On the fate of spacetime singularities

I investigate spacetime singularities from the point of view of the wavefunction of the universe. In order to extend the classical notion of geodesic incompleteness one has to include the proper time of an observer as a degree of freedom in the Wheeler DeWitt equation. This leads to a Schr"odinger equation along the observer worldline. Near the singularity, as in the classical BKL treatment, I ignore spatial gradients and effectively describe the spacetime around the worldline in the mini-superspace approximation. Then the problem proves identical to a spherically symmetric scattering of a quantum particle off a central potential and singularity avoidance is tantamount to unitary evolution for this system. Standard types of matter (dust, radiation) correspond to regular potentials and thus lead to a bounce. The most singular component, spatial anisotropy, is associated to a conserved charge and yields a negative inverse-square potential---like standard angular momentum, but with opposite sign. This potential is critical, in that the unitarity of the evolution depends on the actual numerical factor in front of it, i.e., on the anisotropy charge.