Bogoliubov de Gennes equation on metric graphs

TL;DR

This paper develops a mathematical framework for the Bogoliubov de Gennes equation on metric graphs, deriving boundary conditions, quantization conditions, and transmission matrices, with applications to Majorana wire networks.

Contribution

It introduces self-adjoint vertex boundary conditions for the Bogoliubov de Gennes operator on metric graphs, enabling analysis of quantum states and transport.

Findings

Derived vertex boundary conditions for self-adjointness.

Obtained secular equation for energy quantization.

Discussed applications to Majorana wire networks.

Abstract

We consider Bogoliubov de Gennes equation on metric graphs. The vertex boundary conditions providing self-adjoint realization of the Bogoliubov de Gennes operator on a metric star graph are derived. Secular equation providing quantization of the energy and the vertex transmission matrix are also obtained. Application of the model for Majorana wire networks is discussed.