Estimates of parabolic cylinder functions on the real axis

Details Estimates of parabolic cylinder functions on the real axis Estimates of parabolic cylinder functions on the real axis

2005-10-12

arxiv.org/abs/math/0510249v1

Mathematics

Classical Analysis and ODEs

25 pages, 5 PostScript figures

Scientific paper

We estimate simple combination of the parabolic cylinder functions and their derivatives. These estimates are important for the spectral analysis of non-analytically perturbed quantum harmonic oscillator. The estimates are valid in rather complicated domains and refine there the classical result of Olver. We determine the part of the real x-axis, which is within the domains of the estimates. This requires a detailed study of the image of the real x-axis in the standard quasiclassical variable.

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