Abstract
We consider energy fluctuations for solutions of theSchrödinger equation with an Ornstein-Uhlenbeck random potential when the initialdata is spatially localized. The limit of the fluctuations of the Wigner transformsatisfies a kinetic equation with random initial data. This result generalizes that of [12]where the random potential was assumed to be white noise in time.
Highlights
Solutions of the Schrodinger equation with a weakly random potential ∂φ 1 √i + ∆φ − εV (t, x)φ = 0,∂t 2 and a small parameter ε 1 behave non-trivially on the time scale t ∼ O(ε−1)
0, A convenient tool to study the energy distribution in this long time limit is via the Wigner transform [10, 15] of the solution defined as
In this paper we consider random potentials of Ornstein-Uhlenbeck type that have finite correlation time, and show that the gist of the result is similar to that in [13] – the limit Zis identified as a solution of a deterministic kinetic equation with a random initial data
Summary
∂t 2 and a small parameter ε 1 behave non-trivially on the time scale t ∼ O(ε−1). The corresponding rescaled problem is iε ∂φε ∂t. In this paper we consider random potentials of Ornstein-Uhlenbeck type that have finite correlation time, and show that the gist of the result is similar to that in [13] – the limit Zis identified as a solution of a deterministic kinetic equation with a random initial data. This is because the main contribution to the fluctuations of Zε comes from the initial boundary time layer when the wave energy is very singular, and the fluctuations that are created later are of a smaller size since the wave field becomes spatially distributed. The rest of the paper contains the proof of Theorem 2.8 that is performed via a series of intermediate steps, outlined after the statement of this theorem
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