# Estimates on the number of scattering poles near the real axis for strictly convex obstacles

Johannes Sjöstrand; Maciej Zworski

Annales de l'institut Fourier (1993)

- Volume: 43, Issue: 3, page 769-790
- ISSN: 0373-0956

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topSjöstrand, Johannes, and Zworski, Maciej. "Estimates on the number of scattering poles near the real axis for strictly convex obstacles." Annales de l'institut Fourier 43.3 (1993): 769-790. <http://eudml.org/doc/75019>.

@article{Sjöstrand1993,

abstract = {For the Dirichlet Laplacian in the exterior of a strictly convex obstacle, we show that the number of scattering poles of modulus $\le r$ in a small angle $\theta $ near the real axis, can be estimated by Const $\theta ^\{3/2\}r^n$ for $r$ sufficiently large depending on $\theta $. Here $n$ is the dimension.},

author = {Sjöstrand, Johannes, Zworski, Maciej},

journal = {Annales de l'institut Fourier},

keywords = {resonance; complex scaling; semiclassical problem; Dirichlet Laplacian; exterior of a strictly convex obstacle; scattering poles},

language = {eng},

number = {3},

pages = {769-790},

publisher = {Association des Annales de l'Institut Fourier},

title = {Estimates on the number of scattering poles near the real axis for strictly convex obstacles},

url = {http://eudml.org/doc/75019},

volume = {43},

year = {1993},

}

TY - JOUR

AU - Sjöstrand, Johannes

AU - Zworski, Maciej

TI - Estimates on the number of scattering poles near the real axis for strictly convex obstacles

JO - Annales de l'institut Fourier

PY - 1993

PB - Association des Annales de l'Institut Fourier

VL - 43

IS - 3

SP - 769

EP - 790

AB - For the Dirichlet Laplacian in the exterior of a strictly convex obstacle, we show that the number of scattering poles of modulus $\le r$ in a small angle $\theta $ near the real axis, can be estimated by Const $\theta ^{3/2}r^n$ for $r$ sufficiently large depending on $\theta $. Here $n$ is the dimension.

LA - eng

KW - resonance; complex scaling; semiclassical problem; Dirichlet Laplacian; exterior of a strictly convex obstacle; scattering poles

UR - http://eudml.org/doc/75019

ER -

## References

top- [M1] R. MELROSE, Polynomial bounds on the number of scattering poles, J. Funct. An., 53 (1983), 287-303. Zbl0535.35067MR85k:35180
- [M2] R. MELROSE, Polynomial bounds on the distribution of poles in scattering by an obstacle, Journées équations aux dérivées partielles, Saint Jean de Monts (1984) (published by Centre de Mathématiques, École Polytechnique, Palaiseau, France). Zbl0621.35073
- [R] D. ROBERT, Autour de l'approximation semi-classique, Progress in Math., vol. 68, Birkhäuser (1987). Zbl0621.35001MR89g:81016
- [O] F.W.J. OLVER, The asymptotic expansions of Bessel functions of large order, Phil. Trans. Roy. Soc. London, Ser. A, 247 (1954), 328-368. Zbl0070.30801MR16,696a
- [S] J. SJÖSTRAND, Geometric bounds on the density of resonances for semi-classical problems, Duke Mathematical Journal, 61 (1) (1990), 1-57. Zbl0702.35188
- [SZ1] J. SJÖSTRAND, M. ZWORSKI, Complex scaling and the distribution of scattering poles, Journal of the AMS, 4 (4) (1991), 729-769. Zbl0752.35046MR92g:35166
- [SZ2] J. SJÖSTRAND, M. ZWORSKI, Distribution of scattering poles near the real axis, Comm. P.D.E., 17 (5 & 6) (1992), 1021-1035. Zbl0766.35031MR93h:35152
- [V] G. VODEV, Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys., 146 (1992), 205-216. Zbl0766.35032MR93f:35173

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