Microlocalization of resonant states and estimates of the residue of the scattering amplitude
Bony, Jean-François ; Michel, Laurent
Journées équations aux dérivées partielles, (2003), p. 1-12 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We obtain some microlocal estimates of the resonant states associated to a resonance z 0 of an h-differential operator. More precisely, we show that the normalized resonant states are 𝒪(| Im z 0 |/h +h ∞ ) outside the set of trapped trajectories and are 𝒪(h ∞ ) in the incoming area of the phase space. As an application, we show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally we prove such bound in some examples.

@article{JEDP_2003____A2_0,
     author = {Bony, Jean-Fran\c cois and Michel, Laurent},
     title = {Microlocalization of resonant states and estimates of the residue of the scattering amplitude},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2003},
     pages = {1-12},
     doi = {10.5802/jedp.616},
     mrnumber = {2050588},
     zbl = {02079437},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2003____A2_0}
}

Bony, Jean-François; Michel, Laurent. Microlocalization of resonant states and estimates of the residue of the scattering amplitude. Journées équations aux dérivées partielles,  (2003), pp. 1-12. doi : 10.5802/jedp.616. http://gdmltest.u-ga.fr/item/JEDP_2003____A2_0/

[20] R. Abraham and J. E. Marsden, Foundations of mechanics, Second edition, Advanced Book Program, Benjamin/Cummings Publishing, 1978. | MR 515141 | Zbl 0393.70001

[21] N. Burq, Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math. 124 (2002), no. 4, 677-735. | MR 1914456 | Zbl 1013.35019

[22] M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, 1999. | MR 1735654 | Zbl 0926.35002

[23] S. Fujiié and T. Ramond, Matrice de scattering et résonances associées à une orbite hétérocline, Ann. Inst. H. Poincaré Phys. Théor. 69 (1998), no. 1, 31-82. | Numdam | MR 1635811 | Zbl 0916.34071

[24] S. Fujiié and T. Ramond, Breit-Wigner formula at barrier tops, preprint (2002). | MR 1972758

[25] C. Gérard and A. Martinez, Prolongement méromorphe de la matrice de scattering pour des problèmes à deux corps à longue portée, Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 1, 81-110. | Numdam | MR 1029851 | Zbl 0711.35097

[26] C. Gérard and J. Sjöstrand, Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys. 108 (1987), no. 3, 391-421. | MR 874901 | Zbl 0637.35027

[27] B. Helffer and J. Sjöstrand, Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) (1986), no. 24-25. | Numdam | MR 871788 | Zbl 0631.35075

[28] H. Isozaki and H. Kitada, Scattering matrices for two-body Schrödinger operators, Sci. Papers College Arts Sci. Univ Tokyo 35 (1985), no. 1, 81-107. | MR 847881 | Zbl 0615.35065

[29] N. Kaidi and P. Kerdelhué, Forme normale de Birkhoff et résonances, Asymptot. Anal. 23 (2000), no. 1, 1-21. | MR 1764337 | Zbl 0955.35009

[30] A. Lahmar-Benbernou, Estimation des résidus de la matrice de diffusion associés à des résonances de forme. I, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), no. 3, 303-338. | Numdam | MR 1714347 | Zbl 0944.35060

[31] A. Lahmar-Benbernou and A. Martinez, Semiclassical asymptotics of the residues of the scattering matrix for shape resonances, Asymptot. Anal. 20 (1999), no. 1, 13-38. | MR 1697827 | Zbl 0931.35119

[32] A. Martinez, An introduction to semiclassical and microlocal analysis, Springer-Verlag, New York, 2002. | MR 1872698 | Zbl 0994.35003

[33] L. Michel, Semi-classical behavior of the scattering amplitude for trapping perturbations at fixed energy, Can. J. Math., to appear. | MR 2074047 | Zbl 1084.35067

[34] L. Michel, Semi-classical estimate of the residue of the scattering amplitude for long-range potentials, J. Phys. A 36 (2003), 4375-4393. | MR 1984509 | Zbl 02072849

[35] V. Petkov and M. Zworski, Semi-classical estimates on the scattering determinant, Ann. Henri Poincaré 2 (2001), no. 4, 675-711. | MR 1852923 | Zbl 1041.81041

[36] J. Sjöstrand, Singularités analytiques microlocales, Astérisque, 95, Astérisque, vol. 95, Soc. Math. France, Paris, 1982, pp. 1-166. | MR 699623 | Zbl 0524.35007

[37] J. Sjöstrand, Semiclassical resonances generated by nondegenerate critical points, Pseudodifferential operators (Oberwolfach, 1986), Springer, Berlin, 1987, pp. 402-429. | MR 897789 | Zbl 0627.35074

[38] J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4 (1991), no. 4, 729-769. | MR 1115789 | Zbl 0752.35046

[39] P. Stefanov, Estimates on the residue of the scattering amplitude, Asympt. Anal. 32 (2002), no. 3,4, 317-333. | MR 1993653 | Zbl 1060.35097

[40] P. Stefanov, Sharp upper bounds on the number of resonances near the real axis for trapped systems, Amer. J. Math., 125 (2003), no. 1, 183-224. | MR 1953522 | Zbl 1040.35055

[41] S-H. Tang and M. Zworski, From quasimodes to resonances, Math. Res. Lett. 5 (1998), no. 3, 261-272. | MR 1637824 | Zbl 0913.35101