Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

Abstract

The spatial resolution of eigenfunctions of Sturm–Liouville equations in one-dimension is frequently measured by examining the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster like O(1/N 2) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In this work an alternative measure, l, is proposed for estimating the characteristic length scale of eigenfunctions of Sturm–Liouville equations that does not rely on knowledge of the roots. It is first shown that l is a reasonable measure for evaluating the eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing the minimum distance between Fekete points to l shows that l is a reasonably good measure of the characteristic length scale in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement.