Quantum graph vertices with permutation-symmetric scattering probabilities

Abstract

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 $\mathcal{S}(k)$ 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 $\ell$-th edge is independent of $(j,\ell )$, 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 ${\delta }^{\prime }$ couplings.

9 pages, a few typographical errors corrected

References in corpus (7)

• Approximation of a general singular vertex coupling in quantum graphs
• Index theorems for quantum graphs
• Quantum graphs where back-scattering is prohibited
• Fulop-Tsutsui interactions on quantum graphs
• Spectral filtering in quantum Y-junction
• Hermitian unitary matrices with modular permutation symmetry
• Inverse scattering problem for quantum graph vertices