Self-adjoint extensions of the two-valley Dirac operator with discontinuous infinite mass boundary conditions

Abstract

We consider the four-component two-valley Dirac operator on a wedge in ${\mathbb{R}}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mixing the valleys.