Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2007-09-19
Class.Quant.Grav.25:035001,2008
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
13 pages
Scientific paper
10.1088/0264-9381/25/3/035001
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.
Kaminski Wojciech
Lewandowski Jerzy
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