The structure of Gelfand-Levitan-Marchenko type equations for Delsarte transmutation operators of linear multi-dimensional differential operators and operator pencils. Part 1

TL;DR

This paper develops a new integral equation framework for multi-dimensional Delsarte transmutation operators, revealing their differential-geometric structure and extending the method to affine pencils of differential operators.

Contribution

It introduces an analog of Gelfand-Levitan-Marchenko equations for multi-dimensional operators and extends the approach to affine pencils, advancing the theoretical understanding.

Findings

Constructed integral equations for Delsarte transmutation operators.

Revealed the differential-geometric structure of these operators.

Extended the method to affine pencils of differential operators.

Abstract

An analog of Gelfand-Levitan-Marchenko integral equations for multi- dimensional Delsarte transmutation operators is constructed by means of studying their differential-geometric structure based on the classical Lagrange identity for a formally conjugated pair of differential operators. An extension of the method for the case of affine pencils of differential operators is suggested.