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 $tauinmathbb{R}$; 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.
Download