The Dirac Operator with Complex-Valued Summable Potential

TL;DR

This paper studies the spectral properties of the Dirac operator with complex-valued summable potential on a finite interval, deriving eigenvalue asymptotics and basis properties of eigenfunctions.

Contribution

It provides new asymptotic formulas for eigenvalues and eigenfunctions, and establishes Riesz basis properties depending on regularity conditions.

Findings

Eigenvalues have asymptotic formulas with p-dependent remainders.

Eigen and associated functions form a Riesz basis with parentheses for regular operators.

Strong regularity ensures the eigenfunctions form a usual Riesz basis.

Abstract

The paper deals with the Dirac operator generated on the finite interval $[0,\pi]$ by the differential expression $-B\mathbf{y}'+Q(x)\mathbf{y}$, where $$ B=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad Q(x)=\begin{pmatrix}q_1(x)&q_2(x)\\q_3(x)&q_4(x)\end{pmatrix}, $$ and the entries $q_j(x)$ belong to~$L_p[0,\pi]$ for some $p\geqslant 1$. The classes of regular and strongly regular operators of this form are defined, depending on the boundary conditions. The asymptotic formulas for the eigenvalues and eigenfunctions of such operators are obtained with remainders depending on~$p$. It it is proved that the system of eigen and associated functions of a regular operator forms a Riesz basis with parentheses in the space~$(L_2[0,\pi])^2$ and the usual Riesz basis, provided that the operator is strongly regular.