Theory of Multidimensional Delsarte–Lions Transmutation Operators. I

We present a brief survey of the original results obtained by the authors in the theory of Delsarte–Lions transmutations of multidimensional spectral differential operators based on the classical works by Yu. M. Berezansky, V. A. Marchenko, B. M. Levitan, and R. G. Newton, on the well-known L. D. Faddeev’s survey, the book by L. P. Nyzhnyk, and the generalized de Rham–Hodge theory suggested by I. V. Skrypnik and developed by the authors for the differential-operator complexes. The operator structure of the Delsarte–Lions transformations and the properties of their Volterra factorizations are analyzed in detail. In particular, we study the differential-ga generalized de Rham–Hodge theory.