Eigenvalue curves for generalized MIT bag models

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