On the inverse scattering of star-shape LC-networks

TL;DR

This paper analyzes the inverse scattering problem for star-shaped LC-networks by transforming it into a Schrödinger operator problem on a graph, revealing how high-frequency data determines network structure.

Contribution

It introduces a novel approach linking LC-network scattering data to Schrödinger operators on graphs, enabling network reconstruction from high-frequency reflection coefficients.

Findings

High-frequency reflection coefficients reveal the number of infinite branches.

Wave traveling times for finite branches can be determined from scattering data.

The method ensures a unique self-adjoint extension of the Schrödinger operator on the graph.

Abstract

The study of the scattering data for a star-shape network of LC-transmission lines is transformed into the scattering analysis of a Schr\"odinger operator on the same graph. The boundary conditions coming from the Kirchhoff rules ensure the existence of a unique self-adjoint extension of the mentioned Schr\"odinger operator. While the graph consists of a number of infinite branches and a number finite ones, all joining at a central node, we provide a construction of the scattering solutions. Under non-degenerate circumstances (different wave travelling times for finite branches), we show that the study of the reflection coefficient in the high-frequency regime must provide us with the number of the infinite branches as well as the the wave travelling times for finite ones.