Quantum graph vertices with permutation-symmetric scattering probabilities

TL;DR

This paper explores quantum graph vertices with special boundary conditions where the scattering matrix simplifies, leading to symmetric transmission probabilities and a classification of these couplings, including generalized delta types.

Contribution

It introduces a class of vertex couplings with permutation-symmetric scattering probabilities based on eigenvalue constraints of the boundary matrix.

Findings

Transmission probabilities are independent of edge pairs.

Reflection probabilities are equal across edges.

Classification of couplings includes generalized delta and delta' types.

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 $\delta'$ couplings.