THEORY OF MULTIDIMENSIONAL DELSARTE-LIONS TRANSMUTATION OPERATORS. II

摘要

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand-Levitan-Marchenko equations that describe these operators are studied by using suitable differential de Rham-Hodge-Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang-Mills equations. The soliton solutions are discussed for a certain class of dynamical systems.