Exactly solvable quantum potentials with special functions solutions

Abstract

AbstractIn this work, a canonical point and a gauge transformation applied to the g(x) times differential equations (DE) for special functions (SF) is used to convert it into a Schrödinger‐like equation. This method leads to exactly solvable potentials allowing special functions solutions. For specific applications, keeping in mind the general DE for SF, the choice of the coefficients that identify the SF under study allows to find the equivalent of the Witten superpotential and wavefunctions for each g(x). That is, the choice of a particular g(x) leads straightforwardly to those exactly solvable potentials having as wavefunctions the SF under consideration. As a useful application of the proposed method, we considered explicitly the potentials associated to the standard Hermite, Laguerre, Legendre, and Bessel SF in the case of some particular forms of g(x). However, the proposed approach can be directly applied to any other standard SF with the aim to obtain exactly solvable potentials that can be used in the modelling of quantum chemical applications. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008