A space-time variational formulation for the many-body electronic Schr{\"o}dinger evolution equation

TL;DR

This paper introduces a space-time variational framework for the many-body electronic Schrödinger equation, enabling new discretization methods and low-rank approximations with proven global solutions.

Contribution

It formulates the Schrödinger evolution as a quadratic minimization problem suitable for Galerkin discretization and variational low-rank approximations, applicable to complex many-electron systems.

Findings

Solution expressed as a global space-time quadratic minimization problem.

Applicable to many-electron systems with Coulomb interactions.

Proves global-in-time existence of solutions for low-rank approximations.

Abstract

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.