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, Pawlowski 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 Lambda <= 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 Lambda < 0, then the spectrum is discrete. In the Lambda = 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(-partial derivative(2)/partial derivative y(2) - b/cosh(2)<(y)over cap> (a, b > 0 being some constants) plus a trace class operator.