The Hartree equation with a constant magnetic field: well-posedness theory

We consider the Hartree equation for infinitely many electrons with a constant external magnetic field. For system, we show local well-posedness result when initial data is pertubation of Fermi sea, which non-trace class stationary solution to system. In this case, one particle Hamiltonian Pauli operator, possesses distinct properties from Laplace example, it has discrete spectrum and infinite-dimensional eigenspaces. The new ingredient that use Fourier–Wigner transform asymptotic associated Laguerre polynomials derive collapsing estimate, by establish result.