Bound states and scattering coefficients of self-adjoint Hamiltonians with a mass jump

TL;DR

This paper analyzes self-adjoint Hamiltonians with a mass jump, deriving their spectra and boundary conditions, and applies these findings to model semiconductor heterojunctions within the effective-mass approximation.

Contribution

It provides a comprehensive characterization of self-adjoint extensions for Hamiltonians with a mass jump and links them to physical boundary conditions in semiconductor models.

Findings

Derived spectra and boundary conditions for Hamiltonians with mass jumps.

Connected self-adjoint extensions to physical models of heterojunctions.

Reviewed the case of no mass jump as a limiting scenario.

Abstract

Physical self-adjoint extensions and their spectra of the simplest one-dimensional Hamiltonian operator in which the mass is constant except for a finite jump at one point of the real axis are correctly found. Some self-adjoint extensions are used to model different kinds of semiconductor heterojunctions within the effective-mass approximation. Their properties and relation to different boundary conditions on envelope wave functions are studied. The limiting case of equal masses (with no mass jump) are reviewed.