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On reflectionless equi-transmitting matrices

Kurasov P., Ogik R., Rauf A.

Opuscula Mathematica

2014

Vol. 34, no. 3

483--501

Reflectionless equi-transmitting unitary matrices are studied in connection to matching conditions in quantum graphs. All possible such matrices of size 6 are described explicitly. It is shown that such matrices form 30 six-parameter families intersected along 12 five-parameter families closely connected to conference matrices.

Geometric properties of quantum graphs and vertex scattering matrices

Kurasov P., Nowaczyk M.

Opuscula Mathematica

2010

Vol. 30, no. 3

295-309

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.