2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103643We 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.25 pages, 5 PostScript figuresClassical Analysis and ODEs34M60 (Primary), 34E20 (Secondary)text