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Eigenvalue ratios for the regular Sturm-Liouville system
Author(s):
Yu-Ling
Huang;
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.
References:
- 1.
- 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
- 2.
- 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
- 3.
- M. S. Ashbaugh and R. D. Benguria, Eigenvalue ratios for Sturm-Liouville operators, J. Diff. Eqns. 103 (1993) 205-219. MR 94c:34125
- 4.
- G. Birkhoff and G. C. Rota, ``Ordinary Differential Equations ", 4th ed (1989) Wiley, New York. MR 90h:34001
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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:
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
Copyright of article:
Copyright
1996,
American Mathematical Society
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