  The paper is devoted to the Hamiltonian treatment of classical and quantum
properties of Liouville field theory on a timelike strip in 2d Minkowski space.
We give a complete description of classical solutions regular in the interior
of the strip and obeying constant conformally invariant conditions on both
boundaries. Depending on the values of the two boundary parameters these
solutions may have different monodromy properties and are related to bound or
scattering states. By Bohr-Sommerfeld quantization we find the quasiclassical
discrete energy spectrum for the bound states in agreement with the
corresponding limit of spectral data obtained previously by conformal bootstrap
methods in Euclidean space. The full quantum version of the special vertex
operator $e^{-\phi}$ in terms of free field exponentials is constructed in the
hyperbolic sector.
