  The flat FRW model coupled to the massless scalar field according to the
improved, background scale independent version of Ashtekar, Pawlowski and Singh
is considered. The core of the theory is addressed directly: the APS
construction of the quantum Hamiltonian is analyzed under the assumption that
cosmological constant $\Lambda\le 0$. We prove the essential self-adjointness
of the operator whose square-root defines 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 continues
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
$a(-\frac{\partial^2}{\partial y^2} - \frac{b}{{\rm cosh}^2\hat{y}})$ ($a,b>0$
being some constants) plus a trace class operator.
