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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
UFR de Mathématiques et URA GAT Université des Sciences et Technologies de Lille
, 1999
"... We study the onedimensional random dimer model, with Hamiltonian Hω = ∆ + Vω, where for all x ∈ Z, Vω(2x) = Vω(2x + 1) and where the Vω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V> 0. We show that, for all values of V and with probability one in ω, the spectrum of H is pu ..."
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We study the onedimensional random dimer model, with Hamiltonian Hω = ∆ + Vω, where for all x ∈ Z, Vω(2x) = Vω(2x + 1) and where the Vω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V> 0. We show that, for all values of V and with probability one in ω, the spectrum of H is pure point. If V ≤ 1 and V = 1 / √ 2, the Lyapounov exponent vanishes only at the two critical energies given by E = ±V. For the particular value V = 1 / √ 2, respectively V = √ 2, we show the existence of additional critical energies at E = ±3 / √ 2, resp. E = 0. On any compact interval I not containing the critical energies, the eigenfunctions are then shown to be semiuniformly exponentially localized, and this implies dynamical localization: for all q> 0 and for all ψ ∈ ℓ 2 (Z) with sufficiently rapid decrease: sup t r (q) ψ,I (t) ≡ sup〈PI(Hω)ψt, X t q PI(Hω)ψt 〉 < ∞. Here ψt = e −iHωt ψ, and PI(Hω) is the spectral projector of Hω onto the interval I. In particular if V> 1 and V = √ 2, these results hold on the entire spectrum (so that one can take I = σ(Hω)). 1 1