论文标题
关于量子动力学中哈特里近似的凯奇问题
On the Cauchy problem for the Hartree approximation in quantum dynamics
论文作者
论文摘要
我们证明了量子动力学中产生的时间依赖性的hartree近似的存在和唯一性结果。 Hartree运动方程形成了一个非线性Schr {Ö} dinger方程的耦合系统，用于乘积状态近似的演变。它们是在时间依赖性的dirac-frenkel变化原理的上下文中降低维度的重要例子。得益于Strichartz的估计，我们应对库仑电位的情况。我们的主要结果解决了一个通用设置，在该设置中，非线性耦合不能被视为扰动。证明使用递归结构，该结构灵感来自与对称准线性双曲方程相关的cauchy问题的标准方法。
We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{ö}dinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the the time-dependent Dirac-Frenkel variational principle. We handle the case of Coulomb potentials thanks to Strichartz estimates. Our main result addresses a general setting where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations.
