We study the cut-off resolvent of semiclassical Schr{ö}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $λ^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $χ$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has
\begin{equation*}
\| χ(-h^2Δ+ V-λ^2)^{-1} χ\|_{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 $λ$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2Δ+ V-λ^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $λ$ with real part in a compact interval away from $0$ has imaginary part at most
\begin{equation*}
\mathrm{Im} λ\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 $-Δu = Vu$ 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*}
We first prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued potentials which are H{ö}lder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d $\ge$ 2, and real-valued potentials which are H{ö}lder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be H{ö}lder, Lipschitz or just L $\infty$ .
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.
We give an elementary proof of Burq's resolvent bounds for long range semiclassical Schroedinger operators. Globally, the resolvent norm grows exponentially in the inverse semiclassical parameter, and near infinity it grows linearly. We also weaken the regularity assumptions on the potential.
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.
In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schrödinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.
We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.
We give an elementary proof of weighted resolvent bounds for semiclassical Schrödinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.
In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $α$-Hölder continuity condition, $0\leq α\leq 1$, with sufficient decay of the local radial $C^α$ norm toward infinity. Note, however, that in the Hölder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - α}{3 + α}} [(1-α) \log(h^{-1})+c]\right)$, while …
We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical parameter.
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.
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.
In this article, we shall construct the resolvent of Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.
We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator.
In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.
We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian $h^2Δ_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space $[0,1)_h\times X\times X$ of $h$-dependent integral kernels; …
We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C > 0$. This $E$-depe…
We construct a semiclassical parametrix for the resolvent of the Laplacian acing on functions on non-trapping conformally compact manifolds with variable sectional curvature at infinity, we use it to prove high energy resolvent estimates and to show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to extend these results to conformally compact manifolds with normal hyperbolic trapping.
We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.
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…
In 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 $O(h |\log h|^{-α})$ below the real axis, for some $α\geq 0$, then the cut-off resolvent is actually bounded by $O(|\log h|^{α+1} h^{-1})$ in this strip. As an application, we improve slightly the estimates on the real axis given by Bourgain and Dyatlov in the case of convex co-compact surfaces.
Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.
We prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). In particular, we show that if V (x) = O x --$δ$ with $δ$ > 2, then the resolvent bound is of the form e Ch --4/3 with some constant C > 0. We also get resolvent bounds when 1 < $δ$ $\le$ 2. For slowly decaying $α$-H{ö}lder potentials we get better resolvent bounds of the form e Ch --4/($α$+3) .
In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type estimates by making the commutant also positive. The resolvent estimates, including the weighting of the Sobolev spaces in the estimates, are an immediate consequence.
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.
We study resolvent estimates for non-selfadjoint semiclassical pseudodifferential operators with double characteristics. Assuming that the quadratic approximation along the double characteristics is elliptic, we obtain polynomial upper bounds on the resolvent in a suitable region inside the pseudospectrum.
We extend Vasy's results on semiclassical high energy estimates for the meromorphic continuation of the resolvent for asymptotically hyperbolic manifolds to metrics that are not necessarily even. Vasy's method gives the meromorphic continuation of the resolvent and high energy estimates in strips, assuming that the geodesic flow is non-trapping, without having to construct a parametrix. We prove that the size of the strip to which the resolvent can be extended meromorphically is the same obtained by Guillarmou.
For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an ${\cal O}(h)$--neighborhood of the origin. Moreover, when al…
We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.
We study the linear relaxation Boltzmann equation, a simple semiclassical kinetic model. We provide a resolvent estimate for an associated non-selfadjoint operator as well as an estimate on the return to equilibrium. This is done using a scaling argument and non-semiclassical hypocoercive estimate.
We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$, with generalized Robin boundary conditions.
In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2π\mathbb{Z})^2$ with Hölder dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, …
We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.
We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart.
Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with $C^{1,1}$ boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.
The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics. The occurence of pseudospectra for non-selfadjoint semiclassical differential operators is due to the existence of quasimodes, i.e., approximate local solutions to the eigenvalue problem. For scalar operators,…
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 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 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 prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). We show that if V (x) = O x --$δ$ with $δ$ > 4, then the resolvent bound is of the form exp Ch -- $δ$ $δ$--1 log(h --1) 1 $δ$--1 with some constant C > 0. If V (x) = O e -- C x $α$ with C, $α$ > 0, we get better resolvent bounds of the form exp Ch --1 log(h --1
For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we establish semiclassical hypoelliptic a priori estimates with a loss of the full power of the semiclassical parameter, giving a localization for the low lying spectral values of the…
We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter.
In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $Ω$ with smooth boundary $\partialΩ$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partialΩ$. It will be shown that the resolvent difference of two generalized Rob…
For a second-order symmetric strongly elliptic operator A on a smooth bounded open set Ωin R^n with boundary Σ, the mixed problem is defined by a Neumann-type condition on a part Sigma_+ of the boundary and a Dirichlet condition on the other part Sigma_-. We show a Krein resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Sigma_+. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s_j j^{2/(n-1)}\t…
In this paper we are concerned with resolvent estimates for the Laplacian $Δ$ in Euclidean spaces. Uniform resolvent estimates for $Δ$ were shown by Kenig, Ruiz and Sogge \cite{KRS} who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $L^p$--$L^q$ bounds depending on $z$ has not been considered in a general framework which admits all possible $p,q$. In this paper, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$. We also obtain the sharp resolvent estimates for the f…
We improve the resolvent estimate in the Kreiss matrix theorem for a set of matrices that generate uniformly bounded semigroups. The new resolvent estimate is proved to be equivalent to Kreiss's resolvent condition, and it better describes the behavior of the resolvents at infinity.
We consider the operator ${\mathcal A}_h=-Δ+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.
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…
We study resolvent approximations for elliptic differential nonselfadjoint operators with periodic coefficients in the limit of the small period. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John-Nirenberg space $BMO$ (bounded mean oscillation). We apply the modified method of the first approximation with the usage of Steklov's smoothing.
We consider a semiclassical linear Boltzmann model with a non local collision operator. We provide sharp spectral asymptotics for the small spectrum in the low temperature regime from which we deduce the rate of return to equilibrium as well as a metastability result. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.
