Comparison theorems for the position-dependent mass Schrödinger equation. (English) Zbl 1238.81099
Summary: 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, \(m_0 \leq m(x)\) 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.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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