Strichartz Estimates for Wave equations with Charge Transfer Hamiltonian

TL;DR

This paper establishes Strichartz estimates and related decay properties for wave equations with charge transfer Hamiltonians, facilitating the analysis of scattering states and nonlinear multisoltion systems in three-dimensional space.

Contribution

It provides the first proof of Strichartz estimates for wave equations with charge transfer Hamiltonians, including inhomogeneous versions, and demonstrates scattering behavior to free wave solutions.

Findings

Proved Strichartz estimates for wave equations with charge transfer potentials.

Established energy estimates and local energy decay for scattering states.

Showed scattering states asymptotically behave like free wave solutions.

Abstract

We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in $\mathbb{R}^{3}$: \[ \partial_{tt}u-\Delta u+\sum_{j=1}^{m}V_{j}\left(x-\vec{v}_{j}t\right)u=0. \] The energy estimate and the local energy decay of a scattering state are also established. In order to study nonlinear multisoltion systems, we will present the inhomogeneous generalizations of Strichartz estimates. As an application of our results, we show that scattering states indeed scatter to solutions to the free wave equation.