Wave Equations with Point Interactions in Finite Energy Space

TL;DR

This paper investigates wave equations with point interactions, defining a wave generator in finite energy space, proving phase flow existence, and analyzing symplectic and scattering properties.

Contribution

It introduces a new framework for wave equations with point interactions in finite energy space, including the existence and properties of the wave generator and phase flow.

Findings

Existence of phase flow generated by the wave operator

Description of symplectic structure in the finite energy space

Analysis of scattering theory for wave equations with point interactions

Abstract

Given the abstract wave equation $\ddot\phi-\Delta_\alpha\phi=0$, where $\Delta_\alpha$ 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