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Eigenvalue ratios for the regular Sturm-Liouville system
Authors:
Yu-Ling Huang and C. K. Law
Journal:
Proc. Amer. Math. Soc. 124 (1996), 1427-1436
MSC (1991):
Primary 34B24, 34L15
MathSciNet review:
1328351
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Abstract: Following the method of Ashbaugh-Benguria in Comm. Math. Phys. 124 (1989), 403--415; J. Differential Equations 103 (1993), 205--219, we prove an upper estimate of the arbitrary eigenvalue ratio  for the regular Sturm-Liouville system. This upper estimate is sharp for Neumann boundary conditions. We also discuss the sign of  and include an elementary proof of a useful trigonometric inequality first given in the aforementioned articles.
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Mark
S. Ashbaugh and Rafael
Benguria, Best constant for the ratio of the
first two eigenvalues of one-dimensional Schrödinger operators with
positive potentials, Proc. Amer. Math. Soc.
99 (1987), no. 3,
598–599. MR
875408 (88e:34039), http://dx.doi.org/10.1090/S0002-9939-1987-0875408-4
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Mark
S. Ashbaugh and Rafael
D. Benguria, Optimal bounds for ratios of eigenvalues of
one-dimensional Schrödinger operators with Dirichlet boundary
conditions and positive potentials, Comm. Math. Phys.
124 (1989), no. 3, 403–415. MR 1012632
(91c:34114)
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Mark
S. Ashbaugh and Rafael
D. Benguria, Eigenvalue ratios for Sturm-Liouville operators,
J. Differential Equations 103 (1993), no. 1,
205–219. MR 1218744
(94c:34125), http://dx.doi.org/10.1006/jdeq.1993.1047
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Garrett
Birkhoff and Gian-Carlo
Rota, Ordinary differential equations, 4th ed., John Wiley
& Sons Inc., New York, 1989. MR 972977
(90h:34001)
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- M. S. Ashbaugh and R. D. Benguria, Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials, Proc. Amer. Math. Soc. 99 (1987) 598-599. MR 88e:34039
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- M. S. Ashbaugh and R. D. Benguria, Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials, Comm. Math. Phys. 124 (1989) 403-415. MR 91c:34114
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- M. S. Ashbaugh and R. D. Benguria, Eigenvalue ratios for Sturm-Liouville operators, J. Diff. Eqns. 103 (1993) 205-219. MR 94c:34125
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Additional Information
Yu-Ling Huang
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Republic of China
C. K. Law
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Republic of China
Email:
law@sun1.math.nsysu.edu.tw
DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03396-5
PII:
S 0002-9939(96)03396-5
Keywords:
Regular Sturm-Liouville system,
Neumann boundary conditions,
eigenvalue ratio,
modified Pr\"{u}fer substitution
Received by editor(s):
July 6, 1994
Additional Notes:
This research is partially supported by the National Science Council, Taiwan, R. O. C. under contract number NSC-83-0208-M-110-028
Communicated by:
Hal L. Smith
Article copyright:
© Copyright 1996 American Mathematical Society
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