The MIT Bag Model as an infinite mass limit

TL;DR

This paper demonstrates that the spectrum of a three-dimensional Dirac operator with a large mass outside a bounded region approximates the spectrum of the Dirac operator with MIT bag boundary conditions inside the region, with precise error estimates.

Contribution

It provides a rigorous approximation of the Dirac operator's spectrum in the infinite mass limit using boundary conditions and tubular coordinate analysis.

Findings

Spectrum approximation with error o(1/√m)

Use of tubular coordinates near boundary

Analysis of 1D optimization problems in normal direction

Abstract

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