Wave Equations with Point Interactions in Finite Energy Space

Abstract

Given the abstract wave equation $\ddotϕ-Δ_αϕ=0$, where $Δ_α$ is the Laplace operator with a point interaction of strength $α$, we define and study $\bar W_α$, the associated wave generator in the phase space of finite energy states. We prove the existence of the phase flow generated by $\bar W_α$, and describe its most relevant properties with particular emphasis on the associated symplectic structure and scattering theory