A family of quantum graph vertex couplings interpolating between different symmetries

TL;DR

This paper introduces a new family of quantum graph vertex couplings that smoothly transition between different symmetry types, analyzing their spectral and scattering properties and implications for lattice band spectra.

Contribution

It presents a novel interpolation family of quantum graph vertex couplings using circulant matrices, bridging different symmetry classes and exploring their spectral characteristics.

Findings

Spectral properties of the interpolating vertex couplings analyzed.

Scattering behavior of the new couplings characterized.

Band spectrum of the square lattice graph studied.

Abstract

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the $\delta$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph.