On pointwise decay of waves
On pointwise decay of waves
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DOI：
10.1063/5.0042767
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
W. Schlag
中科院分区：
文献类型：
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作者：
W. Schlag
本文介绍了薛定谔算子在零能量附近的谱测度的行为与相关薛定谔和波演化的长期衰变和色散之间的关系。这些原则说明了作者的工作衰减薛定谔和波动方程下的各种类型的扰动，包括那些基本的度量。特别是，我们考虑局部衰减的线性薛定谔方程和波动方程的弯曲背景表现出捕获的解决方案。一个特别的应用是史瓦西黑洞时空上的波。我们详细说明价格的本地衰减规律，加速角动量，这是最近解决的Hintz，也在更困难的克尔黑洞设置。虽然作者的工作就同一主题进行了十年前，全球半经典表示技术开发有最近被应用的克里格，苗，和作者的非线性问题的稳定性爆破解决方案的临界波映射下的非等变扰动。
This note introduces some of the basic mechanisms relating the behavior of the spectral measure of Schrödinger operators near zero energy to the long-term decay and dispersion of the associated Schrödinger and wave evolutions. These principles are illustrated by means of the author’s work on decay of Schrödinger and wave equations under various types of perturbations including those of the underlying metric. In particular, we consider local decay of solutions to the linear Schrödinger and wave equations on curved backgrounds which exhibit trapping. A particular application are waves on a Schwarzschild black hole space-time. We elaborate on Price’s law of local decay which accelerates with the angular momentum, which has recently been settled by Hintz, also in the much more difficult Kerr black hole setting. While the author’s work on the same topic was conducted ten years ago, the global semiclassical representation techniques developed there have recently been applied by Krieger, Miao, and the author to the nonlinear problem of stability of blowup solutions to critical wave maps under non-equivariant perturbations.