J. J. Peña

Generalization of the Darboux transform

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 ...

On the equivalence of radial potential models for diatomic molecules

AbstractIn this work, we show that various quantum interaction radial potential models used for describing diatomic molecules are particular cases of an exactly solvable multiparameter exponential-type potential. This feature is exploited to find the conditions that potentials should fulfill to be considered as equivalent. As an example of the usefulness of the proposal, the obtained requirements are applied to some of the most common diatomic potential models. Specifically, we have shown the equivalence among the radial potentials of Manning–Rosen, Deng–Fang, Schiöberg, Badawi–Bessis–Bessis, Tietz, Wie, Sun, the negative of the Williams–Poulios and the Möbius square potentials. The proposal to obtain equivalent radial potentials is general and can be directly applied to other interaction models as well as in the study of relativistic molecular physics.

ℓ-State Solutions of Multiparameter Exponential-type Potentials

In the present work, bound state solutions for a class of multiparameter exponential-type potential are obtained in the frame of the Greene and Aldrich approximation for the centrifugal term. The proposal is general and their usefulness is exemplified with the treatment of the Eckart, Manning-Rosen, Hulthen and Deng Fan potentials that are obtained straightforwardly without resorting to specialized methods of solution for each specific potential, as usually is done. Furthermore, the proposal admits other approximations for the centrifugal term indicating an improvement to procedures developed with the same objective. So, our proposal can be considered as an unified treatment of the l-state solutions for exponential-type potentials and can be used to find new solvable potentials.

A Lanczos Potential for Plane Gravitational Waves

We exhibit explicitly a Lanczos generator for the conformal tensor associated with plane gravitational waves.

The Darboux Transform Applied to Schrödinger Equations with a Position-Dependent Mass

Essentially, the Darboux proposition is based on the covariance properties of ordinary and partial differential equations with respect to a gauge transformation in the special case of second order differential equations of the Sturm- Liouville type. In this work, the one-dimensional Schrodinger equation with a position-dependent mass (SEPDM) is transformed into a Schrodinger-like equation with a position-independent mass (SLEPIM) for an effective potential which incorporates the spatially dependent mass. Therefore, taking advantage of the similarity between the SLEPIM and the Sturm-Liouville differential equation it is shown the application of the Darboux transform to the SEPDM problem.

Direct approach to bound-state solutions of the Yukawa potential

In this work, a straightforward approach to finding bound-state solutions of the Yukawa potential is presented. The proposal essentially converts the Schrödinger equation into a hypergeometric differential equation by means of a coordinate transformation together with a function transformation with the aim of finding the bound-state solutions of the multiparameter exponential-type potentials. The usefulness of the proposal is shown with the study of the bound-state solutions of the Yukawa potential in the frame of the Green and Aldrich approximation to the centrifugal term. Besides that the proposal is by far simpler than procedures developed with the same purpose, our algorithm accepts other kind of approximations to the 1/r2 term as well as the treatment of other specific exponential potentials, which can be obtained using a proper selection of the involved parameters. That is, instead of studying a given exponential-type potential with a specialised method, the energy spectra and wavefunctions are directly obtained as a particular case from the proposal.

Kratzer potential algebraic representation and matrix elements recurrence formulae

Generalized recurrence relations for the calculation of multipole matrix elements for Kratzer potential wave functions are obtained operationally. These formulas have been determined by using a non-analytical procedure based on the algebraic representation of the Kratzer eigenfunctions along with the usual ladder properties and commutation relations. For that, the creation and annihilation operators are adequately derived by means of an alternative approach to the factorization method and the exact expressions for matrix elements are achieved with the aid of a relationship between the ladder operators associated with the bra and theket. The proposed algebraic approach as well as the formulas for the calculation of matrix elements thus derived are quite simple and direct when compared with other alternative expressions already obtained analytically or pseudo-algebraically by means of the hypervirial theorem commutator algebra.

Bound state solutions of Dirac equation with radial exponential-type potentials

In this work, a direct approach for obtaining analytical bound state solutions of the Dirac equation for radial exponential-type potentials with spin and pseudospin symmetry conditions within the frame of the Green and Aldrich approximation to the centrifugal term is presented. The proposal is based on the relation existing between the Dirac equation and the exactly solvable Schrodinger equation for a class of multi-parameter exponential-type potential. The usefulness of the present approach is exemplified by considering some known specific exponential-type potentials which are obtained as particular cases from our proposal. That is, instead of solving the Dirac equation for a special exponential potential, by means of a specialized method, the energy spectra and wave functions are derived directly from the proposed approach. Beyond the applications considered in this work, our proposition could be used as an alternative way in the search of bound state solutions of the Dirac equation for other potentials...

Solvable rotating potentials and their generalizations

Abstract An alternative approach to find new solvable symmetric or non-symmetric radial potentials as well as their generalizations is presented. This is based on the application of the supersymmetry and shape invariance methods along with the generalized and standard Darboux transforms to Sturm–Liouville equations including the effective potential. As a useful application of the proposed approach, the procedure is used to obtain the well known isotonic and hydrogen-like potentials as well as to find the solvable rotating Morse and the Hulthen potentials that come from the corresponding Witten superpotentials leading to the Morse and the Hulthen models for s bounded states. In the hydrogen-like case, the results give rise to a justification for which the one-dimensional Coulomb potential should not exist strictly. In the Morse case, the resulting solvable radial potential for l≠0 corresponds to a linear combination of a Morse, a specific Yukawa and a specific hydrogenic potential. Similarly, the solvable rotating Hulthen potential found is given as a linear combination of a specific hydrogenic potential, with a term proportional to the potential 1/( r (exp(− Ar )−1)). For these new potentials the eigenfunctions, the partner isospectral potentials as well as the operators that factorize the associated Hamiltonians are also presented. The proposed method, can be advantageously applied for finding other new solvable rotating potentials that can be useful in the treatment of outstanding quantum mechanical applications.

On the q-deformed exponential-type potentials

In this work, both non-deformed and q-deformed exactly solvable multiparameter exponential-type potentials