The bound state energy eigenvalues and the corresponding wave functions of the “deformed” Rosen-Morse and modified Rosen-Morse potentials have been obtained by the Nikiforov-Uvarov method (Büyükkiliç et al 1997 Theor. Chim Acta. 98 192). The “deformed hyperbolic functions” that were introduced for the first time by Arai (1991 J. Math. Anal. Appl. 158 63) have been used. The energy eigenvalues of these “deformed hyperbolic molecular potentials” bring up the criteria for the shape invariance of the potentials according to the deformation parameter q, as well as bringing a bound for the value of the parameter.

We have investigated the reality of exact bound states of complex and/or PT-symmetric non-Hermitian exponential-type generalized Hulthen potential. The Klein-Gordon equation has been solved by using the Nikiforov-Uvarov method which is based on solving the secondorder linear differential equations by reduction to a generalized equation of hypergeometric type. In many cases of interest, negative and positive energy states have been discussed for different types of complex potentials.

The one-dimensional Dirac equation is solved for the PT-symmetric generalized Hulthén potential. The Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type is used to obtain exact energy eigenvalues and corresponding eigenfunctions.

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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.