Scattering on periodic metric graphs

TL;DR

This paper analyzes the spectral properties of the Laplacian on periodic metric graphs, providing explicit eigenfunction representations, boundedness results, and scattering theory implications for perturbed operators.

Contribution

It introduces a decomposition of the periodic metric graph Laplacian into fiber operators, explicitly relates eigenfunctions to discrete Laplacian eigenfunctions, and studies scattering theory for perturbed operators.

Findings

Eigenfunctions are uniformly bounded.

Wave operators exist and are complete.

Fredholm determinant is analytic in the upper half-plane.

Abstract

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: a) the wave operators exist and are complete, b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, c) the determinant and the corresponding S-matrix satisfy the Birman-Krein identity.