  We study the asymptotic behavior of large data solutions in the energy space
$H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schr\"odinger
equations $i u_t + \Delta u = |u|^{p-1} u + Vu$ in $\R^d$, where $V \in
C^\infty_0(\R^d)$ is a real potential (which could contain bound states), and
$1+\frac{4}{d} < p < 1+\frac{4}{d-2}$ is an exponent which is
energy-subcritical and mass-supercritical. In the spherically symmetric case,
we show that as $t \to +\infty$, these solutions split into a radiation term
that evolves according to the linear Schr\"odinger 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 \emph{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.
