ALTERNATIVE FACTORIZATION OF EIGENVALUE PROBLEMS IN ONE-DIMENSION

We propose an alternative factorization of eigenvalue problems in one dimension. The method is based on simple equations connecting a pair of solutions to two second-order differential equations that differ in the coefficient of the independent variable. Under certain conditions these connection equations play the role of recurrence relations. The method is particularly suitable for the treatment of separable quantum-mechanical problems giving rise to a consistency condition which tells us whether a potential is shape-invariant. From this consistency condition we derive a simple algorithm for the construction of partner potentials and shape-invariant potentials. The present connection method appears to be more general than both the standard factorization method and supersymmetric quantum mechanics. As illustrative examples we consider Bessel's and Legendre's equations, the generalized Kepler problem, inverse quadratic potentials and an asymmetric potential well.