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Dirac operators on the half-line: stability of spectrum and non-relativistic limit

David Kramar, David Krejcirik

TL;DR

This work analyzes Dirac operators on the half-line with generalized infinite-mass boundary conditions and studies spectral stability under additive matrix perturbations $V$ that may be non-self-adjoint. Using Kato's resolvent formula and a polar-decomposition factorization, it derives a weighted $L^1$–type condition $ abla \iint |V(x)| \bigl[1+(q+2m\min(x,y))^2\bigr]|V(y)|dxdy<1$ (with $q=\max(\cot\alpha,\cot\alpha^{-1})$) that ensures $D_V$ is similar to the unperturbed operator $D_0$ and preserves the spectrum, i.e., $\sigma(D_V)=\sigma_c(D_V)=\sigma(D_0)$. The authors also prove optimality in a delta-potential subcase and show that the delta family can saturate the bound for an alternative stability result, computing a critical threshold $t_0 = -\dfrac{\cot(\alpha)}{1+2ma\cot(\alpha)}$ for $V_t=t\delta(x-a)$. Furthermore, they establish a norm-resolvent non-relativistic limit $c\to\infty$ to the Robin Laplacian $S_0$ on $\mathbb{R}_+$ with $\beta=2\cot\alpha$, linking the relativistic model to known non-relativistic stability results and illustrating compatibility between the relativistic and non-relativistic frameworks. The work situates the analysis within a Kato/Birman–Schwinger perturbation context and provides explicit resolvent bounds that underpin the spectral stability results.

Abstract

We consider Dirac operators on the half-line, subject to generalised infinite-mass boundary conditions. We derive sufficient conditions which guarantee the stability of the spectrum against possibly non-self-adjoint potential perturbations and study the optimality of the obtained results. Finally, we establish a non-relativistic limit which makes a relationship of the present model to the Robin Laplacian on the half-line.

Dirac operators on the half-line: stability of spectrum and non-relativistic limit

TL;DR

This work analyzes Dirac operators on the half-line with generalized infinite-mass boundary conditions and studies spectral stability under additive matrix perturbations that may be non-self-adjoint. Using Kato's resolvent formula and a polar-decomposition factorization, it derives a weighted –type condition (with ) that ensures is similar to the unperturbed operator and preserves the spectrum, i.e., . The authors also prove optimality in a delta-potential subcase and show that the delta family can saturate the bound for an alternative stability result, computing a critical threshold for . Furthermore, they establish a norm-resolvent non-relativistic limit to the Robin Laplacian on with , linking the relativistic model to known non-relativistic stability results and illustrating compatibility between the relativistic and non-relativistic frameworks. The work situates the analysis within a Kato/Birman–Schwinger perturbation context and provides explicit resolvent bounds that underpin the spectral stability results.

Abstract

We consider Dirac operators on the half-line, subject to generalised infinite-mass boundary conditions. We derive sufficient conditions which guarantee the stability of the spectrum against possibly non-self-adjoint potential perturbations and study the optimality of the obtained results. Finally, we establish a non-relativistic limit which makes a relationship of the present model to the Robin Laplacian on the half-line.
Paper Structure (5 sections, 12 theorems, 80 equations, 1 figure)

This paper contains 5 sections, 12 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The free Dirac operator and its resolvent
  3. Perturbations and the stability of the spectrum
  4. The Dirac delta potentials and optimality
  5. The non-relativistic limit

Key Result

Theorem 1.1

Let $V \in L^1((0,\infty),\mathbb{C}^{2,2};(1+x) \, dx)$ satisfy where $q := \max(\cot(\alpha),{\cot(\alpha)}^{-1})$. Then $\sigma(D_V) = \sigma_\mathrm{c}(D_V) = \sigma(D_0)$.

Figures (1)

  • Figure 1: The eigenvalue of $D_{V_t}$ as a function of $t$ for $m=1$, $a=2$ and $\alpha=\frac{\pi}{4}$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2: KLS
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1: katoBS
  • Proposition 3.2
  • ...and 8 more