Schur complement dominant operator matrices

TL;DR

This paper introduces a novel spectral analysis method for unbounded operator matrices using Schur complements, avoiding standard perturbation techniques and extending applicability to complex differential operators.

Contribution

It develops a general framework for spectral analysis of unbounded operator matrices via Schur complements, relaxing regularity assumptions and broadening classical results.

Findings

Spectral equivalence established between operator matrices and their Schur complements.

Applicable to damped wave equations with singular damping.

Extended analysis to Dirac operators with Coulomb potentials.

Abstract

We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space $\mathcal{H}$, we extend its action to a suitable distributional triple $\mathcal{D} \subset \mathcal{H} \subset \mathcal{D}_-$ and restrict it to its maximal domain in $\mathcal{H}$. The crucial point in our approach is the choice of the spaces $\mathcal{D}$ and $\mathcal{D}_-$ which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in $\mathcal{H}$ and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns. The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.