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 model. 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 shapeinvariant parameter.

A square potential well with position-dependent mass is studied for bound states. Applying appropriate matching conditions, a transcendental equation is derived for the energy eigenvalues. Numerical results are presented graphically and the variation of the energy of the bound states are calculated as a function of the well-width and mass.

We propose a new approach based on the algebraization of the Associated Lamé equationwithin sl(2,R) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitely obtained for integers m,ℓ. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer ℓ.

Inspired by the possibility of factorizing the second derivative interwining operators using a modified form of the well-known Crum–Darboux transformation, we present a scheme for generating a new pair of isospectral potentials. We also consider the interesting problem of constructing coherent states for such factorizable operators.

We obtain exact solutions of the one-dimensional Schrödinger equation for some families of associated Lamé potentials with arbitrary energy through a suitable ansatz, which may be appropriately extended for other such a families. The formalism of supersymmetric quantum mechanics is used to generate new exactly solvable potentials.