Analysis of Hartree equation with an interaction growing at the spatial infinity

Abstract: We consider nonlinear Schr\"odinger equation with a Hartree-type nonlocal nonlinearity. The case where a nonlinear interaction potential grows at the spatial infinity is studied. By virtue of an effective decomposition of the nonlinearity based on conservation of mass, this kind of growing nonlinear interaction is known to contain an effect like a linear potential. In this paper, a well-posedness result is obtained in a suitable energy space for a class of interaction potential growing at the spatial infinity in at most the quadratic order. If the growth rate of interaction potential is faster than the linear order, a priori information on the center of mass plays a crucial role. When the interaction potential is exactly in the quadratic order, solution are written explicitly.