Weak Bloch property for discrete magnetic Schrödinger operators

Abstract

Abstract. For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties. §0. IntroductionThe spectral analysis of a discrete Laplacian, regarded as a discrete analogue of the Laplace-Beltrami operator on a Riemannian manifold, has been investigated by many authors (cf.One of the main topics is to study the relationship between the spectra of discrete Laplacians and the geometries of graphs from many viewpoints. Many researchers have explored the spectra of discrete magnetic Laplacians, each of which is a purturbed Laplacian on a graph by the magnetic field also. For instance, E. Lieb and M. Loss [11] characterized the ground state (in other words, the bottom of the spectrum) of the discrete magnetic Laplacian called a hopping matrix for a finite bipartite planar graph, and T. Sunada [14] gave some criteria for the spectrum which consists of a union of finite number of closed intervals using a twisted group C * -algebra for a discrete magnetic Laplacian with a weak invariance under a group action on an infinite graph.Our main purpose in this paper is to study the spectral properties of the discrete magnetic Laplacian in terms of a growth function for a graph and the behaviour of the bottom of the spectrum as a function of the magnetic flux.

Magnetic Schrödinger operators on periodic discrete graphs

“…This property for the magnetic Schrödinger operators on a locally finite graph was proved in [LL93], [CTT11], [HS01]. In particular, if the magnetic flux of α is zero for any cycle on Γ * , then the magnetic Schrödinger operator H α = ∆ α + Q is unitarily equivalent to the Schrödinger operator H 0 = ∆ 0 + Q without a magnetic field.…”

“…1) The magnetic Laplacian ∆ α defined by (6.5) has the following factorization (see [HS99a], [HS99b], [HS01]): 2) The quasimomentum ϑ satisfying (6.24) may or may not exist. For example, if #S = d, then such ϑ ∈ T d exists and is unique.…”

“…Higuchi and Shirai [HS01] studied the behaviour of the bottom of the spectrum as a function of the magnetic flux. Colin de Verdière, Torki-Hamza and Truc [CTT11] obtained a condition under which the magnetic Laplacian on an infinite graph is essentially self-adjoint.…”

The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure

“…The bottom * G (%) is continuous for any % and differentiable at %=0 in general without covering structure [5]. Our proof of the analyticity depends on the abelian covering structure of G.…”

“…In this paper, we assume the covering structure of G to prove Theorem 1.1. However, it also holds for more general graphs under the assumption that the boundary area of a graph does not grow exponentially [5].…”