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Huygens principle and complex trajectories or Balian and Bloch twenty years after. (Principe de Huygens et trajectoires complexes ou Balian et Bloch vingt ans après.) (French) Zbl 0805.35024
One sketches, for the Schrödinger equation in three dimensional space with an analytic potential, how the time independent Green’s function could possibly be reconstructed exactly from the dynamics of complex classical trajectories. To start with this program one proves in the complex domain a microlocal convergence theorem for the “multiple scattering expansions” of Balian and Bloch. The real version of this theorem is easily interpreted in terms of Huygens principle, and turns out to be equivalent to a theorem of Hadamard.
Reviewer: F.Pham (Nice)

MSC:
35J10 Schrödinger operator, Schrödinger equation
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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References:
[1] V. ARNOLD, Integrals of rapidly oscillating functions and singularities of the projections of Lagrangean manifolds, Funct. Anal. and its Appl., 6-3 (1972). · Zbl 0278.57010
[2] V. ARNOLD, A. VARCHENKO, S. GUSSEIN-ZADÉ, Singularités des applications différentiables, trad. française, éd. Mir, Moscou, 1986.
[3] M.V. BERRY, Cusped rainbow and incoherence effects..., J. Phys. A, 8-4 (1975).
[4] P. BÉRARD, On the wave equation on a compact Riemannian manifold without conjugate points, Mathematische Zeitschrift, 155 (1977). · Zbl 0341.35052
[5] M.V. BERRY, J.F. NYE, F.R.S. & F.J. WRIGHT, The elliptic umbilic diffraction catastrophe, Phil. Trans. Royal Soc. London, A 291 (1979).
[6] R. BALIAN and C. BLOCH, Distribution of eigenfrequencies for the wave equation in a finite domain I, Ann. of Physics, 60-2, (1970), II, Ann. of Physics, 63-2 (1971), III, Ann. of Physics, 69-1 (1972). · Zbl 0207.40202
[7] R. BALIAN and C. BLOCH, Asymptotic evaluation of the Green’s function for large quantum numbers, Ann. of Physics, 64-1 (1971).
[8] R. BALIAN, C. BLOCH, Solution of the Schrödinger equation in terms of classical paths, Ann. of Physics, 85 (1974). · Zbl 0281.35029
[9] M. BORN, E. WOLF, Principles of optics, Pergamon, 1975.
[10] J. CHAZARAIN, Formule de Poisson pour LES variétés riemanniennes, Inventiones Math., 24 (1974). · Zbl 0281.35028
[11] B. CANDELPERGHER, J.-C. NOSMAS, F. PHAM, Approche de la résurgence, Actualités Mathématiques, Hermann, 1993. · Zbl 0791.32001
[12] Y. COLIN DE VERDIÈRE, Spectre du laplacien et longueurs des géodésiques périodiques, Compositio Mathematica, 27-1 et 27-2 (1973). · Zbl 0281.53036
[13] E. DELABAERE, Résurgence de l’équation de Schrödinger (en préparation).
[14] E. DELABAERE, H. DILLINGER, F. PHAM, Résurgence de voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier, 43-1 (1993). · Zbl 0766.34032
[15] E. DELABAERE, H. DILLINGER, F. PHAM, Exact semi-classical expansions for one-dimensional quantum oscillators (en préparation). · Zbl 0896.34051
[16] E. DELABAERE, H. DILLINGER, F. PHAM, Resurgent methods in semi-classical asymptotics (en préparation). · Zbl 0712.35071
[17] J. DUISTERMAAT and V. GUILLEMIN, The spectrum of positive elliptic operators and periodic bicharacteristics, Inventiones Math., 29 (1975). · Zbl 0307.35071
[18] J. ÉCALLE, Singularités irrégulières et résurgence multiple, in “Cinq applications des fonctions résurgentes”, Prépub. Univ. Orsay, 62 (1984).
[19] V. GUILLEMIN, S. STERNBERG, Geometric asymptotics, A.M.S. Surveys, 14 (1977). · Zbl 0364.53011
[20] J. HADAMARD, Le problème de Cauchy et LES équations aux dérivées partielles hyperboliques, Hermann, 1932. · JFM 58.0519.16
[21] J. LERAY, Le calcul différentiel et intégral sur une variété analytique complexe (problème de Cauchy III), Bull. Soc. Math. France, 87 (1959). · Zbl 0199.41203
[22] V.F. LAZUTKIN, D.Ya. TERMAN, Complexified quantum rules, in Séminaire de théorie spectrale et Géométrie n° 11 (1993), Institut Fourier, Grenoble. · Zbl 0909.58016
[23] M. TAYLOR, Pseudodifferential operators, Princeton Univ. Press, 1981. · Zbl 0453.47026
[24] A. VOROS, The return of the quartic oscillator, Ann. Inst. Henri Poincaré, 29-3 (1983). · Zbl 0526.34046
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