Energy eigenfunctions for position-dependent mass particles in a new class of molecular hamiltonians

Abstract

Based on recent results on quasi-exactly solvable Schrodinger equations, we review a new phenomenological potential class lately reported. In the present paper we consider the quantum differential equations resulting from position dependent mass (PDM) particles. We focus on the PDM version of the hyperbolic potential $V(x)=a{\text{sech}}^{2}x+b{\text{sech}}^{4}x$, which we address analytically with no restrictions on the parameters and the energies. This is the celebrated Manning potential, a double-well widely known in molecular physics, until now not investigated for PDM. We also evaluate the PDM version of the sixth power hyperbolic potential $V(x)=a{\text{sech}}^{6}x+b{\text{sech}}^{4}x$ for which we could find exact expressions under some special settings. Finally, we address a triple-well case $V(x)=a{\text{sech}}^{6}x+b{\text{sech}}^{4}x+c{\text{sech}}^{2}x$ of particular interest for its connection to the new trends in atomtronics. The PDM Schrodinger equations studied in the present paper yield analytical eigenfunctions in terms of local Heun functions in its confluents forms. In all the cases PDM particles are more likely tunneling than ordinary ones. In addition, a merging of eigenstates has been observed when the mass becomes nonuniform.

31 pages, 17 figures, 4 tables. Some references corrected. Accepted for publication in J. Math. Phys

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• Energy eigenstates of position-dependent mass particles in a spherical quantum dot
• Three-dimensional bound states of cylindrical quantum heterostructures with position-dependent mass carriers
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