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The M-matrix inverse problem for the Sturm—Liouville equation on graphs

Published online by Cambridge University Press:  08 July 2009

Sonja Currie
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, South Africa (sonja.currie@wits.ac.za)
Bruce A. Watson
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, South Africa (b.alastair.watson@gmail.com)

Abstract

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2009

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