DARBOUX TRANSFORMATION, FACTORIZATION, AND SUPERSYMMETRY IN ONE-DIMENSIONAL QUANTUM-MECHANICS

We introduce an N-order Darboux transformation operator as a particula r case of general transformation operators. It is shown that this oper ator can always be represented as a product of N first-order Darboux: transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators a re introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of ord er N. For N=2, we have a quadratic superalgebra analogous to the Sklya nin quadratic algebras. The relationship between the transformation in troduced and the inverse scattering problem in quantum mechanics is es tablished. An elementary N-parametric potential that has exactly N pre determined discrete spectrum levels is constructed. The paper conclude s with some examples of new exactly soluble potentials.