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We construct a singular oscillator Hamiltonian with a position-dependent effective mass. We find that an su(1, 1) algebra is the hidden symmetry of this quantum system and the isospectral potentials V(x) depend on the different choices of the m(x). The complete solutions are also presented by using this Lie algebra.

In this article, using an exactly‐solvable multiparameter exponential‐type potential we propose a unified treatment of the analytical bound—state solutions of the Schrödinger equation for exponential‐type potentials in D‐dimensions. Our proposal accepts different approximations to the centrifugal term; however, its usefulness is exemplified in the frame of the Green and Aldrich approach. This fact enables us to compare our results with specific potentials found in the literature and that are obtained here as particular cases of our proposal. That is, instead of solving a specific exponential‐type potential, by resorting each time to a specialized method, the energy spectra and wavefunctions are derived straightforward from the proposed approach. Furthermore, our proposal can be used as an alternative way in the search of solutions to new exponential‐type potentials besides that one can study different approximations to the term 1/r2. © 2014 Wiley Periodicals, Inc.

This article presents a generalization of the standard Darboux transform applied to Sturm–Liouville differential equations. This is achieved with the aid of an ansatz as a particular solution for the Riccati relationship involved, which in turn led us to obtain its generalized Darboux solution that contains, as a particular case, the standard Darboux transform. The proposed generalized Darboux transform (GDT), applied to the quantum mechanical field, gives the opportunity to prove the existence of standard and generalized Darboux potentials that match with the so-called isospectral potentials. This is exemplified by obtaining, through the GDT, a set of standard and generalized Darboux potentials that form the partner of the one-dimensional harmonic oscillator model for any quantum principal number. The worked example indicates how the GDT can be used to obtain the isospectral potentials associated to any known specific potential. We consider also the application of our method as proposed to the theory of solitons in order to show why the GDT will be important in other fields of application where the standard Darboux transform is usually concerned.

ABSTRACT:The solution to a spectral problem involving the Schrödinger equation for a particular class of multiparameter exponential-type potentials is presented. The proposal is based on the canonical transformation method applied to a general second-order differential equation, multiplied by a function g(x), to convert it into a Schrödinger-like equation. The treatment of multiparameter exponential-type potentials comes from the application of the transformed results to the hypergeometric equation under the assumption of a specific g(x). Besides presenting the explicit solutions and their spectral values, it is shown that the problem considered in this article unifies and generalizes several former studies. That is, the proposed exactly solvable multiparameter exponential-type potential can be straightforwardly applied to particular exponential potentials depending on the choice of the involved parameters as exemplified for the Hulthén potential and their isospectral partner. Moreover, depending on the function g(x), the proposal can be extended to find different exactly solvable potentials as well as to generate new potentials that could be useful in quantum chemical calculations.