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Publication# Decay Estimates for the One-dimensional Wave Equation with an Inverse Power Potential

Abstract

We study the wave equation on the real line with a potential that falls off like vertical bar x vertical bar(-alpha) for vertical bar x vertical bar -> infinity where 2 < alpha infinity provided that there are no resonances at zero energy and no bound states. As an application, we consider the l = 0 Price Law for Schwarzschild black holes. This paper is part of our investigations into decay of linear waves on a Schwarzschild background, see [5, 6].

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Wave equation

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics –

Schrödinger equation

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantu

Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational

We consider semiclassical Schr"odinger operators on the real line of the form $H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $f_\pm(\cdot,E;\hbar)$ with error terms that are uniformly controlled for small $E$ and $\hbar$, and construct the scattering matrix as well as the semiclassical spectral measure associated to $H(\hbar)$. This is crucial in order to obtain decay bounds for the corresponding wave and Schr"odinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta $\ell$ where the role of the small parameter $\hbar$ is played by $\ell^{-1}$. It follows from the results in this paper and \cite{DSS2}, that the decay bounds obtained in \cite{DSS1}, \cite{DS} for individual angular momenta $\ell$ can be summed to yield the sharp $t^{-3}$ decay for data without symmetry assumptions.