A Perturbative Approach for the Solution of Sturm-Liouville Problems

Spectral analysis extends the theory of eigenvectors and eigenvalues of a square matrix to a broader theory involving operators. In particular, a branch of spectral analysis is devoted to Sturm-Liouville (SL) problems, which are eigenvalue problems for differential operators. In this study, we propose a numerical method to solve SL problems. This method uses a simple perturbative approach. Starting from an SL problem having differential operator L_0 and known eigensystem, the proposed iterative algorithm considers M SL problems having differential operators L_m, m = 1, 2,...,M, such that L_m is a perturbation of L_{m-1}, and L_M is the differential operator of the SL problem that we want to solve. Each step of this algorithm is based on the well-known Jacobi orthogonal component correction method, which acts on the refinement of approximated eigensystems. Moreover, the proposed method depends on the choice of L0 and the representation basis for the eigenfunctions, thus giving rise to different approximation schemes. We show the performance of the proposed method both in the solution of some selected SL problems and the refinement of approximated eigensystems computed by other numerical methods. In these numerical experiments, the perturbative method is compared with a classical approximation technique and the obtained results are strongly promising in terms of accuracy.