Direct computation of scattering matrices for general quantum graphs

TL;DR

This paper introduces a straightforward algebraic method for computing the scattering matrix of finite quantum graphs, simplifying calculations and extending applicability to graphs with loops, with practical advantages demonstrated on complex examples.

Contribution

The paper presents a novel direct algebraic approach for calculating scattering matrices of quantum graphs, including those with loops, improving upon recursive methods and enabling inverse scattering analysis.

Findings

Method simplifies scattering matrix computation.

Applicable to graphs with loops and tadpoles.

Efficiently handles complex three-dimensional graphs.

Abstract

We present a direct and simple method for the computation of the total scattering matrix of an arbitrary finite noncompact connected quantum graph given its metric structure and local scattering data at each vertex. The method is inspired by the formalism of Reflection-Transmission algebras and quantum field theory on graphs though the results hold independently of this formalism. It yields a simple and direct algebraic derivation of the formula for the total scattering and has a number of advantages compared to existing recursive methods. The case of loops (or tadpoles) is easily incorporated in our method. This provides an extension of recent similar results obtained in a completely different way in the context of abstract graph theory. It also allows us to discuss briefly the inverse scattering problem in the presence of loops using an explicit example to show that the solution is not unique in general. On top of being conceptually very easy, the computational advantage of the method is illustrated on two examples of "three-dimensional" graphs (tetrahedron and cube) for which other methods are rather heavy or even impractical.