We consider semiclassical Schr?dinger operators on the real line of the form, with h{stroke} > 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 _±(·, E; h{stroke}) with error terms that are uniformly controlled for small E and h{stroke}, and construct the scattering matrix as well as the semiclassical spectral measure associated with H(h{stroke}). This is crucial in order to obtain decay bounds for the corresponding wave and Schr?dinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ? where the role of the small parameter h{stroke} is played by ? ~(-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 ? can be summed to yield the sharp t ~(-3) decay for data without symmetry assumptions.
展开▼
机译：我们考虑形式的实线上的半经典Schr？dinger运算符，且h {stroke}> 0小。假定电势V是平滑的，正的并且呈指数衰减到无穷大。我们建立了具有统一误差项的Jost解f _±（·，E; h {stroke}）的半经典全局表示，并为小E和h {stroke}统一控制了误差项，并构造了散射矩阵以及相关的半经典光谱测度与H（h {stroke}）。为了获得相应波和薛定er流的衰减边界，这至关重要。作为一种应用，我们考虑在Schwarzschild背景上针对大角动量的波动方程。小参数h {stroke}的作用是什么？ 〜（-1）。从本文和Donninger等人的结果可以得出。 （Commun Math Phys 2009，arXiv：0911。3179），该衰减范围在Donninger等人（2007年）中获得。 （Adv Math 226（1）：484-540，2011）和Donninger and Wilhelm（Int Math Res Not IMRN 22：4276-4300，2010）中的单个角动量？可以求和以得出没有对称假设的数据的急剧t〜（-3）衰减。
展开▼
