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Zbl 0923.34028
Jager, Lisette
Mathieu functions and eigenfunctions of the relativistic oscillator. (Fonctions de Mathieu et fonctions propres de l'oscillateur relativiste.) (French)
[J] Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7, No.3, 465-495 (1998). ISSN 0240-2963

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)].
[V.Burjan (Praha)]

MSC 2000:: *34B30 Special ODE
83C30 Asymptotic procedures (general relativity)
33E10 Spheroidal wave functions, etc.
34C15 Nonlinear oscillations of solutions of ODE

Keywords: relativistic oscillator; Mathieu functions

Citations: Zbl 0797.58034; Zbl 0912.34035

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