Loop quantum cosmology evolution operator of an FRW universe with a positive cosmological constant

The self-adjointness of an evolution operator ${\Theta }_{\Lambda }$ corresponding to the model of the 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 $\Lambda <{\Lambda }_c\approx 10.3{\ell }_{\mathrm{Pl}}^{-2}$, 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.