275 research outputs found

    Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

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    This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we use semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, along with Mazzeo-Melrose 0-calculus, to construct the high energy resolvent on general non- trapping asymptotically hyperbolic manifolds, generalizing the work due to Melrose, Sa Barreto and Vasy. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.Comment: 49 page

    Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, I

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    We analyze the resolvent R(k)=(P+k2)−1R(k)=(P+k^2)^{-1} of Schr\"odinger operators P=Δ+VP=\Delta+V with short range potential VV on asymptotically conic manifolds (M,g)(M,g) (this setting includes asymptotically Euclidean manifolds) near k=0k=0. We make the assumption that the dimension is greater or equal to 3 and that PP has no L2L^2 null space and no resonance at 0. In particular, we show that the Schwartz kernel of R(k)R(k) is a conormal polyhomogeneous distribution on a desingularized version of M×M×[0,1]M\times M\times [0,1]. Using this, we show that the Riesz transform of PP is bounded on LpL^p for 1<p<n1<p<n and that this range is optimal if VV is not identically zero or if MM has more than one end. We also analyze the case V=0 with one end. In a follow-up paper, we shall deal with the same problem in the presence of zero modes and zero-resonances.Comment: 28 pages, 1 figur
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