Efficient numerical solution of the one-dimensional Schrödinger eigenvalue problem using Magnus integrators

摘要

We discuss a new numerical method, based on a modified Magnus integrator, to solve the Sturm–Liouville eigenvalue problem in its Schrödinger form. A modified Magnus integrator has already been used by Degani & Schiff (2006, J. Comput. Appl. Math., 193, 413–436) to approximate the oscillating solution of a Schrödinger problem over the classically allowed region. Here we show that the technique can be successfully extended to the nonoscillatory classically forbidden region. This means that the modified Magnus integrator can be used to propagate the solution over the whole integration interval and is well suited for application in a shooting process to locate the eigenvalues. Such a shooting procedure is formulated and shown to allow the efficient computation of a range of eigenvalues and eigenfunctions.