Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T19:44:47.136Z Has data issue: false hasContentIssue false
  • English
  • Français

Density of Resonances for Strictly Convex Analytic Obstacles

Published online by Cambridge University Press:  20 November 2018

Johannes Sjöstrand
Johannes Sjöstrand*
Affiliation:
Centre de Mathématiques Ecole Polytechnique F-91128 Palaiseau Cedex France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We estimate the density of resonances close to a critical curve, for strictly convex obstacles with analytic boundary. Contrary to the C∞-case, already treated with Zworski, the estimates are in terms of dynamical quantities. A new feature in the proof is a certain averaging procedure.

Résumé

Résumé

Nous estimons la densité des résonances près d'une courbe critique, pour des obstacles strictement convexes à bord analytique. Contrairement au cas C∞, déjà traité avec Zworski, les estimations font appel aux quantités dynamiques. Une procédure de moyennisation est un aspect nouveau dans la demonstration.

Keywords

35L2035A2235A2735P2535P26resonanceFBI-transform
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 2 , 01 April 1996 , pp. 397 - 447
Copyright
Copyright © Canadian Mathematical Society 1996

References

[BG] Babich, V.M. and Grigoreva, N.S., , Funktsional Anal, i Prilozhen. (1) 8 (1974), 71–74.Google Scholar
[BaLR] Bardos, C., Lebeau, G. and Rauch, J., singularities, Invent. Math. 90 (1987), 77–114.Google Scholar
[Be] Besse, A.,Manifolds all of whose geodesies are closed, Springer Verlag, 1978.Google Scholar
[BoS] Boutet de Monvel, L. and Sjostrand, J., Sur la singularity des noyaux de Bergman et de Szego, Asterisque 34-35 (1976), 12S-164.Google Scholar
[FZ] Filippov, V.B. and Zayev, A.B., Rigorous justification of the asymptotic solutions of sliding wave type, J. Soviet Math. (2) 30 (1985), 2395–2406.Google Scholar
[HaL] Hargé, T. and Lebeau, G., Diffraction par un convexe, Invent. Math. (1) 118 (1994), 161–196.Google Scholar
[HS] Helffer, B. and J.Sjöstrand, Résonances en limite semiclassique, Bull. Soc. Math. France (3) 114, Memoire 24/25, (1986).Google Scholar
[L] Lebeau, G., pour la diffraction, Comment. Partial Differential Equations (15) 9 (1984), 1437–1494.Google Scholar
[P] Popov, G., Asymptotics of Green s functions in the shadow, C. R. Acad. Bulgare Sci. (10) 38 (1985), 1287. 1290.Google Scholar
[S1] Sjöstrand, J., Propagation of analytic singularities for second order Dirichlet problems, Comment. Partial Differential Equationa (1) 5 (1980), 41–94.Google Scholar
[S2] Sjöstrand, J., Singularités analytiques microlocales, Astérisque 95 (1982).Google Scholar
[SZ1] Sjostrand, J. and Zworski, M., Estimates on the number of scattering poles near the real axis for strictly convex obstacles, Ann. Inst. Fourier (3) 43(1993), 169-190.Google Scholar
[SZ2] Sjostrand, J., The complex scaling method for scattering by strictly convex obstacles, Institut Mittag-Leffler 10, Ark. Mat., 1992–1993. preprint, to appear.Google Scholar