Physical interpretation of point-like interactions of one-dimensional Schr\"odinger operator

TL;DR

This paper provides a physical interpretation of specific boundary conditions in one-dimensional Schrödinger operators, linking them to mass-jumps and magnetic flux in nanostructures like Josephson junctions.

Contribution

It offers a novel physical interpretation of a peculiar self-adjoint extension with discontinuities, connecting it to mass-jumps and magnetic flux in quantum nanostructures.

Findings

Discontinuous boundary conditions correspond to mass-jumps and magnetic flux.

The interpretation applies to Josephson junctions with different effective masses.

This extends understanding of boundary conditions in quantum nanostructures.

Abstract

We consider physical interpretations of non-trivial boundary conditions of self-adjoint extensions for one-dimensional Schr\"odinger operator of free spinless particle. Despite its model and rather abstract character this question is worth of investigation due to application for one-dimensional nanostructures. The main result is the physical interpretation of peculiar self-adjoint extension with discontinuity of both the probability density and the derivative of the wave function. We show that this case differs very much from other three which were considered before and corresponds to the presence of mass-jump in a sense of works of Ganella et. al., (Journal of Physics A: Mathematical and Theoretical 42, 465207 (2009)) along with the quantized magnetic flux. Real physical system which can be modeled by such boundary conditions is the localized quantazied flux in the Josephson junction of two superconductors with different effective masses of the elementary excitations.