Reconstruction techniques for inverse Sturm-Liouville problems with complex coefficients

TL;DR

This paper introduces a unified, direct numerical method based on Neumann series of Bessel functions for solving various inverse Sturm-Liouville problems with complex potentials, including boundary condition recovery.

Contribution

It develops a novel, unified approach that efficiently recovers complex potentials and boundary conditions using NSBF representations, with built-in accuracy control.

Findings

Method accurately recovers complex potentials

Efficiently determines boundary conditions

Numerical tests demonstrate high accuracy and efficiency

Abstract

A variety of inverse Sturm-Liouville problems is considered, including the two-spectrum inverse problem, the problem of recovering the potential from the Weyl function, as well as the recovery from the spectral function. In all cases the potential in the Sturm-Liouville equation is assumed to be complex valued. A unified approach for the approximate solution of the inverse Sturm-Liouville problems is developed, based on Neumann series of Bessel functions (NSBF) representations for solutions and their derivatives. Unlike most existing approaches, it allows one to recover not only the complex-valued potential but also the boundary conditions of the Sturm-Liouville problem. Efficient accuracy control is implemented. The numerical method is direct. It involves only solving linear systems of algebraic equations for the coefficients of the NSBF representations, while eventually the knowledge only of the first NSBF coefficients leads to the recovery of the Sturm-Liouville problem. Numerical efficiency is illustrated by several test examples.