Spectral flow and resonance index
Volume 528 / 2017
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 the resonance index outside the essential spectrum.
Among results of this paper are the following:
(1) Investigation of the root space of the compact operator $(H_0+sV-\lambda)^{-1}V$ corresponding to an eigenvalue $(s-r_\lambda)^{-1},$ where $H_0$ is a self-adjoint operator and $r_\lambda \in \mathbb R$ is such that $H_0+r_\lambda V$ belongs to the set $$ \mathscr R(\lambda) = \{H_0+V \colon V=V^* \text{ is $H_0$-compact} \ \text{and} \ \lambda \in \sigma_d(H_0+V)\}. $$
(2) (a) Criteria for a perturbation $V$ to be tangent to the set $\mathscr R(\lambda)$ at a point $H$.
(b) Criteria for the order of tangency of a perturbation $V$ to the set $\mathscr R(\lambda).$
(3) Direct proof of the equality “total resonance index = intersection number”.
(4) Direct proof of the equality “total resonance index = total Fredholm index”.
(5) Total resonance index satisfies the Robbin–Salamon axioms for spectral flow.
(6) Direct proof of the equality “total resonance index = spectral shift function” at a point $\lambda$ not in the essential spectrum $\sigma_{\rm ess}(H_0).$
This analysis gives a finer information about the behaviour of discrete spectrum compared to spectral flow.
Many results of this paper are non-trivial even in finite dimensions, in which case they can be and were tested in numerical experiments.
