Colin Guillarmou
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Consider the wave equation $\partial_t^2u-Δ_xu+V(t,x)u=0$, where $x\in\R^n$ with $n\geq3$ and $V(t,x)$ is $T$-periodic in time and decays exponentially in space. Let $ U(t,0)$ be the associated propagator and let $R(θ)=e^{-D<x>}(U(T,0)-e^{-iθ})^{-1}e^{-D<x>}$ be the resolvent of the Floquet operator $U(T,0)$ defined for $\im(θ)>BT $ with $B>0$ sufficiently large. We establish a meromorphic continuation of $R(θ)$ from which we deduce the asymptotic expansion of $e^{-(D+ε)<x>}U(t,0)e^{-D<x>}f$, where $f\in \dot{H}^1(\R^n)\times L^2(\R^n)$, as $t\to+\infty$ with a remainder term whose energy deca…
Manifolds with infinite cylindrical ends have continuous spectrum of increasing multiplicity as energy grows, and in general embedded resonances (resonances on the real line, embedded in the continuous spectrum) and embedded eigenvalues can accumulate at infinity. However, we prove that if geodesic trapping is sufficiently mild, then the number of embedded resonances and eigenvalues is finite, and moreover the cutoff resolvent is uniformly bounded at high energies. We obtain as a corollary the existence of resonance free regions near the continuous spectrum. We also obtain improved estimates w…
Resolvent estimate for asymptotically hyperbolic surfaces without evenness assumption
本文证明了一类非紧流形上拉普拉斯算子的尖锐截断预解估计，并将其应用于负曲率渐近双曲曲面，得到无需无穷远偶性假设的全局Strichartz与谱投影估计。
We prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. Our point is to use weights that fit the pseudo-differential calculus associated with the asymptotically hyperbolic geometry. Our proof is a combination of a trick using Mourre theory and of results by Froese-Hislop and Cardoso-Vodev.
This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schrödinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schrödinger operators $-Δ+V_1$ and $-Δ+V_2$ have the same poles, with both potentials b…
For geometrically finite hyperbolic manifolds $Γ\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of $Γ$ in large balls of $H^{n+1}$ in terms of the Hausdorff dimension of the limit set of $Γ$.
Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(Δ-z)^{-1}, z\in\C\setminus\R^{+},$ has a meromorphic continuation through $\R^{+}$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $Δ+V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian.
For a class of manifolds that includes quotients of real hyperbolic space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian coincide, with multiplicities, with the poles of the meromorphically continued scattering operator. In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.
We present the Laplace operator associated to a hyperbolic surface $Γ\setminus\mathbb{H}$ and a unitary representation of the fundamental group $Γ$, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of $\mathbb{C}$ by constructing a parametrix for the Laplacian, following the approach by Guillopé and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.
We consider the resolvent on asymptotically Euclidean warped product manifolds in an appropriate 0-Gevrey class, with trapped sets consisting of only finitely many components. We prove that the high-frequency resolvent is either bounded by $C_ε|λ|^ε$ for any $ε>0$, or blows up faster than any polynomial (at least along a subsequence). A stronger result holds if the manifold is analytic. The method of proof is to exploit the warped product structure to separate variables, obtaining a one-dimensional semiclassical Schrödinger operator. We then classify the microlocal resolvent behaviour associat…
Let $A$ be an elliptic operator on a compact manifold with boundary $M$, and let $\wp : \partial\M \to Y$ be a covering map, where $Y$ is a closed manifold. Let $A_C$ be a realization of $A$ subject to a coupling condition $C$ that is elliptic with parameter in the sector $Λ$. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace $\Tr_{L^2} (A_C-λ)^{-N}$ for $N$ sufficiently large has a complete asymptotic expansion as $|λ| \to \infty$, $λ\in Λ$. In particular, the heat trace $\Tr_{L^2}e^{-tA_C}$ ha…
Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting
本文证明了渐近欧氏设置下波方程的局部衰减估计，在偶数维给出大时间渐近轮廓超越最优衰减，在奇数维改进已知最佳估计，结果适用于具短程吸收指数的阻尼波方程。
We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds.
We consider families of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(s) be the Selberg Zeta function of a surface, and let Z_d(s) be the contribution of the pinched geodesics to the Zeta function. Extending a result of Hejhal and Wolpert, we prove that the quotient of these two terms converges to the Zeta function of the limit surface for all arguments s with re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent is sh…
We give a complete classification and present new exotic phenomena of the meromorphic structure of $ζ$-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these $ζ$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give …
We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of Kashiwara-Oshima. From this, we derive that the poles of the corresponding meromorphic continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp--Stein intertwiner. As a by-pro…
We develop the complex scaling method for the Dirichlet Laplacian in a domain with asymptotically cylindrical end. We define resonances as discrete eigenvalues of non-selfadjoint operators, obtained as deformations of the selfadjoint Dirichlet Laplacian $Δ$ by means of the complex scaling. The resonances are identified with the poles of the resolvent matrix elements $((Δ-μ)^{-1}F, G)$ meromorphic continuation in $μ$ across the essential spectrum of $Δ$, where $F$ and $G$ are elements of an explicitly given set of partial analytic vectors. It turns out that the Dirichlet Laplacian has no singul…
We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previously studied by Froese-Hislop-Perry.
Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $Δ$ be the positive Laplacian associated to $g$, and $P = Δ+ V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(λ) = (λ/πi) \big(R(λ+i0) - R(λ- i0) \big)$ of $P_+^{1/2}$, where $R(λ) = (P - λ^2)^{-1}$, as $λ\to 0$, in a manner similar to that done previously by the second author an…
Microlocal analysis and spectral gap for complex hyperbolic manifolds
本文证明凸余紧复双曲流形在极限集不含复圆时存在本质谱隙，通过Quan推广的Vasy方法、Dyatlov-Zahl微局域技术及Bourgain-Dyatlov分形不确定性原理获得新的预解式界。
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.
Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C_0(A,P) in the trace expansion of A(P-λ)^{-N} (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A is extended from the cases of noninteger order, or integer order and even-even parity on odd-dimensional manifolds, to the case of even-odd parity on even-dimensional manifolds. For the ge…
Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent
本研究针对带纤维边界度量的流形上的Hodge拉普拉斯算子低能预解式，通过引入拓展自Grieser和Hunsicker工作的分裂伪微分演算，明确了其参数趋于零时的精确渐近行为，推广了渐近锥形度量的相关结果，为$ϕ$-流形谱不变量研究奠定基础。
We develop the complex scaling for a manifold with an asymptotically cylindrical end under an assumption on the analyticity of the metric with respect to the axial coordinate of the end. We allow for arbitrarily slow convergence of the metric to its limit at infinity, and prove a variant of the Aguilar-Balslev-Combes theorem for the Laplacian $Δ$ on functions. In the case of a manifold with (noncompact) boundary it is either the Dirichlet or the Neumann Laplacian. We introduce resonances as the discrete non-real eigenvalues of non-selfadjoint operators, obtained as deformations of the Laplacia…
Consider a compact manifold with boundary $M$ with a scattering metric $g$ or, equivalently, an asymptotically conic manifold $(M^\circ, g)$. (Euclidean $\mathbb{R}^n$, with a compactly supported metric perturbation, is an example of such a space.) Let $Δ$ be the positive Laplacian on $(M,g)$, and $V$ a smooth potential on $M$ which decays to second order at infinity. In this paper we construct the kernel of the operator $(h^2 Δ+ V - (λ_0 \pm i0)^2)^{-1}$, at a nontrapping energy $λ_0 > 0$, uniformly for $h \in (0, h_0)$, $h_0 > 0$ small, within a class of Legendre distributions on manifolds w…
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.
In this paper, we investigate the long-time structure of the heat kernel on a Riemannian manifold M which is asymptotically conic near infinity. Using geometric microlocal analysis and building on results of Guillarmou and Hassell on the low-energy resolvent, we give a complete description of the asymptotic structure of the heat kernel in all spatial and temporal regimes. We apply this structure to define and investigate a renormalized zeta function and determinant of the Laplacian on M.
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with th…
We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.
Scattering theory of the $p$-form Laplacian on manifolds with generalized cusps
本文研究具有扭曲乘积型类尖端度量奇性流形上的散射理论，证明了 p-形式 Laplace 算子的预解式可延拓为亚纯函数，由此构造广义特征形式并实现散射矩阵的亚纯延拓，发现其满足函数方程。
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…
Resonances on geometrically finite graphs
研究几何有限图上邻接算子的解析延拓及共振态的几何特征，揭示其与双曲面相似的定性现象，且有限字段上代数曲线提供有趣例子。
Analysis on fibred cusp spaces
本文综述了纤维尖点空间的分析与几何结果，涵盖谱几何、分析扭量理论及边界值问题，重点讨论不完整与完整设置的异同。
Resonances and residue operators for pseudo-Riemannian hyperbolic spaces
本文研究实、复、四元数与八元数伪黎曼双曲空间上 Laplace–Beltrami 算子预解式的亚纯延拓，显式确定全部共振，并将留数像刻画为退化主序列的子表示。
Lossless Strichartz and spectral projection estimates on unbounded manifolds
本文研究了在非紧流形上无损斯特里茨茨和谱投影估计的改进方法，并通过新的残差估计获得了相关结果。
We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X$ and prove that $η(D)=\frac{1}{πi}\log Z_{Γ,Σ}^{\rm o}(0)$, thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。