Best constant for the ratio of the first two eigenvalues of onedimensional Schrödinger operators with positive potentials
Authors:
Mark S. Ashbaugh and Rafael Benguria
Journal:
Proc. Amer. Math. Soc. 99 (1987), 598599
MSC:
Primary 34B25
MathSciNet review:
875408
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Abstract: We prove the optimal upper bound for the ratio of the first two eigenvalues of onedimensional 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),
no. 3, 477–478. MR 832589
(87c:34033)
 [2]
M.
M. Crum, Associated SturmLiouville systems, Quart. J. Math.
Oxford Ser. (2) 6 (1955), 121–127. MR 0072332
(17,266g)
 [3]
P.
A. Deift, Applications of a commutation formula, Duke Math. J.
45 (1978), no. 2, 267–310. MR 495676
(81g:47001)
 [4]
E. M. Harrell, unpublished, 1982.
 [5]
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
(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, Bun
Wong, ShingTung
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
(87j:35280)
 [1]
 R. Benguria, A note on the gap between the first two eigenvalues for the Schrödinger operator, J. Phys. A 19 (1986), 477478. MR 832589 (87c:34033)
 [2]
 M. M. Crum, Associated SturmLiouville systems, Quart. J. Math. Oxford (2) 6 (1955), 121127. MR 0072332 (17:266g)
 [3]
 P. A. Deift, Applications of a commutation formula, Duke J. Math. 45 (1978), 267310. 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 E. M. Harrell, unpublished, 1982.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), 695698. 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), 289298. 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), 319333. MR 829055 (87j:35280)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708754084
PII:
S 00029939(1987)08754084
Keywords:
Schrödinger operators,
ratios of eigenvalues,
commutation formula
Article copyright:
© Copyright 1987
American Mathematical Society
