  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.
