Random data final-state problem for the mass-subcritical NLS in $L^2$

Abstract

We study the final-state problem for the mass-subcritical NLS above the Strauss exponent. For $u_{+}\in L^2$, we perform a physical-space randomization, yielding random final states $u_{+}^{\ddot{I}}‰\in L^2$. We show that for almost every $\ddot{I}‰$, there exists a unique, global solution to NLS that scatters to $u_{+}^{\ddot{I}}‰$. This complements the deterministic result of Nakanishi, which proved the existence (but not necessarily uniqueness) of solutions scattering to prescribed $L^2$ final states.

11 pages, updated references and extended to dimensions d=1,2