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The Singular Hartree Equation in Fractional Perturbed Sobolev Spaces

TitleThe Singular Hartree Equation in Fractional Perturbed Sobolev Spaces
Publication TypePreprint
2017
AuthorsMichelangeli, A, Olgiati, A, Scandone, R
Document NumberSISSA;52/2017/MATE

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.

http://preprints.sissa.it/handle/1963/35301

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