Recovering boundary conditions in inverse Sturm-Liouville problems

TL;DR

This paper presents a variational algorithm for solving inverse Sturm-Liouville problems that accurately recovers potentials and boundary conditions from spectral data, demonstrating robustness even with noisy inputs.

Contribution

It introduces a novel variational method that does not require prior knowledge of the potential's mean, improving the reliability of boundary condition recovery.

Findings

Algorithm successfully recovers potential and boundary conditions from two spectra

Works reliably even with noisy spectral data

No strict local minimizers ensure good convergence without initial guesses

Abstract

We introduce a variational algorithm, which solves the classical inverse Sturm-Liouville problem when two spectra are given. In contrast to other approaches, it recovers the potential as well as the boundary conditions without a priori knowledge of the mean of the potential. Numerical examples show that the algorithm works quite reliable, even in the presence of noise. A proof of the absence of strict local minimizers of the functional supports the observation, that a good initial guess is not essential.