ar X iv : h ep - t h / 96 07 06 8 v 1 8 J ul 1 99 6 Remark on Shape Invariant Potential

The usual concept of shape invariance is discussed and one extension of this concept is suggested. Gedenshtein [1] defined the “shape invariant” potentials by the relationship V+(x; a0)− V−(x; a1) = W (x, a0) +W ′(x, a0)−W (x; a1) +W ′(x; a1) = R(a1) (1) where W (x; a) is the superpotential, a0 and a1 stand for parameters of the supersymmetric partner potentials V+ and V−, R(a) is a constant. The supersymmetric partners are related with the supersymmetric Hamiltonian in an usual way [2], V+ = W 2 −W ′ and V− = W +W . The relationship beetween shape invariance and solvable potentials is discussed by several authors (see, for instance, [3] and [4]). Other mathematical aspects of shape invariant potentials are also present in the literature, for example in the supersymmetric WKB approximation, [5], Berry phase, [6], and in the path-integral formulation, [7]. There is a general conclusion about these kind of potentials which is that the concept of shape invariance is a sufficient but not a necessary condition for the potential to become exactly solvable, [4]. In a recent work, [8], the Hulthén potential was studied from the Supersymmetric Quantum Mechanics formalism. This potential has an interesting property, that is when the angular momentum is zero, l = 0, it is not shape invariant in the sense expressed in ref.[1]. However, it is still possible to construct a general form of the potentials in the super-family of Hamiltonians: Vn(r)−E (n) 0 = W 2 n(r)− d dr Wn(r) = n(n− 1)δe 2(1− e−δr)2 − [n(1− n)δ + 2]δe 2(1− e) + 1 2 (− n 2 δ+ 1 n ). (2) Work partially supported by CNPq and FAPESP