Estimates for the cut-off resolvent of the Laplacian for trapping obstacles
Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 2, 12 p.
@article{SEDP_2005-2006____A2_0,
author = {Bony, Jean-Fran\c cois and Petkov, Vesselin},
title = {Estimates for the cut-off resolvent of the Laplacian for trapping obstacles},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2005-2006},
note = {talk:2},
mrnumber = {2276068},
language = {en},
url = {http://www.numdam.org/item/SEDP_2005-2006____A2_0}
}

Bony, Jean-François; Petkov, Vesselin. Estimates for the cut-off resolvent of the Laplacian for trapping obstacles. Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 2, 12 p. http://www.numdam.org/item/SEDP_2005-2006____A2_0/

[1] J.-F. Bony, Résonances dans des domaines de taille h, Inter. Math. Res. Notes, 16 (2001), 817-847. | MR 1853138 | Zbl 1034.35084

[2] J.-F. Bony and L. Michel, Microlocalization of resonant states and estimates of the residue of the scattering amplitude Comm. Math. Phys. 246 (2004), no. 2, 375–402. | MR 2048563 | Zbl 1062.35053

[3] J.-F. Bony and V. Petkov, Resonances for non-trapping time-periodic perturbations, J. Phys. A: Math. Gen. 37 (2004), 9439-9449. | MR 2095429 | Zbl 02149694

[4] J.-F. Bony and V. Petkov, Resolvent estimates and local energy decay of hyperbolic equations, Around Hyberbolic Systems, Conference to the memory of Stefano Benvenutti, Ferrara 2005, to appear. | MR 2273096

[5] N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonances au voisinage du réel, Acta Math. 180 (1998), 1-29. | Zbl 0918.35081

[6] N. Burq, Semi-classical estimates for the resolvent in non-trapping geometries, Inter. Math. Res. Notes, 5 (2002), 221-241. | MR 1876933 | Zbl 01719337

[7] N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by H. Smith and C. Sogge, Comm. Partial Differential Equations, 28 (2003), no. 9-10, 1675–1683. | MR 2001179 | Zbl 1026.35020

[8] N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J. 123 (2004), no. 2, 403–427. | MR 2066943 | Zbl 1061.35024

[9] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1989), 113-146. | Numdam | MR 949013 | Zbl 0636.35045

[10] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Bulletin de SMF, Mémoire no. 31, 116(1), (1988). | Numdam | MR 998698 | Zbl 0654.35081

[11] I. Herbst, Contraction semigroups and the spectrum of ${A}_{1}\otimes I+I\otimes {A}_{2}$, J. Oper. Theory, 7 (1982), 61-78. | MR 650193 | Zbl 0484.47017

[12] R. Melrose and J. Sjöstrand, Singularities of boundary value problems, Comm. Pure Appl. Math. I, 31 (1978), 593-617, II, 35 (1982), 129-168. | Zbl 0546.35083

[13] C. Morawetz, Decay estimates of the exterior problem for the wave equation, Comm. Appl. Math. 28 (1975), 229-264. | MR 372432 | Zbl 0304.35064

[14] P. D. Lax and R.S. Phillips, Scattering Theory, Academic Press, 2nd Edition, 1989. | MR 1037774 | Zbl 0697.35004

[15] V. Petkov and L. Stoyanov, Trapping obstacles and singularities of the scattering kernel, Colloque Franco-Tunisien d’EDP, Hammamet, 2003, Séminaires et Congrès, SMF, to appear.

[16] V. Petkov, Global Strichartz estimates for the wave equation with time-periodic potentials, J. Funct. Anal., to appear. | MR 2225457 | Zbl 1092.35018

[17] G. Popov and G. Vodev, Distribution of the resonances and local energy decay in the transmission problem, Asymptotic Analysis, 19 (1999), 253-265. | MR 1696217 | Zbl 0931.35115

[18] J. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823. | MR 254433 | Zbl 0209.40402

[19] J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), 729-769. | MR 1115789 | Zbl 0752.35046

[20] J. Sjöstrand, A trace formula and review of some estimates for resonances, Microlocal Analysis and spectral theory, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 490 (1997), 377-437. | MR 1451399 | Zbl 0877.35090

[21] J. Sjöstrand, Resonances for bottles and related trace formulae, Math. Nachr. 221 (2001), 95-149. | MR 1806367 | Zbl 0979.35109

[22] S. H. Tang and M. Zworski, Resonances expansions of scattered waves, Comm. Pure Appl. Math. 53 (2000), 1305-1334. | MR 1768812 | Zbl 1032.35148

[23] G. Vodev, On the uniform decay of the local energy, Serdica Math. J. 25 (1999), 191–206. | MR 1742776 | Zbl 0937.35118