# Resonances for transparent obstacles

Journées équations aux dérivées partielles (1999)

- page 1-13
- ISSN: 0752-0360

## Access Full Article

top## Abstract

top## How to cite

topPopov, Georgi, and Vodev, Georgi. "Resonances for transparent obstacles." Journées équations aux dérivées partielles (1999): 1-13. <http://eudml.org/doc/93367>.

@article{Popov1999,

abstract = {This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle $\{\mathcal \{O\}\}$ in $\mathbb \{R\}^n$, $n\ge 2$, with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency support of which coincides with the corresponding gliding manifold $\{\mathcal \{K\}\}$. To construct the quasimode we first find a global symplectic normal form for pairs of glancing hypersurfaces in a neighborhood of $\{\mathcal \{K\}\}$ and then we separate the variables microlocally near the whole glancing manifold $\{\mathcal \{K\}\}$. If the speed of propagation inside $\{\mathcal \{O\}\}$ is bigger than that outside $\{\mathcal \{O\}\}$, than there exists a strip in the upper half plane containing the real axis, which is free of resonances. We also obtain an uniform decay of the local energy for the corresponding mixed problem with an exponential rate of decay when the dimension is odd, and polynomial otherwise. It is well known that such a decay of the local energy holds for the wave equation with Dirichlet (Neumann) boundary conditions for any nontrapping obstacle. In our case, however, $\{\mathcal \{O\}\}$ is a trapping obstacle for the corresponding classical system.},

author = {Popov, Georgi, Vodev, Georgi},

journal = {Journées équations aux dérivées partielles},

keywords = {uniform decay; local energy},

language = {eng},

pages = {1-13},

publisher = {Université de Nantes},

title = {Resonances for transparent obstacles},

url = {http://eudml.org/doc/93367},

year = {1999},

}

TY - JOUR

AU - Popov, Georgi

AU - Vodev, Georgi

TI - Resonances for transparent obstacles

JO - Journées équations aux dérivées partielles

PY - 1999

PB - Université de Nantes

SP - 1

EP - 13

AB - This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle ${\mathcal {O}}$ in $\mathbb {R}^n$, $n\ge 2$, with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency support of which coincides with the corresponding gliding manifold ${\mathcal {K}}$. To construct the quasimode we first find a global symplectic normal form for pairs of glancing hypersurfaces in a neighborhood of ${\mathcal {K}}$ and then we separate the variables microlocally near the whole glancing manifold ${\mathcal {K}}$. If the speed of propagation inside ${\mathcal {O}}$ is bigger than that outside ${\mathcal {O}}$, than there exists a strip in the upper half plane containing the real axis, which is free of resonances. We also obtain an uniform decay of the local energy for the corresponding mixed problem with an exponential rate of decay when the dimension is odd, and polynomial otherwise. It is well known that such a decay of the local energy holds for the wave equation with Dirichlet (Neumann) boundary conditions for any nontrapping obstacle. In our case, however, ${\mathcal {O}}$ is a trapping obstacle for the corresponding classical system.

LA - eng

KW - uniform decay; local energy

UR - http://eudml.org/doc/93367

ER -

## References

top- [1] F. CARDOSO, G. POPOV AND G. VODEV, Distribution of resonances and local energy decay in the transmission problem II, Math. Res. Lett., to appear. Zbl0968.35035
- [2] C. GÉRARD, Asymptotique des poles de la matrice de scattering pour deux obstacles strictement convex. Bull. Soc. Math. France, Mémoire n. 31, 116, 1988. Zbl0654.35081MR91e:35168
- [3] T. GRAMCHEV AND G. POPOV, Nekhoroshev type estimates for billiard ball maps. Ann. Inst. Fourier 45, 859-895 (1995). MR97a:58145
- [4] T. HARGÉ AND G. LEBEAU, Diffraction par un convexe. Invent. Math. 118, 161-196 (1984). Zbl0831.35121MR95h:35167
- [5] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators. Vol. III, IV. Berlin - Heidelberg - New York : Springer, 1985. Zbl0601.35001
- [6] V. KOVACHEV AND G. POPOV, Invariant tori for the billiard ball map, Trans. Am. Math. Soc. 317, 45-81 (1990). Zbl0686.58037MR90e:58050
- [7] P. LAX AND R. PHILLIPS, Scattering Theory. New York : Academic Press. 1967. Zbl0186.16301
- [8] SH. MARVIZI AND R. MELROSE, Spectral invariants of convex planar regions. J. Diff. Geom. 17, 475-502 (1982). Zbl0492.53033MR85d:58084
- [9] R. MELROSE, Equivalence of glancing hypersurfaces. Invent. Math. 37, 165-191 (1976). Zbl0354.53033MR55 #9173
- [10] R. MELROSE AND J. SJÖSTRAND, Singularities of boundary value problems. I, II, Comm. Pure Appl. Math. 31 (1978), 593-617, 35 (1982), 129-168. Zbl0368.35020
- [11] G. POPOV, Quasi-modes for the Laplace operator and glancing hypersurfaces. In : M. Beals, R. Melrose, J. Rauch (eds.) : Proceeding of Conference on Microlocal Analysis and Nonlinear Waves, Minnesota 1989, Berlin-Heidelberg-New York : Springer, 1991. Zbl0794.35030MR92e:35013
- [12] G. POPOV AND G. VODEV, Resonances near the real axis for transparent obstacles, Commun. Math. Phys., to appear. Zbl0951.35036
- [13] G. POPOV AND G. VODEV, Distribution of resonances and local energy decay in the transmission problem, Asympt. Anal., 19, 253-265 (1999). Zbl0931.35115MR2000d:35033
- [14] J. SJÖSTRAND AND M. ZWORSKI, Complex scaling and distribution of scattering poles. J. Amer. Math. Soc. 4, 729-769 (1991). Zbl0752.35046MR92g:35166
- [15] P. STEFANOV, Quasimodes and resonances : Sharp lower bounds. Duke Math. J., to appear. Zbl0952.47013
- [16] P. STEFANOV AND G. VODEV, Neumann resonances in linear elasticity for an arbitrary body. Commun. Math. Phys. 176, 645-659 (1996). Zbl0851.35032MR96k:35122
- [17] S.-H. TANG AND M. ZWORSKI, From quasimodes to resonances. Math. Res. Lett. 5, 261-272 (1998). Zbl0913.35101MR99i:47088
- [18] B. VAINBERG, Asymptotic methods in equations of mathematical physics, Gordon and Breach, New York, 1988.
- [19] G. VODEV, On the uniform decay of the local energy, Serdica Math. J. 25 (1999), to appear. Zbl0937.35118MR2001h:35138

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.