On a minimax principle in spectral gaps

TL;DR

This paper extends the minimax principle for eigenvalues in spectral gaps to more general perturbative settings, including unbounded and off-diagonal cases, and explores related monotonicity and continuity properties.

Contribution

It adapts the minimax principle to abstract perturbations, broadening its applicability and providing new insights into eigenvalue behavior in spectral gaps.

Findings

Extended minimax principle to unbounded and off-diagonal perturbations.

Proved monotonicity and continuity properties of eigenvalues in spectral gaps.

Revisited the Stokes operator as an illustrative example.

Abstract

The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer, Lewis, and Siedentop in [Doc. Math. 4 (1999), 275--283] is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to the spectral gap under consideration. This in part builds upon and extends the considerations in the author's appendix to [J. Spectr. Theory 10 (2020), 843--885]. Several monotonicity and continuity properties of eigenvalues in gaps of the essential spectrum are deduced, and the Stokes operator is revisited as an example.