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

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