Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials

Ashbaugh, Mark S.; Benguria, Rafael

doi:10.1090/S0002-9939-1987-0875408-4

Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials
HTML articles powered by AMS MathViewer

by Mark S. Ashbaugh and Rafael Benguria PDF

Proc. Amer. Math. Soc. 99 (1987), 598-599 Request permission

Abstract:

We prove the optimal upper bound ${\lambda _2}/{\lambda _1} \leq 4$ for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with nonnegative potentials. Equality holds if and only if the potential vanishes identically.

References

R. Benguria, A note on the gap between the first two eigenvalues for the Schrödinger operator, J. Phys. A 19 (1986), no. 3, 477–478. MR 832589, DOI 10.1088/0305-4470/19/3/026
M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121–127. MR 72332, DOI 10.1093/qmath/6.1.121
P. A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), no. 2, 267–310. MR 495676, DOI 10.1215/S0012-7094-78-04516-7
V. A. Marčenko, On reconstruction of the potential energy from phases of the scattered waves, Dokl. Akad. Nauk SSSR (N.S.) 104 (1955), 695–698 (Russian). MR 0075402
L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289–298. MR 84696, DOI 10.1002/sapm1956351289
I. M. Singer, Bun Wong, Shing-Tung Yau, and Stephen S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrödinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 319–333. MR 829055