Application of the modeling potential method to a spectral problem with the $|x|^{N}$ potential

The application of the modeling potential method suggested by the author to the construction of the energy spectrum of the Schr$\stackrel{..}{o}$dinger equation with a potential of the form $|x|^{N}$ is considered. The modeling potential is taken to be the potential of a harmonic oscillator. Estimates are presented for the residual terms in the zeroth-order and first approximations.