The flat FRW model in LQC: self-adjointness

The flat Friedman–Robertson–Walker (FRW) model coupled to the massless scalar field according to the improved, background scale-independent version of Ashtekar, Pawłowski and Singh [1] is considered. The core of the theory is addressed directly: the APS construction of the quantum Hamiltonian is analyzed under the assumption that the cosmological constant Λ ⩽ 0. We prove the essential self-adjointness of the operator whose square-root defines in [1] the quantum Hamiltonian operator and therefore provide the explicit definition. If Λ < 0, then the spectrum is discrete. In the Λ = 0 case, the essential and absolutely continuous 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 , b > 0 being some constants) plus a trace class operator.