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On the exact WKB analysis of operators admitting infinitely many phases. (English) Zbl 1056.34103
The paper is devoted to the analysis of differential operators whose WKB solutions admit infinitely many phases. Therefore, a suitable class of differential operators of WKB type is introduced and their exact WKB theoretic structure near their turning points is analysed. The main tools are techniques and ideas in microlocal analysis, a quantized contact transformation and a Späth-type division theorem. The results are applied to derive a connection formula for WKB solutions near a simple turning point.

MSC:
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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[1] Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d’ordre infini, I. ann. inst. Fourier (Grenoble), 33, 227-250, (1983) · Zbl 0495.58025
[2] Aoki, T., Symbols and formal symbols of pseudodifferential operators, Adv. stud. pure math., 4, 181-208, (1984) · Zbl 0579.58029
[3] Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d’ordre infini, II, Ann. inst. Fourier (Grenoble), 36, 143-165, (1986) · Zbl 0576.58027
[4] Aoki, T., Quantized contact transformations and pseudodifferential operators of infinite-order, Publ. res. inst. math. sci. Kyoto univ., 26, 505-519, (1990) · Zbl 0719.58037
[5] T. Aoki, T. Kawai, Y. Takei, New turning points in the exact WKB analysis for higher order ordinary differential equations. Analyse algébrique des perturbations singulières, I; Méthodes résurgentes, Hermann, Paris, 1994, pp. 69-84. · Zbl 0831.34058
[6] Aoki, T.; Kawai, T.; Takei, Y., Algebraic analysis of singular perturbations—on exact WKB analysis, Sugaku expositions, 8, 217-240, (1995)
[7] Aoki, T.; Kawai, T.; Takei, Y., On the exact WKB analysis for higher order ordinary differential equations with a large parameter, Asian J. math., 2, 625-640, (1998) · Zbl 0963.34045
[8] Aoki, T.; Yoshida, J., Microlocal reduction of ordinary differential operators with a large parameter, Publ. res. inst. math. sci. Kyoto univ., 29, 959-975, (1993) · Zbl 0807.34071
[9] Berk, H.L.; Nevins, M.; Roberts, K.V., New Stokes lines in WKB theory, J. math. phys., 23, 988-1002, (1982) · Zbl 0488.34050
[10] Berk, H.L.; Pfirsch, D., WKB method for systems of integral equations, J. math. phys., 21, 2054-2066, (1980) · Zbl 0457.65083
[11] Berk, H.L.; Rosenbluth, M.N.; Sudan, R.N., Plasma wave propagation in hot inhomogeneous media, Phys. fluids, 9, 1606-1608, (1966)
[12] Cartan, H., Idéaux des fonctions analytiques de n variables complexes, Ann. ecole norm. sup., 61, 3, 149-197, (1944) · Zbl 0035.17103
[13] Delabaere, E.; Dillinger, H.; Pham, F., Résurgence de voros et périodes des courbes hyperelliptiques, Ann. inst. Fourier (Grenoble), 43, 163-199, (1993) · Zbl 0766.34032
[14] Kashiwara, M.; Kawai, T.; Kimura, T., Foundations of algebraic analysis, (1986), Princeton University Press Princeton
[15] B. Malgrange, L’involutivité des caractéristiques des systèmes différentiels et micro-différentiels, Sém. Bourbaki, 1977/78, no. 522.
[16] Pham, F., Multiple turning points in exact WKB analysis (variations on a theme of Stokes), (), 71-85 · Zbl 1017.34091
[17] Sanuki, H., Stability of electrostatic drift waves in bumpy tori, Phys. fluids, 27, 2500-2510, (1984) · Zbl 0547.76128
[18] M. Sato, T. Kawai, M. Kashiwara, Microfunctions and pseudo-differential equations, Lecture Notes in Mathematics, Vol. 287, Springer, Berlin, 1973, pp. 265-529. · Zbl 0277.46039
[19] Voros, A., The return of the quartic oscillator—the complex WKB method, Ann. inst. Henri Poincaré, 39, 211-338, (1983) · Zbl 0526.34046
[20] Watanabe, T.; Sanuki, H.; Watanabe, M., New treatment of eigenmode analysis for an inhomogeneous Vlasov plasma, J. phys. soc. Japan, 47, 286-293, (1979)
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