Journal of Statistical Physics

, Volume 98, Issue 5, pp 1135–1148

Dynamical Localization for the Random Dimer Schrödinger Operator


  • Stephan De Bièvre
    • UFR de Mathématiques et URA GATUniversité des Sciences et Technologies de Lille
  • François Germinet
    • UFR de Mathématiques et URA GATUniversité des Sciences et Technologies de Lille

DOI: 10.1023/A:1018615728507

Cite this article as:
De Bièvre, S. & Germinet, F. Journal of Statistical Physics (2000) 98: 1135. doi:10.1023/A:1018615728507


We study the one-dimensional random dimer model, with Hamiltonian Hω=Δ+Vω, where for all x\(\mathbb{Z}\), 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 Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/\(\sqrt 2\), the Lyapunov exponent vanishes only at the two critical energies given by EV. For the particular value V=1/\(\sqrt 2\), respectively, V=\(\sqrt 2\), we show the existence of new additional critical energies at E=±3/\(\sqrt 2\), respectively, E=0. On any compact interval Inot 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\(\ell\)2(\(\mathbb{Z}\)) with sufficiently rapid decrease
$${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$
Here \(\psi _t = e^{- iH_{\omega ^t}} \psi\), and PI(Hω) is the spectral projector of Hωonto the interval I. In particular, if V>1 and V\(\sqrt 2\), these results hold on the entire spectrum [so that one can take I=σ(Hω)].
Schrödinger operatordimerrandom modelAnderson localizationdynamical localizationLyapunov exponentdelocalization

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© Plenum Publishing Corporation 2000