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ISOSPECTRALITY FOR GRAPH LAPLACIANS UNDER THE CHANGE OF COUPLING AT GRAPH VERTICES: NECESSARY AND SUFFICIENT CONDITIONS

Part of: General theory in ordinary differential equations General mathematical topics and methods in quantum theory General theory of linear operators

Published online by Cambridge University Press:  16 January 2015

Yulia Ershova ,
Irina I. Karpenko  and
Alexander V. Kiselev
Yulia Ershova
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska st. 3, Kiev-4, 01601, Ukraine email julija.ershova@gmail.com
Irina I. Karpenko
Affiliation:
Department of Algebra and Functional Analysis, V.I. Vernadsky Taurida National University, 4 Vernadsky pr., Autonomous Republic of Crimea, Simferopol, 95007, Ukraine email i_karpenko@ukr.net
Alexander V. Kiselev
Affiliation:
Department of Functional Analysis, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3-b Naukova Str., 79060 L’viv, Ukraine Department of Higher Mathematics and Mathematical Physics, St. Petersburg State University, 1 Ulianovskaya Street, St. Petersburg, St. Peterhoff, 198504, Russia email alexander.v.kiselev@gmail.com
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Abstract

Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of types ${\it\delta}$ and ${\it\delta}^{\prime }$. Assuming rational independence of edge lengths, necessary and sufficient conditions for isospectrality of two Laplacians defined on the same graph are derived and scrutinized. It is proved that the spectrum of a graph Laplacian uniquely determines matching conditions for “almost all” graphs.

MSC classification

Primary: 47A10: Spectrum, resolvent
Secondary: 34A55: Inverse problems 81Q35: Quantum mechanics on special spaces
Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016 , pp. 210 - 242
Copyright
Copyright © University College London 2015 

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