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 SchrÃ¶dinger 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 point-like 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.

PB - Taylor & Francis VL - 25 UR - https://doi.org/10.1080/14029251.2018.1503423 ER - TY - RPRT T1 - On some rigorous aspects of fragmented condensation Y1 - 2018 A1 - Daniele Dimonte A1 - Marco Falconi A1 - Alessandro Olgiati UR - https://arxiv.org/abs/1809.03586 ER -