no. 3
On the distribution of scattering poles for perturbations of the Laplacian
Annales de l'Institut Fourier, Volume 42 (1992) no. 3, pp. 625-635.

We consider selfadjoint positively definite operators of the form -Δ+P (not necessarily elliptic) in ℝ n , n≥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 }(Imλ j ≥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 |≤r, ε≤argλ j ≤π-ε}≤C ε r n for any r≥1.

On considère des opérateurs autoadjoints et positifs de la forme -Δ+P (sui ne sont pas nécessairement elliptiques) dans ℝ n , n≥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.

@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},
     pages = {625--635},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {3},
     year = {1992},
     doi = {10.5802/aif.1303},
     zbl = {0738.35054},
     mrnumber = {93i:35098},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1303/}
}
TY  - JOUR
AU  - Vodev, Georgi
TI  - On the distribution of scattering poles for perturbations of the Laplacian
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 625
EP  - 635
VL  - 42
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1303/
DO  - 10.5802/aif.1303
LA  - en
ID  - AIF_1992__42_3_625_0
ER  - 
%0 Journal Article
%A Vodev, Georgi
%T On the distribution of scattering poles for perturbations of the Laplacian
%J Annales de l'Institut Fourier
%D 1992
%P 625-635
%V 42
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1303/
%R 10.5802/aif.1303
%G en
%F AIF_1992__42_3_625_0
Vodev, Georgi. On the distribution of scattering poles for perturbations of the Laplacian. Annales de l'Institut Fourier, Volume 42 (1992) no. 3, pp. 625-635. doi : 10.5802/aif.1303. https://aif.centre-mersenne.org/articles/10.5802/aif.1303/

[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

[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 | Zbl

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

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

[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

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

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

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

Cited by Sources: