Michelangeli, AlessandroOlgiati, AlessandroScandone, Raffaele2017-11-142017-11-142017https://openscience.sissa.it/handle/1963/35301We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed pointlike impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.enPoint interactionsSingular perturbations of the LaplacianRegular and singular Hartree equationFractional singular Sobolev spacesStrichartz estimates for point interactionHamiltonians Fractional Leibniz ruleKato-Ponce commutator estimatesPreprint