Propagation through trapped sets and semiclassical resolvent estimates
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2347-2377
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.
Motivé par l’étude des estimations de la résolvante dans la présence de capture, on démontre un théorème de propagation semiclassique dans un voisinage d’un sous-ensemble compact et invariant du flôt bicaractéristique, qui est isolé dans un sens convenable. Les exemples incluent un ensemble capté global et une trajectoire périodique isolée. Ceci est appliqué pour obtenir des estimations microlocales de la résolvante sans perte par rapport au cas non-captif.
DOI : https://doi.org/10.5802/aif.2751
Classification:  58J47,  35L05
Keywords: Resolvent estimates, trapping, propagation of singularities. 
@article{AIF_2012__62_6_2347_0,
     author = {Datchev, Kiril and Vasy, Andr\'as},
     title = {Propagation through trapped sets and semiclassical resolvent estimates},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2347-2377},
     doi = {10.5802/aif.2751},
     zbl = {1271.58014},
     mrnumber = {3060760},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_6_2347_0}
}

Propagation through trapped sets and semiclassical resolvent estimates. Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 2347-2377. doi : 10.5802/aif.2751. https://aif.centre-mersenne.org/item/AIF_2012__62_6_2347_0/

[1] Bony, J.-F.; Burq, N.; Ramond, T. Minoration de la résolvante dans le cas captif. [Lower bound on the resolvent for trapped situations], C. R. Math. Acad. Sci. Paris., Tome 348 (2010) no. 23-24, pp. 1279-1282 | MR 2745339 | Zbl 1206.35182

[2] Bony, J.-F.; Petkov, V. Resolvent estimates and local energy decay for hyperbolic equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., Tome 52 (2006) no. 2, pp. 233-246 | MR 2273096 | Zbl 1142.35059

[3] Burq, N. Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math., Tome 124 (2002) no. 4, pp. 677-735 | MR 1914456 | Zbl 1013.35019

[4] Burq, N.; Guillarmou, C.; Hassell, H. Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics, Geom. Funct. Anal, Tome 20 (2010), pp. 627-656 | MR 2720226 | Zbl 1206.58009

[5] Burq, N.; Zworski, M. Geometric control in the presence of a black box, J. Amer. Math. Soc., Tome 17 (2004) no. 2, pp. 443-471 | MR 2051618 | Zbl 1050.35058

[6] Cardoso, F.; Popov, G.; Vodev, G. Semi-classical resolvent estimates for the Schrödinger operator on non-compact complete Riemannian manifolds, Bull. Braz. Math. Soc. (N.S.), Tome 35 (2004) no. 3, pp. 333-344 | MR 2106308 | Zbl 1159.58308

[7] Cardoso, F.; Vodev, G. Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II, Ann. Henri Poincaré, Tome 3 (2002) no. 4, pp. 673-691 | MR 1933365 | Zbl 1021.58016

[8] Christianson, H. Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Tome 246 (2007) no. 2, pp. 145-195 | MR 2321040 | Zbl 1119.58018

[9] Christianson, H. Dispersive estimates for manifolds with one trapped orbit, Comm. Partial Differential Equations, Tome 33 (2008) no. 7, pp. 1147-1174 | MR 2450154 | Zbl 1152.58024

[10] Christianson, H.; Wunsch, J. Local smoothing for the Schrödinger equation with a prescribed loss (Preprint available at arXiv:1103.3908)

[11] Datchev, K. Local smoothing for scattering manifolds with hyperbolic trapped sets, Comm. Math. Phys., Tome 286 (2009) no. 3, pp. 837-850 | MR 2472019 | Zbl 1189.58016

[12] Datchev, K.; Vasy, A. Gluing semiclassical resolvent estimates via propagation of singularities (Preprint available at arXiv:1008.3964, 2010)

[13] Dereziński, J.; Gérard, C. Scattering theory of classical and quantum N -particle systems, Springer-Verlag, Berlin, Texts and Monographs in Physics. (1997) | MR 1459161 | Zbl 0899.47007

[14] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semiclassical limit, Cambridge University Press, London Math. Soc. Lecture Note Ser. 268 (1999) | MR 1735654 | Zbl 0926.35002

[15] Evans, L. C.; Zworski, M. Semiclassical analysis (Lecture notes available online at http://math.berkeley.edu/~zworski/semiclassical.pdf)

[16] Gérard, C. Semiclassical resolvent estimates for two and three-body Schrödinger operators, Comm. Partial Differential Equations, Tome 15 (1990) no. 8, pp. 1161-1178 | MR 1070240 | Zbl 0711.35095

[17] Gérard, C.; Sjöstrand, J. Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Partial Differential Equations, Tome 108 (1987), pp. 391-421 | MR 874901 | Zbl 0637.35027

[18] Greenleaf, A.; Uhlmann, G. Estimates for singular radon transforms and pseudodifferential operators with singular symbols, J. Func. Anal., Tome 89 (1990) no. 1, pp. 202-232 | MR 1040963 | Zbl 0717.44001

[19] Guillarmou, C. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds, Duke Math. J., Tome 129 (2005) no. 1, pp. 1-37 | MR 2153454 | Zbl 1099.58011

[20] Melrose, R. B.; Sá Barreto, A.; Vasy, A. Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces (Preprint available at arXiv:1103.3507, 2011)

[21] Nonnenmacher, S.; Zworski, M. Quantum decay rates in chaotic scattering, Acta Math., Tome 203 (2009) no. 2, pp. 149-233 | MR 2570070 | Zbl 1226.35061

[22] Petkov, Vesselin; Stoyanov, Luchezar Singularities of the scattering kernel related to trapping rays, Advances in phase space analysis of partial differential equations (volume 78 of Progr. Nonlinear Differential Equations Appl.) (2009), pp. 235-251 | MR 2664614 | Zbl 1197.35183

[23] Ralston, J. V. Trapped rays in spherically symmetric media and poles of the scattering matrix, Comm. Pure Appl. Math., Tome 24 (1971), pp. 571-582 | MR 457962 | Zbl 0206.39603

[24] Sigal, I. M.; Soffer, A. N-particle scattering problem: asymptotic completeness for short range systems, Ann. Math., Tome 125 (1987), pp. 35-108 | MR 898052 | Zbl 0646.47009

[25] Vasy, A. Geometry and analysis in many-body scattering, Inside out: inverse problems and applications (volume 47 of Math. Sci. Res. Inst. Publ.) (2003), pp. 333-379 | MR 2029685 | Zbl 1086.35508

[26] Vasy, A. Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (Preprint available at arXiv: 1012.4391, 2010)

[27] Vasy, A. Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates (Preprint available at arXiv:1104.1376, 2011)

[28] Vasy, A.; Zworski, M. Semiclassical estimates in asymptotically Euclidean scattering, Comm. Math. Phys., Tome 212 (2000) no. 1, pp. 205-217 | MR 1764368 | Zbl 0955.58023

[29] Wang, X. P. Semiclassical resolvent estimates for N-body Schrödinger operators, J. Funct. Anal., Tome 97 (1991) no. 2, pp. 466-483 | MR 1111191 | Zbl 0739.35047

[30] Wunsch, J.; Zworski, M. Resolvent estimates for normally hyperbolic trapped sets, Ann. Inst. Henri Poincaré (A)., Tome 12 (2011) no. 7, pp. 1349-1385 | MR 2846671 | Zbl 1228.81170