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Register a new userResolvent estimates with mild trapping
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Authors
Jared Wunsch
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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.
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In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order emph{complex absorbing potential} gives an operator enjoying polynomial resolvent bounds on the real axis, then the resolvent associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.
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 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 $C_epsilon |lambda|^epsilon$ for any $epsilon>0$, 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 Schrodinger operator. We then classify the microlocal resolvent behaviour associated to every possible type of critical value of the potential, and translate this into the associated resolvent estimates. Weakly stable trapping admits highly concentrated quasimodes and fast growth of the resolvent. Conversely, using a delicate inhomogeneous blowup procedure loosely based on the classical positive commutator argument, we show that any weakly unstable trapping forces at least some spreading of quasimodes. As a first application, we conclude that either there is a resonance free region of size $| Im lambda | leq C_epsilon | Re lambda |^{-1-epsilon}$ for any $epsilon>0$, or there is a sequence of resonances converging to the real axis faster than any polynomial. Again, a stronger result holds if the manifold is analytic. As a second application, we prove a spreading result for weak quasimodes in partially rectangular billiards.
In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the degenerated directions of the subelliptic structure. First, for any $gammageq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $Delta_gamma=partial_x^2+|x|^{2gamma}partial_y^2$, which has step $gamma+1$. We then derive consequences for the observability of the Schrodinger type equation $ipartial_tu-(-Delta_gamma)^{s}u=0$ where $sin N$. We identify three different cases: depending on the value of the ratio $(gamma+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $partial_tu+(-Delta_gamma)^su=0$ and establish a decay rate for the damped wave equation associated with $Delta_{gamma}$.
We consider manifolds with conic singularites 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 Schr{o}dinger equation.
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