Applied Mathematics and Computation 274 (2016) 531–538 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Klein–Gordon particles in symmetrical well potential Nuray Candemir ∗ Department of Physics, Anadolu University 26470, Eski ¸ sehir, Turkey article info MSC: 03.65.Pm 03.65.Ge 03.65.Fd Keywords: Klein–Gordon equation Symmetrical well potential Approximation schemes Nikiforov–Uvarov method abstract An approximate analytical solution of the Klein–Gordon equation for the equal scalar and vec- tor symmetrical well potential is presented in this study. The energy eigenvalue equation and corresponding wave functions for arbitrary n and l quantum states are obtained using the Nikiforov–Uvarov method in the closed form. Some special cases are also investigated. © 2015 Elsevier Inc. All rights reserved. 1. Introduction The analytical solutions of the Klein–Gordon equation for a given potential model have received much attention in nuclear and high energy physics [1–5]. There have been continuous research efforts in finding solutions of this equation for various potentials using different methods and techniques, e.g., supersymmetric quantum mechanics (SUSYQM) [6], quasilinearization method (QLM) [7,8], asymptotic iteration method (AIM) [9,10], Nikiforov–Uvarov (NU) method [11–21], so on [22,23]. In most of the works, the analytical solution of the Klein–Gordon equation is examined by equal scalar S(r) and vector V(r) potentials [9–12,24–26]. Moreover, it is considered in the case of the unequal scalar potential S(r) and vector V(r) potentials [27–31]. The equal scalar S(r) and vector V(r) coupling potentials have been mainly used in studies by two reasons. First, for the case S(r) ≥ V(r) there exist bound state solutions for a relativistic spin-zero particle. Second, it simplifies the solution of the relativistic problem. Particularly, the wave equation is reduced to a second-order differential equation. Afterwards, the problem can be solved using a variety of well known analytic tools and techniques [32,33]. It is impossible to reach an exact solution of the Klein–Gordon equation in most potentials for the l = 0 states. Therefore, one has to resort to some approximation scheme [34,35] which deal with the orbital centrifugal term, such as the Pekeris [36], the Greene and Aldrich [37,38]. The symmetrical well, which can be used to describe the vibration of polyatomic molecules, is an important potential. The potential was proposed by Büyükkılıç et al. [39]. They obtained the bound state energy and corresponding wave function for the Schrödinger equation by using the NU method. They also analyzed the ammonia molecule (NH 3 ) under this potential. Zhao et al. found the s-wave solutions of Klein–Gordon equation and Dirac equation for generalized symmetrical well potential within the framework of supersymmetric quantum mechanics, using the shape invariance method [40]. Setare and Haidari determined the bound state Schrödinger equation in generalized symmetrical well potential [41]. They also obtained the ladder operators for this potential with the factorization method. Wei et al. determined the energy eigenvalues and the corresponding wave functions of Dirac equation for this potential, assuming pseudospin symmetry and spin symmetry, and by using the Greene and Aldrich approximation [42,43]. On the other hand, Klein–Gordon equation has not been studied with the symmetrical well potential ∗ Tel.: +90 222 3350580; fax: +90 222 3204910. E-mail address: ncandemi@anadolu.edu.tr http://dx.doi.org/10.1016/j.amc.2015.11.031 0096-3003/© 2015 Elsevier Inc. All rights reserved.