Polynomial Solutions of the Schrödinger Equation for the “Deformed” Hyperbolic Potentials by Nikiforov–Uvarov Method

Abstract

In this study, the bound state energy eigenvalues and the corresponding wave functions of the deformed Po schlÈTeller and Hyperbolic Kratzer-like potentials have been obtained by the NikiforovÈUvarov method using the deformed hyperbolic functions (sinh q (x) 4 1 2 (ex [ qe~x), cosh q (x) 4 and that 1 2(ex ] qe~x), sech q (x) 4 1/cosh q (x) tanh q (x) 4 sinh q (x)/cosh q (x)) were introduced for the Ðrst time by Arai [J. Math. Anal. Appl. 158, 63 (1991)]. It is also observed that, the energy eigenvalues of these ""newÏÏ type deformed hyperbolic potentials bring up the criteria for the shape invariance of the potentials according to the deformation parameter q.

The Klein–Gordon equation of generalized Hulthén potential in complex quantum mechanics

“…We shall study the time-independent KG equation with a family of exponential potentials, (17) which is called generalized Hulthen potential [25]. We have to note that, for some specific q this potential reduces to the well-known types : such as for 0 = q , to the exponential potential;…”

Exact solution of Schrödinger equation with deformed ring-shaped potential

“…The method is based on the solution of differential equation transformed into the hypergeometric type [21,22].…”

Analytical solution of the local fractional Klein--Gordon equation for generalized Hulthen potential

“…Then the reduced equation, which is called the basic equation of the method, can be solved systematically by means of special orthogonal functions and eigenstate solutions can be achieved completely [20][21][22][23].…”