Numerical solution of singular ODE eigenvalue problems in electronic structure computations

Abstract

We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.keywords: electronic structure computation, polynomial collocation, fullpotential core solver, singular eigenvalue problems MSC 65L15, 65L60, 65N25, 81-08 PACS 31.15ae, 31.15es, 32.60.+i 1

Convergence of collocation schemes for boundary value problems in nonlinear index 1 DAEs with a singular point

“…For α = 1 the problem is called singular with a singularity of the first kind, for α > 1 it is essentially singular (singularity of the second kind). Research activities in related fields, like the computation of connecting orbits in dynamical systems [33], or singular Sturm-Liouville problems [4,12], also benefit from new findings for problems of the form (1.1). Both the analysis and the numerical treatment of this problem class considerably differ from the regular case [16,17,18,19,25].…”

A Spectral Transform Method for Singular Sturm--Liouville Problems with Applications to Energy Diffusion in Plasma Physics

“…In addition, [13] contains a wealth of information on the history of the problem and provides an insightful review of the literature. There are also many papers on computing eigenvalues and eigenfunctions (rather than continuous spectra and density functions) for singular Sturm-Liouville problems; see, e.g., [23,31].…”

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“…Therefore, results derived for equation (1.1a) also apply for the modified equation (t −a v ′ (t)) ′ = g(t, v(t), v ′ (t)). Such type of models arises in the study of phase transitions of Van der Waals fluids [3], [8], [12], [14], [18], in population genetics, in models for the spatial distribution of the genetic composition of a population [6], [7], in the homogenenous nucleation theory [1], in relativistic cosmology in description of particles which can be treated as domains in the universe [15], and in the nonlinear field theory [9], in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [5].…”