We prove in this paper that the solution of the time-dependent Schr{\"o}dinger equation can be expressed as the solution of a global space-time quadratic minimization problem that is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation. The present analysis can be applied to the electronic many-body time-dependent Schr{\"o}dinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities. We motivate the interest of the present approach with two goals: first, the design of Galerkin space-time discretization methods; second, the definition of dynamical low-rank approximations following a variational principle different from the classical Dirac-Frenkel principle, and for which it is possible to prove the global-in-time existence of solutions.

翻译：本文证明，含时薛定谔方程的解可表述为一个全局时空二次最小化问题的解，该问题通过适当的极小二乘公式适用于伽辽金时空离散化方案。本分析可应用于具有任意电子数及库仑奇点相互作用势的电子多体含时薛定谔方程。我们阐述本方法的价值基于两个目标：其一，设计伽辽金时空离散化方法；其二，依据不同于经典狄拉克-弗伦克尔原理的变分原理定义动态低秩近似，并能够证明该近似解在全时间范围内的存在性。