Semiclassical low energy scattering for one-dimensional Schr\"odinger operators with exponentially decaying potentials

TL;DR

This paper develops semiclassical analysis techniques for one-dimensional Schrödinger operators with exponentially decaying potentials, providing uniform representations of solutions and spectral data, which are applied to wave decay on Schwarzschild backgrounds.

Contribution

It introduces uniform semiclassical representations of Jost solutions and the scattering matrix for exponentially decaying potentials, enabling decay analysis of wave equations in curved spacetime.

Findings

Uniform control of Jost solutions for small energy and semiclassical parameter

Construction of the scattering matrix and spectral measure with error bounds

Derivation of sharp decay bounds for wave equations on Schwarzschild backgrounds

Abstract

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.