Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. II: Proofs of theorems on reducibility

In this paper, we continue to study factorization of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with nonsingular coefficients. We define a class of potentials that are invariant under the Darboux-Crum transformations and prove a number of lemmas and theorems substantiating the formulated conjectures on reducibility of differential operators for spectral equivalence transformations. Analysis of the general case is performed with all the necessary proofs. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 214–237.