2011/08/13 by D. A. Kulikov
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #quant-ph
paper · pdf · doi:10.5402/2012/461452
published as ISRN Mathematical Physics, Vol. 2012 (2012) 461452 · 11 pages, 2 figures
arxiv created 2011/08/13 · openalex publication_date 2011/12/06 · arxiv updated 2012/06/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established. (i) If a constant mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> and a PDM <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtext mathvariant="bold">x</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> are ordered everywhere, that is either, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtext mathvariant="bold">x</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtext mathvariant="bold">x</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>∇</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtext mathvariant="bold">x</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.