairs with non-trivial singular spectral shift functions, a novel result in spectral analysis.

Findings

Existence of such operator pairs is established.

The singular spectral shift function can be non-zero on the absolutely continuous spectrum.

Provides new examples in spectral theory.

Abstract

In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$