Results 1 
2 of
2
Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
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 ..."
Abstract
 Add to MetaCart
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