Unified QLM for regular and arbitrary singular potentials

摘要

A parameterless numerical implementation of the Quasilinearization Method (QLM) is constructed and tested to 21–25 digits precision to give quadratically convergent energies E of the Klauder effect exhibiting spiked harmonic oscillator with the λ/rα or exp(λ/rα) type spikes in a Riccati reformulation of the Schrödinger equation. The radial solution is uniformly quadratic convergent to the same precision as E, except in the small minorization interval where the self-correcting property of QLM assures geometric convergence like in the Picard algorithm to about 12–16 digits, sufficient not to affect the convergence of E, confirming what is expected on physical grounds. It was shown before that for regular potentials, immediate onset of quadratic convergence is guaranteed by the initial iteration of the WKB form, and that for quadratic convergence of E for power-type spikes it suffices to augment this by a nonlinear integration point distribution and by minorization of (negative) solution values. The form of the Riccati equation used allows the minorization function to easily be formally defined over the entire interval, without the need for a cutoff radius of application, and dependence on its scale factor is plateau-like and negligible.