Asymptotic Properties of Difference Equations for Isotropic Loop Quantum Cosmology

TL;DR

This paper investigates the asymptotic behavior of difference equations in isotropic loop quantum cosmology, revealing conditions for bounded solutions and small-scale oscillations using continued fraction methods.

Contribution

It introduces a novel analysis of the difference equations' properties, providing explicit initial conditions for bounded solutions in various matter models.

Findings

Bounded solutions exist under specific initial conditions.

Explicit formulas for initial values leading to bounded solutions.

Small-scale oscillations are characterized in the solutions.

Abstract

In loop quantum cosmology, a difference equation for the wave function describes the evolution of a universe model. This is different from the differential equations that arise in Wheeler-DeWitt quantizations, and some aspects of general properties of solutions can appear differently. Properties of particular interest are boundedness and the presence of small-scale oscillations. Continued fraction techniques are used to show in different matter models the presence of special initial conditions leading to bounded solutions, and an explicit expression for these initial values is derived.