Solutions of the Bogoliubov-de Gennes equation with position dependent Fermi--velocity and gap profiles

TL;DR

This paper investigates bound and scattering states in the one-dimensional Bogoliubov-de Gennes equation with position-dependent Fermi velocity and gap profiles, revealing conditions for bound states, BIC-like solutions, and scattering phenomena.

Contribution

It introduces the analysis of the BdG equation with spatially varying Fermi velocity and gap, highlighting new bound and scattering solutions not previously characterized.

Findings

Bound state solutions exist with position-dependent Fermi velocity.

Presence of bound states in continuum (BIC)-like solutions in both phases.

Step-like profiles lead to scattering with normal reflection and transmission.

Abstract

It is shown that bound state solutions of the one dimensional Bogoliubov-de Gennes (BdG) equation may exist when the Fermi velocity becomes dependent on the space coordinate. The existence of bound states in continuum (BIC) like solutions has also been confirmed both in the normal phase as well as in the super-conducting phase. We also show that a combination of Fermi velocity and gap parameter step-like profiles provides scattering solutions with normal reflection and transmission.