Eigenvalue Ratios for Sturm-Liouville Operators

In this paper we prove various optimal bounds for eigenvalue ratios for the Sturm-Liouville equation − [p(x) y′]′ + q(x)y = λw(x)y and certain specializations. Our results primarily concern the regular case with Dirichlet boundary conditions though various extensions and generalizations to other situations are possible. Our results here extend the result λm/λ1 ≤ m2 obtained in a previous paper for the one-dimensional Schrödinger equation, − y″ + q(x)y = λy, on a finite interval with Dirichlet boundary conditions and nonnegative potential (q ≥ 0). In particular, we obtain λm/λ1 ≤ Km2/k, where the constants k, K satisfy 0 < k ≤ p(x) w(x) ≤ K for all x. If q ≡ 0, lower bounds can also be obtained. Our methods involve a slight modification of the Prüfer variable techniques employed in the Schrödinger case. We also examine the consequences of our recent proof of the Payne-Pólya-Weinberger conjecture in the one-dimensional (Sturm-Liouville) setting. Finally, we compare our general bounds to the detailed analyses of Keller and of Mahar and Willner for the special case of the inhomogeneous stretched string.