Schrödinger picture of quantum gravitational collapse

The functional Schrödinger equation is used to study the quantum collapse of a gravitating, spherical domain wall and a massless scalar field coupled to the metric. The approach includes backreaction of pre-Hawking radiation on the gravitational collapse. Truncating the degrees of freedom to a minisuperspace leads to an integro-differential Schrödinger equation. We define a 'black hole' operator and find its eigenstates. The black hole operator does not commute with the Hamiltonian, leading to an energy-black holeness uncertainty relation. We discuss energy eigenstates and also obtain a partial differential equation for the time-dependent gravitational collapse problem.