The LQC evolution operator of FRW universe with positive cosmological constant

Abstract

The self-adjointness of an evolution operator ${\Theta }_{\Lambda }$ corresponding to the model of flat FRW universe with massless scalar field and cosmological constant quantized in the framework of Loop Quantum Cosmology is studied in the case $\Lambda >0$. It is shown, that for $Λ<Λ_c\approx 10.3 \lPl^{-2}$ the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for $\Lambda \ge {\Lambda }_c$ the operator is essentially self-adjoint, however the physical Hilbert space of the model does not contain any physically interesting states.

RevTex4, 8 pages

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