Bound states in the Kratzer plus polynomial potentials and their new exact tractability via nonlinear algebraic equations

Schrödinger equation with potentials of the Kratzer plus polynomial type (say, quartic V (r) = Ar +B r + C r +D r + F/r +G/r etc) is considered and a new method of exact construction of some of its bound states is presented. Our approach is made feasible via a combination of the traditional use of the infinite series ψ(r) (terminated rigorously after N+1 terms at certain specific couplings and energies) with several new ideas. We proceed in two steps. Firstly, in the strongcoupling regime with G → ∞, we find the exact, complete and compact unperturbed solution of our N + 2 coupled and nonlinear algebraic conditions of the termination. Secondly, we adapt the current Rayleigh-Schrödinger perturbation theory to our nonlinear equations and define the general G < ∞ bound states via an innovated, triple perturbation series. In its tests we show how all the corrections appear in integer arithmetics and remain, therefore, exact.