Propagation through trapped sets and semiclassical resolvent estimates
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 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: 10.5802/aif.2751
Classification: 58J47,  35L05
Keywords: Resolvent estimates, trapping, propagation of singularities.
Datchev, Kiril 1; Vasy, András 2

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4397, U.S.A.
2 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.
@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},
     pages = {2347--2377},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     doi = {10.5802/aif.2751},
     mrnumber = {3060760},
     zbl = {1271.58014},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2751/}
}
TY  - JOUR
AU  - Datchev, Kiril
AU  - Vasy, András
TI  - Propagation through trapped sets and semiclassical resolvent estimates
JO  - Annales de l'Institut Fourier
PY  - 2012
DA  - 2012///
SP  - 2347
EP  - 2377
VL  - 62
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2751/
UR  - https://www.ams.org/mathscinet-getitem?mr=3060760
UR  - https://zbmath.org/?q=an%3A1271.58014
UR  - https://doi.org/10.5802/aif.2751
DO  - 10.5802/aif.2751
LA  - en
ID  - AIF_2012__62_6_2347_0
ER  - 
%0 Journal Article
%A Datchev, Kiril
%A Vasy, András
%T Propagation through trapped sets and semiclassical resolvent estimates
%J Annales de l'Institut Fourier
%D 2012
%P 2347-2377
%V 62
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2751
%R 10.5802/aif.2751
%G en
%F AIF_2012__62_6_2347_0
Datchev, Kiril; Vasy, András. 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/articles/10.5802/aif.2751/

[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., Volume 348 (2010) no. 23-24, pp. 1279-1282 | MR | Zbl

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

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

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

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

[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.), Volume 35 (2004) no. 3, pp. 333-344 | MR | Zbl

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

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

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

[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., Volume 286 (2009) no. 3, pp. 837-850 | MR | Zbl

[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, Texts and Monographs in Physics., Springer-Verlag, Berlin, 1997 | MR | Zbl

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

[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, Volume 15 (1990) no. 8, pp. 1161-1178 | MR | Zbl

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

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

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

[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., Volume 203 (2009) no. 2, pp. 149-233 | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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., Volume 212 (2000) no. 1, pp. 205-217 | MR | Zbl

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

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

Cited by Sources: