Dirac operators with infinite mass boundary conditions on unbounded domains with infinite corners

Abstract

We investigate the self-adjointness of the two dimensional Dirac operator with infinite mass boundary conditions on an unbounded domain with an infinite number of corners. We prove that if the domain has no concave corners, then the operator is self-adjoint. On the other hand, when concave corners are present, the operator is no longer self-adjoint and self-adjoint extensions can be constructed. Among these, we characterize the distinguished extension as the unique one whose domain is included in the Sobolev space $H^{s}$, where $s>1/2$ depends on the amplitude of the corners. Lastly, we study the spectrum of this distinguished self-adjoint extension.