title: Pseudo numerical ranges, Schur complement dominant operator matrices, Schrödinger operators with accretive
potentials in weighted spaces and applications to damped wave equations
Creators: name: Gerhát, Borbála Mercédesz
Publisher: name: Universität Bern
Year: 2024
Subjects: subject: 510 Mathematics

description: This cumulative dissertation consists of three research papers, which are put into a common context by an introductory chapter. The general research theme is the (spectral) analysis of non-selfadjoint operator matrices in the absence of standard dominance patterns. While in the first (single authored) work we develop a general weak framework to relate the spectra of an operator matrix and its Schur complement, in the second article (with C. Tretter) we introduce new pseudo numerical ranges for the study of the spectrum and resolvent norm of both matrix and Schur complement. In the third contribution (with P. Siegl) we implement and study Schrödinger type operators with accretive potentials which appear as Schur complements of certain matrix problems. All three works contain applications of the derived abstract results to damped wave equations. More details on the particular articles are listed below.
Schur complement dominant operator matrices. In mathematical physics, matrix differential operators arise naturally in applications as coupled systems of partial differential equations. Up to now, the spectral analysis of such problems has commonly been tackled assuming certain patterns of relative boundedness within the matrix entries. We propose to view operator matrices in a more general setting, which allows our results to abstain from perturbative arguments of this type. Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D ⊂ H ⊂ D− and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D− which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, eventually implying closedness and non-empty resolvent set of the operator matrix. Finally, we apply our abstract results to the damped wave equation with possibly unbounded and/or singular damping, as well as to second order matrix differential operators with certain minimal restrictions on their coefficients. By means of our methods, the previously used regularity assumptions can be weakened substantially in both cases.
Pseudo numerical ranges and spectral enclosures. We introduce the new concepts of pseudo numerical range for operator functions and families of sesquilinear forms as well as the pseudo block numerical range for n × n operator matrix functions. While these notions are new even in the bounded case, we cover operator polynomials with unbounded coefficients, unbounded holomorphic form families of type (a) and associated operator families of type (B). Our main results include spectral inclusion properties of pseudo numerical ranges and pseudo block numerical ranges. For diagonally dominant and off-diagonally dominant operator matrices they allow us to prove spectral enclosures in terms of the pseudo numerical ranges of Schur complements that no longer require dominance order 0 and not even < 1. As an application, we establish a new type of spectral bounds for linearly damped wave equations with possibly unbounded and/or singular damping. Schrödinger operators with accretive potentials in weighted spaces.
We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.