The aim of this talk is to present novel global space-time methods for the approximation of the time-dependent Schrödinger equation on low-complexity manifolds. The backbone of the approach is the use of a least-square formulation of the time-dependent Schrödinger equation with many-body Coulomb interaction potentials, which can be obtained using Kato theory. The latter can be used in conjunction with low-rank tensor formats (such as Tensor Trains for instance)  gaussian wavepackets to derive new variational principles to compute dynamical low-complexity approximations of the solution. These new approximatons are different from the ones obtained through the classical Dirac-Frenkel principle. One significant advantage of this new variational formulation is that the existence of a dynamical low-rank approximation for any finite-time horizon can be proved with low-rank tensor formats, whereas dynamical low-rank approximations constructed with the Dirac-Frenkel principle can usually be proved to exist only locally in time. Illustrative numerical results will be presented to highlight the differences between the dynamical low-rank approximations obtained with these different approaches.

Joint work with Clément Guillot and Mi-Song Dupuy.