Jager, Lisette Mathieu functions and eigenfunctions of the relativistic oscillator. (Fonctions de Mathieu et fonctions propres de l’oscillateur relativiste.) (French) Zbl 0923.34028 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) Cited in 2 Documents 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 PDF BibTeX XML Cite \textit{L. Jager}, Ann. Fac. Sci. Toulouse, Math. (6) 7, No. 3, 465--495 (1998; Zbl 0923.34028) Full Text: DOI Numdam EuDML OpenURL Digital Library of Mathematical Functions: 7th item ‣ §28.33(iii) Stability and Initial-Value Problems ‣ §28.33 Physical Applications ‣ Applications ‣ Chapter 28 Mathieu Functions and Hill’s Equation 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.