Space-time scattering for the Schrödinger equation (Q1810327)

The author studies the space-time scattering for the nonlinear Schrödinger equation \[ i\partial_tu(t,x)=-\Delta_xu(t,x)+V(t,x)u(t,x)+F(u(t,x)), \] where \(V\) is an explicitly time-dependent potential satisfying the condition \[ \hat{V}(t,\xi)\in L^1(\mathbb{R},M(\mathbb{R}^d)), \] where \(\hat{V}(t,\xi)\) is the Fourier transform with respect to the \(x\)-variable and \(M(\mathbb{R}^d)\) is the Banach space of finite regular complex measures on \(\mathbb{R}^d\). The function \(F\) satisfies the conditions \(F(0)=0\) and \[ |F'(\zeta)|=|\partial_xF(\zeta)|+|\partial_yF(\zeta)|\leq M|\zeta|^{k-1}, \;\;k>1, \] where \(k\) satisfies the inequalities \[ {d+2+\sqrt{d^2+12d+4}\over 2d}<k<1+{4\over d-2}. \] The corresponding initial data is chosen small in a certain sense.