Microlocalization of resonant states and estimates of the residue of the scattering amplitude
Journées équations aux dérivées partielles (2003), article no. 2, 12 p.

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.

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     title = {Microlocalization of resonant states and estimates of the residue of the scattering amplitude},
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     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {2},
     pages = {1--12},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.616},
     mrnumber = {2050588},
     zbl = {02079437},
     language = {en},
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Bony, Jean-François; Michel, Laurent. Microlocalization of resonant states and estimates of the residue of the scattering amplitude. Journées équations aux dérivées partielles (2003), article  no. 2, 12 p. doi : 10.5802/jedp.616. http://www.numdam.org/articles/10.5802/jedp.616/

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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