Mixed regularity and sparse grid approximations of $N$-body Schr\"odinger evolution equation

In this paper, we present a mathematical analysis of time-dependent $N$-body electronic systems and establish mixed regularity for the corresponding wavefunctions. Based on this, we develop sparse grid approximations to reduce computational complexity, including a sparse grid Gaussian-type orbital (GTO) scheme. We validate the approach on the Helium atom (${\rm He}$) and Hydrogen molecule (${\rm H}_2$), showing that sparse grid GTOs offer an efficient alternative to full grid discretizations.