  We propose new methods for calculation of the discrete spectrum, the
reflection amplitude and the correlation functions of boundary Liouville theory
on a strip with Lorentzian signature. They are based on the structure of the
vertex operator $V=e^{-\phi}$ in terms of the asymptotic operators. The methods
first are tested for the particle dynamics in the Morse potential, where
similar structures appear. Application of our methods to boundary Liouville
theory reproduces the known results obtained earlier in the bootstrap approach,
but there can arise a certain extension when the boundary parameters are near
to critical values. Namely, in this case we have found up to four different
equidistant series of discrete spectra, and the reflection amplitude is
modified respectively.
