2 6 A pr 1 99 9 COSMOLOGICAL LEVINSON THEOREM

If at least some Wheeler-DeWitt solutions can be interpreted as zero-energy resonances then the total s-wave cross section of the corresponding quantum universes is infinite. The present, common 'quantum' cosmological framework is based on the Wheeler-DeWitt equation, which being a 'stationary', zero-energy differential equation appears to have no obvious time dependence and therefore forcing one to invent clock models, a feature known in the literature as the problem of time. Here I would like to draw attention to the fact that for a more advantageous study of some important issues, like quantum cosmological irreversibility, it may prove convenient to think of some Wheeler-DeWitt 'wavefunction of the universe' solutions as similar to zero-energy resonances (also known as half-bound states, and possible only for zero angular momentum [1]) in ordinary quantum scattering. The zero-energy resonance wavefunctions are zero at the arbitrary chosen cosmological origin, are finite at infinity, and are not normalizable. For such a zero-energy resonance, taking into account that there are no cosmological 'bound states', one may write a cosmological nonperturbative Levinson theorem of the form [1] η u (0) = π 2 sin 2 η u (0) (1) for the s-wave phase shift of the quantum universe at zero energy, with the obvious result η u (0) = π 2. Thus, one can claim that the total s-wave scattering cross section of the 'quantum' universes with zero-energy resonances is infinite.