For any weight parameter in the improved-dynamics LQC Hamiltonian, negative weights give essentially self-adjoint operators, while positive weights require U(1)-labeled self-adjoint extensions, which the paper implements in a propagator.
The LQC evolution operator of FRW universe with positive cosmological constant
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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 $\Lambda<\Lambda_c\approx 10.3 \lPl^{-2}$ the operator admits many self-adjoint extensions, each of the purely discrete spectrum. On the other hand for $\Lambda\geq\Lambda_c$ the operator is essentially self-adjoint, however the physical Hilbert space of the model does not contain any physically interesting states.
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Unitary evolution and cosmic acceleration in Loop Quantum Cosmology
For any weight parameter in the improved-dynamics LQC Hamiltonian, negative weights give essentially self-adjoint operators, while positive weights require U(1)-labeled self-adjoint extensions, which the paper implements in a propagator.