  A general quantum constraint of the form $C= - \partial_T^2 \otimes B -
I\otimes H$ (realized in particular in Loop Quantum Cosmology models) is
studied. Group Averaging is applied to define the Hilbert space of solutions
and the relational Dirac observables. Two cases are considered. In the first
case, the spectrum of the operator $(1/2)\pi^2 B - H$ is assumed to be
discrete. The quantum theory defined by the constraint takes the form of a
Schroedinger-like quantum mechanics with a generalized Hamiltonian
$\sqrt{B^{-1} H}$. In the second case, the spectrum is absolutely continuous
and some peculiar asymptotic properties of the eigenfunctions are assumed. The
resulting Hilbert space and the dynamics are characterized by a continuous
family of the Schroedinger-like quantum theories. However, the relational
observables mix different members of the family. Our assumptions are motivated
by new Loop Quantum Cosmology models of quantum FRW spacetime. The two cases
considered in the paper correspond to the negative and, respectively, positive
cosmological constant. Our results should be also applicable in many other
general relativistic contexts.
