Eigenvalue ratios for the regular Sturm-Liouville system
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- by Yu-Ling Huang and C. K. Law PDF
- Proc. Amer. Math. Soc. 124 (1996), 1427-1436 Request permission
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
- 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, DOI 10.1090/S0002-9939-1987-0875408-4
- 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, DOI 10.1007/BF01219657
- Mark S. Ashbaugh and Rafael D. Benguria, Eigenvalue ratios for Sturm-Liouville operators, J. Differential Equations 103 (1993), no. 1, 205–219. MR 1218744, DOI 10.1006/jdeq.1993.1047
- Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 4th ed., John Wiley & Sons, Inc., New York, 1989. MR 972977
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
- 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 1996 American Mathematical Society
- 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