A Survey of the Spectral and Differential Geometric Aspects of the De Rham-hodge-skrypnik Theory Related with Delsarte Transmutation Operators in Multidimension and Its Applications to Spectral and Soliton

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions 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 multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual YangMills eqautions. The soliton like solutions to the related set of nonlinear dynamical systemare discussed. 1. De Rham-Hodge-Skrypnik theory aspects and related Delsarte-Darboux type binary transformations 1.1 A differential-geometric analysis of Delsarte-Darboux type transformations that was done in Chapter 3 of Part 1 for differential operator expressions acting in a functional space H = L1(T;H), where T = R 2 and H := L2(R ;C), appears to have a deep relationship with classical de Rham-Hodge-Skrypnik theory [3, 4, 5, 6] devised in the midst of the past century for a set of commuting differential operators defined, in general on a smooth compactm-dimensional metric space M. Concerning our problem of describing the spectral structure of Delsarte-Darboux type transmutaions acting in H, we preliminarily consider following Part 1 some backgrounds of the de-Rham-Hodge-Skrypnik differential complex theory devised for studying transformations of differential operators. Consider a smooth metric space M being a suitably compactified form of the space R, m ∈ Z+. Then one can define on MT := T ×M the standard Grassmann algebra Λ(MT;H) of differential forms on T×M and consider a generalized external I.V. Skrypnik [3, 4] anti-differentiation operator dL : Λ(MT;H) → Λ(MT;H) acting as follows: for 1991 Mathematics Subject Classification. Primary 34A30, 34B05 Secondary 34B15 PACS: 02.30.Jr, 02.30.Uu, 02.30.Zz, 02.40.Sf.