A Poincaré–Steklov map for the MIT bag model

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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 ⊂ ‫ޒ‬ 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 − U zthe MIT bag boundary value of U z -to + U z , where ± are projections along the boundary ∂ and 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 + Mβ1 ‫ޒ‬ 3 \ for large masses M > 0 in terms of the resolvent of the MIT bag operator on . With its help, the large coupling convergence with a convergence rate of O(M −1 ) is shown. 1. Introduction 2923 2. Preliminaries 2927 3. Basic properties of the MIT bag model 2933 4. Poincaré-Steklov operators as pseudodifferential operators 2939 5. Approximation of the Poincaré-Steklov operators for large masses 2945 6. Resolvent convergence to the MIT bag model 2955 Appendix: Dirac algebra and applications 2966 Acknowledgements 2968 References 2968