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

TL;DR

This paper establishes local well-posedness for the Hartree equation with a constant magnetic field, considering initial data as a perturbation of a Fermi sea, using Fourier-Wigner transform and Laguerre polynomial properties.

Contribution

It introduces a novel approach employing Fourier-Wigner transform and Laguerre polynomials to prove well-posedness for the Hartree equation under magnetic fields.

Findings

Proves local well-posedness for the system with magnetic field.

Utilizes Fourier-Wigner transform and Laguerre polynomials.

Handles non-trace class stationary solutions like the Fermi sea.

Abstract

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.