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Approximation of Dirac operators with confining electrostatic and Lorentz scalar $δ$-shell potentials

Christian Stelzer-Landauer

TL;DR

The work analyzes how Dirac operators with confining $δ$-shell potentials on a smooth hypersurface can be obtained as norm-resolvent limits of Dirac operators with strongly localized regular potentials. By introducing a scaling framework $V_ε = ηI_N + τβ$ supported in a tubular neighborhood and using boundary integral operators $Φ_z$ and $𝒞_z$, the authors prove that $H_{f(ε)V_ε}$ converges in the norm-resolvent sense to $H_{ ilde{V}δ_Σ}$ with $ ilde{V} = (2/\,√{|d|})V$ for $d = η^2 - τ^2 < 0$, and that choosing $f(ε)$ so that $ ilde{d} = -4$ yields confining interaction strengths. They establish explicit error bounds: $ig\\|(H_{ ilde{V}δ_Σ}-z)^{-1} - (H_{ ilde{V}_ε δ_Σ}-z)^{-1}ig\\| \\le C e^{-f(ε)√{|d|}}$ and $ig\\|(H_{ ilde{V}_ε δ_Σ}-z)^{-1} - (H_{f(ε)V_ε}-z)^{-1}ig\\| \\le C f(ε)^{3/2} ε^{\gamma}$ with $\\gamma obreakin (0,1/2)$. Consequently, $H_{f(ε)V_ε}$ converges to $H_{ ilde{V}δ_Σ}$ in norm-resolvent sense, and in the $d=-4$ case one recovers the target confining δ-shell operator, providing a constructive bridge from regularized, localized potentials to singular Dirac-interface interactions.

Abstract

In this paper we study the approximation of Dirac operators with $δ$-shell potentials in the norm resolvent sense. In particular, we consider the approximation of Dirac operators with confining electrostatic and Lorentz scalar $δ$-shell potentials, where the support of the $δ$-shell potentials is impermeable to particles modelled by such Dirac operators.

Approximation of Dirac operators with confining electrostatic and Lorentz scalar $δ$-shell potentials

TL;DR

The work analyzes how Dirac operators with confining -shell potentials on a smooth hypersurface can be obtained as norm-resolvent limits of Dirac operators with strongly localized regular potentials. By introducing a scaling framework supported in a tubular neighborhood and using boundary integral operators and , the authors prove that converges in the norm-resolvent sense to with for , and that choosing so that yields confining interaction strengths. They establish explicit error bounds: and with . Consequently, converges to in norm-resolvent sense, and in the case one recovers the target confining δ-shell operator, providing a constructive bridge from regularized, localized potentials to singular Dirac-interface interactions.

Abstract

In this paper we study the approximation of Dirac operators with -shell potentials in the norm resolvent sense. In particular, we consider the approximation of Dirac operators with confining electrostatic and Lorentz scalar -shell potentials, where the support of the -shell potentials is impermeable to particles modelled by such Dirac operators.

Paper Structure

This paper contains 4 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Notations and Assumptions
  3. A bound for (\ref{['eq_res_dif_1']})
  4. A bound for (\ref{['eq_res_dif_2']})

Key Result

Theorem 1.1

Let $q$ be as in eq_q, $V = \eta I_N + \tau \beta$, $\eta,\tau \in\mathbb{R}$, satisfy eq_conf_V, $V_\varepsilon$ be defined by eq_V_eps, $f$ be as in eq_conf_f(with $\gamma \in (0,1/2)$) and $z \in \mathbb{C}\setminus\mathbb{R}$. Moreover, set $\widetilde{V} = (2/\sqrt{|d|})V$, where $d = \eta^2 - for all $\varepsilon \in (0,\delta)$. In particular, $H_{f(\varepsilon) V_\varepsilon}$ converges

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more