Eigenvalue curves for generalized MIT bag models

Abstract

We study spectral properties of Dirac operators on bounded domains $\hat{I}\text{c◯}\subset {\mathbb{R}}^3$ with boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter $\ddot{I}„\in \mathbb{R}$; the case $\ddot{I}„=0$ corresponds to the MIT bag model. We show that the eigenvalues are parametrized as increasing functions of $\ddot{I}„$, and we exploit this monotonicity to study the limits as $\ddot{I}„\to \pm \infty$. We prove that if $\hat{I}\text{c◯}$ is not a ball then the first positive eigenvalue is greater than the one of a ball with the same volume for all $\ddot{I}„$ large enough. Moreover, we show that the first positive eigenvalue converges to the mass of the particle as $\ddot{I}„\downarrow -\infty$, and we also analyze its first order asymptotics.

49 pages, 5 figures. v2: version after referee report (Conjecture 1.8 and Remark 1.9 added) v3: Final version