Soft-core Coulomb potentials and Heun’s differential equation

Abstract

Schrödinger's equation with the attractive potential V (r) = −Z/(r q + β q ) 1 q , Z > 0, β > 0, q ≥ 1, is shown, for general values of the parameters Z and β, to be reducible to the confluent Heun equation in the case q = 1, and to the generalized Heun equation in case q = 2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases this factor becomes a polynomial. The Asymptotic Iteration Method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detail solutions for both cases are provided.

Thed-dimensional softcore Coulomb potential and the generalized confluent Heun equation

“…The relation between Schrödinger's equation (2) for ≠ 0 and the confluent Heun equation was pointed earlier by H. Exton [1], (see also [2]). In the present work we study this connection in more detail.…”

Polynomial solutions of certain differential equations arising in physics

“…In this section, we illustrate the effectiveness of the algorithmic approach of Section 2 by applying it to different types of differential equations. Examples include the general Heun's equation as well as some physically significant differential equations that arise in the study of solutions to Schrödinger equation and radial Schrödinger equation with shifted potential .…”

“…Nevertheless, for any given n and a , the algorithmic procedure of Proposition 4 can be easily implemented to determine ε , f for which ODE admits polynomial solutions of degree n as well as to compute the corresponding polynomial solution. As an illustration, if we take and look for a solution of degree n = 6 then the implementation of Proposition 4 determines ε = 15, f = 3(750) 1 ∕ 4 and computes the corresponding polynomial solution of degree 6 of ODE as Some other lower degree solutions are displayed in Table . Example In this example, we consider a differential equation related to the investigation of the radial Schrödinger equation with shifted Coulomb potential and which has been discussed recently in . The anstaz of ,Eq.35 that the radial Schrödinger equation admits a solution which vani...…”

“…In this section, we illustrate the effectiveness of the algorithmic approach of Section 2 by applying it to different types of differential equations. Examples include the general Heun's equation as well as some physically significant differential equations that arise in the study of solutions to Schrödinger equation [2,13] and radial Schrödinger equation with shifted potential [2,14,15]. These examples show how to implement the method algorithmically to determine the conditions for the existence of polynomial solutions and also to calculate the corresponding polynomial solutions.…”

“…In this example, we consider a differential equation related to the investigation of the radial Schrödinger equation with shifted Coulomb potential and which has been discussed recently in [2,14,15]. The anstaz of [2,Eq.35] that the radial Schrödinger equation admits a solution which vanishes at the origin and at infinity leads to the question of obtaining solutions of the following differential equation:…”