Eigenvalue curves for generalized MIT bag models

Abstract

We study spectral properties of Dirac operators on bounded domains O¿R3 with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter t¿R; the case t=0 corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of t, and we exploit this monotonicity to study the limits as t¿±8. We prove that if O is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all t large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as t¿-8, and we also analyze its first order asymptotics.