Mathieu functions and eigenfunctions of the relativistic oscillator (Q1284682)

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 \textit{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)].