Operator theoretic methods for the eigenvalue counting function in spectral gaps

TL;DR

This paper introduces an operator theoretic approach leveraging spectral flow and shift functions to analyze eigenvalues in spectral gaps, offering new proofs and generalizations of existing results.

Contribution

It presents a novel, simplified method for studying eigenvalues in spectral gaps, extending previous results and providing a new proof of the Birman-Schwinger principle.

Findings

Generalized and streamlined proofs of existing spectral gap results

New proof of the generalized Birman-Schwinger principle

Framework applicable to asymptotic eigenvalue problems in spectral gaps

Abstract

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift function theory. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman-Schwinger principle.