渐近双曲流形上的逆散射

Radiation Fields, Scattering and Inverse Scattering on Asymptotically Hyperbolic Manifolds

In analogy with radiation fields for asymptotically Euclidean manifolds, introduced by F.G. Friedlander, we define radiation fields for asymptotically hyperbolic manifolds. We use them to give a translation representation of the wave group and to obtain the scattering matrix for such manifolds. Furthermore, we prove a support theorem for the radiation fields which generalizes to this setting the well known support theorem of Helgason and Lax and Phillips for the horocyclic Radon transform. This is used to show that the manifold and the metric, up to invariants, are determined by the scattering…

Fractal Weyl laws for asymptotically hyperbolic manifolds

For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them.

渐近双曲流形上薛定谔算子负特征值的分布

The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

本文研究渐近双曲流形上薛定谔算子负特征值的计数函数渐近性，给出势条件判定有限或无限个负特征值，并在无限情形下得到区间$(-\infty,-E)$内特征值计数当$E\to 0$时的渐近行为。

Absence of resonance near the critical line on asymptotically hyperbolic spaces

As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with constant curvature near infinity is also considered: there exists a strip with at most a finite number of resonances.

On the spectrum of the Laplace-Beltrami operator for p-forms on asymptotically hyperbolic manifolds

Under suitable conditions on the asymptotic decay of the metric, we compute the essential spectrum of the Laplace-Beltrami operator acting on $p$-forms on asymptotically hyperbolic manifolds.

Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace-Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in particular, does not require any global assumptions on the topology or the curvature, unlike previous papers on the same topic.

Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

We show that the resolvent of the Laplacian on asymptotically hyperbolic spaces extends meromorphically with finite rank poles to the complex plane if and only if the metric is `even' (in a sense). If it is not even, there exist some cases where the resolvent has an essential singularity in the non-physical sheet.

Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

关于渐近双曲流形反散射问题与Conformal Laplacian Calderón问题的相互作用

On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal Laplacian

本文通过渐近双曲流形与Conformal Laplacian的关系，证明了特定能量下的散射矩阵能确定边界度量的jet结构。

Resonances and scattering poles on asymptotically hyperbolic manifolds

On an asymptotically hyperbolic manifold (X,g), we show that the resolvent resonances coincide, with multiplicities, with the poles of the renormalized scattering operator, except for the special points n/2-k (with k>0 integer) where an additional term appears: this is the dimension of the kernel of the k-conformal Laplacian on the boundary when (X,g) is Einstein.

渐近双曲流形的散射极点

Scattering poles for asymptotically hyperbolic manifolds

对于一类包含实双曲空间的凸余紧离散群的投影的流形，我们证明了Laplacian算子的亚纯连续预解核的共振与亚纯连续散射算子的极点重合，并具有重数.为了进行证明，我们使用Shmuel Agmon的扰动理论的共振，以显示这两个预解共振和散射极点是简单的一般潜在的扰动。

Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds

We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting absorption principle for the resolvent and the absolute continuity of the continuous spectrum. (4)Construction of the generalized Fourier transform. (5)symptotic completeness of time-dependent wave operators. (6)Characterization of the space of scattering solutions to the Helmhotz equation in terms of the generalized Fourier transform. (7)Asymptotic expansion o…

On the spectral theory and dynamics of asymptotically hyperbolic manifolds

We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.

渐近双曲流形及其子流形的第一特征值估计

First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds

本文研究渐近双曲流形及其子流形上p-Laplacian第一特征值的估计，证明一类共形紧子流形渐近双曲，并给出极小情形下的精确特征值及有界平均曲率情形下的下界，推广了Cheung-Leung的结果。

Dynamics of Asymptotically Hyperbolic Manifolds

We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex co-compact hyperbolic manifolds which we use to prove a growth estimate for the length spectrum counting function. We next define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a pr…

Inverse scattering results for manifolds hyperbolic near infinity

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.