Scattering on periodic metric graphs

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.

42 pages, 5 figures

References in corpus (8)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• On the spectra of carbon nano-structures
• Spectral Properties of Schrödinger Operators on Perturbed Lattices
• Spectral and scattering theory for Schrödinger operators on perturbed topological crystals
• Inverse scattering for Schrödinger operators on perturbed lattices
• Spectral estimates for Schrödinger operators on periodic discrete graphs
• Modified wave operators for discrete Schrödinger operators with long-range perturbations
• Spectral band bracketing for Laplacians on periodic metric graphs

Cited by in corpus (5)

• Trace formulas for Schrödinger operators on periodic graphs
• Weighted estimates for the discrete Laplacian on the cubic lattice
• Inverse scattering on the quantum graph for graphene
• Inverse scattering on the quantum graph -- Edge model for graphen
• Eigenvalues of periodic difference operators on lattice octant