The spectral and differential geometric aspects of a generalized De Rham-Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems

A review on spectral and differential-geometric properties of Delsarte transmutation operators in multidimension is given. Their differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand–Levitan–Marchenko type equations are studied making use of the generalized de Rham–Hodge–Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmutated operators are stated. Some applications to integrable dynamical systems theory in multidimension are presented.