Essential Self-adjointness of Symmetric First-Order Differential Systems and Confinement of Dirac Particles on Bounded Domains in $${\mathbb {R}}^d$$

Abstract

We prove essential self-adjointness of Dirac operators with Lorentz scalar potentials which grow sufficiently fast near the boundary $\partial\Omega$ of the spatial domain $\Omega\subset\mathbb R^d$. On the way, we first consider general symmetric first order differential systems, for which we identify a new, large class of potentials, called scalar potentials, ensuring essential self-adjointness. Furthermore, using the supersymmetric structure of the Dirac operator in the two dimensional case, we prove confinement of Dirac particles, i.e. essential self-adjointness of the operator, solely by magnetic fields $\mathcal{B}$ assumed to grow, near $\partial\Omega$, faster than $1/\big(2\text{dist} (x, \partial\Omega)^2\big)$.