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On the eigenvalue ratio for vibrating strings
Author(s):
Min-Jei
Huang
Journal:
Proc. Amer. Math. Soc.
127
(1999),
1805-1813.
MSC (1991):
Primary 34L15
Posted:
February 17, 1999
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Abstract:
For vibrating strings with concave densities or symmetric single-barrier densities, the ratio  of the first two eigenvalues is minimized when the density is constant; while, for vibrating strings with symmetric single-well densities, the ratio  is maximized when the density is constant.
References:
- [1]
- M. Ashbaugh and R. Benguria, On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials, Differential Equations and Mathematical Physics (I. W. Knowles and Y. Saito, eds.), Lecture Notes in Math., vol. 1285, Springer-Verlag, Berlin, 1987, pp. 16-25. MR 89a:35151
- [2]
- M. Ashbaugh and R. Benguria, Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials, Proc. Amer. Math. Soc. 105 (1989), 419-424. MR 89f:81028
- [3]
- M. Ashbaugh and R. Benguria, Eigenvalue ratios for Sturm-Liouville operators, J. Differential Equations 103 (1993), 205-219. MR 94c:34125
- [4]
- R. D. Gentry and D. O. Banks, Bounds for functions of eigenvalues of vibrating systems, J. Math. Anal. Appl. 51 (1975), 100-128. MR 51:8528
- [5]
- J. B. Keller, The minimum ratio of two eigenvalues, SIAM J. Appl. Math. 31 (1976), 485-491. MR 54:10737
- [6]
- R. Lavine, The eigenvalue gap for one-dimensional convex potentials, Proc. Amer. Math. Soc. 121 (1994), 815-821. MR 94i:35144
- [7]
- T. J. Mahar and B. E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math. 29 (1976), 517-529. MR 54:13201
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Additional Information:
Min-Jei
Huang
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043
Email:
mjhuang@math.nthu.edu.tw
DOI:
10.1090/S0002-9939-99-05015-7
PII:
S 0002-9939(99)05015-7
Keywords:
Eigenvalue ratio,
eigenfunction,
concave density,
symmetric single-well density
Received by editor(s):
September 19, 1997
Posted:
February 17, 1999
Communicated by:
Hal L. Smith
Copyright of article:
Copyright
1999,
American Mathematical Society
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