Cut-off Resolvent Research Articles

Wave Asymptotics for Waveguides and Manifolds with Infinite Cylindrical Ends

Abstract We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty $. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty $. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sjöstrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.

Resolvent estimates on asymptotically cylindrical manifolds and on the half line

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 when the resolvent is cut off away from part of the trapping, and along the way we prove some resolvent estimates for repulsive potentials on the half line which may be of independent interest.

Poisson wave trace formula for Dirac resonances at spectrum edges and applications

We study the self-adjoint Dirac operators D = D0 + V (x), where D0 is the free three-dimensional Dirac operator and V (x) is a smooth compactly supported Hermitian matrix potential. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges ±m, we establish a generalized Birman-Krein formula, taking into account possible resonances at ±m. As an application of the new Birman-Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on V , the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schr¨odinger operators.

Exponential lower resolvent bounds far away from trapped sets

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.

Semiclassical resolvent estimates for bounded potentials

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi \|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\Delta + V-\lambda^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} \lambda \leq - C^{-1} \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\Delta u = Vu$ with $0\not\equiv V\in L^{\infty}(\mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int_{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} \mathrm{e}^{-M \|V\|_{\infty}^{2/3} R^{4/3}}\|u\|^2_2. \end{equation*}

Local energy decay for Lipschitz wavespeeds

We prove a logarithmic local energy decay rate for the wave equation with a wavespeed that is a compactly supported Lipschitz perturbation of unity. The key is to establish suitable resolvent estimates at high and low energy for the meromorphic continuation of the cutoff resolvent. The decay rate is the same as that proved by Burq for a smooth perturbation of the Laplacian outside an obstacle.

Sharp resolvent bounds and resonance-free regions

ABSTRACTIn this note, we consider semiclassical scattering on a manifold which is Euclidean near infinity or asymptotically hyperbolic. We show that, if the cut-off resolvent satisfies polynomial estimates in a strip of size O(h|logh|−α) below the real axis, for some α≥0, then the cut-off resolvent is actually bounded by in this strip. As an application, we improve slightly the estimates on the real axis given recently by Bourgain and Dyatlov in the case of convex co-compact surfaces.

Resonance expansions of massless Dirac fields propagating in the exterior of a de Sitter–Reissner–Nordström black hole

We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter–Reissner–Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schrödinger operators from [4]. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.

Poisson wave trace formula for perturbed Dirac operator

We consider self-adjoint Dirac operators D = D0 + V(x), where D0 is the free three-dimensional Dirac operator and V(x) is a smooth compactly supported Hermitian matrix. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of D.

Spectral gaps for normally hyperbolic trapping

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 one-sided cutoff resolvent on the real line.

Resonances and lower resolvent bounds

We show how the presence of resonances close to the real axis implies exponential lower bounds on the norm of the cut-off resolvent on the real axis.

Resolvent estimates and local decay of waves on conic manifolds

We consider manifolds with conic singularities that are isometric to $\mathbb{R}^n$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a “very weak” Huygens’ principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schrödinger equation.

Semiclassical estimates of the cut-off resolvent for trapping perturbations

This paper is devoted to the study of the cut-off resolvent of a semiclassical “black box” operator P . We estimate the norm of \varphi ( P - z )^{- 1} \varphi , for any \varphi \in C^{\infty}_{0} ( \mathbb R^{n} ) , by the norm of {1\!\!1}_{\mathcal C_{a,b}} ( P - z )^{- 1} {1\!\!1}_{\mathcal C_{a,b}} where \mathcal C_{a,b} = \{ x \in \mathbb R^{n} ; \ a < \vert x \vert < b \} and a \gg 1 . For z in the unphysical sheet with - M h \vert \ln h \vert \leq \operatorname{Im} z \leq 0 , we prove that this estimate holds with a constant \frac{h}{\vert \operatorname{Im} z \vert} e^{C \vert \operatorname{Im} z \vert / h} . We also study the resonant states u of the operator P and we obtain bounds for \Vert \varphi u \Vert by \Vert 1\!\!1_{\mathcal C_{a,b}} u \Vert . These results hold without any assumption on the trapped set nor any assumption on the multiplicity of the resonances.

Resolvent estimates with mild trapping

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.

Local energy decay for the wave equation with a time-periodic non-trapping metric and moving obstacle

Consider the mixed problem with Dirichelet condition associated to the wave equation ∂ 2 u - divx(a(t,x)∇xu) = 0, where the scalar metric a(t,x) is T-periodic in t and uniformly equal to 1 outside a compact set in x, on a T-periodic domain. Let U(t,0) be the associated propagator. Assuming that the perturbations are non-trapping, we prove the meromorphic continuation of the cut-off resolvent of the Floquet operator U(T,0) and we establish sufficient conditions for local energy decay.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Abstract Let s 0 0 be the abscissa of absolute convergence of the dynamical zeta function Z ( s ) for several disjoint strictly convex compact obstacles K i ⊂ R N , i = 1 , … , κ 0 , κ 0 ⩾ 3 and let R χ ( z ) = χ ( − Δ D − z 2 ) −1 χ , χ ∈ C 0 ∞ ( R N ) , be the cut-off resolvent of the Dirichlet Laplacian − Δ D in Ω = R N ∖ ⋃ i = 1 κ 0 K i ¯ . We prove that there exists σ 2 s 0 such that Z ( s ) is analytic for R ( s ) ⩾ σ 2 and the cut-off resolvent R χ ( z ) has an analytic continuation for ℑ ( z ) − i σ 2 , | R ( z ) | ⩾ C . To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Applications of Cutoff Resolvent Estimates to the Wave Equation

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.

Cutoff resolvent estimates and the semilinear Schrödinger equation

This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrodinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrodinger equation.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Let s 0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z ( s ) for several disjoint strictly convex compact obstacles K i ⊂ R N , i = 1 , … , κ 0 , κ 0 ⩾ 3 and let R χ ( z ) = χ ( − Δ D − z 2 ) −1 χ , χ ∈ C 0 ∞ ( R N ) , be the cut-off resolvent of the Dirichlet Laplacian − Δ D in Ω = R N ∖ ⋃ i = 1 κ 0 K i ¯ . We prove that there exists σ 2 < s 0 such that Z ( s ) is analytic for R ( s ) ⩾ σ 2 and the cut-off resolvent R χ ( z ) has an analytic continuation for ℑ ( z ) < − i σ 2 , | R ( z ) | ⩾ C . To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Resolvent estimates and local energy decay for hyperbolic equations

We examine the cut-off resolvent R χ (λ) = χ (–Δ D – λ2)–1χ, where Δ D is the Laplacian with Dirichlet boundary condition and $\chi \in C_0^{\infty}(\mathbb{R}^n)$ equal to 1 in a neighborhood of the obstacle K. We show that if R χ (λ) has no poles for $\Im \lambda \geq -\delta,\: \delta > 0$ , then $\|R_{\chi}(\lambda)\|_{L^2 \to L^2} \leq C|\lambda|^{n-2},\: \lambda \in \mathbb{R},\: |\lambda| \geq C_0.$ This estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray. Keywords: Trapping obstacles, Resonances, Local energy decay, Cut-off resolvent