Inverse Problems and Imaging (IPI)

Inverse problems for quantum trees II: Recovering matching conditions for star graphs

Pages: 579 - 598, Volume 4, Issue 4, November 2010      doi:10.3934/ipi.2010.4.579

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Sergei Avdonin - Department of Mathematics and Statistics, University of Alaska, Fairbanks, AK 99775-6660, United States (email)
Pavel Kurasov - Dept. of Mathematics, LTH, Lund Univ., Box 118, 221 00 Lund, Sweden (email)
Marlena Nowaczyk - Institute of Mathematics, PAN, ul. Św.Tomasza 30, 31-027 Kraków, Poland (email)

Abstract: The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.

Keywords:  quantum graphs, inverse problems, matching conditions.
Mathematics Subject Classification:  Primary: 81C05, 35R30, 35L05, 93B05, 49E15.

Received: November 2009;      Revised: May 2010;      Available Online: September 2010.