Singular Hartree equation in fractional perturbed Sobolev spaces
We 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.
Point interactions, Singular perturbations of the Laplacian, Regular and singular Hartree equation, Fractional singular Sobolev spaces, Strichartz estimates for point interaction, Hamiltonians Fractional Leibniz rule, Kato-Ponce commutator estimates