A space-time variational formulation for the many-body electronic Schr{ö}dinger evolution equation
This provides a new variational formulation for quantum dynamics, potentially advancing computational methods in quantum chemistry and physics, though it appears incremental as it builds on existing principles.
The paper tackles the problem of solving the time-dependent Schrödinger equation for many-electron systems by proving it can be expressed as a global space-time quadratic minimization problem, enabling Galerkin discretization and new dynamical low-rank approximations with proven global-in-time existence of solutions.
We prove in this paper that the solution of the time-dependent Schr{ö}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{ö}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.