  The inverse scale factor, which in classical cosmological models diverges at
the singularity, is quantized in isotropic models of loop quantum cosmology by
using techniques which have been developed in quantum geometry for a
quantization of general relativity. This procedure results in a bounded
operator which is diagonalizable simultaneously with the volume operator and
whose eigenvalues are determined explicitly. For large scale factors (in fact,
up to a scale factor slightly above the Planck length) the eigenvalues are
close to the classical expectation, whereas below the Planck length there are
large deviations leading to a non-diverging behavior of the inverse scale
factor even though the scale factor has vanishing eigenvalues. This is a first
indication that the classical singularity is better behaved in loop quantum
cosmology.
