Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Plamen Stefanov

Journées équations aux dérivées partielles (2001)

  • page 1-16
  • ISSN: 0752-0360

Abstract

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We study semiclassical resonances in a box Ω ( h ) of height h N , N ≫ 1 . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set 𝒯 of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # ( h ) with discrete spectrum the number of resonances in Ω ( h ) is bounded by the number of eigenvalues of P # ( h ) in an interval a bit larger than the projection of Ω ( h ) on the real line. As an application, we prove a Weyl type estimate of the number of resonances in Ω ( h ) in terms of the measure of 𝒯 . We prove a similar estimate in case of classical scattering by a metric and obstacle.

How to cite

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Stefanov, Plamen. "Weyl type upper bounds on the number of resonances near the real axis for trapped systems." Journées équations aux dérivées partielles (2001): 1-16. <http://eudml.org/doc/93410>.

@article{Stefanov2001,
abstract = {We study semiclassical resonances in a box $\Omega (h)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $\mathcal \{T\}$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^\#(h)$ with discrete spectrum the number of resonances in $\Omega (h)$ is bounded by the number of eigenvalues of $P^\#(h)$ in an interval a bit larger than the projection of $\Omega (h)$ on the real line. As an application, we prove a Weyl type estimate of the number of resonances in $\Omega (h)$ in terms of the measure of $\mathcal \{T\}$. We prove a similar estimate in case of classical scattering by a metric and obstacle.},
author = {Stefanov, Plamen},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-16},
publisher = {Université de Nantes},
title = {Weyl type upper bounds on the number of resonances near the real axis for trapped systems},
url = {http://eudml.org/doc/93410},
year = {2001},
}

TY - JOUR
AU - Stefanov, Plamen
TI - Weyl type upper bounds on the number of resonances near the real axis for trapped systems
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 16
AB - We study semiclassical resonances in a box $\Omega (h)$ of height $h^N$, $N\gg 1$. We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set $\mathcal {T}$ of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator $P^\#(h)$ with discrete spectrum the number of resonances in $\Omega (h)$ is bounded by the number of eigenvalues of $P^\#(h)$ in an interval a bit larger than the projection of $\Omega (h)$ on the real line. As an application, we prove a Weyl type estimate of the number of resonances in $\Omega (h)$ in terms of the measure of $\mathcal {T}$. We prove a similar estimate in case of classical scattering by a metric and obstacle.
LA - eng
UR - http://eudml.org/doc/93410
ER -

References

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  1. [B2] N. Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, preprint. Zbl1013.35019MR1914456
  2. [B3] N. Burq, Semi-classical estimates for the resolvent in non-trappimg geometries, preprint. MR1876933
  3. [G] C. Gérard, Asymptotique des poles de la matrice de scattering pour deux obstacles strictement convex, Bull. Soc. Math. France, Mémoire n. 31, 116, 1988. Zbl0654.35081MR998698
  4. [DSj] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Math. Society Lecture Notes Series, No. 268, Cambridge Univ. Press, 1999. Zbl0926.35002MR1735654
  5. [HSj] B. Helffer, J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.), 24-25, 1986. Zbl0631.35075MR871788
  6. [H] L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin, 1985. Zbl0601.35001MR404822
  7. [I] V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, Berlin, 1998. Zbl0906.35003MR1631419
  8. [K] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1966. Zbl0148.12601MR203473
  9. [LZ] K. Lin and M. Zworski, Resonances in chaotic scattering, www.math.berkeley.edu/~kkylin/resonances/. 
  10. [MSj] R. Melrose and J. Sjöstrand, Singularities of boundary value problems. I, II. Comm. Pure Appl. Math. 31 1978, no. 5, 593-617, and 35 1982, no. 2, 129-168. Zbl0546.35083
  11. [Po] G. Popov, Quasi-modes for the Laplace operator and Glancing hypersurfaces, in: Proceedings of Conference on Microlocal analysis and Nonlinear Waves, Minnesota 1989, M. Beals, R. Melrose and J. Rauch eds., Springer Verlag, Berlin-Heidelberg-New York, 1991. Zbl0794.35030MR1120290
  12. [Sj1] J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60(1) 1990, 1-57. Zbl0702.35188MR1047116
  13. [Sj2] J. Sjöstrand, A trace formula and review of some estimates for resonances, in Microlocal analysis and Spectral Theory (Lucca, 1996), 377-437, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, Kluwer Acad. Publ., Dordrecht, 1997. Zbl0877.35090MR1451399
  14. [SjV] J. Sjöstrand and G. Vodev, Asymptotics of the number of Rayleigh resonances, Math. Ann. 309 1997, 287-306. Zbl0890.35098MR1474193
  15. [SjZ] J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, Journal of AMS 4(4) 1991, 729-769. Zbl0752.35046MR1115789
  16. [St1] P. Stefanov, Quasimodes and resonances: sharp lower bounds, Duke Math. J. 99 1999, 75-92. Zbl0952.47013MR1700740
  17. [St2] P. Stefanov, Lower bound of the number of the Rayleigh resonances for arbitrary body, Indiana Univ. Math. J. 49(2)( 2000), 405-426. Zbl0961.35098MR1777025
  18. [St3] P. Stefanov, Resonance expansions and Rayleigh waves, Math. Res. Lett., 8(1-2)( 2001), 105-124. Zbl0987.35097MR1825264
  19. [StV1] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body, Duke Math. J. 78 1995, 677-714. Zbl0846.35139MR1334206
  20. [TZ1] S.-H. Tang and M. Zworski, From quasimodes to resonances, Math. Res. Lett., 5 1998, 261-272. Zbl0913.35101MR1637824
  21. [TZ2] S.-H. Tang and M. Zworski, Resonance expansions of scattered waves, Comm. Pure Appl. Math. 53(10)( 2000), 1305-1334. Zbl1032.35148MR1768812
  22. [V] G. Vodev, Sharp polynomial bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys. 146 1992, 39-49. Zbl0754.35105MR1125006
  23. [Ze] M. Zerzeri, Majoration du nombre des résonances près de l'axe reél pour une perturbation, à support compacte, abstraite, du laplacien, preprint, 2000-2001, Univ. Paris 13. MR1876413
  24. [Z1] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59 1989, 311-323. Zbl0705.35099MR1016891
  25. [Z2] M. Zworski, private communication, 1992. 

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