A Poincaré–Steklov map for the MIT bag model

Benhellal, Badreddine; Bruneau, Vincent; Zreik, Mahdi

doi:10.2140/apde.2024.17.2923

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A Poincaré–Steklov map for the MIT bag model

Badreddine Benhellal, Vincent Bruneau and Mahdi Zreik

Vol. 17 (2024), No. 8, 2923–2970

DOI: 10.2140/apde.2024.17.2923

Abstract

The purpose of this paper is to introduce and study Poincaré–Steklov (PS) operators associated to the Dirac operator $D_{m}$ with the so-called MIT bag boundary condition. In a domain $Ω \subset ℝ^{3}$ , for a complex number $z$ and for $U_{z}$ a solution of $(D_{m} - z) U_{z} = 0$ , the associated PS operator maps the value of $Γ_{-} U_{z}$ — the MIT bag boundary value of $U_{z}$ — to $Γ_{+} U_{z}$ , where $Γ_{\pm}$ are projections along the boundary $\partial Ω$ and $(Γ_{-} + Γ_{+}) = t_{\partial Ω}$ is the trace operator on $\partial Ω$ .

In the first part of this paper, we show that the PS operator is a zeroth-order pseudodifferential operator and give its principal symbol. In the second part, we study the PS operator when the mass $m$ is large, we prove that it fits into the framework of $1 ∕ m$ -pseudodifferential operators, and we derive some important properties, especially its semiclassical principal symbol. Subsequently, we apply these results to establish a Krein-type resolvent formula for the Dirac operator $H_{M} = D_{m} + M β 1_{ℝ^{3} ∖ \bar{Ω}}$ for large masses $M > 0$ in terms of the resolvent of the MIT bag operator on $Ω$ . With its help, the large coupling convergence with a convergence rate of $𝒪 (M^{- 1})$ is shown.

Keywords

Poincaré–Steklov operators, Dirac operator, the MIT bag model, h-pseudodifferential operators, large coupling limit

Mathematical Subject Classification

Primary: 35Q40

Secondary: 35P05, 81Q10, 81Q20

Milestones

Received: 13 September 2022

Revised: 7 March 2023

Accepted: 18 July 2023

Published: 12 October 2024

Authors

Badreddine Benhellal
Institut für Mathematik Carl von Ossietzky Universität Oldenburg Oldenburg Germany

Vincent Bruneau
Institut de Mathematiques Université de Bordeaux I Talence France

Mahdi Zreik
Institut de Mathematiques Université de Bordeaux Talence France	Departamento de Matemáticas Universidad del País Vasco Leioa Spain

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Vol. 17, No. 8, 2024