One-Dimensional Tunnel-Junction Formula for Schrodinger Particle

TL;DR

This paper classifies all boundary conditions for a one-dimensional Schrödinger operator with a junction, deriving a tunnel-junction formula that suggests potential applications in quantum computing devices.

Contribution

It provides a comprehensive analysis of self-adjoint extensions for the Schrödinger operator in a junction setting and introduces a tunnel-junction formula for non-relativistic electrons.

Findings

Derived a tunnel-junction formula for Schrödinger particles

Classified all boundary conditions for self-adjoint extensions

Proposed a mathematical model for a qubit tunnel-junction device

Abstract

We handle all the self-adjoint extensions of the minimal Schroedinger operator for the non-relativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the non-relativistic electron. We find a tunnel-junction formula for the non-relativistic electron passing through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.