Quantum geometrodynamics for black holes and wormholes

Abstract

The geometrodynamics of spherical gravity with a self-gravitating thin dust shell as a source is constructed. The shell Hamiltonian constraint is derived and the corresponding Schr\"odinger equation is obtained. This equation appears to be a finite difference equation. Its solutions are required to be analytic functions on the relevant Riemannian surface. The method of finding discrete spectra is suggested based on the analytic properties of the solutions. The large black hole approximation is considered and the discrete spectra for bound states of quantum black holes and wormholes are found. They depend on two quantum numbers and are, in fact, quasicontinuous.