Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top 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.

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.

[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 2429063 | Zbl 1163.35004

[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 | Article | MR 2357351 | Zbl 1157.35127

[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 | Article | MR 2426141 | Zbl 1159.35007

[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 | Article | MR 2048563 | Zbl 1062.35053

[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 | Article | MR 900530 | Zbl 0629.47043

[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 | Article | MR 876987 | Zbl 0622.47047

[7] Burq, Nicolas; Zworski, Maciej Resonance expansions in semi-classical propagation, Comm. Math. Phys., Volume 223 (2001) no. 1, pp. 1-12 | Article | MR 1860756 | Zbl 1042.81582

[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 | Article | MR 2051618 | Zbl 1050.35058

[9] Christiansen, T.; Zworski, M. Resonance wave expansions: two hyperbolic examples, Comm. Math. Phys., Volume 212 (2000) no. 2, pp. 323-336 | Article | MR 1772249 | Zbl 0955.58024

[10] Christianson, Hans Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Volume 246 (2007) no. 2, pp. 145-195 | MR 2321040 | Zbl 1119.58018

[11] De Bièvre, Stephan; Robert, Didier Semiclassical propagation on $|log\hslash |$ time scales, Int. Math. Res. Not. (2003) no. 12, pp. 667-696 | Article | MR 1951402 | Zbl 1127.81328

[12] Dereziński, Jan; Gérard, Christian Scattering theory of classical and quantum $N$-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | MR 1459161 | Zbl 0899.47007

[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 1735654 | Zbl 0926.35002

[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 998698 | Zbl 0654.35081

[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 1029851 | Zbl 0711.35097

[16] Gérard, C.; Sigal, I. M. Space-time picture of semiclassical resonances, Comm. Math. Phys., Volume 145 (1992) no. 2, pp. 281-328 | Article | MR 1162800 | Zbl 0755.35102

[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 | Article | MR 874901 | Zbl 0637.35027

[18] Guillarmou, Colin; Naud, Frédéric Wave decay on convex co-compact hyperbolic manifolds, Comm. Math. Phys., Volume 287 (2009) no. 2, pp. 489-511 | Article | MR 2481747 | Zbl 1196.58011

[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 | Article | MR 2472888 | Zbl 1152.35454

[20] Helffer, B.; Martinez, André Comparaison entre les diverses notions de résonances, Helv. Phys. Acta, Volume 60 (1987) no. 8, pp. 992-1003 | MR 929933

[21] Helffer, B.; Sjöstrand, J. Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells, Math. Nachr., Volume 124 (1985), pp. 263-313 | Article | MR 827902 | Zbl 0597.35023

[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 871788 | Zbl 0631.35075

[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 2288739 | Zbl 1172.35085

[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 880742 | Zbl 0619.46068

[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 783182 | Zbl 0582.35036

[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 847881 | Zbl 0615.35065

[27] Kaidi, Nourredine; Kerdelhué, Philippe Forme normale de Birkhoff et résonances, Asymptot. Anal., Volume 23 (2000) no. 1, pp. 1-21 | MR 1764337 | Zbl 0955.35009

[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 1714347 | Zbl 0944.35060

[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 1697827 | Zbl 0931.35119

[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 1037774 | Zbl 0697.35004

[31] Martinez, A. Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré, Volume 3 (2002) no. 4, pp. 739-756 | Article | MR 1933368 | Zbl 1026.81012

[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 | Article | MR 1984509 | Zbl 1113.81119

[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 1002815 | Zbl 0686.35090

[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 1002816 | Zbl 0686.35091

[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 | Article | MR 2020213 | Zbl 1037.35064

[36] Ramond, Thierry Semiclassical study of quantum scattering on the line, Comm. Math. Phys., Volume 177 (1996) no. 1, pp. 221-254 | Article | MR 1382227 | Zbl 0848.34074

[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 897789 | Zbl 0627.35074

[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 1451399 | Zbl 0877.35090

[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 | Article | MR 1115789 | Zbl 0752.35046

[40] Stefanov, Plamen Estimates on the residue of the scattering amplitude, Asymptot. Anal., Volume 32 (2002) no. 3-4, pp. 317-333 | MR 1993653 | Zbl 1060.35097

[41] Tang, Siu-Hung; Zworski, Maciej From quasimodes to reasonances, Math. Res. Lett., Volume 5 (1998) no. 3, pp. 261-272 | MR 1637824 | Zbl 0913.35101

[42] Tang, Siu-Hung; Zworski, Maciej Resonance expansions of scattered waves, Comm. Pure Appl. Math., Volume 53 (2000) no. 10, pp. 1305-1334 | Article | MR 1768812 | Zbl 1032.35148

[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 1054376 | Zbl 0743.35001