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Long time Anderson localization for the nonlinear random Schrödinger equation. (English) Zbl 1193.82022

The paper studies the lattice nonlinear equation in one dimension \[ i\dot q_j = v_j q_j + \varepsilon_1 (q_{j-1}+q_{j+1}) + \varepsilon_2 |q_j|^2 q_j, \quad j\in \mathbb Z, \] where \(v_j\) is a collection of i.i.d. uniform random variables on \([0,1]\) and \(\epsilon_1\), \(\epsilon_2\) are positive parameters. It proves long-time Anderson localisation for arbitrary \(\ell^2\) initial data in the form inspired by the RAGE theorem. Namely, it shows that given \(A>1\), \(\delta >0\), initial datum \(q_j(0)\in \ell^2\) and \(j_0\) satisfying \[ \sum_{|j|>j_0}|q_j(0)|^2< \delta, \] there exist \(C\), \(\epsilon\) and \(N\) depending on \(A\) such that for all \(t\leq (\delta/C)\epsilon^{-A}\) one has, with a high probabililty, \[ \sum_{|j|>j_0+N}|q_j(t)|^2< 2\delta. \]
The proof uses a Birkhoff normal type transform to create a barrier where there is essentially no propagation. One of the novelties of this transform is that it is in small neighbourhoods only, enabling to treat “rough” data without any moment condition.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
47B80 Random linear operators
35R60 PDEs with randomness, stochastic partial differential equations
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