Bounds on the non-real spectrum of differential operators with indefinite weights

TL;DR

This paper investigates the spectral properties of differential operators with indefinite weights, showing that their non-real spectrum remains confined within a compact set under certain conditions, with applications to Sturm-Liouville and elliptic PDEs.

Contribution

It provides new bounds on the non-real spectrum of indefinite-weight differential operators, extending spectral analysis in Krein spaces with quantitative estimates.

Findings

Non-real spectrum is contained in a compact set.

Real spectrum outside this set has definite type.

Results apply to Sturm-Liouville and elliptic PDEs with indefinite weights.

Abstract

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty$ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.