Inverse problem for singular diffusion operator

TL;DR

This paper investigates the inverse problem for a singular diffusion operator with jump conditions, deriving integral representations, analyzing eigenproperties, and reconstructing the operator using the Weyl function.

Contribution

It introduces a method to reconstruct the singular diffusion operator from the Weyl function, including eigenvalue and eigenfunction asymptotics.

Findings

Eigenvalues and eigenfunctions analyzed and their properties established.

Asymptotic formulas for eigenvalues and eigenfunctions derived.

Reconstruction of the operator demonstrated via the Weyl function.

Abstract

In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions are investigated. Asymtotic representation of eigenvalues and eigenfunction have been obtained. Reconstruction of the singular diffusion operator have been shown by the Weyl function.