1999/07/07 by Stephan De Bièvre, S. De Bièvre, François Germinet +1 · 2 citations
Mathematics · Physics and Astronomy · #Combinatorics #Dimer #Eigenfunction #Eigenvalues and eigenvectors #Exponent #Hamiltonian (control theory) #Mathematical physics #Mathematics #Nuclear magnetic resonance #Omega #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP
paper · pdf · doi:10.1023/a:1018615728507
14 pages
arxiv created 1999/07/07 · openalex publication_date 1999/07/07 · arxiv updated 2015/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the one-dimensional random dimer model, with Hamiltonian Hω=Δ+ 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 semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ℓ2(\Z) with sufficiently rapid decrease: supt r(q)ψ,I(t) ≡ supt < PI(Hω)ψt, |X|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)).