Self-adjointness for the MIT bag model on an unbounded cone

TL;DR

This paper proves the self-adjointness of the massless Dirac operator with MIT bag boundary conditions on convex unbounded cones, using spectral estimates and separation of variables, and extends results to non-convex cones and quantum dot conditions.

Contribution

It establishes self-adjointness for the Dirac operator on unbounded cones and provides a Hardy inequality, extending understanding of boundary conditions in quantum models.

Findings

Self-adjointness proven for convex cones using spectral estimates.

Numerical evidence suggests self-adjointness extends to non-convex cones.

Hardy inequality established, implying stability under potential perturbations.

Abstract

We consider the massless Dirac operator with the MIT bag boundary conditions on an unbounded three-dimensional circular cone. For convex cones, we prove that this operator is self-adjoint defined on four-component $H^1$--functions satisfying the MIT bag boundary conditions. The proof of this result relies on separation of variables and spectral estimates for one-dimensional fiber Dirac-type operators. Furthermore, we provide a numerical evidence for the self-adjointness on the same domain also for non-convex cones. Moreover, we prove a Hardy-type inequality for such a Dirac operator on convex cones, which, in particular, yields stability of self-adjointness under perturbations by a class of unbounded potentials. Further extensions of our results to Dirac operators with quantum dot boundary conditions are also discussed.