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

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 ℝn,n≧3, odd, are obtained. In particular, in the elliptic case the estimateN(r)≦Crn+C is proved.

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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

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