Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation (Q1735481)

The authors study the Cauchy problem for the cubic non-linear Schrödinger equation (NLS) with the spatial variable living in \(\mathbb{R}^4\). Smooth solutions of the equation enjoy invariance by a scaling transformation and also conservation of energy. However, the scaling transformation leaves the norm of the Sobolev space \(H^s(\mathbb{R}^4)\) invariant when \(s < s_c:=1\); that is the reason behind considering the exponent \(s_c=1\) as a critical exponent; the Cauchy problem is then studied for initial data in the Sobolev space \(H^s(\mathbb{R}^4)\) with \(s<1\). Starting with an initial data \(f\in H^s(\mathbb{R}^4)\), the authors introduce a randomization of \(f\), yielding a new initial data \(f^\omega\). For the randomized initial data \(f^\omega\), the authors establish local well-posedness of the Cauchy problem for \(\frac13 < s <1\) (i.e. local in time existence and uniqueness of the solution); that is the content of Theorem~1.3. The other main result, Theorem~1.9, establishes a global solution which scatters at infinite time, in the sense that it behaves like \(e^{i\Delta}(f^\omega+ v^\pm)\) as \(t\to\pm \infty\), for some states \(v^\pm\in \dot H^1(\mathbb{R}^4)\); this scattering result is valid under the hypothesis that the initial data \(f\) is radially symmetric.