Essential self-adjointness of symmetric first-order differential systems and confinement of Dirac particles on bounded domains in $\mathbb{R}^d$

TL;DR

This paper establishes conditions under which Dirac operators with certain potentials are essentially self-adjoint, ensuring well-defined quantum dynamics and confinement of particles in bounded domains.

Contribution

It introduces new criteria for essential self-adjointness of symmetric first-order systems and Dirac operators, including growth conditions on potentials and magnetic fields.

Findings

Dirac operators with rapidly growing potentials are essentially self-adjoint.

A new class of scalar potentials guarantees self-adjointness of first-order systems.

Magnetic fields growing faster than a specific rate confine Dirac particles within domains.

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 $1/\big(2\text{dist} (x, \partial\Omega)^2\big)$.