  We show that the existence of the family of self-adjoint Lyapunov operators
introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition
of the state of a quantum mechanical system into two parts: A past time
asymptote, which is asymptotic to the state of the system at t goes to minus
infinity and vanishes at t goes to plus infinity, and a future time asymptote,
which is asymptotic to the state of the system at t goes to plus infinity and
vanishes at t goes to minus infinity. We demonstrate the usefulness of this
decomposition for the description of resonance phenomena by considering the
resonance scattering of a particle off a square barrier potential. We show that
the past time asymptote captures the behavior of the resonance. In particular,
it exhibits the expected exponential decay law and spatial probability
distribution.
