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

A review on spectral and differential-geometric properties of Delsarte transmutation operators in multidimension is given. Their differential geometrical and topological structure in multidimension is analyzed, the relationships with De Rham-Hodge-Skrypnik theory of generalized differential complexes are stated. Some applications to integrable dynamical systems theory in multidimension are presented. 1. Spectral operators and generalized eigenfunctions expansions 1.1. Let H be a Hilbert space in which there is defined a linear closable operator L ∈ L(H) with a dense domain D(L) ⊂ H. Consider the standard quasi-nucleous Gelfand rigging [12] of this Hilbert space H with corresponding positive H+ and negative H− Hilbert spaces as follows: (1.1) D(L) ⊂ H+ ⊂ H ⊂ H− ⊂ D(L), being suitable for proper analyzing the spectral properties of the operator L in H. We shall use below the following definition motivated by considerations from [12], Section 5. Definition 1.1. An operator L ∈ L(H) will be called spectral if for all Borel subsets ∆ ⊂ σ(L) of the spectrum σ(L) ⊂ C and for all pairs (u, v) ∈ H+ × H+ there are defined the following expressions: