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. 本年度学院报告总序号： 141