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Semiclassical resonances generated by a closed trajectory of hyperbolic type. (English) Zbl 0637.35027
Authors’ summary: All the resonances in certain rectangular regions of the complex plane of the Schrödinger operator \(-h^ 2\Delta +V\) when \(h\to 0\) (and more general semiclassical operators) are determined under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding Poincaré map is hyperbolic.
Reviewer: G.Derfel

35J10 Schrödinger operator, Schrödinger equation
35S05 Pseudodifferential operators as generalizations of partial differential operators
37-XX Dynamical systems and ergodic theory
Full Text: DOI
[1] Abraham, R., Marsden, J.: Foundations of mechanics. New York: Benjamin/Cummings 1978 · Zbl 0393.70001
[2] Boutet de Monvel, L., Grigis, A., Helffer, B.: Parametrixes d’opérateurs pseudodifferentiels à caractéristiques multiples. Astérisque34-35, 93-121 (1976) · Zbl 0344.32009
[3] Cordoba, A., Fefferman, C.: Wavepackets and Fourier integral operators. Commun. Partial Differ. Equations3, 979-1005 (1978) · Zbl 0389.35046 · doi:10.1080/03605307808820083
[4] Gérard, C.: Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes (preprint) · Zbl 0654.35081
[5] Guillope, L.: Private communication
[6] Helffer, B., Sjöstrand, J.: Résonances en limite semi-classique. Mémoire de la S.M.F., n\(\deg\) 24/25, Suppl. an Bull. de la S.M.F., 114 (1986), fasc. 3 · Zbl 0631.35075
[7] Ikawa, M.: On the poles of the scattering matrix for two convex obstacles. J. Math. Kyoto Univ.23, 127-194 (1983). An addendum in J. Math. Kyoto Univ.23, 795-802 (1983) · Zbl 0561.35060
[8] Ikawa, M.: Precise information on the poles of the scattering matrix for two strictly convex obstacles. Preprint, also in the proceedings of the Journées des E.D.P. at St. Jean de Monts, 1985 · Zbl 0587.35057
[9] Melin, A., Sjöstrand, J.: Fourier integral operators with complex valued phase functions. Lecture Notes in Mathematics, Vol. 459. Berlin, Heidelberg, New York: Springer, pp. 120-223 · Zbl 0306.42007
[10] Reed, M., Simon, B.: Methods of modern mathematical physics, I?IV. New York: Academic Press, pp. 1974-82
[11] Sjöstrand, J.: Analytic wavefront sets and operators with multiple characteristics. Hokkaido Math. J.12, 392-433 (1983) · Zbl 0531.35022
[12] Sjöstrand, J.: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Math.12, 85-120 (1974) · Zbl 0317.35076 · doi:10.1007/BF02384749
[13] Sjöstrand, J.: Singularités analytiques microlocales. Astérisque95 (1982)
[14] Sjöstrand, J.: Semiclassical resonances generated by non-degenerated critical points. Preprint of the University of Lund (1986) · Zbl 0609.35069
[15] Ralston, J.V.: On the construction of quasimodes associated with stable periodic orbits. Commun. Math. Phys.51, 219-242 (1976) · Zbl 0333.35066 · doi:10.1007/BF01617921
[16] Colin de Verdière, Y.: Quasimodes sur les variétés riemanniennes. Invent. Math.43, 15-52 (1977) · Zbl 0449.53040 · doi:10.1007/BF01390202
[17] Candelpergher, B., Nosmas, J.C.: Quantification et approximations semiclassiques. Colloque de Saint Jean de Monts (1982) · Zbl 0497.58023
[18] Voros, A.: The W.K.B. method for non-separable systems. Actes du congrès international de géométrie symplectique et physique mathématique, Aix (1974)
[19] Duistermaat, J.J.: Oscillatory integrals, lagrangian immersions and unfolding of singularities. Commun. Pure Appl. Math.27, 207-281 (1974) · Zbl 0285.35010 · doi:10.1002/cpa.3160270205
[20] Pollak, E.: A quasiclassical model for resonance widths in quantal colinear reactive scattering. J. Chem. Phys.76, 5843-5848 (1982) · doi:10.1063/1.442983
[21] Gutzwiller, M.C.: Periodic orbits and classical quantization conditions. J. Math. Phys.12, 343-358 (1971) · doi:10.1063/1.1665596
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