On the numerical integration of the Schrodinger equation for symmetrical potentials: a highly stable shooting method with the Numerov integrator

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Abstract
The problem of the stability of the numerical solution of the one-dimensional Schrodinger equation with symmetrical potential V(x) is considered. This problem is illustrated by the example y"+(E-x2)y=0 with y(0)=1 and y(x) to 0 when x to infinity , for which the conventional shooting method using the Numerov integrator fails for E=-1, to find y(x) beyond x=5. It is shown that these values are reached by using a different procedure for shooting with the same Numerov integrator. This procedure starts the integration at any 'large' value L of x, and steps backward towards x=0. The method is applied to another value of E for which an exact value of y(x) is available. This tests shows that the accuracy of the computed values of y(x) is independent of the choice of L. Thus the method does not improve the eigenvalue computation, but it allows the determination of the solution y(x) for large x when such values are needed.