Small analytic solutions to the Hartree equation

We consider the Cauchy problem for the Hartree equation in space dimension d≥3. We assume that the interaction potential V belongs to the weak Ld/2 space. We prove that if the initial data ϕ is sufficiently small in the L2-sense and either ϕ or its Fourier transform Fϕ satisfies a real-analytic condition, then the solution u(t) is also real-analytic for any t≠0. We also prove that if ϕ and V satisfy some strong condition, then u(t) can be extended to an entire function on Cd for any t≠0. A part of our method is applicable to the final value problem. We remark that no L2 smallness condition is imposed on first and higher order partial derivatives of ϕ and Fϕ.