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

Authors:
Mark S. Ashbaugh and Rafael Benguria

Journal:
Proc. Amer. Math. Soc. **99** (1987), 598-599

MSC:
Primary 34B25

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875408-4

MathSciNet review:
875408

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the optimal upper bound 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.

**[1]**R. Benguria,*A note on the gap between the first two eigenvalues for the Schrödinger operator*, J. Phys. A**19**(1986), 477-478. MR**832589 (87c:34033)****[2]**M. M. Crum,*Associated Sturm-Liouville systems*, Quart. J. Math. Oxford (2)**6**(1955), 121-127. MR**0072332 (17:266g)****[3]**P. A. Deift,*Applications of a commutation formula*, Duke J. Math.**45**(1978), 267-310. MR**495676 (81g:47001)****[4]**E. M. Harrell, unpublished, 1982.**[5]**V. A. Marchenko,*The construction of the potential energy from the phases of the scattered waves*, Dokl. Akad. Nauk SSSR**104**(1955), 695-698. MR**17**, 740. MR**0075402 (17:740e)****[6]**L. E. Payne, G. Pólya, and H. F. Weinberger,*On the ratio of consecutive eigenvalues*, J. Math. and Phys.**35**(1956), 289-298. MR**0084696 (18:905c)****[7]**I. M. Singer, B. Wong, S.-T. Yau, and S. S.-T. Yau,*An estimate of the gap of the first two eigenvalues in the Schrödinger operator*, Ann. Scuola Norm. Sup. Pisa (4)**12**(1985), 319-333. MR**829055 (87j:35280)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0875408-4

Keywords:
Schrödinger operators,
ratios of eigenvalues,
commutation formula

Article copyright:
© Copyright 1987
American Mathematical Society