Eigenvalue ratios for Sturm--Liouville operators
This paper deals with estimates for eigenvalue ratios for regular Sturm-Liouville problems \(-[p(x)y']'+q(x)y=\lambda w(x)y\) on a finite interval with Dirichlet boundary conditions. In the general case \(q \geq 0\), the authors prove the upper estimate \(\lambda_ m/ \lambda_ l \leq K \{m/l\} ^ 2/k\) for \(m>l \geq 1\) where \(k\), \(K \geq 0\) are such that \(k \leq pw \leq K\) and \(\{ m/l \}\) denotes the least integer greater or equal \(m/l\). This result generalizes estimates for the Dirichlet eigenvalues of the one-dimensional Schrödinger equation \(-y''+q(x)y=\lambda y\). For \(q=0\), it can be improved and supplemented by a lower estimate to \(km^ 2/Kl^ 2 \leq \lambda_ m/ \lambda_ l \leq Km^ 2/kl^ 2\). In addition the upper bound for the eigenvalue ratio of the first two eigenvalues is improved using a different approach based on the authors' proof of the Payne-Pólya-Weinberger conjecture. Finally, the equation of motion of the inhomogeneous stretched string \(-y''=\lambda w(x)y\) is used to compare the best bounds obtained for \(\lambda_ 2/ \lambda_ 1\) to numerically computed results of Keller and of Mahar and Willner.
scientific article; zbMATH DE number 425453