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