Bound state solutions of the Klein-Gordon equation under a non-central potential: the Eckart plus a ring-shaped potential

We report the bound state solutions of a novel non-central potential, the Eckart plus a ring-shaped potential, in the Klein-Gordon equation by using the Nikiforov-Uvarov method. We apply a developed scheme to overcome the centrifugal term. The energy eigenvalues and the corresponding normalised wave function of a mentioned system are presented in a closed and compact form. We also discuss various special cases related to our considered potential which are utility for other physical systems, consistent with previous studies. We then compute the eigenvalues and eigenfunctions numerically for arbitrary quantum numbers to show the accuracy of our results. It is shown that our results are sensitive to the parameter delta as well as the quantum numbers n(r) and N or l, as expected.