Quantum superposition principle and gravitational collapse: Scattering   times for spherical shells
  A quantum theory of spherically symmetric thin shells of null dust and their gravitational field is studied. In Nucl. Phys. 603 (2001) 515 (hep-th/0007005), it has been shown how superpositions of quantum states with different geometries can lead to a solution of the singularity problem and black hole information paradox: the shells bounce and re-expand and the evolution is unitary. The corresponding scattering times will be defined in the present paper. To this aim, a spherical mirror of radius R_m is introduced. The classical formula for scattering times of the shell reflected from the mirror is extended to quantum theory. The scattering times and their spreads are calculated. They have a regular limit for R_m\to 0 and they reveal a resonance at E_m = c^4R_m/2G. Except for the resonance, they are roughly of the order of the time the light needs to cross the flat space distance between the observer and the mirror. Some ideas are discussed of how the construction of the quantum theory could be changed so that the scattering times become considerably longer. 
M. Ambrus and P. Hajicek