Geometric bounds on the density of resonances for semiclassical problems

Duke Mathematical Journal

Volumn 60, Issue 1, 1990, Pages 1-57

  Sjöstrand, Johannes ^a,b  

^a Université de Paris Sud ^*  (France)

^b Équipe CNRS NO D0760 ^*  (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84974003886     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-90-06001-6     Document Type: Article

References (40)

1. J. AGUILAR, J. M. COMBES, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269-279.
2. E. BALSLEV, J. M. COMBES, Spectral properties of many-body Schrödinger operators with dilation analytic interactions, Comm. Math. Phys. 22 (1971), 280-294.
3. C. BARDOS, G. LEBEAU, J. RAUCH, Scattering frequences and Gevrey 3 singularities, Inv. Math. 90 1 (1987), 77-114.
4. L. BOUTET DE MONVEL, V. GUILLEMIN, The spectral theory of Toeplitz operators, Ann. of Math. Studies, 99 (1981), Princeton Univ. Press.
5. P. BRIET, J. M. COMBES, P. DUCLOS, On the location of resonances in the semiclassical limit I. Resonance free domains, J. Math. Anal. Appl. 126 (1987), 90-99.
6. P. BRIET, J. M. COMBES, P. DUCLOS, On the location of resonances in the semiclassical limit II, Comm. PDE 12 2 (1987), 201-222.
7. A. CORDOBA, C. FEFFERMAN, Wave packets and Fourier integral operators, Comm. P.D.E., 3 11 (1978), 979-1006.
8. N. DUNFORD, J. SCHWARTZ, Linear Operators, part II. Pure and Appl. Math., Wiley (1964).
9. K. J. FALCONER, The geometry of fractal sets, Cambridge tracts in mathematics, 95, Cambridge University press (1985).
10. C. GÉRARD, Asymptotique des pôles de la matrice de scattering pour deux obstacles stricte- ment convexes, Bulletin de la S.M.F., Mémoire n°31,116 (1) (1988).
11. C. GÉRARD, J. SJÖSTRAND, Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys. 108 (1987), 391-421.
12. C. GÉRARD, J. SJÖSTRAND, Resonances en limite semiclassique et exposants de Lyapunov, Comm. Math. Phys. 116 (1988), 193-213.
13. W. GOODHUE, Scattering theory for hyperbolic systems with coefficients of Gevrey type. Trans. Am. Math. Soc. 180 (1973), 337-346.
14. B. HELFFER, J. SJÖSTRAND, Résonances en limite semiclassique, Bull, de la S.M.F., mémoire n°24/25, 114 (3).
15. B. HELFFER, A. MARTINEZ, Comparaison entre les diverses notions de résonances, Helv. Phys. Acta 60 (1987), 992-1003.
16. L. HÖRMANDER, The analysis of linear partial differential operators III, Grundlehren der math. Wiss. 274 (1985), Springer Verlag.
17. M. IKAWA, (a) On the poles of the scattering matrix for two strictly convex obstacles, J. Math. Kyoto Univ. 23 1 (1983), 127-194.
18. M. IKAWA, (b) On the poles of the scattering matrix for two strictly convex obstacles: An addendum, J. Math. Kyoto Univ. 23 4 (1983), 795-802.
19. M. IKAWA, (c) On the distribution of the poles of the scattering matrix for two strictly convex obstacles, Hokkaido Math. J. 12 (1988), 343-359.
20. M. IKAWA, (a) Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis, Preprint.
21. M. IKAWA, (b) "On the Scattering matrix for two convex obstacles," Proc of the Taniguchi Int. Symp., on hyperbolic eq. and related topics, Katata and Kyoto, 1984, Kinokuniya Co. LTD Tokyo 1986.
22. M. IKAWA, Decay of solutions of the wave equations in the exterior of several convex bodies, Preprint.
23. M. IKAWA, Proc. of Japan Academy (1988).
24. A. INTISSAR, A polynomial bound on the number of the scattering poles for a potential in even dimensions, Comm. P.D.E. 11 (1986), 367-396.
25. A. KELLEY, "The stable, center-stable, center, center-unstable and unstable manifolds", Appendix C in Transversal mappings and flows by R. Abraham and J. Robbin, W. A. Benjamin Inc., New York, Amsterdam 1967.
26. M. KLEIN, On the absence of resonances for Schrödinger operators in the classical limit, Comm. Math. Phys. 106 (1986), 485-494.
27. P. LAX, R. PHILLIPS, Scattering theory, Pure and Appl. Math., Academic Press (1967).
28. G. LEBEAU, Régularité Gevrey 3 pour la diffraction, Comm. P.D.E. 9 15 (1984), 1437-1494.
29. R. MELROSE, "Polynomial bound on the distribution of poles in scattering by an obstacle", Proceedings of the Journées "Equations aux dérivées partielles" à St Jean de Monts, 4-8 Juin 1984, Société Mathématique de France.
30. S. NAKAMURA, A note on the absence of resonances for Schrödinger operators, Lett. Math. Phys. 16(1988), 217-233.
31. S. NAKAMURA, Shape resonances for distortion analytic Schrödinger operators, Preprint.
32. V. PETKOV, STOJANOV, Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles, Preprint (1987).
33. M. SHUB, Global stability of dynamical systems, Springer Verlag, (1987).
34. YA. G. SINAI, "Development of Krylov’s ideas," Addendum to Works on the foundations of statistical physics, by N, S. Krylov, Princeton Univ. Press, 1979.
35. J. SJÖSTRAND, Singularités analytiques microlocales, Astérisque 95 (1982).
36. J. SJÖSTRAND, Propagation of singularities for operators with multiple characteristics, Ann. Inst. Fourier, 26, (1) (1976), 141-155.
37. J. SJÖSTRAND, Semiclassical resonances generated by a non-degenerate critical point, Springer LNM, 1256,402-429.
38. J. SJÖSTRAND, "Estimates on the number of resonances for semiclassical Schrödinger operators," Proc. of the VIII Latin American School of Mathematics (1986), Springer Lecture Notes in Math. 1324,286-292.
39. C. TRICOT, TWO definitions of fractional dimension, Math. Proc. Camb. Phil. Soc, 91 (1982), 57-78.
40. M. ZWORSKI, Sharp polynomial bounds on the number of scattering poles, Preprint.