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 , 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 , 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.
Keywords: Schrödinger operator, quantum resonances, semiclassical analysis, resolvent estimate
Mot clés : opérateur de Schrödinger, résonances quantiques, analyse semiclassique, estimation de la résolvante
@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}, zbl = {1246.35033}, mrnumber = {2951496}, language = {en}, url = {http://www.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 SP - 1351 EP - 1406 VL - 61 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/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 http://www.numdam.org/articles/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://www.numdam.org/articles/10.5802/aif.2643/
[1] Semiclassical scattering amplitude at the maximum of the potential, Asymptot. Anal., Volume 58 (2008) no. 1-2, pp. 57-125 | MR | Zbl
[2] 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] 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] 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] 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] 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] Resonance expansions in semi-classical propagation, Comm. Math. Phys., Volume 223 (2001) no. 1, pp. 1-12 | DOI | MR | Zbl
[8] 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] Resonance wave expansions: two hyperbolic examples, Comm. Math. Phys., Volume 212 (2000) no. 2, pp. 323-336 | DOI | MR | Zbl
[10] Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Volume 246 (2007) no. 2, pp. 145-195 | MR | Zbl
[11] Semiclassical propagation on time scales, Int. Math. Res. Not. (2003) no. 12, pp. 667-696 | DOI | MR | Zbl
[12] Scattering theory of classical and quantum -particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | MR | Zbl
[13] Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999 | MR | Zbl
[14] 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] 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] Space-time picture of semiclassical resonances, Comm. Math. Phys., Volume 145 (1992) no. 2, pp. 281-328 | DOI | MR | Zbl
[17] 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] Wave decay on convex co-compact hyperbolic manifolds, Comm. Math. Phys., Volume 287 (2009) no. 2, pp. 489-511 | DOI | MR | Zbl
[19] 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] Comparaison entre les diverses notions de résonances, Helv. Phys. Acta, Volume 60 (1987) no. 8, pp. 992-1003 | MR
[21] Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells, Math. Nachr., Volume 124 (1985), pp. 263-313 | DOI | MR | Zbl
[22] Résonances en limite semi-classique, Mém. Soc. Math. France (N.S.) (1986) no. 24-25, pp. iv+228 | Numdam | MR | Zbl
[23] Diophantine tori and spectral asymptotics for nonselfadjoint operators, Amer. J. Math., Volume 129 (2007) no. 1, pp. 105-182 | MR | Zbl
[24] Distortion analyticity and molecular resonance curves, Ann. Inst. H. Poincaré Phys. Théor., Volume 45 (1986) no. 4, pp. 339-358 | Numdam | MR | Zbl
[25] 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] 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] Forme normale de Birkhoff et résonances, Asymptot. Anal., Volume 23 (2000) no. 1, pp. 1-21 | MR | Zbl
[28] 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] 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] 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] Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré, Volume 3 (2002) no. 4, pp. 739-756 | DOI | MR | Zbl
[32] 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] 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] 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] 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] Semiclassical study of quantum scattering on the line, Comm. Math. Phys., Volume 177 (1996) no. 1, pp. 221-254 | DOI | MR | Zbl
[37] 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] 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] Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., Volume 4 (1991) no. 4, pp. 729-769 | DOI | MR | Zbl
[40] Estimates on the residue of the scattering amplitude, Asymptot. Anal., Volume 32 (2002) no. 3-4, pp. 317-333 | MR | Zbl
[41] From quasimodes to reasonances, Math. Res. Lett., Volume 5 (1998) no. 3, pp. 261-272 | MR | Zbl
[42] Resonance expansions of scattered waves, Comm. Pure Appl. Math., Volume 53 (2000) no. 10, pp. 1305-1334 | DOI | MR | Zbl
[43] 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: