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

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 $H$.

Made changes according to referee suggestions and fixed typos to improve the exposition. Added more detail to the sections discussing the weighted dispersive bound and the bounds when zero is not regular

References in corpus (5)

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Cited by in corpus (5)

• Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy
• The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions
• A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy
• Dispersive estimates for massive Dirac operators in dimension two
• The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates