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On the existence of eigenvalues of a one-dimensional Dirac operator

Daniel Sánchez-Mendoza, Monika Winklmeier

TL;DR

The paper addresses the existence and quantitative description of eigenvalues in the spectral gap of the one-dimensional Dirac operator with bounded potentials. It develops a generalized variational principle for block operator matrices, using the Schur complement to characterize eigenvalues in the gap and to obtain existence criteria, upper and lower bounds, and counts of discrete eigenvalues. By applying these abstract results to the Dirac operator on the real line and on the half-line, the authors derive Sturm-Liouville comparisons, global and local eigenvalue estimates, and concrete examples (analytic toy models and a hydrogenic case) that align with prior work. The approach provides a rigorous framework for predicting how potential wells create bound states in the Dirac gap and yields practical bounds and qualitative behavior useful for mathematical physics and spectral theory.

Abstract

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.

On the existence of eigenvalues of a one-dimensional Dirac operator

TL;DR

The paper addresses the existence and quantitative description of eigenvalues in the spectral gap of the one-dimensional Dirac operator with bounded potentials. It develops a generalized variational principle for block operator matrices, using the Schur complement to characterize eigenvalues in the gap and to obtain existence criteria, upper and lower bounds, and counts of discrete eigenvalues. By applying these abstract results to the Dirac operator on the real line and on the half-line, the authors derive Sturm-Liouville comparisons, global and local eigenvalue estimates, and concrete examples (analytic toy models and a hydrogenic case) that align with prior work. The approach provides a rigorous framework for predicting how potential wells create bound states in the Dirac gap and yields practical bounds and qualitative behavior useful for mathematical physics and spectral theory.

Abstract

The aim of this paper is to study the existence of eigenvalues in the gap of the essential spectrum of the one-dimensional Dirac operator in the presence of a bounded potential. We employ a generalized variational principle to prove existence of such eigenvalues, estimate how many eigenvalues there are, and give upper and lower bounds for them.
Paper Structure (5 sections, 17 theorems, 105 equations, 4 figures)

This paper contains 5 sections, 17 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The one-dimensional Dirac operator
  3. Variational principle for eigenvalues of a block operator matrix in a gap of the essential spectrum
  4. The Dirac operator on the real line
  5. The Dirac operator on the halfline

Key Result

Proposition 2.1

Assume that item:A1, item:A2, item:A3 hold.

Figures (4)

  • Figure 1: Graphs of $M_1$ and $-M_2$ from Example \ref{['ex:Analytic']} for $\gamma\in\{0,1,-1\}$.
  • Figure 2: Eigenvalues (continuous) and their corresponding upper bounds (dashed) from Example \ref{['ex:Analytic']} for $M=4$ and $\gamma\in\{0,1,-1\}$.
  • Figure 3: Numerically computed eigenvalue trajectories from Example \ref{['ex:Shadi']} for $M=1$. The parameter $t$ in (a) corresponds to Gamma in (b).
  • Figure 4: Plot of the left hand side (black) and right hand side (colors) of \ref{['eq:neweigenvalue']} for various values of $\alpha$ and $\gamma\in\{0,1,-1\}$. At any $t$ where a curve for fixed $\alpha$ intersects the black curve, a new eigenvalue enters the spectral gap from $M$.

Theorems & Definitions (43)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • ...and 33 more