Space and time discontinuities in Liouville theory and its discrete analogue

We consider the deformed harmonic oscillator as a discrete version of the Liouville theory and study this model in the presence of local integrable defects. From this, the time evolution of the defect degrees of freedom are determined, found in the form of the local equations of motion. We also revisit the continuous Liouville theory, deriving its local integrals of motion and comparing these with previous results from the sine-Gordon point of view.Finally, the generic Backlund type relations are presented, corresponding to the implementation of time-like and space-like impurities in the continuum model. Finally, we consider the interface of the Liouville theory with the free massless theory. With the appropriate choice of the defect (Darboux) matrix we are able to derive the hetero-Backlund transformation for the Liouville theory.