On the ratio of the first two eigenvalues of perturbed harmonic oscillators (Q1353509)

In 1956, L. E. Payne, G. Pólya and H. F. Weinberger conjectured that, among all homogeneous membranes with fixed boundary, the circular membrane gives the maximal ratio \(\lambda_2/ \lambda_1\) of the two lowest eigenvalues. This conjecture was later proved by M. S. Ashbaugh and R. D. Benguria. Here the ratio of the two lowest eigenvalues of the Schrödinger operator \(-\Delta +V(x)\) in \(\mathbb{R}^n\) with discrete spectrum is considered. The potential is assumed to have the form \(V(x)=|x|^2 +q(x)\), and an essential restriction on the ``perturbation'' \(q(x)\) is its radial symmetry \(q(x)= q(|x|)\). Let further \(V(r)\) be non-decreasing, \(q(r)\) be smooth \(\geq 0\) and \(q'(r)\leq 0\). Among such potentials the ratio \(\lambda_2/ \lambda_1\) is a maximum [equal to \(1+(2/n)]\) if and only if \(q(r) \equiv 0\), i.e. in the absence of perturbations.