Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere

Abstract

We consider a family of extensions of the Kepler-Coulomb potential on a ddimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function method, wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W + (r) [and its accompanying function W − (r)]. From the results obtained for the latter for the first three family members, we propose some formulas for W ± (r) valid for the mth family member, depending on m + 1 constants a 0 , a 1 , . . . , a m . Such constants satisfy a system of m + 1 linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the a i constants in terms of the parameters of the problem.