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.

1 aMichelangeli, Alessandro1 aOlgiati, Alessandro1 aScandone, Raffaele uhttps://doi.org/10.1080/14029251.2018.150342300361nas a2200109 4500008004100000245005600041210005300097100002100150700001900171700002400190856003700214 2018 eng d00aOn some rigorous aspects of fragmented condensation0 asome rigorous aspects of fragmented condensation1 aDimonte, Daniele1 aFalconi, Marco1 aOlgiati, Alessandro uhttps://arxiv.org/abs/1809.03586