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Partial inverse problems for Sturm–Liouville operators on trees

Part of: Ordinary differential operators Operator theory: Ordinary differential operators Boundary value problems General theory in ordinary differential equations

Published online by Cambridge University Press:  15 August 2017

Natalia Bondarenko  and
Chung-Tsun Shieh
Natalia Bondarenko
Affiliation:
Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia ([email protected]) and Department of Applied Mathematics, Samara National Research University, Moskovskoye Shosse 34, Samara 443086, Russia
Chung-Tsun Shieh
Affiliation:
Department of Mathematics, Tamkang University, 151 Ying-chuan Road, Tamsui, Taipei County, Taiwan 25137, People's Republic of China ([email protected])
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Extract

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

Keywords

quantum graphsSturm–Liouville operatorsinverse spectral problemsmethod of spectral mappings

MSC classification

Secondary: 34A55: Inverse problems 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47) 34B45: Boundary value problems on graphs and networks 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 5 , October 2017 , pp. 917 - 933
Copyright
Copyright © Royal Society of Edinburgh 2017 

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