The de Rham-Hodge-Skrypnik theory of Delsarte transmutation operators in multidimension and its applications

We study differential-geometric and topological structures related with Delsarte transmutations of multi-dimensional differential operators in Hilbert spaces. Based on the naturally defined de Rham-Hodge-Skrypnik differential complex the relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multi-dimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills equations. The soliton-like solutions to the related set of nonlinear dynamical systems are discussed.