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

Abstract

We consider semiclassical Schrödinger operators on the real line of the form $$H(\hslash )=-{\hslash }^2\frac{d^2}{d{x}^2}+V(\cdot ;\hslash )$$ with $\hslash >0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $f_{\pm }(\cdot ,E;\hslash )$ with error terms that are uniformly controlled for small $E$ and $\hslash$, and construct the scattering matrix as well as the semiclassical spectral measure associated to $H(\hslash )$. This is crucial in order to obtain decay bounds for the corresponding wave and Schrödinger 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 $\hslash$ 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.

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

References in corpus (1)

• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Cited by in corpus (5)

• On pointwise decay of waves
• Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility
• Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schrödinger equation
• Full semiclassical asymptotics near transition points
• SL(3,C) structure of one-dimensional Schrödinger equation