Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?

TL;DR

This study investigates whether the shape of a system can be uniquely determined by its scattering properties, using microwave networks as experimental models, and finds that different shapes can produce indistinguishable scattering signatures.

Contribution

First experimental investigation of the relationship between shape and scattering properties in microwave networks, showing non-uniqueness in shape identification from scattering data.

Findings

Scattering matrices of isospectral networks are indistinguishable within experimental error.

Scattering matrices are related by transplantation, indicating non-uniqueness.

Experimental results suggest shape cannot be uniquely determined from scattering properties.

Abstract

The famous question of Mark Kac "Can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator can be legitimately extended to scattering systems. In the modified version one asks whether the geometry of a vibrating system can be determined by scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the, so called, transplantation relation.