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We describe the mathematical underpinnings of the biorthogonal von Neumannmethod for quantum mechanical simulations (PvB). In particular, we present adetailed discussion of the important issue of non-orthogonal projection ontosubspaces of biorthogonal bases, and how this differs from orthogonalprojection. We present various representations of the Schr\"odinger equation inthe reduced basis and discuss their relative merits. We conclude withillustrative examples and a discussion of the outlook and challenges ahead forthe PvB representation.

Quantum Dynamics in Phase Space using Projected von Neumann Bases