Semiclassical Low Energy Scattering for One-Dimensional Schrodinger Operators with Exponentially Decaying Potentials

We consider semiclassical Schrodinger operators on the real line of the form H((h) over bar = -(h) over bar (2) d(2)/dx(2) + V(.;(h) over bar) with small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions with error terms that are uniformly controlled for small E and , and construct the scattering matrix as well as the semiclassical spectral measure associated with . This is crucial in order to obtain decay bounds for the corresponding wave and Schrodinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta a"" where the role of the small parameter is played by a"" (-1). It follows from the results in this paper and Donninger et al. (Commun Math Phys 2009, arXiv:0911.3179), that the decay bounds obtained in Donninger et al. (Adv Math 226(1):484-540, 2011) and Donninger and Wilhelm (Int Math Res Not IMRN 22:4276-4300, 2010) for individual angular momenta a"" can be summed to yield the sharp t (-3) decay for data without symmetry assumptions.