Fulop-Tsutsui interactions on quantum graphs

TL;DR

This paper investigates Fulop-Tsutsui scale-invariant couplings on quantum graphs, showing how to replicate free coupling scattering amplitudes with specific subfamilies and proposing an approximation scheme using delta potentials.

Contribution

It identifies parameter subfamilies that mimic free coupling scattering amplitudes and develops an approximation method with delta potentials for general Fulop-Tsutsui vertices.

Findings

Same scattering amplitudes as free coupling achieved in specific subfamilies

Approximation scheme using only n delta potentials

Parameter subfamilies depend on the parity of n

Abstract

We examine scale invariant Fulop-Tsutsui couplings in a quantum vertex of a general degree $n$. We demonstrate that essentially same scattering amplitudes as for the free coupling can be achieved for two $(n-1)$-parameter Fulop-Tsutsui subfamilies if $n$ is odd, and for three $(n-1)$-parameter Fulop-Tsutsui subfamilies if $n$ is even. We also work up an approximation scheme for a general Fulop-Tsutsui vertex, using only $n$ $\delta$ function potentials.