A polynomial approach to the spectral corrections for Sturm-Liouville problems

A technique based on the evaluation of the zeros of a polynomial is proposed to estimate the spectral errors and set up a correcting procedure in Sturm-Liouville problems. The method suggested shows its effectiveness both in the regular and nonregular case and can be successfully applied also to those problems containing the eigenvalue parameter rationally. Some numerical experiments clearly confirm the theoretical results. In the case of an eigenvalue embedded in the essential spectrum the correcting procedure seems to be particularly helpful because the inner singularity gives rise to possible decay of the performance of some classical methods for the numerical integration.