Note on the role of symmetry in scattering from isospectral graphs and drums

TL;DR

This paper examines how symmetry influences scattering properties of isospectral quantum graphs and compares these findings with conjectures about isospectral domains in Euclidean space, highlighting the role of symmetry in spectral characteristics.

Contribution

It demonstrates that symmetry-preserving attachments lead to identical scattering spectra in isospectral graphs and discusses differences with conjectures on isospectral domains.

Findings

Scattering matrices of isospectral graphs share the same spectrum and polar structure when symmetry is preserved.

Comparison with conjectures suggests differences in pole distributions between graphs and planar domains.

Symmetry plays a crucial role in the spectral properties of isospectral structures.

Abstract

We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in [1, 2]. It was shown in [3] that scattering matrices of such graphs have the same spectrum and polar structure, provided that infinite leads are attached in a way which preserves the symmetry of isospectral construction. In the current paper we compare this result with the conjecture put forward by Okada et al. [4] that the pole distribution of scattering matrices in the exterior of isospectral domains in R^2 are different.