Journal of Statistical Physics

, Volume 155, Issue 2, pp 237–276 | Cite as

One-Dimensional Disordered Quantum Mechanics and Sinai Diffusion with Random Absorbers

  • Aurélien Grabsch
  • Christophe Texier
  • Yves Tourigny


We study the one-dimensional Schrödinger equation with a disordered potential of the form
$$\begin{aligned} V (x) = \phi (x)^2+\phi '(x) + \kappa (x) \end{aligned}$$
where \(\phi (x)\) is a Gaussian white noise with mean \(\mu g\) and variance \(g\), and \(\kappa (x)\) is a random superposition of delta functions distributed uniformly on the real line with mean density \(\rho \) and mean strength \(v\). Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers: \(\phi (x)\) models the force field acting on the diffusing particle and \(\kappa (x)\) models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent \( \varOmega (E) = \gamma (E) - \mathrm{i}\pi N(E) \), where \(N\) is the integrated density of states per unit length and \(\gamma \) the reciprocal of the localisation length. By using the continuous version of the Dyson–Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean \(v=2g\). Building on this result, we then solve the general case— in the low-energy limit— in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour
$$\begin{aligned} N(E) \underset{E\rightarrow 0+}{\sim } E^\nu \quad \hbox {where } \quad \nu =\sqrt{\mu ^2+2\rho /g}. \end{aligned}$$
This confirms and extends several results obtained previously by approximate methods.


Disordered 1D quantum mechanics Anderson localisation Classical diffusion in random environment Sinai problem 

Mathematics Subject Classification

Primary 82B44 Secondary 60G51 



AG thanks the University of Bristol for its hospitality while part of this work was carried out. CT would like to thank Christian Hagendorf, with whom his interest in this topic began, as well as Alberto Rosso for some helpful suggestions. YT is grateful to the Laboratoire de Physique Théorique et Modèles Statistiques for generously funding his visits to Orsay during the project. We also thank Alain Comtet for many stimulating discussions, and for allowing us to include in this paper the calculation presented in Sect. 5.3.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Aurélien Grabsch
    • 1
  • Christophe Texier
    • 2
  • Yves Tourigny
    • 3
  1. 1.École normale supérieure de CachanCachan CedexFrance
  2. 2.Univ. Paris-Sud, CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626OrsayFrance
  3. 3.School of MathematicsUniversity of BristolBristolUnited Kingdom

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