Bounds for the eigenfunctions of multiparameter Sturm–Liouville systems

Article type: Research Article

Authors: Schäfke, Reinhard | Volkmer, Hans

Affiliations: Fachbereich Mathematik, Universität Essen GHS, Universitätsstraße 3, 4300 Essen 1, Fed. Rep. Germany

Abstract: We derive bounds for the values of L2-normalized eigenfunctions of Sturm–Liouville systems containing k spectral parameters λ1,…,λk. In the case of continuous coefficients this bound is of order O(|λ|k/4 as |λ|→∞, where λ=(λ1,…,λk) is the corresponding eigenvalue and |λ|=max|λi|. If the coefficients are continuously differentiable then we obtain a better bound of order O(|λ|(k−1)/6). The latter result generalizes a bound of order O(|λ|1/6) given by Faierman (1980) in the two-parameter case k = 2 under the assumption of analytic coefficients. We show by examples that the powers ¼k and ⅙(k−1) of |λ| in the above bounds are optimal.

DOI: 10.3233/ASY-1989-2203

Journal: Asymptotic Analysis, vol. 2, no. 2, pp. 139-159, 1989