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.

我们考虑具有恒定外部磁场的无限多电子的 Hartree 方程。对于系统，当初始数据是费米海的扰动时，我们显示了局部适定结果，这是系统的非迹类平稳解。在这种情况下，单粒子哈密顿算符是泡利算符，它具有与拉普拉斯算符不同的性质，例如，它具有离散谱和无限维特征空间。新的成分是我们使用傅里叶-维格纳变换和相关拉盖尔多项式的渐近性质来推导出一个折叠估计，通过它我们建立局部适定性结果。