Quantum graph vertices with permutation-symmetric scattering probabilities

Details Quantum graph vertices with permutation-symmetric scattering probabilities Quantum graph vertices with permutation-symmetric scattering probabilities

2011-08-03

arxiv.org/abs/1108.0856v2

Phys. Lett. A 375 (2011), 3775-3780

Physics

Mathematical Physics

9 pages, a few typographical errors corrected

Scientific paper

10.1016/j.physleta.2011.09.006

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 $\delta'$ couplings.

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