Reconstruction techniques for complex potentials

TL;DR

This paper introduces a novel numerical approach using Neumann series of Bessel functions to solve inverse coefficient problems for Sturm-Liouville equations with complex potentials, including various inverse problems.

Contribution

It presents a new method based on Bessel function series for reconstructing complex potentials, reducing inverse problems to linear algebra and enabling efficient numerical solutions.

Findings

Method effectively reconstructs complex potentials from spectral data.

Numerical examples demonstrate high efficiency and accuracy.

Applicable to multiple inverse Sturm-Liouville problems.

Abstract

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y={\lambda}y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectra Sturm-Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.