Abstract
We study the onedimensional Schrödinger equation with a disordered potential of the form V(x)=ϕ(x)2+ϕ′(x)+κ(x)where 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
exponent Ω(E)=γ(E)−iπN(E), where 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 exponentiallydistributed with mean v=2g.
Building on this result, we then solve the general case— in the
lowenergy 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 N(E)∼E→0+Eνwhere ν=μ2+2ρ/g−−−−−−−−√.This confirms and extends several results obtained previously by approximate methods.
Original language  English 

Pages (fromto)  237–276 
Number of pages  40 
Journal  Journal of Statistical Physics 
Volume  155 
Issue number  2 
Early online date  28 Feb 2014 
DOIs  
Publication status  Published  1 Apr 2014 
Keywords
 Primary 82B44
 Secondary 60G51
 Disordered 1D quantum mechanics
 Anderson localisation
 Classical diffusion in random environment
 Sinai problem
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Dr Yves J M Tourigny
 Probability, Analysis and Dynamics
 School of Mathematics  Senior Lecturer in Numerical Analysis
 Applied Mathematics
 Mathematical Physics
Person: Academic , Member