A fast converging method for generating solutions to the Riccati equation H. Eleuch a,b,⇑ , H. Bahlouli c,d , Y.V. Rostovtsev e a Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montreal, QC H3C 3J7, Canada b Ecole Polytechnique, C.P. 6079, Montreal, QC H3C 3A7, Canada c Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia d Saudi Center for Theoretical Physics, Dhahran 31261, Saudi Arabia e Department of Physics, University of North Texas, 1155 Union Circle 311427, Denton, TX 76203-5017, USA article info Keywords: Riccati equation Nonlinear equations Schrödinger equation Supersymmetric quantum mechanics abstract We devise an approximate and fast converging iterative approach to solve the Riccati equa- tion. This solution provides a systematic way to generate isospectral potentials in quantum mechanics. The proposed solutions derived by our approach are compared with exact solu- tions for various potentials. Ó 2013 Elsevier Inc. All rights reserved. Since the advent of quantum mechanics, exactly solvable Schrödinger equations have been the subject of intense re- search. Exactly solvable potentials in non-relativistic quantum mechanics have more than a pedagogical value. The associ- ated physical quantities (e.g., eigenvalues, eigenfunctions, scattering phase shift, etc.) give insight into the physical concepts of quantum mechanics. In some cases, these potentials could provide good testing models for numerical routines or compu- tational algorithms designed to solve complicated systems. Hence the search for exactly solvable potentials has been a sub- ject of intense research [1,2]. Numerous methods have been devised for solving the stationary Schrödinger equation; however, most of these methods were based on symmetry considerations. The factorization method, originally introduced by Schrödinger [3,4], is the stan- dard method for solving quantum mechanical problems. However, historically, this factorization formalism for obtaining ex- act solutions for second order differential equations was achieved before the era of quantum mechanics by Darboux [5]. Based on the factorization method, Infeld and Hull have succeeded in classifying most of the exactly solvable Hamiltonians [6,7]. In recent years, there have been tremendous efforts in classifying all types of solvable quantum problems based on their symmetry. First, the idea of shape invariance played a major role in classifying exactly solvable non-relativistic quan- tum problems in distinct classes [8]. Second, new methods were developed to generate solutions of solvable models, the supersymmetric quantum mechanics approach (SUSY) is one of the most recent achievements in this field [9–11]. Later it was used to generate exactly solvable quantum mechanical problems in one dimension [8,12] and for problems in higher dimensions [13]. A comprehensive review on supersymmetric quantum mechanics was published in [14,15]. In our previous work, two of us have developed a new approach (ERS-method) for deriving an analytical solution for the Schrödinger equation for an arbitrary potential [16–21] and recently extended it to Dirac equation [22,23]. It was shown that the developed method gives better accuracy than the JWKB method. This approach was applied to the scattering problem. In this work we customize this approach to develop an approximate solution of the Riccati equation for the pseudo-potential, which allows us to generate the associated isospectral potentials in quantum mechanics. The necessary conditions for the validity of our approach and its fast convergence are explicitly investigated. 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.08.004 ⇑ Corresponding author at: Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montreal, QC H3C 3J7, Canada. E-mail address: heleuch@fulbrightmail.org (H. Eleuch). Applied Mathematics and Computation 222 (2013) 548–552 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc