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

## TL;DR

This paper analyzes the self-adjoint extensions of a two-valley Dirac operator with infinite mass boundary conditions on a wedge, revealing the mathematical structure and physical implications of valley mixing.

## Contribution

It provides a complete parametrization of all self-adjoint extensions of the two-valley Dirac operator with specific boundary conditions, highlighting the impossibility of valley decomposition.

## Key findings

- Deficiency indices of the operator are (1,1).
- Explicit defect element is computed.
- No self-adjoint extension decomposes into separate valleys.

## 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.