Numerical solution of Sturm–Liouville problems via Fer streamers

Abstract

We address the numerical challenge of solving regular Sturm–Liouville problems in Liouville’s normal form, with a continuous and piecewise analytic potential and self-adjoint separated boundary conditions. The novelty of our approach, which is based on a non-standard truncation of Fer expansions, which we call ‘Fer streamers’, lies in the construction of a new numerical method, which (1) does not impose any restriction on the step size for eigenvalues which are greater than or equal to the minimum of the potential, (2) requires only a mild restriction on the step size for the remaining finite number of eigenvalues, (3) can attain any convergence rate, which grows exponentially with the number of terms, and is uniform for every eigenvalue, and (4) lends itself to a clear understanding of the manner in which the potential affects the local and global errors. We provide our numerical method with its analytical underpinning, but emphasize that it is at an early stage of development and that much remains to be done. In particular, we comment on our investigation of efficient discretization schemes for the integrals which arise in Fer streamers.