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} }
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/item/AIF_2012__62_6_2347_0/
[1] 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] 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] 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] 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] 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] 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] 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] Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Tome 246 (2007) no. 2, pp. 145-195 | MR 2321040 | Zbl 1119.58018
[9] 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] Local smoothing for the Schrödinger equation with a prescribed loss (Preprint available at arXiv:1103.3908)
[11] 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] Gluing semiclassical resolvent estimates via propagation of singularities (Preprint available at arXiv:1008.3964, 2010)
[13] Scattering theory of classical and quantum -particle systems, Springer-Verlag, Berlin, Texts and Monographs in Physics. (1997) | MR 1459161 | Zbl 0899.47007
[14] Spectral asymptotics in the semiclassical limit, Cambridge University Press, London Math. Soc. Lecture Note Ser. 268 (1999) | MR 1735654 | Zbl 0926.35002
[15] Semiclassical analysis (Lecture notes available online at http://math.berkeley.edu/~zworski/semiclassical.pdf)
[16] 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] 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] 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] 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] Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces (Preprint available at arXiv:1103.3507, 2011)
[21] Quantum decay rates in chaotic scattering, Acta Math., Tome 203 (2009) no. 2, pp. 149-233 | MR 2570070 | Zbl 1226.35061
[22] 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] 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] -particle scattering problem: asymptotic completeness for short range systems, Ann. Math., Tome 125 (1987), pp. 35-108 | MR 898052 | Zbl 0646.47009
[25] 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] Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (Preprint available at arXiv: 1012.4391, 2010)
[27] Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates (Preprint available at arXiv:1104.1376, 2011)
[28] Semiclassical estimates in asymptotically Euclidean scattering, Comm. Math. Phys., Tome 212 (2000) no. 1, pp. 205-217 | MR 1764368 | Zbl 0955.58023
[29] Semiclassical resolvent estimates for -body Schrödinger operators, J. Funct. Anal., Tome 97 (1991) no. 2, pp. 466-483 | MR 1111191 | Zbl 0739.35047
[30] 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