On the complexity of the inverse Sturm-Liouville problem

TL;DR

This paper investigates the computational complexity of solving the inverse Sturm-Liouville problem with Robin boundary conditions, establishing conditions for finite or single-limit solutions and providing explicit algorithms with error control.

Contribution

It characterizes the number of limits needed for algorithms to recover potentials and boundary conditions, and offers explicit algorithms with error bounds based on the problem's spectral data.

Findings

Finite limits suffice when eigenvalues match zero potential for all but finitely many.

Single limit is required otherwise, with error control possible under certain conditions.

Explicit algorithms with numerical examples are provided.

Abstract

This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples.