Opuscula Math. 30, no. 3 (2010), 295-309
http://dx.doi.org/10.7494/OpMath.2010.30.3.295
Opuscula Mathematica
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.
- 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
- Received: 2010-02-08.
- Accepted: 2010-02-21.
