Spectral flow and resonance index

TL;DR

This paper extends the concept of spectral flow to the essential spectrum, introduces the resonance index, and establishes its properties and relationships with other spectral invariants, providing new insights into spectral theory.

Contribution

It introduces the resonance index outside the essential spectrum and proves its key properties, linking it to spectral flow, intersection number, and Fredholm index.

Findings

Total resonance index satisfies Robbin-Salamon axioms.

Proved equality between total resonance index and intersection number.

Established criteria for tangency and order of tangency of perturbations.

Abstract

It has been shown recently that spectral flow admits a natural integer-valued extension to essential spectrum. This extension admits four different interpretations; two of them are singular spectral shift function and total resonance index. In this work we study resonance index outside essential spectrum. Among results of this paper are the following. 1. Total resonance index satisfies Robbin-Salamon axioms for spectral flow. 2. Direct proof of equality "total resonance index = intersection number". 3. Direct proof of equality "total resonance index = total Fredholm index". 4. (a) Criteria for a perturbation~$V$ to be tangent to the~resonance set at a point~$H,$ where the resonance set is the infinite-dimensional variety of self-adjoint perturbations of the initial self-adjoint operator~$H_0$ which have~$\lambda$ as an eigenvalue. (b) Criteria for the order of tangency of a perturbation~$V$ to the resonance set. 5. Investigation of the root space of the compact operator $(H_0+sV-\lambda)^{-1}V$ corresponding to an eigenvalue $(s-r_\lambda)^{-1},$ where $H_0+r_\lambda V$ is a point of the resonance set. This analysis gives a finer information about behaviour of discrete spectrum compared to spectral flow. Finally, many results of this paper are non-trivial even in finite dimensions, in which case they can be and were tested in numerical experiments.