The singular Hartree equation in fractional perturbed Sobolev spaces

TL;DR

This paper develops local and global well-posedness theories for a singular Hartree equation modeling Bose gases with point impurities, using fractional Sobolev spaces to handle the singular perturbation of the Laplacian.

Contribution

It introduces a novel analysis framework for the singular Hartree equation with point interactions, extending well-posedness results to fractional Sobolev spaces.

Findings

Established local well-posedness in fractional Sobolev spaces.

Proved global well-posedness in mass and energy spaces.

Handled the singular perturbation of the Laplacian with self-adjoint extensions.

Abstract

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 Schr\"odinger 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.