Numerical methods for the eigenvalue determination of second-order ordinary differential equations

An accurate method for the numerical solution of the eigenvalue problem of second-order ordinary differential equation using the shooting method is presented. The method has three steps. Firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method. Secondly the initial-value problem is solved using new, highly accurate formulas of the linear multistep method. Thirdly the eigenvalue is properly corrected at the matching point. The efficiency of the proposed methods is demonstrated by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Pöschl–Teller potential in quantum mechanics.