On some polynomial potentials in d-dimensions

The d-dimensional Schrödinger's equation is analyzed with regard to the existence of exact solutions for polynomial potentials. Under certain conditions on the interaction parameters, we show that the polynomial potentials V8(r) = ∑k = 18αkrk,α8>0 and V10(r) = ∑k = 110αkrk,α10>0 are exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for the existence of these exact solutions are discussed. Finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.