A Poincar\'e-Steklov map for the MIT bag model

TL;DR

This paper introduces and analyzes Poincaré-Steklov operators for the Dirac operator with MIT bag boundary conditions, revealing their pseudodifferential nature and deriving resolvent formulas for large mass regimes.

Contribution

It establishes the pseudodifferential properties of the Poincaré-Steklov operator for the Dirac operator with MIT boundary conditions and derives a Krein-type resolvent formula for large mass parameters.

Findings

PS operator is a zero-order pseudodifferential operator with explicit principal symbol.

For large mass m, the PS operator is a 1/m-pseudodifferential operator with a semiclassical principal symbol.

A Krein-type resolvent formula for the Dirac operator with large mass coupling is derived.

Abstract

The purpose of this paper is to introduce and study Poincar\'e-Steklov (PS) operators associated to the Dirac operator $D_m$ with the so-called MIT bag boundary condition. In a domain $\Omega\subset\mathbb{R}^3$, 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 $\Gamma_- U_z$, the MIT bag boundary value of $U_z$, to $\Gamma_+ U_z$, where $\Gamma_\pm$ are projections along the boundary $\partial\Omega$ and $(\Gamma_ - + \Gamma_+) = t_{\partial\Omega}$ is the trace operator on $\partial\Omega$. In the first part of this paper, we show that the PS operator is a zero-order pseudodifferential operator and give its principal symbol. In the second part, we study the PS operator when the mass $m$ is large, and 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 $H_M= D_m+ M\beta 1_{\mathbb{R}^3\setminus\overline{\Omega}}$ for large masses $M>0$, in terms of the resolvent of the MIT bag operator on $\Omega$. With its help, the large coupling convergence with a convergence rate of $\mathcal{O}(M^{-1})$ is shown.