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

Abstract

We study the one-dimensional Schrödinger equation with a disordered potential of the formwhere φ(x) is a Gaussian white noise with mean µg and variance g, and κ(x) is a random superposition of delta functions distributed uniformly on the real line with mean density ρ 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 : φ(x) models the force field acting on the diffusing particle and κ(x) models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponentwhere N is the integrated density of states per unit length and γ 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

Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model

“…[19] for a discussion of the S-matrix symmetry). The study of the strictly 1D case (N = 1) has emphasized the role of a Riccati variable for providing the spectral and localisation informations [20][21][22][23]. We extend here this analysis to the multichannel case and emphasize the connection with the scattering problem.…”

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

“…A famous example is the Herbert-Jones-Thouless relation [63,64,65] (see also the appendix of Ref. [66] for a discussion of the continuous case). This remark will allow us to make a precise connection with the study of the one-dimensional Schrödinger equation for a random (time-independent) potential.…”

“…The contribution of the jumps is estimated by writing z jump (τ ) = n h n (τ − τ n ) where h n (τ ) is a narrow function describing the jump at time τ n . Over large scale, [66] and references therein). This simple argument shows that the strongest dependence of the rate in E is mostly controled by N (E), hence presents the non-analytic behaviour…”

“…we deduce that P (z) = Φ R 0 (z; 0) is the stationary solution of the FPE for a uniform current J = −N (E), where N (E) is the integrated DoS of the random Schrödinger operator (25) introduced in Subsection 7.2.2 (see also Refs. [73,65,71,70,66]). The inspection of its asymptotic behaviour…”

Regular Expansion for the Characteristic Exponent of a Product of 2 × 2 Random Matrices

“…In this limit, the product of random matrices becomes a stochastic differential equation system. An exactly solvable structure emerges from that SDE and the analog of (1.3) and (1.4) has been shown to hold (see [13] for the case α ∈ (0, 2) and [4] for the general case). As pointed out in [4], it is rather remarkable that the structure of (1.3) and (1.4) holds also in the weak disorder limit and this appears to be a rather deep fact.…”