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

Jager, Lisette

Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7, No. 3, 465-495 (1998).

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

Keywords:

relativistic oscillator; Mathieu functions

Citations:

Zbl 0797.58034; Zbl 0912.34035

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