Communications in Mathematical Physics

, Volume 146, Issue 1, pp 205–216 | Cite as

Sharp bounds on the number of scattering poles for perturbations of the Laplacian

  • Georgi Vodev
Article
Received:
Revised:

Abstract

Sharp bounds on the numberN(r) of the scattering poles in the disc |z|≦r for a large class of compactly supported perturbations (not necessarily selfadjoint) of the Laplacian in ℝ n ,n≧3, odd, are obtained. In particular, in the elliptic case the estimateN(r)≦Crn+C is proved.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Large Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gohberg, I., Krein, M.: Introduction to the theory of linear non-selfadjoint operators. Providence, RI: AMS, 1969Google Scholar
  2. 2.
    Intissar, A.: A polynomial bound on the number of scattering poles for a potential in even dimensional space ℝn. Commun. Partial Differ. Eqs.11, 367–396 (1986)Google Scholar
  3. 3.
    Lax, P.D., Phillips, R.S.: Scattering theory. New York: Academic Press 1967Google Scholar
  4. 4.
    Melrose, R.B.: Polynomial bounds on the number of scattering poles. J. Funct. Anal.53, 287–303 (1983)CrossRefGoogle Scholar
  5. 5.
    Melrose, R.B.: Polynomial bounds on the distribution of poles in scattering by an obstacle. Journées “Equations aux Dérivées Partielle,” Saint-Jean-de-Montes, 1984Google Scholar
  6. 6.
    Melrose, R.B.: Weyl asymptotics for the phase in obstacle scattering. Commun. Partial Differ. Eqs.13, 1431–1439 (1988)Google Scholar
  7. 7.
    Menikoff, A., Sjöstrand, J.: On the eigenvalues of a class of hypoelliptic operators. Math. Ann.235, 55–85 (1978)CrossRefGoogle Scholar
  8. 8.
    Sjöstrand, J.: Geometric bounds on the number of resonances for semiclassical problems. Duke Math. J.60, 1–57 (1990)CrossRefGoogle Scholar
  9. 9.
    Sjöstrand, J., Zworski, M.: Complex scaling and distribution of scattering poles. J. Am. Math. Soc. (to appear)Google Scholar
  10. 10.
    Titchmarsh, E.C.: The theory of functions, Oxford: Oxford University Press 1968Google Scholar
  11. 11.
    Vainberg, B.: Asymptotic methods in equations of mathematical physics. New York: Gordon and Breach 1988Google Scholar
  12. 12.
    Vodev, G.: Polynomial bounds on the number of scattering poles for symmetric systems. Ann. Inst. H. Poincaré (Physique Théorique)54, 199–208 (1991)Google Scholar
  13. 13.
    Vodev, G.: Polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in ℝn,n≧3, odd. Osaka. J. Math.28, 441–449 (1991)Google Scholar
  14. 14.
    Vodev, G.: Sharp polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in ℝn. Math. Ann.291, 39–49 (1991)CrossRefGoogle Scholar
  15. 15.
    Zworski, M.: Distribution of poles for scattering on the real line. J. Funct. Anal.73, 277–296 (1987)CrossRefGoogle Scholar
  16. 16.
    Zworski, M.: Sharp polynomial bounds on the number of scattering poles of radial potentials. J. Funct. Anal.82, 370–403 (1989)CrossRefGoogle Scholar
  17. 17.
    Zworski, M.: Sharp polynomial bounds on the number of scattering poles. Duke Math. J.59, 311–323 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Georgi Vodev
    • 1
  1. 1.Section of Mathematical Physics, Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations