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Spectrality of the Dirac Operator with Complex-Valued Periodic Coefficients

O. A. Veliev

Abstract

In this paper, we study the spectrality of the non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. We establish a condition on the off-diagonal elements of the matrix Q under which L(Q) is an asymptotically spectral operator. Moreover, we derive a condition on Q that ensures the spectrality of this operator. Finally, we consider the spectral expansion in these cases.

Spectrality of the Dirac Operator with Complex-Valued Periodic Coefficients

Abstract

In this paper, we study the spectrality of the non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. We establish a condition on the off-diagonal elements of the matrix Q under which L(Q) is an asymptotically spectral operator. Moreover, we derive a condition on Q that ensures the spectrality of this operator. Finally, we consider the spectral expansion in these cases.
Paper Structure (3 sections, 6 theorems, 79 equations)

This paper contains 3 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction and Preliminary Facts
  2. Asymptotic Formulas for Bloch eigenvalues and Bloch functions
  3. Spectrality and Spectral Expansion

Key Result

Theorem 1

If (2) holds, then: $(a)$The eigenvalues of $L_{t}(Q),$ for $t\in(-1,1),$ consist of two sequences $\left\{ \lambda_{n,1}(t);\text{ }n\in\mathbb{Z}\right\}$ and $\left\{ \lambda_{n,2}(t);\text{ }n\in\mathbb{Z}\right\}$ which satisfy the following asymptotic formulas These formulas are uniform with respect to $t\in(-1,1].$ There exists $N$ such that $\lambda_{n,j}(t)$ are the simple eigenvalues o

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • Remark 2