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The von Neumann lattice refers to a discrete basis of Gaussians located on alattice in phase space. It provides an attractive approach for solving quantummechanical problems, allowing the pruning of tensor-product basis sets usingphase space considerations. In a series of recent articles Shimshovitz et al.[Phys. Rev. Lett. 109 7 (2012)], Takemoto et al. [Journal of Chemical Physics137 1 (2012)] Machnes et al. [Journal of Chemical Physics, accepted (2016)]),we have introduced two key new elements into the method: a formalism forconverging the basis and for efficient pruning by use of the biorthogonalbasis. In this paper we review the key components of the theory and thenpresent new, efficient and parallelizable iterative algorithms for solving thetime-independent and time-dependent Schr\"odinger equations. The algorithmsdynamically determine the active reduced basis iteratively without resorting toclassical analogs. These algorithmic developments, combined with the previousformal developments, allow quantum dynamics to be performed directly andeconomically in phase space. We provide two illustrative examples: double-welltunneling and double ionization of helium.

Quantum Dynamics in Phase space using the Biorthogonal von Neumann bases: Algorithmic Considerations