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On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials. (English) Zbl 0623.34027

Differential equations and mathematical physics, Proc. Int. Conf., Birmingham/Ala. 1986, Lect. Notes Math. 1285, 16-25 (1987).
[For the entire collection see Zbl 0619.00011.]
We survey current knowledge on the ratio, \(\lambda_ 2/\lambda_ 1\), of the first two eigenvalues of the Schrödinger operator \(H_ V=-\Delta +V(x)\) on the region \(\Omega \subset {\mathbb{R}}^ n\) with Dirichlet boundary conditions and nonnegative potentials. We discuss the Payne- Pólya-Weinberger conjecture for \(H_ 0=-\Delta\) and generalize the conjecture to Schrödinger operators. Lastly, we present our recent result giving the best possible upper bound \(\lambda_ 2/\lambda_ 1\leq 4\) for one-dimensional Schrödinger operators with nonnegative potentials and discuss some extensions of this result.

MSC:

34L99 Ordinary differential operators
47E05 General theory of ordinary differential operators

Citations:

Zbl 0619.00011