  The k=0 Friedmann Lemaitre Robertson Walker model with a positive
cosmological constant and a massless scalar field is analyzed in detail. If one
uses the scalar field as relational time, new features arise already in the
Hamiltonian framework of classical general relativity: In a finite interval of
relational time, the universe expands out to infinite proper time and zero
matter density. In the deparameterized quantum theory, the true Hamiltonian now
fails to be essentially self-adjoint both in the Wheeler DeWitt (WDW) approach
and in LQC. Irrespective of the choice of the self-adjoint extension, the big
bang singularity persists in the WDW theory while it is resolved and replaced
by a big bounce in loop quantum cosmology (LQC). Furthermore, the quantum
evolution is surprisingly insensitive to the choice of the self-adjoint
extension. This may be a special case of an yet to be discovered general
property of a certain class of symmetric operators that fail to be essentially
self-adjoint.
