Stochastic Hartree NLS in 3d coming from a Many-Body Quantum System with White Noise Potential

Abstract

In this paper, we consider the defocusing Hartree NLS with white noise external potential on T^3 i.e. the Hartree NLS whose linear part is given by the Anderson Hamiltonian. A Strichartz-type estimate is established for the Anderson Hamiltonian using perturbative arguments and the local and global well-posedness of the NLS is considered with initial data in the domain and form-domain of the Anderson Hamiltonian under different regularity assumptions on the Hartree interaction. Furthermore, we establish the Anderson Hartree NLS as an effective equation describing many-body Bosonic systems and, in particular, we prove the convergence of the linear Schr\"odinger equation for the many body system to the BBGKY hierarchy for the Coulomb interaction.