Perturbations of spectral operators, and applications. I. Bounded perturbations

Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators

We derive a spectral representation for the oblate spheroidal wave operator, which is holomorphic in the aspherical parameter Ω in a neighborhood of the real line. For real Ω, estimates are derived for all eigenvalue gaps uniformly in Ω. The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex Ω is deriv...

The Spectral Shift Function for Compactly Supported Perturbations of Schrödinger Operators on Large Bounded Domains

We study the asymptotic behavior as L → ∞ of the finitevolume spectral shift function for a positive, compactly-supported perturbation of a Schrödinger operator in d-dimensional Euclidean space, restricted to a cube of side length L with Dirichlet boundary conditions. The size of the support of the perturbation is fixed and independent of L. We prove that the finite-volume spectral shift functi...

Perturbations of Operators, Ii

Perturbations of Hermitian Matrices and Applications to Spectral Symbols

It is often observed in practice that matrix sequences {An}n generated by discretization methods applied to linear differential equations, possess a Spectral Symbol, that is a measurable function describing the asymptotic distribution of the eigenvalues of An. Sequences composed by Hermitian matrices own real spectral symbols, that can be derived through the axioms of Generalized Locally Toepli...

Positive Perturbations of Unbounded Operators

This work studies the spectral properties of certain unbounded selfadjoint operators by considering positive perturbations of such operators and the unitary equivalence of the perturbed and unperturbed transformations. Conditions are obtained on the unitary operators implementing this equivalence which guarantee that the selfadjoint operators have an absolutely continuous part.