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On the complexity of the inverse Sturm–Liouville problem

Jonathan Ben-Artzi, Marco Marletta and Frank Rösler

Vol. 5 (2023), No. 4, 895–925
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.

Keywords
inverse Sturm–Liouville problem, solvability complexity index hierarchy, computational complexity
Mathematical Subject Classification
Primary: 34A55, 34B24, 65F18, 65L09, 68Q25
Milestones
Received: 29 September 2022
Revised: 19 February 2023
Accepted: 26 May 2023
Published: 15 December 2023
Authors
Jonathan Ben-Artzi
School of Mathematics
Cardiff University
Cardiff
United Kingdom
Marco Marletta
School of Mathematics
Cardiff University
Cardiff
United Kingdom
Frank Rösler
Mathematisches Institut
Universität Bern
Bern
Switzerland