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Abstract
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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.
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Keywords
inverse Sturm–Liouville problem, solvability complexity
index hierarchy, computational complexity
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Mathematical Subject Classification
Primary: 34A55, 34B24, 65F18, 65L09, 68Q25
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Milestones
Received: 29 September 2022
Revised: 19 February 2023
Accepted: 26 May 2023
Published: 15 December 2023
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© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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