Abstract. This paper is devoted to the mathematical investigation of the MIT bag model, that is the Dirac operator on a smooth and bounded domain of R 3 with certain boundary conditions. We prove that the operator is self-adjoint and, when the mass m goes to˘8, we provide spectral asymptotic results.

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass m ą 0 lies outside a smooth and bounded open set Ω Ă R 3 , it is proved that its spectrum is approximated by the one of the Dirac operator on Ω with the MIT bag boundary condition. The approximation, which is developed up to and error of order op1{? mq, is carried out by introducing tubular coordinates in a neighborhood of BΩ and analyzing the corresponding one dimensional optimization problems in the normal direction.2010 Mathematics Subject Classification. 35J60, 35Q75, 49J45, 49S05, 81Q10, 81V05, 35P15, 58C40.Key words and phrases. Dirac operator, relativistic particle in a box, MIT bag model, spectral theory.

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set with Dirichlet conditions on the boundary. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal–Bargmann and Hardy spaces associated with the magnetic field.