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

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).