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The canonic operator (real case). (English) Zbl 0311.35079

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47F05 General theory of partial differential operators
35B40 Asymptotic behavior of solutions to PDEs
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[1] V. I. Arnol’d, Lectures on Classical Mechanics [in Russian], Izd. MGU, Moscow (1968).
[2] V. I. Arnol’d, ?On the characteristic class occurring in the quantization condition,? Funktsional’. Analiz i Ego Prilozhen.,1, No. 1, 1?14 (1967). · doi:10.1007/BF01075861
[3] V. I. Arnol’d, ?Integrals of rapidly oscillating functions and the singularities of projections of Lagrange manifolds,? Funktsional’. Analiz i Ego Prilozhen.,6, No. 3, 61?62 (1972).
[4] V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short Wave Diffraction Problems. Method of Standard Problems [in Russian], Nauka, Moscow (1972), 456 pp. · Zbl 0255.35002
[5] V. A. Borovikov, Diffraction by Polygons and Polyhedrons [in Russian], Nauka, Moscow (1966), 455 pp.
[6] V. S. Buslaev, ?Generating integral and Maslov’s canonic operator in the WKB method,? Funktsional’. Analiz i Ego Prilozhen.,3, No. 3, 17?31 (1969).
[7] L. A. Vainshtein, Open Resonators and Open Waveguides [in Russian], Sov. Radio, Moscow (1966).
[8] M. I. Vishik and L. A. Lyus-ternik, ?Regular degeneration and boundary layer for linear differential equations with a small parameter,? Uspekhi Matem. Nauk,12, No. 5, 3?122 (1957).
[9] M. I. Vishik and G. I. Éskin, ?Convolution equations in a bounded region,? Uspekhi Matem. Nauk,20, No. 3, 89?152 (1965).
[10] F. R. Gantmakher, Lectures on Analytical Mechanics [in Russian], Fizmatgiz, Moscow (1960). · Zbl 0212.56701
[11] I. M. Gel’fand and S. V. Fomin, Calculus of Variations [in Russian], Fizmatgiz, Moscow (1961), 228 pp.
[12] H. Goldstein, Classical Mechanics, Addison-Wesley (1950).
[13] V. A. Dubrovskii and G. A. Skuridin, ?Asymptotic expansions in wave mechanics,? Zh. Vychisl. Mat. i Mat. Fiz.,4, No. 5, 848?870 (1964).
[14] V. R. Kogan, ?Asymptotic behavior of the Laplace-Beltrami operator on the unit sphere Sn,? Izv. Vuzov, Radiofizika,12, No. 11, 1675?1680 (1969).
[15] V. R. Kogan, ?Asymptotic behavior of the Laplace-Beltrami operator in the n-dimensional ball En,? Izv. Vuzov, Radiofizika,12, No. 11, 1681?1689 (1969).
[16] J. J. Cohn and L. Nirenberg, ?Algebra of pseudodifferential operators,? in: Pseudodifferential Operators [Russian translation], Mir, Moscow (1967), pp. 9?62.
[17] Yu. A. Kravtsov, ?On a modification of a method in geometric optics,? Izv. Vuzov, Radiofizika,7, No. 4 (1964).
[18] R. Courant, Partial Differential Equations [Russian translation], Mir, Moscow (1964), 830 pp.
[19] V. V. Kucherenko, ?Quasiclassical a symptotics of the point source function for the stationary Schrödinger equation,? Teor. i Mat. Fiz.,1, No. 3, 384?406 (1969).
[20] V. V. Kucherenko, ?On a method for computing the terms in the asymptotic expansion of the integral ?e i?S(x)? (x)dx,x ? Rn, as ? ? ?,? Tr. Mosk. Inst. Élektron. Mashinostr., No. 4, 189?217 (1968).
[21] J. Leray, L. Gärding, and T. Kotake, The Cauchy Problem [Russian translation], Mir, Moscow (1967), 152 pp.
[22] V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Izd. MGU, Moscow (1965).
[23] V. P. Maslov, ?The WKB method in the multidimensional case.? Appendix in: J. Hadding, Introduction to the Method of Phase Integrals (the WKB Method) [Russian translation], Mir, Moscow (1965), pp. 175?237.
[24] V. P. Maslov, ?Quasiclassical asymptotics of solutions of certain mathematical physics problems,? Zhurn. Vychisl. Mat. i Mat. Fiz.,1, No. 1, 113?128; No. 4, 638?663 (1961).
[25] V. P. Maslov, ?On the regularization of the Cauchy problem for pseudodifferential equations,? Dokl. Akad. Nauk SSSR,177, No. 6, 1277?1280 (1967).
[26] J. Milnor, Morse Theory, Princeton Univ. Press (1963).
[27] V. I. Smirnov, Course in Higher Mathematics [in Russian], Vol. 4, Fizmatgiz, Moscow (1958).
[28] M. V. Fedoryuk, ?The stationary phase method and pseudodifferential operators,? Uspekhi Matem. Nauk,26, No. 1, 67?112 (1971). · Zbl 0221.47036
[29] V. A. Fok, Diffraction Problems and the Propagation of Electromagnetic Waves [in Russian], Sov. Radio, Moscow (1970).
[30] V. A. Fok, ?On a canonic transformation in classical and quantum mechanics,? Vestn. LGU,16, 67?71 (1959).
[31] F. Frank and R. Mizes, Differential and Integral Equations of Mathematical Physics [in Russian], ONTI, Moscow (1937).
[32] D. B. Fuks, ?On Maslov-Arnol’d characteristic classes,? Dokl. Akad. NaukSSSR,178, No. 2, 303?306 (1968).
[33] L. Hörmander, ?Pseudodifferential operators,? in: Pseudodifferential Operators [Russian translation], Mir, Moscow (1967), pp. 63?87.
[34] L. Hörmander, ?Integral Fourier operators,? Matematika (Periodic. Collection of Translations of Foreign Articles),16, No. 1, 17?61: No. 2, 67?136 (1972).
[35] G. D. Birkhoff, ?Quantum mechanics and asymptotic series,? Bull. Amer. Math. Soc.,39, 681?700 (1933). · Zbl 0008.08902 · doi:10.1090/S0002-9904-1933-05716-6
[36] R. Courant and P. Lax, ?The propagation of discontinuities in wave motion,? Proc. Nat. Acad. Sci. USA,42, No. 11, 872?876 (1956). · Zbl 0072.30803 · doi:10.1073/pnas.42.11.872
[37] J. B. Keller, ?Diffraction by a convex cylinder,? Trans. IRE, Ant. and Prop.,4, No. 3, 312 (1956). · doi:10.1109/TAP.1956.1144427
[38] J. B. Keller, ?Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,? Ann. Phys.,4, No. 2, 180?188 (1958). · Zbl 0085.43103 · doi:10.1016/0003-4916(58)90032-0
[39] J. B. Keller, R. M. Lewis, and B. D. Seckler, ?Asymptotic solution of some diffraction problems,? Commun. Pure and Appl. Math.,9, No. 2, 207?265 (1956). · Zbl 0073.44105 · doi:10.1002/cpa.3160090205
[40] J. B. Keller and S. Rubinow, ?Asymptotic solution of eigenvalue problems,? Ann. Phys.,9, No. 1, 24?75 (1960). · Zbl 0087.43002 · doi:10.1016/0003-4916(60)90061-0
[41] P. Lax, ?Asymptotic solutions ot oscillatory initial-value problems,? Duke Math. J.,24, No. 4, 627?646 (1957). · Zbl 0083.31801 · doi:10.1215/S0012-7094-57-02471-7
[42] D. Ludwig, ?Exact and asymptotic solutions of the Cauchy problem,? Commun. Pure and Appl. Math.,13, No. 3, 473?508 (1960). · Zbl 0098.29601 · doi:10.1002/cpa.3160130310
[43] D. Ludwig, ?Uniform asymptotic expansion of the field scattering by a convex object at high frequencies,? Commun. Pure and Appl. Math.,20, No. 1, 103?138 (1967). · Zbl 0154.12802 · doi:10.1002/cpa.3160200103
[44] S. P. Novikov, ?Algebraic construction and properties of hermitian analogs of K-theory over rings with involution from the viewpoint of hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. 1, 2,? Math. USSR Izv.,4, No. 3, 257?292, 479?505 (1970). · Zbl 0216.45003 · doi:10.1070/IM1970v004n02ABEH000903
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