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Mathieu functions and eigenfunctions of the relativistic oscillator. (Fonctions de Mathieu et fonctions propres de l’oscillateur relativiste.) (French) Zbl 0923.34028

The differential operator of the relativistic oscillator \[ L= -{1\over 4\pi} \Biggl({d^2\over dx^2}- 4\pi^2x^2+ {1\over c^2} \Biggl(x{d\over dx}\Biggr)^2\Biggr) \] is studied as a relativistic deformation of the harmonic oscillator, using the symbolic calculus of Klein-Gordon. An approximate calculus of the \(L^\beta(\ln L)^d\) is developed (\(\beta\) is complex, \(d\) integer). Some exact and some asymptotic results are given on Mathieu functions in association with the relativistic oscillator. The Klein-Gordon calculus by A. Unterberger [Bull. Soc. Math. Fr. 121, No. 4, 479-508 (1993; Zbl 0797.58034)] is used as a substitute for the Weyl calculus. The zeta function of the oscillator is determined. The present article sums up some parts of the paper by the author [C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 7, 713-716 (1997; Zbl 0912.34035)].
Reviewer: V.Burjan (Praha)

MSC:

34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
83C30 Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory
33E10 Lamé, Mathieu, and spheroidal wave functions
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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References:

[1] Campbell ( R. ) . - Théorie générale de l’équation de Mathieu et de quelques autres équations différentielles de la mécanique , Masson , Paris , 1955 . MR 73760 | Zbl 0066.31702 · Zbl 0066.31702
[2] Dieudonné ( J. ) .- Calcul infinitésimal , Hermann , Paris , 1980 . MR 226971 | Zbl 0497.26004 · Zbl 0497.26004
[3] Helffer ( B. ) et Robert ( D. ) .- Propriétés asymptotiques du spectre d’opérateurs pseudo-différentiels sur RN , Comm. in P.D.E. 7 ( 1982 ), pp. 795 - 882 . MR 662451 | Zbl 0501.35081 · Zbl 0501.35081
[4] Jager ( L. ) . - Fonctions de Mathieu et calcul de Klein-Gordon , Thèse, Univ. de Reims-Champagne-Ardenne ( 1994 ).
[5] Magnus ( W. ), Oberhettinger ( F. ) et Soni ( R.P. ) .- Formulas and theorems for the special functions of mathematical physics , Springer-Verlag , Berlin , 3e éd. ( 1966 ). MR 232968 | Zbl 0143.08502 · Zbl 0143.08502
[6] Seeley ( R. ) .- Complex powers of an elliptic operator , Proc. Symp. A.M.S. Pure Math. 10 ( 1967 ), pp. 288 - 307 . MR 237943 | Zbl 0159.15504 · Zbl 0159.15504
[7] Titchmarsh ( E.C. ) .- Eigenfunction expansions associated with second-order differential equations , Clarendon Press , Oxford , Part. 1, 2e éd. ( 1962 ). MR 176151 | Zbl 0099.05201 · Zbl 0099.05201
[8] Unterberger ( A. ) .- Quantification relativiste , Mémoires de la Soc. Math. de France (nouvelle série 44 -45), 1991 . Numdam | MR 1101973 | Zbl 0745.35057 · Zbl 0745.35057
[9] Unterberger ( A. ) .- L’oscillateur relativiste et les fonctions de Mathieu , Bull. Soc. Math. de France 121 , n^\circ 4 ( 1993 ), pp. 479 - 508 . Numdam | MR 1254750 | Zbl 0797.58034 · Zbl 0797.58034
[10] Unterberger ( A. ) . - Relativity, spherical functions and the hypergeometric equation , Ann. Inst. Henri-Poincaré, Phys. Théor. 62 , n^\circ 2 ( 1995 ), pp. 103 - 144 . Numdam | MR 1317183 | Zbl 0834.43011 · Zbl 0834.43011
[11] Whittaker ( E.T. ) et Watson ( G.N. ) .- A course of modern analysis , Cambridge Univ. Press , 4e éd. ( 1965 ). JFM 45.0433.02 · JFM 45.0433.02
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