Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations

E. I. Dinaburg; Ya G. Sinai; A. B. Soshnikov

doi:10.1007/s002200050217

Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations

E. I. Dinaburg, Ya G. Sinai, A. B. Soshnikov

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Abstract

We study the spectral properties of a two-dimensional Schrödinger operator with a uniform magnetic field and a small external periodic field: (formula presented) where V(x,y) = V₀(y) + ε₁V₁(x,y), and ε₀, ε₁ are small parameters. Representing L_ε0 as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume 2π/B to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue measure of the low bands is positive and proportional in the main order to ε₀.

Original language	English (US)
Pages (from-to)	559-575
Number of pages	17
Journal	Communications In Mathematical Physics
Volume	189
Issue number	2
DOIs	https://doi.org/10.1007/s002200050217
State	Published - 1997

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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title = "Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations",

abstract = "We study the spectral properties of a two-dimensional Schr{\"o}dinger operator with a uniform magnetic field and a small external periodic field: (formula presented) where V(x,y) = V0(y) + ε1V1(x,y), and ε0, ε1 are small parameters. Representing Lε0 as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume 2π/B to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue measure of the low bands is positive and proportional in the main order to ε0.",

author = "Dinaburg, {E. I.} and Sinai, {Ya G.} and Soshnikov, {A. B.}",

year = "1997",

doi = "10.1007/s002200050217",

language = "English (US)",

volume = "189",

pages = "559--575",

journal = "Communications In Mathematical Physics",

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T1 - Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations

AU - Dinaburg, E. I.

AU - Sinai, Ya G.

AU - Soshnikov, A. B.

PY - 1997

Y1 - 1997

N2 - We study the spectral properties of a two-dimensional Schrödinger operator with a uniform magnetic field and a small external periodic field: (formula presented) where V(x,y) = V0(y) + ε1V1(x,y), and ε0, ε1 are small parameters. Representing Lε0 as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume 2π/B to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue measure of the low bands is positive and proportional in the main order to ε0.

AB - We study the spectral properties of a two-dimensional Schrödinger operator with a uniform magnetic field and a small external periodic field: (formula presented) where V(x,y) = V0(y) + ε1V1(x,y), and ε0, ε1 are small parameters. Representing Lε0 as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume 2π/B to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue measure of the low bands is positive and proportional in the main order to ε0.

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DO - 10.1007/s002200050217

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JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

ER -

Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations

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