$L^2$ Analysis of the Multi-Configuration Time-Dependent Hartree-Fock Equations
Norbert J. Mauser, Saber Trabelsi
备注：
This work was supported by the Viennese Science Foundation (WWTF) via the project "TDDFT" (MA-45), the Austrian Science Foundation (FWF) via the Wissenschaftkolleg "Differential equations" (W17) and the START Project (Y-137-TEC) and the EU funded Marie Curie Early Stage Training Site DEASE (MEST-CT-2005-021122)
多组态方法被量子物理学家和化学家广泛地用于多电子薛定谔方程的数值逼近。最近，在不定常模型上获得了第一个数学上严格的结果，例如索伯列夫空间中有界相互作用$H^2$的短时适定性（C.卢比坎德岛Koch}，初始数据在$H^2$中，在库仑相互作用的能量空间中，初始数据在相同的空间中（Trabelsi，Bardos等人），以及在初始数据能量的充分条件下的全局适定性（Bardos et al.）。目前的贡献扩展了分析设置一个$L^2$理论的MCTDHF的一般相互作用，包括库仑的情况下。这类结果也是当密度矩阵的修正（“正则化”）破坏能量守恒性质而保持质量不变时，用于数值计算的ad-hoc方法的理论基础。
英文摘要：
The multiconfiguration methods are widely used by quantum physicists and chemists for numerical approximation of the many electron Schrödinger equation. Recently, first mathematically rigorous results were obtained on the time-dependent models, e.g. short-in-time well-posedness in the Sobolev space $H^2$ for bounded interactions (C. Lubichand O. Koch} with initial data in $H^2$, in the energy space for Coulomb interactions with initial data in the same space (Trabelsi, Bardos et al.}, as well as global well-posedness under a sufficient condition on the energy of the initial data (Bardos et al.). The present contribution extends the analysis by setting an $L^2$ theory for the MCTDHF for general interactions including the Coulomb case. This kind of results is also the theoretical foundation of ad-hoc methods used in numerical calculation when modification ("regularization") of the density matrix destroys the conservation of energy property, but keeps invariant the mass.
