PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 5, Pages 1425–1434 S 0002-9939(05)08100-1 Article electronically published on October 13, 2005 A BOUND FOR RATIOS OF EIGENVALUES OF SCHR ¨ ODINGER OPERATORS WITH SINGLE-WELL POTENTIALS MIKL ´ OS HORV ´ ATH AND M ´ ARTON KISS (Communicated by Carmen C. Chicone) Abstract. For Schr¨odinger operators with nonnegative single-well potentials ratios of eigenvalues are extremal only in the case of zero potential. To prove this, we investigate some monotonicity properties of Pr¨ ufer-type variables. 1. Introduction Consider the Schr¨ odinger operator (1.1) −y ′′ + q(x)y = λy on the interval [0,π] with Dirichlet boundary conditions. If q ∈ L 1 (0,π) is real- valued, then the spectrum consists of a growing sequence of infinitely many points, λ 1 , λ 2 , ...; see for example in [3]. Moreover, if q(x) is nonnegative, λ n ≥ n 2 (as it is seen later, for example, from (2.7) and Lemma 2.1). Ashbaugh and Benguria in [2] proved the bound (1.2) λ n λ 1 ≤ n 2 for nonnegative potentials. They also examined the ratio of two arbitrary eigenval- ues, and found (1.3) λ n λ m ≤  n m  2 where ⌈x⌉ denotes the smallest integer greater than or equal to x. To show that this estimate is optimal, they constructed multiple-well examples which came arbitrarily near to attain the bound. They formulated the conjecture that if the potential is nonnegative and convex, then (1.4) λ n λ m ≤ n 2 m 2 , n ≥ m, holds. In this paper we prove more. Namely, we only need that the potential q ≥ 0 be single-well. This means that there is a point a ∈ [0,π] such that q is decreasing Received by the editors December 5, 2003 and, in revised form, December 10, 2004 and De- cember 14, 2004. 2000 Mathematics Subject Classification. Primary 34L15, 34B24. Key words and phrases. Schr¨odinger operator, eigenvalues. This research was supported by the Hungarian NSF Grants OTKA T 32374, T 37491 and T 47035. c 2005 American Mathematical Society 1425 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use