A Global Compact Attractor for High-dimensional Defocusing Non-linear Schrödinger Equations with Potential

We study the asymptotic behavior of large data solutions in the energy space H := H(R) in very high dimension d ≥ 11 to defocusing Schrödinger equations iut + ∆u = |u| u + V u in R, where V ∈ C∞ 0 (R ) is a real potential (which could contain bound states), and 1 + 4 d < p < 1 + 4 d−2 is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H to a compact attractor K, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H . The main novelty of this result is that K is a global attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit.