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Complex scaling and the distribution of scattering poles


Authors: Johannes Sjöstrand and Maciej Zworski
Journal: J. Amer. Math. Soc. 4 (1991), 729-769
MSC: Primary 35P25; Secondary 35B20, 58G25
MathSciNet review: 1115789
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  • [1] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279. MR 0345551
  • [2] E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys. 22 (1971), 280–294. MR 0345552
  • [3] Yves Colin de Verdière, Pseudo-laplaciens. II, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 87–113 (French). MR 699488
  • [4] I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • [5] Bernard Helffer and André Martinez, Comparaison entre les diverses notions de résonances, Helv. Phys. Acta 60 (1987), no. 8, 992–1003 (French). MR 929933
  • [6] B. Helffer and J. Sjöstrand, Résonances en limite semi-classique, Bull. Soc. Math. France, mémoire no. 24/25 114 (1986).
  • [7] -, Analyse semi-classique pour l'équation de Harper, Bull. Soc. Math. France 116 (1988).
  • [8] B. Helffer and J. Sjöstrand, On diamagnetism and de Haas-van Alphen effect, Ann. Inst. H. Poincaré Phys. Théor. 52 (1990), no. 4, 303–375 (English, with French summary). MR 1062904
  • [9] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • [10] W. Hunziker, Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré Phys. Théor. 45 (1986), no. 4, 339–358 (English, with French summary). MR 880742
  • [11] Ahmed Intissar, A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces 𝑅ⁿ, Comm. Partial Differential Equations 11 (1986), no. 4, 367–396. MR 829322, 10.1080/03605308608820428
  • [12] Ahmed Intissar, A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces 𝑅ⁿ, Comm. Partial Differential Equations 11 (1986), no. 4, 367–396. MR 829322, 10.1080/03605308608820428
  • [13] Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
  • [14] Gilles Lebeau, Fonctions harmoniques et spectre singulier, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 269–291 (French). MR 584087
  • [15] André Martinez, Prolongement des solutions holomorphes de problèmes aux limites, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 93–116 (French). MR 781780
  • [16] Anders Melin, Trace distributions associated to the Schrödinger operator, J. Anal. Math. 59 (1992), 133–160. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226956, 10.1007/BF02790222
  • [17] Richard Melrose, Scattering theory and the trace of the wave group, J. Funct. Anal. 45 (1982), no. 1, 29–40. MR 645644, 10.1016/0022-1236(82)90003-9
  • [18] -, Polynomial bounds on the number of scattering poles, J. Funct. Anal. 53 (1983), 287-303.
  • [19] -, Growth estimates for the poles in potential scattering, unpublished, 1984.
  • [20] -, Polynomial bounds on the distribution of poles in scattering by an obstacle, Journées ``Equations aux Dérivées Partielle,'' Saint-Jean-de Montes, 1984.
  • [21] Richard B. Melrose, Weyl asymptotics for the phase in obstacle scattering, Comm. Partial Differential Equations 13 (1988), no. 11, 1431–1439. MR 956828, 10.1080/03605308808820582
  • [22] W. Müller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, preprint.
  • [23] Johannes Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J. 60 (1990), no. 1, 1–57. MR 1047116, 10.1215/S0012-7094-90-06001-6
  • [24] -, Estimates on the number of resonances for semi-classical Schrödinger operators, (Proc. VIII Latin American School of Mathematics, 1986) Lecture Notes in Math., vol. 1324, Springer-Verlag, Berlin-New York, 1988, pp. 286-292.
  • [25] B. R. Vaĭnberg, Asymptotic methods in equations of mathematical physics, Gordon & Breach Science Publishers, New York, 1989. Translated from the Russian by E. Primrose. MR 1054376
  • [26] Georgi Vodev, Sharp polynomial bounds on the number of scattering poles for metric perturbations of the Laplacian in 𝑅ⁿ, Math. Ann. 291 (1991), no. 1, 39–49. MR 1125006, 10.1007/BF01445189
  • [27] Maciej Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (1987), no. 2, 277–296. MR 899652, 10.1016/0022-1236(87)90069-3
  • [28] Maciej Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal. 82 (1989), no. 2, 370–403. MR 987299, 10.1016/0022-1236(89)90076-1
  • [29] Maciej Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59 (1989), no. 2, 311–323. MR 1016891, 10.1215/S0012-7094-89-05913-9

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DOI: http://dx.doi.org/10.1090/S0894-0347-1991-1115789-9
Article copyright: © Copyright 1991 American Mathematical Society