On the distribution of scattering poles for perturbations of the Laplacian

Vodev, Georgi

Vodev, Georgi

Annales de l'Institut Fourier, Tome 42 (1992), p. 625-635 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On considère des opérateurs autoadjoints et positifs de la forme $- Δ + P$ (sui ne sont pas nécessairement elliptiques) dans $ℝ^{n}$ , $n \geq 3$ , où $P$ est un opérateur différentiel du deuxième ordre, à coefficients à support compact. On montre que le nombre des pôles de la diffusion en dehors d’un voisinage conique de l’axe réel admet des estimations semblables au cas elliptique.

We consider selfadjoint positively definite operators of the form $- Δ + P$ (not necessarily elliptic) in $ℝ^{n}$ , $n \geq 3$ , odd, where $P$ is a second-order differential operator with coefficients of compact supports. We show that the number of the scattering poles outside a conic neighbourhood of the real axis admits the same estimates as in the elliptic case. More precisely, if ${λ_{j}} (Im λ_{j} \geq 0_{)}$ are the scattering poles associated to the operator $- Δ + P$ repeated according to multiplicity, it is proved that for any $ε > 0$ there exists a constant $C_{ε} > 0$ so that $# {λ_{j} : | λ_{j} | \leq r$ , $ε \leq arg λ_{j} \leq π - ε} \leq C_{ε} r^{n}$ for any $r \geq 1$ .

Publié le : 1992-01-01

MR 93i:35098 | Zbl 0738.35054 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1303

@article{AIF_1992__42_3_625_0,
     author = {Vodev, Georgi},
     title = {On the distribution of scattering poles for perturbations of the Laplacian},
     journal = {Annales de l'Institut Fourier},
     volume = {42},
     year = {1992},
     pages = {625-635},
     doi = {10.5802/aif.1303},
     mrnumber = {93i:35098},
     zbl = {0738.35054},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1992__42_3_625_0}
}

Vodev, Georgi. On the distribution of scattering poles for perturbations of the Laplacian. Annales de l'Institut Fourier, Tome 42 (1992) pp. 625-635. doi : 10.5802/aif.1303. http://gdmltest.u-ga.fr/item/AIF_1992__42_3_625_0/

Bibliographie

[1] C. Bardos, J. Guilot, J. Ralston, La relation de Poisson pour l'équation des ondes dans un ouvert non borné. Application à la théorie de la diffusion, Commun. Partial Differ. Equations, 7 (1982), 905-958. | Zbl 0496.35067

[2] A. Intissar, A polynomial bound on the number of scattering poles for a potential in even dimensional space Rn, Commun. Partial Differ. Equations, 11 (1986), 367-396. | MR 87j:35286 | Zbl 0607.35069

[3] P. D. Lax, R. S. Phillips, Scattering Theory, Academic Press, 1967. | Zbl 0186.16301

[4] R. B. Melrose, Polynomial bounds on the number of scattering poles, J. Func. Anal., 53 (1983), 287-303. | MR 85k:35180 | Zbl 0535.35067

[5] R. B. Melrose, Polynomial bounds on the distribution of poles in scattering by an obstacle, Journées "Équations aux dérivées partielles", Saint-Jean-de-Monts (1984). | Numdam | Zbl 0621.35073

[6] R. B. Melrose, Weyl asymptotics for the phase in obstacle scattering, Commun. Partial Differ. Equations, 13 (1988), 1431-1439. | MR 90a:35183 | Zbl 0686.35089

[7] J. Sjöstrand, Geometric bounds on the number of resonances for semiclassical problems, Duke Math. J., 60 (1990), 1-57. | Zbl 0702.35188

[8] J. Sjöstrand, M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., (1991), 729-769. | MR 92g:35166 | Zbl 0752.35046

[9] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1968.

[10] B. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Sci. Publ., 1988. | Zbl 0743.35001

[11] G. Vodev, Polynomial bounds on the number of scattering poles for symmetric systems, Ann. Inst. Henri Poincaré (Physique théorique), 54 (1991), 199-208. | Numdam | MR 92j:81349 | Zbl 0816.35101

[12] G. Vodev, Polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in Rn, n ≥ 3, odd, Osaka J. Math., 28 (1991), 441-449. | MR 93b:35106 | Zbl 0754.35102

[13] G. Vodev, Sharp polynomial bounds on the number of scattering poles for metric perturbation of the Laplacian in Rn, Math. Ann., 291 (1991), 39-49. | MR 93f:47060 | Zbl 0754.35105

[14] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Lapacian, Commun. Math. Phys., 145 (1992), to appear. | MR 1191913 | Zbl 0766.35032

[15] M. Zworski, Distribution of poles for scattering in the real line, J. Func. Anal., 73 (1987), 227-296. | MR 88h:81223 | Zbl 0662.34033

[16] M. Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Func. Anal., 82 (1989), 370-403. | MR 90d:35233 | Zbl 0681.47002

[17] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J., 59 (1989), 311-323. | MR 90h:35190 | Zbl 0705.35099