A bound for ratios of eigenvalues of Schrödinger operators on the real line (Q934430)

Consider the one-dimensional Schrödinger equation \[ -y''+q(x)y=\lambda y \quad{ on }(-\infty,+\infty) \] with the boundary condition \(\lim_{| x| \to\infty}y(x)=0\). Let the potential \(q\) be nonnegative, single-well (meaning that there is a point \(a\in(-\infty,\infty)\) such that \(q\) is decreasing in \((-\infty,a)\) and increasing in \((a,+\infty)\)) and \(\lim_{| x| \to\infty}=+\infty\). The authors prove that in this case the ratio of the \(n\)th and \(m\)th eigenvalue \((m<n)\) satisfies the estimate \(\lambda_n/\lambda_m<n^2/m^2\). Thus, they modify their result from [Proc. Am. Math. Soc. 134, 1425--1434 (2006; Zbl 1098.34067)] which applied for operators defined on a finite segment.