On G-convergence of positive Self-adjoint operators

TL;DR

This paper investigates the asymptotic behavior of eigenvalue problems for positive definite self-adjoint operators using G-convergence, providing convergence results and characterizations of the G-limit as the parameter h approaches infinity.

Contribution

It extends G-convergence theory to analyze the eigenvalue problems of h-dependent operators, including elliptic and general linear operators, with new convergence characterizations.

Findings

Convergence of eigenvalues as h→∞

Characterization of G-limit for operators with perturbations

Spectral convergence results

Abstract

We apply G-convergence theory to study the asymptotic of the eigenvalue problems of positive definite bounded self-adjoint $h$-dependent operators as $h\to\infty$. Two operators are considered; a second order elliptic operator and a general linear operator. Using the definition of G-convergence of elliptic operator, we review convergence results of the elliptic eigenvalue problem as $h\to\infty$. Also employing the general definition of G-convergence of positive definite self-adjoint operator together with $\Gamma$-convergence of the associated quadratic form, we characterize the G-limit as $h\to\infty$ of the general operator with some classes of perturbations. As a consequence, we also prove the convergence of the corresponding spectrum.