Johannes Sjoestrand
We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application we determine all resonances generated by a saddle point in a fixed disc around the critical energy.
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In this note we explicitly compute the resonances on hyperbolic cones. These are hyperbolic manifolds with a conic singularity equipped with a warped product metric. The calculation is based on separation of variables and Kummer's connection formulae for hypergeometric functions. To our knowledge this is the one of the few explicit calculations of resonances that does not rely on the resolvent being a two-point function.
Resonance counting and spectral gaps for hyperbolic surfaces via Fourier decay
该研究针对无限面积凸余紧双曲曲面，利用极限集的豪斯多夫维数与Patterson–Sullivan测度的傅里叶衰减指数，证明了改进的分形魏尔上界，进而得到其本质谱隙的更尖锐上界。
Upper bound on the number of resonances for even asymptotically hyperbolic manifolds with real-analytic ends
本文证明了具有实解析端的偶渐近双曲流形在半径趋于无穷大的圆盘内共振数量的多项式上界，并为Schwarzschild-de Sitter时空的拟正规频率数量提供了类似上界。
For manifolds Euclidian at infinity and compact perturbations of the Laplacian, we show that under assumptions involving hyperbolicity of the classical flow on the trapped set and its period spectrum, there are strips below the real axis where the resonance counting function grows sub-linearly. We also provide an inverse result, showing that the knowledge of the scattering poles can give some information about the Hausdorff dimension of the trapped set when the classical flow satisfies the Axiom-A condition.
The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimesional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. And, for large time, a new period for the dynamics appears : the wave packets revivals.
We prove an asymptotic formula for the number of scattering resonances in a strip near the real axis when the trapped set is r-normally hyperbolic with r large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and motivated by the setting of black holes. The key tool is a Fourier integral operator which microlocally projects onto the resonant states in the strip. In addition to Weyl law, this operator provides new information about microlocal concentration of resonant states.
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.
Existence of resonances for Schrodinger operators on hyperbolic space
本文利用热迹与波迹展开及散射相的渐近性，证明了双曲空间上薛定谔算子在光滑紧支撑势下的共振存在性与下界。
We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.
Let $Λ$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=Λ\backslash\H$ has infinitely many resonances in an effective strip depending on the dimension of the limit set $δ$. Applications to lower bounds for the hyperbolic lattice point counting problem are derived.
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 $X$ be a convex co-compact hyperbolic surface and let $δ$ be the Hausdorff dimension of the limit set of the underlying discrete group. We show that the density of the resonances of the Laplacian in strips ${σ\leq \re(s) \leq δ}$ with $|\im(s)| \leq T$ is less than $O(T^{1+δ-ε(σ)})$ with $ε>0$ as long as $σ>δ/2$. This improves the fractal Weyl upper bounds of Zworski and supports numerical results obtained for various models of quantum chaotic scattering.
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.
For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.
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.
For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.
We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectrums of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary con…
Let $Γ$ be a convex co-compact discrete group of isometries of the hyperbolic plane $\mathbb{H}^2$, and $X=Γ\backslash \mathbb{H}^2$ the associated surface. In this paper we investigate the behaviour of resonances of the Laplacian for large degree covers of $X$ given by a finite index normal subgroup of $Γ$. Using various techniques of thermodynamical formalism and representation theory, we prove two new existence results of "sharp non-trivial resonances" close to $\Re(s)=δ_Γ$, both in the large degree limit, for abelian covers and also infinite index congruence subgroups of $SL2(\mathbb{Z})$.
We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.
Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.
We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $δ$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-β,0]$ is $O(R^{m(β,δ)+})$, where the exponent $m(β,δ)=\min(2δ+2β+1-n,δ)$ changes its behavior at $β=(n-1-δ)/2$. In the case $δ<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ λ > δ-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view rece…
We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\mathrm{Im}\,λ\leq -\frac{1}{2}$ which is optimal in dimension $2$. In odd dimensions we also show that $\mathrm{Im}\,λ\leq -\fracμρ$ for a universal constant $μ$, where $ρ$ is the radius of a ball containing the obstacle; this gives an improvement for small obstacles. In dimensions $3$ and higher the proofs follow the classical vector field approach of Morawetz, while in dimension $2$ we obtain our bound by working with spaces coming from general relativity. We also show tha…
On geometrically finite hyperbolic manifolds $Γ\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bound is $N(R)= O(R^d (\log R)^{d+1})$.
Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions
本文研究了凸余紧双曲曲面，通过结合转移算子机制与振荡积分新估计，证明了改进的分形 Weyl 界，并推导出存在大无共振区域。
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.
We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hol…
For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.
The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigenvalues, relevant to estimate the long time asymptotics of the wave. In order to understand distribution of these resonances at high frequency, we employ semiclassical tools, which leads to considering the classical scattering problem, and in particular the set of trapped trajectories. We focus on "chaotic" situations, where this set is a hyperbolic repeller, generally with a fractal geometry. In this context, we derive fractal Weyl upp…
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.
Low-lying resonances for infinite-area hyperbolic surfaces with long closed geodesics
本文研究无限面积双曲曲面覆盖序列的低层共振分布，在最短闭测地线长度增长足够快的条件下证明低层共振的丰富性，并改进同余覆盖情形下的已有界。
In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on …
Counting Pollicott--Ruelle resonances for Axiom A flows
本文研究了开双曲系统和公理A流的Pollicott--Ruelle共振计数问题，证明了在横截性条件下模小于$r$的共振数量具有多项式上界和次线性下界。
Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders
本文证明当相关施赖埃尔图构成双侧扩展图时，凸余紧双曲曲面的覆盖族具有均匀谱隙，并给出新的通用无共振区域，推广了 Brooks、Bourgain-Gamburd-Sarnak 等先前结果。
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。
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.
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。
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with …
This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schrödinger operators in one dimension. We study the case where the two underlying classical Hamiltonian trajectories cross tangentially in the phase space. In the setting that one of the classical trajectories is a simple closed curve whereas the other one is non-trapping, we show that the imaginary part of the resonances is of order $h^{(m_0+3)/(m_0+1)}$, where $m_0$ is the maximal contact order of the crossings. This principal order comes from the subprincipal terms of the…
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard fl…
Given a closed orientable hyperbolic manifold of dimension $\neq 3$ we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numb…
For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $Γ$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $Γ$ is greater than $n/2$.
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