Opuscula Mathematica
Opuscula Math. 30, no. 3 (2010), 295-309

Geometric properties of quantum graphs and vertex scattering matrices

Pavel Kurasov
Marlena Nowaczyk
Abstract. Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph.
Keywords: scattering theory, quantum graphs, matching (boundary) conditions.
Mathematics Subject Classification: 35R30, 47A10, 81U40, 81Q10.
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About the authors ▼
Pavel Kurasov
LTH, Lund University Dept. of Mathematics, Box 118, 221 00 Lund, Sweden; Stockholm Univ., Dept. of Mathematics, 106 91 Stockholm, Sweden; St. Petersburg Univ., Institute of Physics, 198904 St. Petersburg, Russia
Marlena Nowaczyk
Institute of Mathematics, PAN ul. Sw.Tomasza 30, 31-027 Kraków, Poland
About the article ▼
Received: 2010-02-08.
Accepted: 2010-02-21.
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Pavel Kurasov, Marlena Nowaczyk, Geometric properties of quantum graphs and vertex scattering matrices, Opuscula Math. 30, no. 3 (2010), 295-309, http://dx.doi.org/10.7494/OpMath.2010.30.3.295.
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