  Coupling any interacting quantum mechanical system to gravity in one (time)
dimension requires the cosmological constant to belong to the matter energy
spectrum and thus to be quantised, even though the gravity sector is free of
any quantum dynamics. Furthermore, physical states are also confined to the
subspace of the matter quantum states for which the energy coincides with the
value of the cosmological constant. These general facts are illustrated through
some simple examples. The physical projector quantisation approach readily
leads to the correct representation of such systems, whereas other approaches
relying on gauge fixing methods are often plagued by Gribov problems in which
case the quantisation rule is not properly recovered. Whether such a
quantisation of the cosmological constant as well as the other ensuing
consequences in terms of physical states extend to higher dimensional
matter-gravity coupled quantum systems is clearly a fascinating open issue.
