The Birman-Schwinger principle on the essential spectrum

TL;DR

This paper extends the Birman-Schwinger principle to the essential spectrum of self-adjoint operators and relates the spectral projection index to the scattering matrix spectrum, advancing spectral analysis methods.

Contribution

It introduces an extension of the Birman-Schwinger principle to the essential spectrum and connects the spectral projection index with the scattering matrix spectrum.

Findings

Extended Birman-Schwinger principle to the essential spectrum.

Established a relation between spectral projection index and scattering matrix spectrum.

Provided new tools for spectral analysis of self-adjoint operators.

Abstract

Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. We consider the spectral projections of $H_0$ and $H$ corresponding to a semi-infinite interval of the real line. We discuss the index of this pair of spectral projections and prove an identity which extends the Birman-Schwinger principle onto the essential spectrum. We also relate this index to the spectrum of the scattering matrix for the pair $H_0$, $H$.