Remark on small analytic solutions to the Schr̈odinger equation with cubic convolution

We consider the Cauchy problem for the Schrödinger equation with cubic convolution in space dimension d ≥ 3. We assume that the interaction potentialV belongs to the weakL space with2 ≤ σ < d. We prove that if the initial data φ is sufficiently small in the sense of the Sobolev space Hσ/2−1 and either φ or its Fourier transform Fφ satisfies a real-analytic condition, then the solutionu(t) is also real-analytic for anyt ̸= 0. We also prove that ifφ and V satisfy some strong condition, then u(t) can be extended to an entire function onC for any t ̸= 0. We remark that no Hσ/2−1 smallness condition is imposed on first and higher order partial derivatives of φ and Fφ.

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