Eigenvalue Curves for Generalized MIT Bag Models

Abstract

We study spectral properties of Dirac operators on bounded domains Ω ⊂ R 3 with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter τ ∈ R; the case τ = 0 corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of τ , and we exploit this monotonicity to study the limits as τ → ±∞. We prove that if Ω is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all τ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as τ ↓ −∞, and we also analyze its first order asymptotics.