Klein-Gordon Potentials Solvable in Terms of the General Heun Functions

We study the potentials for which the one-dimensional stationary Klein-Gordon equation can be solved in terms of the general Heun function, a special function of the new generation that generalizes the ordinary hypergeometric function. We reveal that, under the assumption of independent variability of all potential parameters, there are only 35 possible forms of the independent variable transformation that allow reduction of the Klein-Gordon equation to the general Heun equation, each of these transformations generating an unconditionally exactly solvable potential. Due to the symmetry of the Heun equation with respect to the transposition of its singularities, only 11 out of 35 admissible potentials are independent; the others can be derived from these 11 by specifying the involved parameters to particular (real or complex) values. Since one of these latter potentials is a constant, there are just 10 nontrivial independent potentials. Four of these potentials are six-parametric; the other six are five-parametric. Four of the potentials possess ordinary hypergeometric sub-potentials. We present explicit solutions of the Klein-Gordon equation using the general Heun functions for these 10 independent potentials, and provide a comprehensive representation of the involved parameters.