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Asymptotic distribution of resonances for convex obstacles

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References

  1. Babich, V. M. &Grigoreva, N. S., The analytic continuation of the resolvent of the exterior three-dimensional problem for the Laplace operator to second sheet.Funktsional. Anal. i Prilozhen., 8 (1974), 71–74.

    Article  Google Scholar 

  2. Bardos, C., Lebeau, G. &Rauch, J., Scattering frequencies and Gevrey 3 singularities.Invent. Math., 90 (1987), 77–114.

    Article  MathSciNet  MATH  Google Scholar 

  3. Delort, J.-M.,F.B.I. Transformation. Second Microlocalization and Semi-Linear Caustics. Lecture Notes in Math. 1522. Springer-Verlag, Berlin, 1992.

    Google Scholar 

  4. Dimassi, M. &Sjöstrand, J. Spectral Asymptotics in the Semi-Classical Limit. London Math. Soc. Lecture Note Ser., 268. Cambridge Univ. Press, Cambridge, 1999.

    MATH  Google Scholar 

  5. Filippov, V. B. &Zayaev, A. B., Rigorous justification of the asymptotic solutions of sliding wave type.J. Soviet Math., 30 (1985), 2395–2406.

    Article  MATH  Google Scholar 

  6. Guillopé, L. &Zworski, M., Scattering asymptotics for Riemann surfaces.Ann. of Math., 145 (1997), 597–660.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hargé, T. &Lebeau, G., Diffraction par un convexe.Invent. Math., 118, (1994), 161–196.

    Article  MathSciNet  MATH  Google Scholar 

  8. Helffer, B. &Sjöstrand, J.,Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.), 24/25. Bordas, Paris, 1986.

    Google Scholar 

  9. Hörmander, L.,The Analysis of Linear Partial Differential Operators, Vols. III, IV. Grundleheren Math. Wiss., 274, 275. Springer-Verlag, Berlin-New York, 1985.

    Google Scholar 

  10. Lax, P. &Phillips, R.,Scattering Theory. Academic Press, New York-London, 1967.

    MATH  Google Scholar 

  11. — Decaying modes for the wave equation in the exterior of an obstacle.Comm. Pure Appl. Math., 22 (1969), 737–787.

    MathSciNet  MATH  Google Scholar 

  12. Lascar, B. &Lascar, R., FBI transforms in Gevrey classes.J. Anal. Math., 72 (1997), 105–125.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lebeau, G., Deuxième microlocalisation sur les sous-variétés isotropes.Ann. Inst. Fourier (Grenoble), 35 (1985), 145–216.

    MATH  MathSciNet  Google Scholar 

  14. Levin, B. Ya.,Lectures on Entire Functions. Transl. Math. Monographs, 150. Amer. Math. Soc., Providence, RI, 1996.

    MATH  Google Scholar 

  15. Melrose, R. B. Polynomial bound on the distribution of poles in scattering by an obstacle, inJournées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1984), Exp. No. III. Soc. Math. France, Paris, 1984.

    Google Scholar 

  16. —Geometric Scattering Theory. Stanford Lectures. Cambridge Univ. Press, Cambridge, 1995.

    MATH  Google Scholar 

  17. Melrose, R. B., Sá Barreto, A. &Zworski, M.,Semi-Linear Diffraction of Conormal Waves. Astérisque, 240. Soc. Math. France, Paris, 1997.

    Google Scholar 

  18. Nussenzveig, H. M., High-frequency scattering by an impenetrable sphere.Ann. Physics, 34 (1965), 23–95.

    Article  MathSciNet  Google Scholar 

  19. Olver, F. W. J., The asymptotic expansion of Bessel functions of large order.Philos. Trans. Roy. Soc. London Ser. A, 247 (1954), 328–368.

    MathSciNet  Google Scholar 

  20. Popov, G., Asymptotics of Green's functions in the shadow.C. R. Acad. Bulgare Sci., 38 (1985), 1287–1290.

    MATH  MathSciNet  Google Scholar 

  21. Robert, D.,Autour de l'approximation semi-classique. Progr. Math., 68. Birkhäuser Boston, Boston, MA, 1987.

    MATH  Google Scholar 

  22. Shubin, M. &Sjöstrand, J., Appendix to: Weak Bloch property and weight estimates for elliptic operators, inSeminaire sur les Équations aux Dérivées Partielles 1989–1990, Exp. No. V. Ecole Polytechnique, Palaiseau, 1990.

    Google Scholar 

  23. Sjöstrand, J.,Singularité analytiques microlocales. Astérisque, 95. Soc. Math. France, Paris, 1982.

    Google Scholar 

  24. —, Geometric bounds on the density of resonances for semi-classical problems.Duke Math. J., 60 (1990), 1–57.

    Article  MATH  MathSciNet  Google Scholar 

  25. —, Density of resonances for strictly convex analytic obstacles.Canad. J. Math., 48 (1996), 397–447.

    MATH  MathSciNet  Google Scholar 

  26. —, A trace formula and review of some estimates for resonances, inMicrolocal Analysis and Spectral Theory (Lucca, 1996), pp. 377–437. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490. Kluwer Acad. Publ., Dordrecht, 1997.

    Google Scholar 

  27. —, A trace formula for resonances and application to semi-classical Schrödinger operators, inSeminaire sur les Équations aux Dérivées Partielles 1996–1997 Exp. No. II. École Polytechnique, Palaiseau, 1997.

    Google Scholar 

  28. Sjöstrand, J. &Zworski, M., Complex scaling and the distribution of scattering poles.J. Amer. Math. Soc., 4 (1991), 729–769.

    Article  MathSciNet  MATH  Google Scholar 

  29. —, Lower bounds on the number of scattering poles.Comm. Partial Differential Equations, 18 (1993), 847–857.

    MathSciNet  MATH  Google Scholar 

  30. —, Lower bounds on the number of scattering poles, II.J. Funct. Anal., 123 (1994), 336–367.

    Article  MathSciNet  MATH  Google Scholar 

  31. —, Estimates on the number of scattering poles for strictly convex obstacles near the real axis.Ann. Inst. Fourier (Grenoble), 43 (1993), 769–790.

    MathSciNet  MATH  Google Scholar 

  32. —, The complex scaling method for scattering by strictly convex obstacles.Ark. Mat., 33 (1995), 135–172.

    MathSciNet  MATH  Google Scholar 

  33. Stefanov, P., Quasimodes and resonances: sharp lower bounds.Duke Math. J., 99 (1999), 75–92.

    Article  MATH  MathSciNet  Google Scholar 

  34. Tang, S.-H. &Zworski, M., From quasimodes to resonances.Math. Res. Lett., 5 (1998), 261–272.

    MathSciNet  MATH  Google Scholar 

  35. Vodev, G., Sharp bounds on the number of scattering poles in even-dimensional spaces.Duke Math. J., 74 (1994), 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  36. Watson, G. N., The diffraction of electric waves by the earth.Proc. Roy. Soc. London Ser. A, 95 (1918), 83–99.

    MATH  Google Scholar 

  37. Zworski, M., Counting scattering poles, inSpectral and Scattering Theory (Sanda, 1992), pp. 301–331. Lecture Notes in Pure and Appl. Math., 161. Dekker, New York, 1994.

    Google Scholar 

  38. —, Poisson formulæ for resonances, inSéminaire sur les Équations aux Derivées Partielles 1996–1997 Exp. No. XIII. École Polytechnique, Palaiseau, 1997.

    Google Scholar 

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Author notes

  1. Maciej Zworski

    Present address: Department of Mathematics, University of California, 94720, Berkeley, CA, USA

Authors and Affiliations

  1. Centre de Mathématiques, École Polytechnique UMR 7640, CNRS, FR-91128, Palaiseau Cedex, France

    Johannes Sjöstrand

  2. Department of Mathematics, University of Toronto, M5S 3G3, Toronto, Canada

    Maciej Zworski

Authors
  1. Johannes Sjöstrand
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  2. Maciej Zworski
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Sjöstrand, J., Zworski, M. Asymptotic distribution of resonances for convex obstacles. Acta Math. 183, 191–253 (1999). https://doi.org/10.1007/BF02392828

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