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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 {\it 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:
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
83C30Asymptotic procedures (general relativity)
33E10Lamé, Mathieu, and spheroidal wave functions
34C15Nonlinear oscillations, coupled oscillators (ODE)
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