Solution of the discrete Wheeler-DeWitt equation in the vicinity of small scale factors and quantum mechanics in the space of negative constant curvature

TL;DR

This paper finds asymptotic solutions to the discrete Wheeler-DeWitt equation near small scale factors, showing its equivalence to a Schrödinger equation in negative curvature space and identifying the spectrum's start point.

Contribution

It introduces a novel asymptotic analysis of the Wheeler-DeWitt equation near small scale factors and links it to quantum mechanics in negatively curved space.

Findings

Asymptotic solutions near small scale factors are derived.

The problem is shown to be equivalent to a Schrödinger equation in negative curvature space.

The minimum positive eigenvalue and the onset of the continuous spectrum are identified.

Abstract

The asymptotic of the solution of the discrete Wheeler-DeWitt equation is found in the vicinity of small scale factors. It is shown that this problem is equivalent to the solution of the stationary Schr\"{o}dinger equation in the (super) space of negative constant curvature. The minimum positive eigenvalue is found from which a continuous spectrum begins.