Spectrum of the Laplacian on asymptotically Euclidean spaces.
Spectrum of the Laplacian on asymptotically Euclidean spaces.
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渐近欧几里得空间上的拉普拉斯谱。
DOI：
10.1307/mmj/1030132362
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发表时间：
1999
影响因子：
0.9
通讯作者：
H. Donnelly
中科院分区：
文献类型：
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作者：
H. Donnelly
欧几里得空间器的拉普拉斯1具有以下性质：(a)−1的本质谱为[0，∞]；(b)1无点谱；(c) 1没有奇异连续谱。如果(x1, x2，…， xn)为r上的标准全局坐标，则耗尽函数b(x) = （x2 1 + x2 2 +···+ x2 n）满足(i)当x 6= 0时|∇b| = 1, (ii) Hessb2 = 2g。这里表示欧几里德度规。设M是一个完备的黎曼流形，它允许一个适当的耗尽函数。如果(i)和（ii）在弱或近似意义上被满足，那么我们想证明m的拉普拉斯算子1具有类似于欧几里德拉普拉斯算子的性质。这个程序在我们之前的论文b[6]中开始。在|1b|和||∇| - 1|的一般平均L2条件下，我们证明了- 1的本质谱为[0，∞]。为了消除1的点谱的可能性，|Hessb2 - 2g|和||∇| - 1|需要更严格的点衰减条件。在[6]中没有讨论奇异连续谱。本文扩展了前人关于点谱的工作，给出了关于奇异连续谱的新结果。如果M允许一个耗尽函数b具有性质2.1，则定理2.3表明1没有平方可积的特征函数。对于某些ε > 0，类似的结果需要更强的假设||∇| - 1|≤cb - ε和|Hessb2 - 2g|≤cb - ε，而性质2.1对这些量没有特定的衰减率。然而，性质2.1（iv）限制了b的三阶导数，而[6]中没有这样的条件。对于非负Ricci曲率、欧氏体积增长和二次曲率衰减的流形，Cheeger and Colding[3]和Colding and Minicozzi[4]构造了一个性质为2.1的耗尽函数。第3节研究了奇异连续谱。如果b满足性质3.1（比2.1更严格），则定理3.5表明−1没有奇异连续谱。[1]的渐近欧几里德空间支持具有性质3.1的耗尽函数。对于这些空间，曲率可能具有可变符号，但曲率衰减速度比二次型快。我们对奇异连续谱的处理是抽象摩尔定理的一个应用
The Laplacian1 for Euclidean spaceR has the following properties: (a) the essential spectrum of −1 is [0,∞); (b)1 has no point spectrum; and (c) 1 has no singular continuous spectrum. If (x1, x2, . . . , xn) are the standard global coordinates onR, then theexhaustion functionb(x) = (x2 1 + x2 2 + · · · + x2 n) satisfies (i)|∇b| = 1 for x 6= 0 and (ii) Hessb2 = 2g. Hereg denotes the Euclidean metric. Let M be a complete Riemannian manifold that admits a proper exhaustion functionb. If (i) and (ii) above are satisfied in a weak or approximate sense, then we would like to show that the Laplacian 1 ofM has properties similar to those of the Euclidean Laplacian. This program was started in our earlier paper [6]. Under general averaged L2 conditions on|1b| and ||∇b| − 1|, we showed that the essential spectrum of −1 is [0,∞). More stringent pointwise decay conditions for |Hessb2 − 2g| and||∇b| − 1| were needed to eliminate the possibility of a point spectrum for1. The singular continuous spectrum was not discussed in [6]. The present paper extends the earlier work concerning the point spectrum and provides new results about the singular continuous spectrum. If M admits an exhaustion functionb having Properties 2.1, then Theorem 2.3 states that 1 has no square integrable eigenfunctions. The analogous result in [6] required the stronger hypotheses||∇b|−1| ≤ cb−ε and|Hessb2−2g| ≤ cb−ε for someε > 0,whereas Properties 2.1 impose no specific decay rate on these quantities. However, Property 2.1(iv) restricts the third derivatives of b, whereas no such condition was imposed in [6]. For manifolds with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay, Cheeger and Colding [3] and Colding and Minicozzi [4] constructed an exhaustion function with Properties 2.1. The singular continuous spectrum is studied in Section 3. If b satisfies Properties 3.1 (which are more restrictive than 2.1) then Theorem 3.5 states that −1 has no singular continuous spectrum. The asymptotically Euclidean spaces of [1] support exhaustion functions with Properties 3.1. For these spaces, the curvature may have variable sign but the curvature decay is faster than quadratic. Our treatment of the singular continuous spectrum is an application of the abstract Mourre