Relative convergence estimates for the spectral asymptotic in the Large Coupling Limit

TL;DR

The paper establishes optimal convergence estimates for eigenvalues and eigenvectors in singular perturbation problems, introducing constructive techniques applicable even in spectral gaps and identifying classes of regular perturbations with favorable asymptotic behavior.

Contribution

It provides new constructive methods for spectral asymptotics in large coupling limits, including analysis of regular and singular perturbations with practical examples.

Findings

Optimal convergence estimates for eigenvalues and eigenvectors.

Identification of regular perturbation classes with good asymptotic properties.

Application to elasticity models and Schrödinger operators with singular potentials.

Abstract

We prove optimal convergence estimates for eigenvalues and eigenvectors of a class of singular/stiff perturbed problems. Our profs are constructive in nature and use (elementary) techniques which are of current interest in computational Linear Algebra to obtain estimates even for eigenvalues which are in gaps of the essential spectrum. Further, we also identify a class of "regular" stiff perturbations with (provably) good asymptotic properties. The Arch Model from the theory of elasticity is presented as a prototype for this class of perturbations. We also show that we are able to study model problems which do not satisfy this regularity assumption by presenting a study of a Schroedinger operator with singular obstacle potential.