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Local smoothing for scattering manifolds with hyperbolic trapped sets. (English) Zbl 1189.58016
For a noncompact complete Riemannian manifold, an effective way to understand its spectrum is to study its resolvent. A particular nice class of manifolds to which the method of microlocal analysis can be applied is the class of scattering manifolds, introduced by Richard Melrose. The author proves an estimate for the resolvent of a scattering manifold with a hyperbolic trapped set and deduces local smoothing for \(L^2\)-functions.

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
35P25 Scattering theory for PDEs
Full Text: DOI arXiv
[1] Burq N.: Lower bounds for shape resonance widths of long range Schrödinger operators. Amer. J. Math. 124, 677–735 (2002) · Zbl 1013.35019 · doi:10.1353/ajm.2002.0020
[2] Burq N.: Smoothing effect for Schrödinger boundary value problems. Duke Math. J. 123, 403–427 (2004) · Zbl 1061.35024 · doi:10.1215/S0012-7094-04-12326-7
[3] Burq N., Gérard P., Tzvetkov N.: On nonlinear Schrödinger equations in exterior domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 295–318 (2004) · Zbl 1061.35126 · doi:10.1016/S0294-1449(03)00040-4
[4] Cardoso F., Vodev G.: Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds II. Ann. Henri Poincaré 3, 673–691 (2002) · Zbl 1021.58016 · doi:10.1007/s00023-002-8631-8
[5] Christianson H.: Cutoff resolvent estimates and the semilinear Schrödinger equation. Proc. Am. Math. Soc. 136, 3513–3520 (2008) · Zbl 1156.35085 · doi:10.1090/S0002-9939-08-09290-3
[6] Constantin P., Saut J.-C.: Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1, 413–439 (1988) · Zbl 0667.35061 · doi:10.1090/S0894-0347-1988-0928265-0
[7] Doi S.-I.: Smoothing effects of Schrödinger evolution groups on Riemannian manifolds. Duke Math. J. 82, 679–706 (1996) · Zbl 0870.58101 · doi:10.1215/S0012-7094-96-08228-9
[8] Evans, L.C., Zworski, M.: Lectures on semiclassical analysis. Lecture notes, available at http://math.berkeley.edu/\(\sim\)zworski/semiclassical.pdf , 2003
[9] Guillarmou C., Hassell A.: The resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, Part I. Math. Ann. 341, 859–896 (2008) · Zbl 1141.58017 · doi:10.1007/s00208-008-0216-5
[10] Hörmander L.: On the existence and the regularity of solutions of linear pseudo-differential equations. Enseign. Math. 2, 99–163 (1971) · Zbl 0224.35084
[11] Joshi M., Sá Barreto A.: Recovering asymptotics of metrics from fixed energy scattering data. Inv. Math. 137, 127–143 (1999) · Zbl 0953.58025 · doi:10.1007/s002220050326
[12] Kato T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1966) · Zbl 0139.31203 · doi:10.1007/BF01360915
[13] Melrose, R.: Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. In: Spectral and Scattering Theory, M. Ikawa, ed., New York: Marcel Dekker, 1994, pp. 85–130 · Zbl 0837.35107
[14] Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering, preprint, 2007, available at http://math.berkeley.edu/\(\sim\)zworski/nz3.pdf , 2007 · Zbl 1226.35061
[15] Schrohe, E.: Spaces of weighted symbols and weighted Sobolev spaces on manifolds. In: Pseudodifferential Operators, Lecture Notes in Mathematics 1256, Berlin: Springer-Verlag, 1987, pp. 360–377 · Zbl 0638.58026
[16] Sjölin P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987) · Zbl 0631.42010 · doi:10.1215/S0012-7094-87-05535-9
[17] Vasy A., Zworski M.: Semiclassical estimates in asymptotically Euclidean scattering. Commun. Math. Phys. 212, 205–217 (2000) · Zbl 0955.58023 · doi:10.1007/s002200000207
[18] Vega L.: Schrödinger equations: pointwise convergence to the initial data. Proc. Am. Math. Soc. 102, 874–878 (1988) · Zbl 0654.42014
[19] Wang X.P.: Asymptotic expansion in time of the Schrödinger group on conical manifolds. Ann. Inst. Fourier (Grenoble) 56, 1903–1945 (2006) · Zbl 1118.35022
[20] Wunsch J., Zworski M.: Distribution of resonances for asymptotically Euclidean manifolds. J. Diff. Geom. 55, 43–82 (2000) · Zbl 1030.58024
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