Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schr\"odinger operator in one dimension

TL;DR

This paper explores how specific boundary conditions for the one-dimensional Schrödinger operator can be physically realized through position-dependent effective mass profiles, revealing novel effects like quantized magnetic flux and Zeeman-like splitting.

Contribution

It introduces mass-jump and mass-bump boundary conditions for singular self-adjoint extensions and demonstrates their physical realization in systems with position-dependent mass profiles.

Findings

Mass-jump boundary conditions can produce quantized magnetic flux.

Zeeman-like splitting occurs for states with opposite angular momentum projections.

Different effective mass inhomogeneity profiles lead to distinct boundary conditions.

Abstract

Physical realizations of non-standard singular self-adjoint extensions for one-dimensional Schr\"odinger operator in terms of the mass-jump are considered. It is shown that corresponding boundary conditions can be realized for the Hamiltonian with the position-dependent effective mass in two qualitatively different profiles of the effective mass inhomogeneity: the mass-jump and the mass-bump. The existence of quantized magnetic flux in a case of the mass-jump is proven by explicit demonstration of the Zeeman-like splitting for states with the opposite projections of angular momentum.