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The relativistic oscillator and the Mathieu functions. (L’oscillateur relativiste et les fonctions de Mathieu.) (French) Zbl 0797.58034

The author studies the operator \(L\), corresponding to the relativistic oscillator, defined by \[ -4\pi L = \sum {\partial^ 2 \over \partial x^ 2_ j} - 4\pi^ 2 \sum x^ 2_ j + c^{-2} \biggl[\biggl(\sum x_ j {\partial \over \partial x_ j}\biggr)^ 2 + (n - 1) \sum x_ j {\partial\over \partial x_ j}\biggr]. \] He uses the Klein-Gordon symbolic calculus to obtain the symbols of the families of operators that commute with \(L\). Then he derives a Feynman integral type representation for \(e^{-\varepsilon L}\) and finds some interesting properties of this representation. In the case of dimension 1, some new properties and formulas of the Mathieu functions are found.
Reviewer: V.Oproiu (Iaşi)

MSC:

53D50 Geometric quantization
58D30 Applications of manifolds of mappings to the sciences
83A05 Special relativity
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References:

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