  Attempts to consider evolution across space-time singularities often lead to
quantum systems with time-dependent Hamiltonians developing an isolated
singularity as a function of time. Examples include matrix theory in certain
singular time-dependent backgounds and free quantum fields on the
two-dimensional compactified Milne universe. Due to the presence of the
singularities in the time dependence, the conventional quantum-mechanical
evolution is not well-defined for such systems. We propose a natural way,
mathematically analogous to renormalization in conventional quantum field
theory, to construct unitary quantum evolution across the singularity. We carry
out this procedure explicitly for free fields on the compactified Milne
universe and compare our results with the matching conditions considered in
earlier work (which were based on the covering Minkowski space).
