Spectral Properties of Schrödinger Operators on Perturbed Lattices

Abstract

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice.

Finiteness of discrete spectrum of the two-particle Schrӧdinger operator on diamond lattices

“…We first give descriptions of the d−dimensional diamond lattice and a discrete Schrödinger operator on the d−dimensional diamond lattice [12].…”

Long-range scattering theory for discrete Schrödinger operators on graphene

“…The derivation of H 0 is found in e.g. [2] and [3]. As is seen later, H 0 has purely absolutely continuous spectrum and σ(H 0 ) = σ ac (H 0 ) = [ −3, 3].…”

“…we learn that U ′ (ξ) is a unitary matrix-valued smooth function on T 2 and 3] by the condition of κ. Thus we obtain the following lemma.…”

“…For a shortrange scattering theory for discrete Schrödinger operators on general lattices, including the hexagonal lattice, see [14]. See also [3] and [4] for spectral properties of discrete Schrödinger operators on general lattices. Let H = ℓ 2 (Z 2 ; C 2 ).…”

Spectral and scattering theory for Gauss–Bonnet operators on perturbed topological crystals

“…This perturbation is encoded by a non periodic measure that converges at infinity to a periodic measure. This kind of metric perturbation has received some attention lately for the Laplacian acting on vertices, but only considering compactly supported perturbations [AIM15] or restricting the study to the essential spectrum [SS15].…”