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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
DOI: https://doi.org/10.1090/S0002-9939-96-03396-5
MathSciNet review: 1328351
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Abstract | References | Similar Articles | Additional Information

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 $ ( \mu_m / \mu_n ) $ for the regular Sturm-Liouville system. This upper estimate is sharp for Neumann boundary conditions. We also discuss the sign of $ \mu_1 $ and include an elementary proof of a useful trigonometric inequality first given in the aforementioned articles.


References [Enhancements On Off] (What's this?)

  • 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
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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: https://doi.org/10.1090/S0002-9939-96-03396-5
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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