Abstract
A new explicit hybrid eighth algebraic order two-step method with phase-lag of order twelve is developed for computing eigenvalues and resonances of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the resonance problem for the well known case of the Woods-Saxon potential and for the integration of the eigenvalue problem for the well known case of the Morse potential show that this new method is better than other variable-step methods.
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