Scattering theory for Riemannian Laplacians
Scattering theory for Riemannian Laplacians
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DOI：
10.1016/j.jfa.2013.02.002
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发表时间：
2011-09
影响因子：
1.7
通讯作者：
Kenichi Ito;E. Skibsted
中科院分区：
文献类型：
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作者：
Kenichi Ito;E. Skibsted
我们引入了Laplace-Beltrami算子在非紧连通完备黎曼流形上的散射理论的概念。利用无穷远处角子流形的第二基本形式的某个正下界给出了一个主要条件。另一个条件是导数的某些界限，直到这个量的迹的阶之一。这些条件被证明是最佳的存在性和完整性的波算子。我们的理论不涉及规定的渐近行为的度量在无穷远（如渐近欧几里德或双曲度量研究以前在文献中）。该理论的一个结果是Laplace-Beltrami算子的谱理论，包括连续谱的识别和奇异连续谱的不存在。
We introduce a notion of scattering theory for the Laplace–Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behavior of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace–Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.