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Various algorithms for the numerical investigation of the spectrum of the one-dimensional Schrödinger operator are presented, based on established correlations between the asymptotics of solutions of the corresponding differential equations and spectral properties. The algorithms enable the spectral contribution of specific l-points in R to be assessed, and the location of isolated points of the discrete spectrum to be determined. Implementation of the algorithms is carried out using series solutions in conjunction with the programming and computer algebra facilities of Mathematica. As illustration, the displacement of eigenvalues of the perturbed harmonic oscillator on the half-line is considered.
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