Shape-invariant quantum Hamiltonian with position-dependent effective mass through second-order supersymmetry

Abstract

A second-order supersymmetric approach is taken to the system describing motion of a quantum particle in a potential endowed with position-dependent effective mass. It is shown that the intertwining relations between second-order partner Hamiltonians may be exploited to obtain a simple shape-invariant condition. Indeed, a novel relation between potential and mass functions is derived, which leads to a class of exactly solvable models. As an illustration of our procedure, two examples are given for which one obtains whole spectra algebraically. Both shape-invariant potentials exhibit harmonic-oscillator-like or singular-oscillator-like spectra depending on the values of the shape-invariant parameter.