Time decay estimates for the wave equation with potential in dimension two

TL;DR

This paper investigates decay estimates for solutions to the wave equation with a potential in two dimensions, establishing decay rates under spectral assumptions and analyzing the impact of zero being a regular point.

Contribution

It provides new dispersive decay estimates for the wave equation with potential in two dimensions, including cases where zero is or isn't a regular spectral point.

Findings

Dispersive estimate with decay rate |t|^{-1/2}

Faster decay rate |t|^{-1}(log|t|)^{-2} for large |t|

Dispersive estimates when zero is not a regular spectral point

Abstract

We study the wave equation with potential $u_{tt}-\Delta u+Vu=0$ in two spatial dimensions, with $V$ a real-valued, decaying potential. With $H=-\Delta+V$, we study a variety of mapping estimates of the solution operators, $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$ under the assumption that zero is a regular point of the spectrum of $H$. We prove a dispersive estimate with a time decay rate of $|t|^{-\frac{1}{2}}$, a polynomially weighted dispersive estimate which attains a faster decay rate of $|t|^{-1}(\log |t|)^{-2}$ for $|t|>2$. Finally, we prove dispersive estimates if zero is not a regular point of the spectrum of $H$.