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We consider the deformed harmonic oscillator as a discrete version of theLiouville theory and study this model in the presence of local integrabledefects. From this, the time evolution of the defect degrees of freedom aredetermined, found in the form of the local equations of motion. We also revisitthe continuous Liouville theory, deriving its local integrals of motion andcomparing these with previous results from the sine-Gordon point ofview.Finally, the generic Backlund type relations are presented, correspondingto the implementation of time-like and space-like impurities in the continuummodel. Finally, we consider the interface of the Liouville theory with the freemassless theory. With the appropriate choice of the defect (Darboux) matrix weare able to derive the hetero-Backlund transformation for the Liouville theory.

Space & time discontinuities in Liouville theory and the deformed oscillator model