On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials

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We survey current knowledge on the ratio, λ21, of the first two eigenvalues of the Schrödinger operator HV=-Δ+V(x) on the region Ω ⊂ ℝn with Dirichlet boundary conditions and non-negative potentials. We discuss the Payne-Pólya-Weinberger conjecture for H0=−Δ and generalize the conjecture to Schrödinger operators. Lastly, we present our recent result giving the best possible upper bound λ21≤4 for one-dimensional Schrödinger operators with nonnegative potentials and discuss some extensions of this result.