De Rham-Hodge-Skrypnik theory. 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 problems. Part 1

TL;DR

This paper surveys the spectral and differential-geometric properties of Delsarte transmutation operators in multiple dimensions, exploring their connections with De Rham-Hodge-Skrypnik theory and applications to integrable systems.

Contribution

It provides a comprehensive review of the multidimensional De Rham-Hodge-Skrypnik theory and its relation to Delsarte transmutation operators, highlighting new geometric and topological insights.

Findings

Analysis of the geometric and topological structure of transmutation operators

Relationships established between transmutation operators and generalized differential complexes

Applications demonstrated in multidimensional integrable dynamical systems

Abstract

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.