Resolvent estimates for normally hyperbolic trapped sets
Jared Wunsch, Maciej Zworski
备注：
Further changes to erratum correcting small problems with Section 3.5 and Lemma 4.1; this now also corrects hypotheses, explicitly requiring trapped set to be symplectic. Erratum follows references in this version
We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schrödinger propagator as well as local energy decay results for the wave equation.
We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs microlocally supported away from the trapping, a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs to be supported at the trapped set, giving estimates which are intermediate between the trapping and non-trapping ones.
This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.
Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.
We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal size of the strip and give an $o(h^{-2})$ resolvent bound there. We next show existence of deeper resonance free strips under the $r$-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the ne-sided cutoff resolvent on the real line.
We give examples of semiclassical Schrödinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the resolvent kernel in detail using ordinary differential equation techniques. In particular, we identify a threshold spatial radius where the resolvent behavior changes. We apply these results to wave equations with radial wavespeed, identifying a corresponding threshold radius at which wave decay properties change.
We establish a mixed-norm Strichartz type estimate for the wave equation on Riemannian manifolds $(Ω,g)$, for the case that $Ω$ is the exterior of a smooth, normally hyperbolic trapped obstacle in $n$ dimensional Euclidean space, and $n$ is a positive odd integer. As for the normally hyperbolic trapped obstacles, we will some loss of derivatives for data in the local energy decay estimate. Hence the global Strichartz estimate has a derivative loss. However, we can show that the forcing term is bounded by the sum of no more than two Lebesgue $(p,q)$ mixed norms.
We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.
In this paper we prove semiclassical resolvent estimates for operators with normally hyperbolic trapping which are lossless relative to non-trapping estimates but take place in weaker function spaces. In particular, we obtain non-trapping estimates in standard $L^2$ spaces for the resolvent sandwiched between operators which localize away from the trapped set $Γ$ in a rather weak sense, namely whose principal symbols vanish on $Γ$.
We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normal hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green's functions (cut-off resolvents) and polynomial bounds on the growth sof those functions…
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.
We prove resolvent estimates for semiclassical operators such as $-h^2 Δ+V(x)$ in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by $h^{-M}$ in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrödinger propagation and to energy decay of solutions to wave equations.
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.
We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension.
We show that the resolvent grows at most exponentially with frequency for the wave equation on a class of stationary spacetimes which are bounded by non-degenerate Killing horizons, without any assumptions on the trapped set. Correspondingly, there exists an exponentially small resonance-free region, and solutions of the Cauchy problem exhibit logarithmic energy decay.
We prove a local smoothing result for the Schrödinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm}) depending on the accumulation rate of the critical points of the profile curve. The proof uses an h-dependent version of semiclassical propagation of singularities, and a result on gluing an h-dependent number of cutoff resolvent estimates.
Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.
We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a norm…
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…
We consider a family of spherically symmetric, asymptotically Euclidean manifolds with two trapped sets, one which is unstable and one which is semi-stable. The phase space structure is that of an inflection transmission set. We prove a sharp local smoothing estimate for the linear Schrödinger equation with a loss which depends on how flat the manifold is near each of the trapped sets. The result interpolates between the family of similar estimates in \cite{ChWu-lsm}. As a consequence of the techniques of proof, we also show a sharp high energy resolvent estimate with a polynomial loss dependi…
For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: \int_0^T || ρ_s e^{it(Δ_g-V)} u_0 ||_{H^{1/2 - ε}(M)}^2 dt \leq C_T || u_0 ||_{L^2(M)}^2, where $ρ_s(x) \in \Ci(M)$ satisfies $ρ_s = <\dist_g(x,x_0)>^{-s}$, $s> \half$, and $V \in \Ci(M)$, $0 \leq V \leq C$ satisfies $|\nabla V| \leq C <\dist(x,x_0)>^{-1-δ}$ for some $δ>0$. From the local smoothing estimate, we…
The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not imposed. The first key assumption guarantees the result on propagation of singularities, the second is the smallness of the Lebesgue measure of the portions of the trapped sets in the fibers of cosphere bundle, and the third concerns the upper bound of the sojourn time. As a by-product of these estimates, the local energy decay estimate for solutions to the …
This note places primary emphasis on improving the asymptotic behavior of a multi-dimensional delayed wave equation in the absence of any displacement term. In the first instance, the delay is assumed to occur in the boundary. Then, invoking a geometric condition BLR on the domain, the exponential convergence of solutions to their equilibrium state is proved. The strategy adopted of the proof is based on an interpolation inequality combined with a resolvent method. In turn, an internal delayed wave equation is considered in the second case, where the domain possesses trapped ray and hence (BLR…
This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--inc…
We address the decay rates of the energy for the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate $1/\sqrt{t}$ (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expec…
We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we…
