The flat FRW model in LQC: the self-adjointness

TL;DR

This paper proves the essential self-adjointness of the quantum Hamiltonian in the flat FRW model within Loop Quantum Cosmology, analyzing spectral properties for different cosmological constant values.

Contribution

It provides a rigorous mathematical analysis of the quantum Hamiltonian's self-adjointness and spectrum in the flat FRW model with a massless scalar field, extending previous work.

Findings

Proves the essential self-adjointness of the quantum Hamiltonian operator.

Derives the spectrum for negative and zero cosmological constant cases.

Establishes a unitary relation to a known quantum mechanics operator.

Abstract

The flat FRW model coupled to the massless scalar field according to the improved, background scale independent version of Ashtekar, Pawlowski and Singh is considered. The core of the theory is addressed directly: the APS construction of the quantum Hamiltonian is analyzed under the assumption that cosmological constant $\Lambda\le 0$. We prove the essential self-adjointness of the operator whose square-root defines the quantum Hamiltonian operator and therefore, provide the explicit definition. If $\Lambda<0$, then the spectrum is discrete. In the $\Lambda=0$ case, the essential and absolutely continues spectra of the operator are derived. The latter operator is related in the unitary way to the absolutely continuous part of the Quantum Mechanics operator $a(-\frac{\partial^2}{\partial y^2} - \frac{b}{{\rm cosh}^2\hat{y}})$ ($a,b>0$ being some constants) plus a trace class operator.