Asymptotics of the Solutions of the Random Schrödinger Equation

Abstract

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transformζ ε (t, ξ) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit ofζ ε (t, ξ) has the formζ 0 (ξ ) exp(i B κ (t, ξ)) where B κ (t, ξ) is a fractional Brownian motion.

Inverse scattering for a random potential

“…This approach can be poorly justified if the scatterer changes slowly during the measurements or is independent of time. In this paper, the data is assumed to be a weighted average of far-field patterns at a given separation τ > 0: (3) M (τ, θ) = lim coincides with M (τ, θ) almost surely, whereas the contribution from 2nd order and multiple backscattering becomes negligible at the limit:…”

The 1D Schrödinger equation with a spacetime white noise: the average wave function

“…Here z 1 is the same constant as in Theorem 1.1. If we instead consider a high frequency initial data φ(0, x) = φ 0 (x), which varies on the same scale as the random media, a kinetic equation was derived on the time scale of t ε in [2]:…”

The random Schrödinger equation: Slowly decorrelating time-dependent potentials

“…The initial profile φ 0 is assumed to be of the Schwartz class: φ 0 ∈ S(R d ). As in [2,7], in order to eliminate the large phase coming simply from the deterministic evolution, we consider the compensated wave function…”

“…A posteriori, this limit justifies the basic assumption of the informal computation we have shown above: the effects of the Laplacian operator are suppressed via the phase compensation, the dynamics is essentially reduced to an ODE, and the random potential behaves as a fractional Gaussian noise in the limit. We should mention that the restriction α ≥ κ − α c for β ∈ (1/2, 1) is a limitation of the technique of the proof, and is the analog of the restriction β ≤ 1/2 that was needed in [2] in the case α = 0 considered there.…”

“…We mention that the case ℓ = 1, that is, when the the probing signal varies on a scale comparable to that of the random potential, was investigated in [2]. It was shown that, after the phase compensation, the wave field behaves on an anomalous time scale like a lognormal distribution.…”