Quantum graph vertices with permutation-symmetric scattering probabilities

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix U. Observing that if U has at most two eigenvalues, then the scattering matrix S(k) of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta k, the transmission probability from the j-th edge to l-th edge is independent of (j, l), and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the concept of generalized delta- and delta'-couplings. (C) 2011 Elsevier B.V. All rights reserved.