A Poincaré-Steklov map for the MIT bag model

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 $\Omega \subset {\mathbb{R}}^3$, 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.

In this version, we have added section 2.4 on the extrinsically defined Dirac operator on $\S$. We have made a few minor changes to the statements of Theorem 4.1 and Theorem 5.1, corrected several typos, and added additional references

References in corpus (3)

• Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions
• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• Eigenvalue curves for generalized MIT bag models