An overview of periodic elliptic operators

In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break PT symmetry and (b) the existence of topologically protected edge states-states that propagate parallel to and are localized transverse to a line defect or "edge"-for a large class of rational edges, and that are robust to a class of large transverse-localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit.We begin with a review of Floquet-Bloch theory. See, for example, [11,37,38,49] and [3,23,24,34,53]. Fourier Analysis on L 2 .R=ƒ/Let fv 1 ; v 2 g be a linearly independent set in R 2 , and introduce the following:Lattice: ƒ D Zv 1˚Z v 2 D fm 1 v 1 C m 2 v 2 W m 1 ; m 2 2 ZgI Dual lattice: ƒ D ZK 1˚Z K 2 D fm E K

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“…See, for example, [11,37,38,49] and [3,23,24,34,53]. See, for example, [11,37,38,49] and [3,23,24,34,53].…”

mentioning

confidence: 99%

Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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“…For a smooth ϵ it is possible to use a so called Liouville‐Green transformation and to show that σ ( A TE ) coincide with the spectrum of a Schrödinger operator. Thomas' approach can therefore be used to prove that the spectrum of a self‐adjoint realization of A TE in L 2 (R 2 ) is absolutely continuous . However, the material properties are in most applications piecewise constant in space.…”

Section: Two‐dimensional Photonic Crystalsmentioning

confidence: 99%

“…Thomas' approach [24] can therefore be used to prove that the spectrum of a self-adjoint realization of A TE in L 2 (R 2 ) is absolutely continuous. [4,6] However, the material properties are in most applications piecewise constant in space. For a piecewise constant , Figotin and Kuchment [27] used forms to construct a self-adjoint realization of A TE in L 2 (R 2 ).…”

Section: Two-dimensional Photonic Crystalsmentioning

confidence: 99%

“…The literature on conservative photonic crystals that can be described by periodic (elliptic) operators is extensive and our bibliography is therefore far from comprehensive. Our main references for the mathematics of conservative photonic crystals are Reed et al, [2][3][4][5][6] which contain further information and point to original sources. While the applied research on dispersive and absorptive photonic crystals is progressing rapidly, the mathematical research in this direction is still in its infancy.…”

Section: Introductionmentioning

confidence: 99%

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Spectral properties of conservative, dispersive, and absorptive photonic crystals

Engström

Torshage

2018

GAMM-Mitteilungen

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“…Even earlier, Floquet [19] applied the same argument to decompose ordinary differential equations that are invariant with respect to discrete translation groups in a similar manner. The recent survey article [31] gives an overview of the theory and applications. The Zak transforms (3.6) and (4.11) provide the possibility to generalize the theory to operators that are invariant with respect to more general group actions.…”

Section: A Bloch-floquet Theorem For Group Actionsmentioning

confidence: 99%

The Zak transform on strongly proper G-spaces and its applications

Jüstel

2017

J. London Math. Soc.

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Abstract. The Zak transform on R d is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact G-spaces, where G is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. It is shown that the Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of L 2 -functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for G-spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.

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An overview of periodic elliptic operators

Cited by 185 publications

References 271 publications

Honeycomb Schrödinger Operators in the Strong Binding Regime

Honeycomb Schrödinger Operators in the Strong Binding Regime

Spectral properties of conservative, dispersive, and absorptive photonic crystals

The Zak transform on strongly proper G-spaces and its applications

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