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发表时间:2017-06-19 阅读次数:219次
报告题目: Sharp spectral transitions and growth of eigenfunctions of Laplacians on Riemmannian manifolds
报 告 人:刘文才
报告人所在单位:University of California, Irvine
报告日期:2017-06-19 星期一
报告时间:15:00-16:00
报告地点:光华东主楼1801
  
报告摘要:

We study the eigenvalues or singular continuous spectrum of the free Laplacian embedded in the essential spectrum(absolutely continuous spectrum) on either asymptotically flat or asymptotically hyperbolic manifolds. The essential spectrum is $[\frac{c^2}{4},\infty]$ if $\Delta r \to c$ as $r$ goes to infinity, where $r(x)$ is the distance function. Kumura proved that there are no eigenvalues embedded in the essential spectrum $\sigma_{{\rm ess}}(-\Delta)=\left[\frac{1}{4}(n-1)^2,\infty\right)$ of Laplacians on asymptotically hyperbolic manifolds, where asymptotic  hyperbolicity is characterized by the radial curvature, i.e., $K_{\rm rad}=-1+o(r^{-1})$. He also constructed a manifold for which an eigenvalue $\frac{(n-1)^2}{4} + 1$ is embedded  into its essential spectrum $[ \frac{(n-1)^2}{4} , \infty )$ with the radial curvature $K_{\rm rad}(r)  = -1+O(r^{-1})$.

The first part of the talk, based on a joint work with S.Jitomirskaya, is devoted to construction of manifolds with embedded eigenvaluesand singular continous spectrum. Given any finite (countable)  positive energies $\{\lambda_n\}\in [\frac{K_0}{4}(n-1)^2,\infty)$, we construct     Riemannian manifolds  with the decay of order $K_{\rm rad}+K_0=O(r^{-1})$   with $K_0\geq 0$ ($K_{\rm rad}+K_0=\frac{C(r)}{r}$, where $C(r)\geq 0$ and $C(r)\to \infty $ arbitrarily slowly) such that  the eigenvalues $\{\lambda_n\}$  are embedded  in the essential spectrum  $\sigma_{{\rm ess}}(-\Delta)=\left[\frac{K_0}{4}(n-1)^2,\infty\right)$.

We also construct Riemannian manifolds with the decay of order $K{\rm rad}+K_0=\frac{C(r)}{r}$, where $C(r)\geq 0$ and $C(r)\to \infty $ arbitrarily slowly such that there is singular continous spectrum    embedded in the essential spectrum $\sigma_{{\rm ess}}(-\Delta)=\left[\frac{K_0}{4}(n-1)^2,\infty\right)$.

In the second part, I discuss criteria for the absence of eigenvalues embedded into essential spectrum in terms of the asymptotic behavior of $\Delta r$ . Under a weaker  convexity o.

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本年度学院报告总序号:141

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