Comparison theorems for the position-dependent mass Schroedinger equation

TL;DR

This paper establishes comparison theorems for the eigenvalues of the position-dependent mass Schrödinger equation, providing conditions under which eigenvalues can be ordered based on mass distribution and ambiguity parameters.

Contribution

It introduces new comparison rules for the discrete spectrum of PDM Schrödinger equations, extending spectral ordering results to variable mass and ambiguity parameter scenarios.

Findings

Eigenvalues are ordered when PDM and constant mass are pointwise ordered.

Eigenvalues are ordered for different ambiguity parameters if the Laplacian of inverse mass has a definite sign.

The theorems are demonstrated with practical examples.

Abstract

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schroedinger equation are established. (i) If a constant mass $m_0$ and a PDM $m(x)$ are ordered everywhere, that is either $m_0\leq m(x)$ or $m_0\geq m(x)$, then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if $\nabla^2 (1/m(x))$ has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.