Estimates of parabolic cylinder functions on the real axis

Research paper by Alexis Pokrovski

Indexed on: 12 Oct '05Published on: 12 Oct '05Published in: Mathematics - Classical Analysis and ODEs


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.