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

O. A. Veliev

TL;DR

The paper addresses the spectral expansion for the one-dimensional Dirac operator $L(Q)$ with a complex periodic potential by analyzing the Floquet fibers $L_t(Q)$ and their Bloch eigenpairs. It develops uniform asymptotic formulas for Bloch eigenvalues and eigenfunctions, establishing that the eigenvalue sequences near $2n\pm t$ have controlled $O(1/n)$ corrections and that the corresponding root functions form a Riesz basis away from $t=0,1$. It introduces and analyzes essential spectral singularities (ESS) and shows that ESS occur only at $t=0$ or $t=1$, with multiple eigenvalues at other values yielding spectral singularities but not ESS. Finally, it constructs a spectral expansion for $L(Q)$ by carefully integrating over the Floquet parameter, grouping ESS-related terms into total projections, and obtaining an explicit expansion formula that converges in $L_2^2$ for compactly supported data.

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).

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

TL;DR

The paper addresses the spectral expansion for the one-dimensional Dirac operator with a complex periodic potential by analyzing the Floquet fibers and their Bloch eigenpairs. It develops uniform asymptotic formulas for Bloch eigenvalues and eigenfunctions, establishing that the eigenvalue sequences near have controlled corrections and that the corresponding root functions form a Riesz basis away from . It introduces and analyzes essential spectral singularities (ESS) and shows that ESS occur only at or , with multiple eigenvalues at other values yielding spectral singularities but not ESS. Finally, it constructs a spectral expansion for by carefully integrating over the Floquet parameter, grouping ESS-related terms into total projections, and obtaining an explicit expansion formula that converges in for compactly supported data.

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).
Paper Structure (4 sections, 12 theorems, 138 equations)

This paper contains 4 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction and Preliminary Facts
  2. Uniform Asymptotic Formulas for $t\in D_{h}(0,\pi)$
  3. Spectral singularities and ESS
  4. Spectral Expansions

Key Result

Theorem 1

The eigenvalues of $L_{t}(Q),$ for $t\in D_{h}(0,1),$ consist of two sequences $\left\{ \lambda_{n,1}(t);\text{ }n\in\mathbb{Z}\right\}$ and $\left\{ \lambda_{n,1}(t);\text{ }n\in\mathbb{Z}\right\}$ which satisfy the following asymptotic formulas where $D_{h}(0,1)=D_{h}\backslash\left( U_{h}(0)\cup U_{h}(1)\right) ,$$D_{h}=\{t\in\mathbb{C}:|\operatorname{Im}t|\leq2h,-h\leq\operatorname{Re}t\leq

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • ...and 7 more