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

Abstract

AbstractWe 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 H1‐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.