  The Lax-Phillips scattering theory is an appealing abstract framework for the
analysis of scattering resonances. Quantum mechanical adaptations of the theory
have been proposed. However, since these quantum adaptations essentially retain
the original structure of the theory, assuming the existence of incoming and
outgoing subspaces for the evolution and requiring the spectrum of the
generator of evolution to be unbounded from below, their range of applications
is rather limited. In this paper it is shown that if we replace the assumption
regarding the existence of incoming and outgoing subspaces by the assumption of
the existence of Lyapunov operators for the quantum evolution (the existence of
which has been proved for certain classes of quantum mechanical scattering
problems) then it is possible to construct a structure analogous to the
Lax-Phillips structure for scattering problems for which the spectrum of the
generator of evolution is bounded from below.
