Physics Letters A 349 (2006) 297–301 www.elsevier.com/locate/pla On some classes of exactly-solvable Klein–Gordon equations A. de Souza Dutra a,b,∗ , G. Chen c,d a Abdus Salam ICTP, Strada Costiera 11, 34014 Trieste, Italy b UNESP-Campus de Guaratinguetá-DFQ 1 , Av. Dr. Ariberto Pereira da Cunha, 333 C.P. 205, 12516-410 Guaratinguetá, SP, Brazil c Department of Physics, Shaoxing College of Arts and Sciences, Shaoxing 312000, PR China d Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, PR China Received 29 July 2005; received in revised form 20 September 2005; accepted 22 September 2005 Available online 3 October 2005 Communicated by R. Wu Abstract In this work we discuss some exactly solvable Klein–Gordon equations. We basically discuss the existence of classes of potentials with different nonrelativistic limits, but which shares the intermediate effective Schroedinger differential equation. We comment about the possible use of relativistic exact solutions as approximations for nonrelativistic inexact potentials.  2005 Elsevier B.V. All rights reserved. PACS: 03.65.Ge; 03.65.Pm As a consequence of the physical importance of exact solu- tions of relativistic equations for the study of systems under the influence of strong potentials, in nuclear physics and other ar- eas, an increasing interest in the study of the Klein–Gordon and Dirac equations has appeared in the last few years [1–13]. How- ever, as asserted by one of the authors in a recent paper [6], it is remarkable that in the most part of the works in this area, the scalar and vector potentials are equal. In some few other cases [2,5–7], it is considered the case where the scalar potential is greater than the vector potential (in order to guarantee the ex- istence of bound states). Here we report an extension of this idea to a larger class of potentials where, particularly, we deal with systems with arbitrary angular momentum quantum num- bers, in contrast with the majority of the cases considered in the literature which, as far as we know, are restricted to calculate the exact solutions for the s -wave states. Finally, we trace some comments about the possibility of using the exact relativistic solutions, in order to get an analytical approximation for the inexact nonrelativistic potentials, and this becomes more inter- * Corresponding author. E-mail addresses: dutra@feg.unesp.br (A. de Souza Dutra), chengang@zscas.edu.cn (G. Chen). 1 Permanent institution. esting if one consider that usually the Schroedinger equation to be solved is for fermionic particles and that, on the other hand, its relativistic counterpart is the Dirac equation. So, the idea is to use the bosonic particle Klein–Gordon equation as a mathematical tool to reach the goal of obtaining approximate solutions for nonrelativistic fermionic particles eigenstates. In this work we note that a more general imposition about the relationship between the vector and scalar potentials could be done by choosing (1) V(r) ≡ V 0 + βS(r), with V 0 and β being arbitrary constants and, usually it is re- quired that S>V in order to grant the existence of bound states. The case where these potentials are such that S 2 = V 2 must be considered separately as we will see along this work. It is interesting to note that this restriction includes the case where V(r) = 0 when both constants vanish, the situation where the potentials are equal (V 0 = 0; β = 1), and also the case where the potentials are proportional [6] when V 0 = 0. Substituting this constraint among the potentials into the time-independent Klein–Gordon equation for a s -wave ( ¯ h = c = 1), (2) − d 2 u(r) dr 2 + ( m + S(r) ) 2 u(r) = ( E − V(r) ) 2 u(r), 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.09.056