Quasiclassical mass spectrum of the black hole model with selfgravitating dust shell

TL;DR

This paper derives a discrete quasiclassical mass spectrum for a quantum black hole model with a selfgravitating dust shell using complex domain analysis and Bohr-Sommerfeld quantization conditions.

Contribution

It introduces a novel quasiclassical approach to quantize the mass spectrum of a black hole model with a selfgravitating dust shell, extending previous work with new quantization conditions.

Findings

Discrete mass spectrum derived for the model.

Quantization conditions include Bohr-Sommerfeld and wave function regularity.

Unbound motion spectrum follows a  spectrum for collapsing null shells.

Abstract

We consider a quantum mechanical black hole model introduced in {\it Phys.Rev.}, {\bf D57}, 1118 (1998) that consists of the selfgravitating dust shell. The Schroedinger equation for this model is a finite difference equation with the shift of the argument along the imaginary axis. Solving this equation in quasiclassical limit in complex domain leads to quantization conditions that define discrete quasiclassical mass spectrum. One of the quantization conditions is Bohr-Sommerfeld condition for the bound motion of the shell. The other comes from the requirement that the wave function is unambiguously defined on the Riemannian surface on which the coefficients of Schroedinger equation are regular. The second quantization condition remains valid for the unbound motion of the shell as well, and in the case of a collapsing null-dust shell leads to $m\sim\sqrt{k}$ spectrum.