Comparison Theorems for the Position-Dependent Mass Schrödinger Equation

Abstract EN

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established.

(i) If a constant mass m0 and a PDM m(x) are ordered everywhere, that is either, m0≤m(x) or m0≥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.