Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials

Details Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials Semiclassical low energy scattering for one-dimensional Schrödinger operators with exponentially decaying potentials

2011-05-21

arxiv.org/abs/1105.4221v2

Mathematics

Spectral Theory

44 pages, minor modifications in order to match the published version, will appear in Annales Henri Poincare

Scientific paper

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.

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