G-Convergence of Dirac Operators

TL;DR

This paper studies the behavior of Dirac operators with locally periodic potentials as a parameter varies, establishing G-compactness and spectral convergence results in the context of operator theory.

Contribution

It introduces G-convergence analysis for Dirac operators with periodic potentials, demonstrating spectral gap convergence and operator compactness.

Findings

Proves G-compactness in the strong resolvent sense.

Establishes convergence of the point spectrum in the spectral gap.

Provides a framework for analyzing Dirac operators with periodic potentials.

Abstract

We consider the linear Dirac operator with a (-1)-homogeneous locally periodic potential that varies with respect to a small parameter. Using the notation of G-convergence for positive self-adjoint operators in Hilbert spaces we prove G-compactness in the strong resolvent sense for families of projections of Dirac operators. We also prove convergence of the corresponding point spectrum in the spectral gap.