Spectral Expansion for the One-Dimensional Dirac Operator with a Complex-Valued Periodic Potential

TL;DR

This paper develops a spectral expansion for a non-self-adjoint one-dimensional Dirac operator with complex periodic potential, analyzing asymptotics of eigenvalues and spectral singularities to advance understanding of its spectral properties.

Contribution

It introduces a detailed spectral expansion framework for the non-self-adjoint Dirac operator with complex periodic potential, including asymptotic analysis of eigenvalues and spectral singularities.

Findings

Asymptotic formulas for Bloch eigenvalues and functions established

Spectral singularities of the operator characterized

Spectral expansion constructed for the non-self-adjoint operator

Abstract

In this paper, we construct the spectral expansion for the one dimensional non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. To this end, we study in detail asymptotic formulas for the Bloch eigenvalues and Bloch functions that are uniform with respect to the complex quasimomentum, as well as the essential spectral singularities of L(Q).