Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1351-1406.

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrödinger group in terms of these resonances.

On étudie le projecteur spectral associé aux résonances engendrées par le sommet du potentiel d’un opérateur de Schrödinger semiclassique. On démontre d’abord une estimation de la résolvante pour les énergies complexes proches de ces résonances. À l’aide de cette estimation et d’une représentation explicite des états résonants, on prouve que le projecteur spectral admet un développement asymptotique en puissances entières de h, dont on donne le terme principal. Ce résultat nous permet alors de calculer le résidu de l’amplitude de diffusion en ces résonances. Finalement, on décrit le comportement en temps grand du groupe de Schrödinger en fonction des résonances.

DOI: 10.5802/aif.2643
Classification: 35B34,  35B38,  35C20,  35P25,  81Q20,  81U20
Keywords: Schrödinger operator, quantum resonances, semiclassical analysis, resolvent estimate
Bony, Jean-François 1; Fujiié, Setsuro 2; Ramond, Thierry 3; Zerzeri, Maher 4

1 Université Bordeaux 1 IMB (UMR CNRS 5251) 33405 Talence (France)
2 University of Hyogo Graduate School of Material Science (Japan)
3 Université Paris Sud 11 LMO (UMR CNRS 8628) 91405 Orsay (France)
4 Université Paris 13 LAGA (UMR CNRS 7539) 93430 Villetaneuse (France)
@article{AIF_2011__61_4_1351_0,
     author = {Bony, Jean-Fran\c{c}ois and Fujii\'e, Setsuro and Ramond, Thierry and Zerzeri, Maher},
     title = {Spectral projection, residue of the scattering amplitude and {Schr\"odinger} group expansion for barrier-top resonances},
     journal = {Annales de l'Institut Fourier},
     pages = {1351--1406},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     doi = {10.5802/aif.2643},
     mrnumber = {2951496},
     zbl = {1246.35033},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2643/}
}
TY  - JOUR
AU  - Bony, Jean-François
AU  - Fujiié, Setsuro
AU  - Ramond, Thierry
AU  - Zerzeri, Maher
TI  - Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
JO  - Annales de l'Institut Fourier
PY  - 2011
DA  - 2011///
SP  - 1351
EP  - 1406
VL  - 61
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2643/
UR  - https://www.ams.org/mathscinet-getitem?mr=2951496
UR  - https://zbmath.org/?q=an%3A1246.35033
UR  - https://doi.org/10.5802/aif.2643
DO  - 10.5802/aif.2643
LA  - en
ID  - AIF_2011__61_4_1351_0
ER  - 
%0 Journal Article
%A Bony, Jean-François
%A Fujiié, Setsuro
%A Ramond, Thierry
%A Zerzeri, Maher
%T Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
%J Annales de l'Institut Fourier
%D 2011
%P 1351-1406
%V 61
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2643
%R 10.5802/aif.2643
%G en
%F AIF_2011__61_4_1351_0
Bony, Jean-François; Fujiié, Setsuro; Ramond, Thierry; Zerzeri, Maher. Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1351-1406. doi : 10.5802/aif.2643. http://archive.numdam.org/articles/10.5802/aif.2643/

[1] Alexandrova, Ivana; Bony, Jean-François; Ramond, Thierry Semiclassical scattering amplitude at the maximum of the potential, Asymptot. Anal., Volume 58 (2008) no. 1-2, pp. 57-125 | MR | Zbl

[2] Bony, Jean-François; Fujiié, Setsuro; Ramond, Thierry; Zerzeri, Maher Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point, J. Funct. Anal., Volume 252 (2007) no. 1, pp. 68-125 | DOI | MR | Zbl

[3] Bony, Jean-François; Häfner, Dietrich Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric, Comm. Math. Phys., Volume 282 (2008) no. 3, pp. 697-719 | DOI | MR | Zbl

[4] Bony, Jean-François; Michel, Laurent Microlocalization of resonant states and estimates of the residue of the scattering amplitude, Comm. Math. Phys., Volume 246 (2004) no. 2, pp. 375-402 | DOI | MR | Zbl

[5] Briet, Ph.; Combes, J.-M.; Duclos, P. On the location of resonances for Schrödinger operators in the semiclassical limit. I. Resonances free domains, J. Math. Anal. Appl., Volume 126 (1987) no. 1, pp. 90-99 | DOI | MR | Zbl

[6] Briet, Ph.; Combes, J.-M.; Duclos, P. On the location of resonances for Schrödinger operators in the semiclassical limit. II. Barrier top resonances, Comm. Partial Differential Equations, Volume 12 (1987) no. 2, pp. 201-222 | DOI | MR | Zbl

[7] Burq, Nicolas; Zworski, Maciej Resonance expansions in semi-classical propagation, Comm. Math. Phys., Volume 223 (2001) no. 1, pp. 1-12 | DOI | MR | Zbl

[8] Burq, Nicolas; Zworski, Maciej Geometric control in the presence of a black box, J. Amer. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471 | DOI | MR | Zbl

[9] Christiansen, T.; Zworski, M. Resonance wave expansions: two hyperbolic examples, Comm. Math. Phys., Volume 212 (2000) no. 2, pp. 323-336 | DOI | MR | Zbl

[10] Christianson, Hans Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Volume 246 (2007) no. 2, pp. 145-195 | MR | Zbl

[11] De Bièvre, Stephan; Robert, Didier Semiclassical propagation on |log| time scales, Int. Math. Res. Not. (2003) no. 12, pp. 667-696 | DOI | MR | Zbl

[12] Dereziński, Jan; Gérard, Christian Scattering theory of classical and quantum N -particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | MR | Zbl

[13] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[14] Gérard, C. Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes, Mém. Soc. Math. France (N.S.) (1988) no. 31, pp. 146 | Numdam | MR | Zbl

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

[16] Gérard, C.; Sigal, I. M. Space-time picture of semiclassical resonances, Comm. Math. Phys., Volume 145 (1992) no. 2, pp. 281-328 | DOI | MR | Zbl

[17] Gérard, C.; Sjöstrand, J. Semiclassical resonances generated by a closed trajectory of hyperbolic type, Comm. Math. Phys., Volume 108 (1987) no. 3, pp. 391-421 | DOI | MR | Zbl

[18] Guillarmou, Colin; Naud, Frédéric Wave decay on convex co-compact hyperbolic manifolds, Comm. Math. Phys., Volume 287 (2009) no. 2, pp. 489-511 | DOI | MR | Zbl

[19] Hassell, Andrew; Melrose, Richard; Vasy, András Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, Volume 1 (2008) no. 2, pp. 127-196 | DOI | MR | Zbl

[20] Helffer, B.; Martinez, André Comparaison entre les diverses notions de résonances, Helv. Phys. Acta, Volume 60 (1987) no. 8, pp. 992-1003 | MR

[21] Helffer, B.; Sjöstrand, J. Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells, Math. Nachr., Volume 124 (1985), pp. 263-313 | DOI | MR | Zbl

[22] Helffer, B.; Sjöstrand, J. Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) (1986) no. 24-25, pp. iv+228 | Numdam | MR | Zbl

[23] Hitrik, Michael; Sjöstrand, Johannes; Vũ Ngọc, San Diophantine tori and spectral asymptotics for nonselfadjoint operators, Amer. J. Math., Volume 129 (2007) no. 1, pp. 105-182 | MR | Zbl

[24] Hunziker, W. Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré Phys. Théor., Volume 45 (1986) no. 4, pp. 339-358 | Numdam | MR | Zbl

[25] Isozaki, Hiroshi; Kitada, Hitoshi Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 32 (1985) no. 1, pp. 77-104 | MR | Zbl

[26] Isozaki, Hiroshi; Kitada, Hitoshi Scattering matrices for two-body Schrödinger operators, Sci. Papers College Arts Sci. Univ. Tokyo, Volume 35 (1986) no. 2, pp. 81-107 | MR | Zbl

[27] Kaidi, Nourredine; Kerdelhué, Philippe Forme normale de Birkhoff et résonances, Asymptot. Anal., Volume 23 (2000) no. 1, pp. 1-21 | MR | Zbl

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

[29] Lahmar-Benbernou, Amina; Martinez, André Semiclassical asymptotics of the residues of the scattering matrix for shape resonances, Asymptot. Anal., Volume 20 (1999) no. 1, pp. 13-38 | MR | Zbl

[30] Lax, Peter D.; Phillips, Ralph S. Scattering theory, Pure and Applied Mathematics, 26, Academic Press Inc., Boston, MA, 1989 (With appendices by Cathleen S. Morawetz and Georg Schmidt) | MR | Zbl

[31] Martinez, A. Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré, Volume 3 (2002) no. 4, pp. 739-756 | DOI | MR | Zbl

[32] Michel, Laurent Semi-classical estimate of the residues of the scattering amplitude for long-range potentials, J. Phys. A, Volume 36 (2003) no. 15, pp. 4375-4393 | DOI | MR | Zbl

[33] Nakamura, Shu Scattering theory for the shape resonance model. I. Nonresonant energies, Ann. Inst. H. Poincaré Phys. Théor., Volume 50 (1989) no. 2, pp. 115-131 | Numdam | MR | Zbl

[34] Nakamura, Shu Scattering theory for the shape resonance model. II. Resonance scattering, Ann. Inst. H. Poincaré Phys. Théor., Volume 50 (1989) no. 2, pp. 133-142 | Numdam | MR | Zbl

[35] Nakamura, Shu; Stefanov, Plamen; Zworski, Maciej Resonance expansions of propagators in the presence of potential barriers, J. Funct. Anal., Volume 205 (2003) no. 1, pp. 180-205 | DOI | MR | Zbl

[36] Ramond, Thierry Semiclassical study of quantum scattering on the line, Comm. Math. Phys., Volume 177 (1996) no. 1, pp. 221-254 | DOI | MR | Zbl

[37] Sjöstrand, J. Semiclassical resonances generated by nondegenerate critical points, Pseudodifferential operators (Oberwolfach, 1986) (Lecture Notes in Math.), Volume 1256, Springer, Berlin, 1987, pp. 402-429 | MR | Zbl

[38] Sjöstrand, J. A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 490, Kluwer Acad. Publ., Dordrecht, 1997, pp. 377-437 | MR | Zbl

[39] Sjöstrand, Johannes; Zworski, Maciej Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., Volume 4 (1991) no. 4, pp. 729-769 | DOI | MR | Zbl

[40] Stefanov, Plamen Estimates on the residue of the scattering amplitude, Asymptot. Anal., Volume 32 (2002) no. 3-4, pp. 317-333 | MR | Zbl

[41] Tang, Siu-Hung; Zworski, Maciej From quasimodes to reasonances, Math. Res. Lett., Volume 5 (1998) no. 3, pp. 261-272 | MR | Zbl

[42] Tang, Siu-Hung; Zworski, Maciej Resonance expansions of scattered waves, Comm. Pure Appl. Math., Volume 53 (2000) no. 10, pp. 1305-1334 | DOI | MR | Zbl

[43] Vaĭnberg, B. R. Asymptotic methods in equations of mathematical physics, Gordon & Breach Science Publishers, New York, 1989 (Translated from the Russian by E. Primrose) | MR | Zbl

Cited by Sources: