The Structure of Gelfand-Levitan-Marchenko Type Equations for Delsarte Transmutation Operators of Linear Multi-Dimensional Differential Operators and Operator Pencils. Part 2

The differential-geometric and topological structure of Delsarte transmutation operators their associated Gelfand-Levitan-Marchenko type equations are studied making use of the 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 the three-dimensional Laplace operator and the two-dimensional classical Dirac operator and its multi-dimensional affine extension, related with self-dual Yang-Mills equations. The soliton like solutions to the related set of nonlinear dynamical systems are discussed.