Asymptotics of the solutions of the random Schrödinger equation (Q717479)

This article studies the behavior of solutions of the following Schrödinger equation in \(\mathbb{R}^d\) (\(d \geq 1\)): \[ i \frac{\partial \phi}{\partial t} + \frac{1}{2} \Delta \phi - \gamma V(t,x) \phi = 0, \qquad \phi(0,x) = \phi_0(x), \] for a random potential \(V(t,x)\), homogeneous in space and time, with mean zero, and where \(\gamma \ll 1\) is a small parameter. This gives more detailed information than limiting results on the Wigner transform associated with \(\phi\). A hyperbolic scaling is considered, so that the function under study is \(\phi_\varepsilon(t,x) = \phi(t/\varepsilon,x/\varepsilon)\), the solution of \[ i \varepsilon \frac{\partial \phi_\varepsilon}{\partial t} + \frac{\varepsilon^2}{2} \Delta \phi_\varepsilon - \gamma V\left(\frac{t}{\varepsilon}, \frac{x}{\varepsilon}\right) \phi_\varepsilon = 0. \] The scaling of \(\varepsilon\) as a function of \(\gamma\) depends on the statistics of the random perturbation. More precisely, when the two-point correlation function \[ R(t,x) = \mathbb{E}(V(s,y)V(t+s,x+y)) \] decays rapidly enough, then \(\varepsilon = \gamma^2\) and, as \(\varepsilon \to 0\), the appropriately rescaled spatial Fourier transform of \(\phi_\varepsilon\) (with the divergent phase term arising from the free evolution subtracted) \[ \zeta_\varepsilon(t,\xi) = \frac{1}{\varepsilon^d} \widehat{\phi}\left(t,\frac{\xi}{\varepsilon}\right) \, \mathrm{e}^{i |\xi|^2 t /(2\varepsilon)} \] converges in law (for fixed \(t,x\)) to a stochastic complex Gaussian limit. On the other hand, when the correlations are slowly decaying, then \(\varepsilon = \gamma^{1/\kappa}\) where \(\kappa\) depends explicitly on the noise statistics, and the pointwise limit of \(\zeta_\varepsilon(t,\xi)\) is \(\widehat{\phi}_0(\xi) \, \mathrm{e}^{i B_\kappa(t,\xi)}\) where \(B_\kappa\) is a fractional Brownian motion. These two results, presented in Section~1, are proved respectively in Sections~3 and~4. The proof is based on an expansion of \(\phi_\varepsilon\) in powers of \(\gamma\) (see Section~2), and an identification of the limits of all moments of \(\zeta_\varepsilon\).