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Register a new userResolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach
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Andras Vasy
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We use a Lagrangian regularity perspective to discuss resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. In addition to the Lagrangian perspective we introduce and use a resolved pseudodifferential algebra to deal with zero energy degeneracies in a robust manner.
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We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum, as well as the `off-spectrum, spectral family is Fredholm in function spaces which encode the Lagrangian regularity of generalizations of `outgoing spherical waves of scattering theory, and indeed this persists in the `physical half plane.
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.
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.
We prove that both the Laplacian on functions, and the Lichnerowicz Laplacian on symmetric 2-tensors with respect to asymptotically hyperbolic metrics, are sectorial maps in weighted Holder spaces. As an application, the machinery of analytic semigroups then applies to yield well-posedness results for parabolic evolution equations in these spaces.
In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-.
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