Bounds for the eigenfunctions of multiparameter Sturm–Liouville systems

Abstract

We derive bounds for the values of L²-normalized eigenfunctions of Sturm–Liouville systems containing k spectral parameters λ₁,…,λ_k. In the case of continuous coefficients this bound is of order O(|λ|^k/4 as |λ|→∞, where λ=(λ₁,…,λ_k) is the corresponding eigenvalue and |λ|=max|λ_i|. 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.