The Selberg Zeta Function for Convex Co-Compact Schottky Groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ⁿ⁺¹: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|) where is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|ⁿ⁺¹) , and it gives new bounds on the number of resonances (scattering poles) of \ⁿ⁺¹ . The proof of this result is based on the application of holomorphic L²-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider \ⁿ⁺¹ as the simplest model of quantum chaotic scattering.     

Rigidity of Asymptotically Hyperbolic Manifolds

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.The first author’s research is partially supported by NSF grant of China.The second author’s research is partially supported by an NSF grant and a Simon fund.     

Special Temporal Functions on Globally Hyperbolic Manifolds

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.     

Hyperbolic Systems and Propagation on Causal Manifolds

In this paper, which is essentially a survey, we solve the global Cauchy problem on causal manifolds for hyperbolic systems of linear partial differential equations in the framework of hyperfunctions. Besides the classical Cauchy–Kowalevsky theorem, our proofs only use tools and ideas of purely algebraic and geometric nature from the microlocal theory of sheaves.     

Asymptotically Hyperbolic Manifolds with Small Mass

For asymptotically hyperbolic manifolds of dimension n with scalar curvature at least equal to ?n(n ? 1) the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.     

Large Isoperimetric Regions in Asymptotically Hyperbolic Manifolds

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres are uniquely isoperimetric in manifolds that are Schwarzschild–anti-deSitter at infinity. These results have important repercussions for our understanding of spacelike hypersurfaces in Lorentzian space-times which are asymptotic to null infinity. In fact, as an application of our results, we verify the asymptotically hyperbolic Penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes.     

Scarring on Invariant Manifolds for Perturbed Quantized Hyperbolic Toral Automorphisms

We exhibit scarring for the quantization of certain nonlinear ergodic maps on the torus. We consider perturbations of hyperbolic toral automorphisms preserving certain co-isotropic submanifolds. The classical dynamics is ergodic, hence, in the semiclassical limit almost all quantum eigenstates converge to the volume measure of the torus. Nevertheless, we show that for each of the invariant submanifolds, there are also eigenstates which localize and converge to the volume measure of the corresponding submanifold.     

A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least ?6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter–Schwarzschild metrics.     

Local Smoothing for Scattering Manifolds with Hyperbolic Trapped Sets

We prove a resolvent estimate for the Laplace-Beltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate near the trapped region, a result of Burq and Cardoso-Vodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two.     

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author’s version (Liverani in Ann Math 159:1275–1312, 2004) of Dolgopyat’s estimates for contact flows and the first author’s work with Gouëzel (J Mod Dyn 4:91–137, 2010) on piecewise hyperbolic discrete-time dynamics.     

Resonance Wave Expansions:¶Two Hyperbolic Examples

For scattering on the modular surface and on the hyperbolic cylinder, we show that the solutions of the wave equations can be expanded in terms of resonances, despite the presence of trapping. Expansions of this type are expected to hold in greater generality but have been understood only in non-trapping situations. Received: 1 October 1999 / Accepted: 24 January 2000     

Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

Let {M _k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M _∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M _k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d ? 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).     

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.     

Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

守恒律方程组的大时间步长叠波格式

大时间步长叠波格式最初思想为LeVeque提出的大时间步长Godunov格式,通过叠加间断分解发出的强波来构造数值格式.原方法只给出了间断强波的穿越叠加方法,文章对其进行了完善,并推广到多维.针对膨胀波提出了一种网格单元分解法可以自动满足熵条件,避免出现非物理解.给出了格式的具体计算公式,并用单个守恒律方程、一维/多维Euler方程组进行了数值计算.计算结果表明,新格式除了可以采用大时间步长的优点外,在一定范围内随CFL数增加其耗散反而更低,因而对激波接触间断膨胀波的分辨率更高.     

Nonresonant Parametric Decay of a Finite Extent Lower Hybrid Wave

The decay of lower hybrid waves into nonresonant quasimodes inside the lower hybrid resonance cone is observed. Sideband frequency shifts of up to 30 percent from the pump frequency occur. The low-frequency mode is identified as a nonresonant quasi-mode satisfying the approximate conditions ?/k? ? ?e. The sideband wave-number and the sideband and quasi-mode growth rates were measured. The transition between ion-acoustic mode and quasi-mode decay spectra is observed as plasma density or pump frequency is varied.     

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior as long as the source term is bounded in the norm . In particular this gives scattering at small amplitudes for non-linear scalar fields of the form for all 2 < p. This paper is dedicated to the memory of Hope Machedon The second author would like thank MSRI and Princeton University, where a portion of this research was conducted during the Fall of 2005. The second author was also supported by a NSF postdoctoral fellowship.     

Energy Decay for the Damped Wave Equation Under a Pressure Condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This result holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.     

Parametric Decay of Electron Electrostatic Waves Pumped by an Electromagnetic Wave

In a computer simulation study, we have observed the parametric decay of an electromagnetic wave propagating parallel to the DC magnetic field into two electron electrostatic waves. We have identified the mechanism as three-wave parametric interaction. The growth constants of the decay waves have been measured. After saturation, the decay waves are observed to trap electrons. Consequently, the velocity distribution function develops a high velocity tail. The details of these processes are discussed. We have found that the kinetic energy of the total assembly of electrons will increase by about a factor of four.