Wave Equations with Point Interactions in Finite Energy Space

Given the abstract wave equation $\ddot{\varphi }-{\mathrm{\Delta }}_{\alpha }\varphi =0$, where ${\mathrm{\Delta }}_{\alpha }$ is the Laplace operator with a point interaction of strength $\alpha$, we define and study ${\bar{W}}_{\alpha }$, the associated wave generator in the phase space of finite energy states. We prove the existence of the phase flow generated by ${\bar{W}}_{\alpha }$, and describe its most relevant properties with particular emphasis on the associated symplectic structure and scattering theory