The MIT Bag Model as an infinite mass limit

The Dirac operator, acting in three dimensions, is considered. Assuming that a large mass $m>0$ lies outside a smooth and bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^3$, it is proved that its spectrum is approximated by the one of the Dirac operator on $\mathrm{\Omega }$ with the MIT bag boundary condition. The approximation, which is developed up to and error of order $o(1\mathrm{/}\sqrt{m})$, is carried out by introducing tubular coordinates in a neighborhood of $\mathrm{\partial }\mathrm{\Omega }$ and analyzing the corresponding one dimensional optimization problems in the normal direction.