Comparison Theorems for the Position-Dependent Mass Schrödinger Equation

Abstract

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established. (i) If a constant mass m 0 and a PDM m ( x ) are ordered everywhere, that is either, m 0 ≤ m ( x ) or m 0 ≥ 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 ∇ 2 ( 1 / m ( x ) ) has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.