本文主要討論有勢能數列(potential sequence)的離散型Sturm-Liouville 算子，研究其固有值及固有向量。首先證明一些第一個固有向量的基本定理；接著運用這些定理，和一些排序不等式，來描述在勢能數列之和及上下界相同的情況下，當第一固有值有極大極小時，其勢能數列的形式，進而討論當第二固有值有極大值時，其勢能數列的形式。

[1] Ashbaugh, M. S>, and Benguria, R. (1989) Optimal lower bounded for the gap bewteen the rst
two eigenvalues of one-dimensional Schrodinger operator with symmetric single-well potential,
Proc. Amer. Math. Soc., 105, 419{424.
[2] Ashbaugh,Mark S. (1990) Some eigenvalue inequlities for a class of Jacobi matrices, Linear Al-
gebra and its applications.136,215-234.
[3] Beesack, P. R. (1957) A note on an integral inequlity, Amer. Math. Soc. 8, 875-897.
[4] Chern, H.-H. and Shen, Chao-Liang (1994) On the maximum and minimum of functionals for the
eigenvalue problem of Sturm-Liouville type, Journal of Dierential Equations, 107, No. 1, 68-79.
[5] Gantmacher, F. P. and Kren, M. G. (1950) Oscillation Matrices and Kernels and Small Vibrations
of Mechanical Systems, U. S. Atomic Energy Comission.
[6] Gladwell, G. M. L. (1986) Inverse Problems in Vibration, Martinus Nijho Publishers, Dordrecht.
[7] Hardy, G.H. Littlewood, J.E. and Polya, G. (1951) Inequalities, second edition.
[8] Huang, M.-J. (1999) On the eigenvalue ration with vibrating string, Proc. Amer. Math. Soc., 127,
1805{1813.
[9] Keller, J. B. (1976) The minimum ratio of twp eogenvalues, SIAM J. Appl. Math., 31, 485{491.
[10] Krein, M. G. (1955) On certain problems on the maximum and minimum of characteristic values
and on Lyapunov zones of stability, Amer. Math. Soc. Transl. ser. 2, 1, 163-187.
[11] Mahar, T. J. and Willner, B. E. (1976) An extremal eigenvalue problem, Comm. Pure Appl.
Math., XIX, 517{529.
[12] Pei, Y.-K. (1996) On some eigenvalue problems of Jacobi matrices and selfaddjoint second order
linear dierential systems with Jacobi matrix-valued coecients, Ph. D. thesis, Department of
Mathematics, National Tsing Hua University.
[13] Huang, Y. L. and Law, C.-K. (1996) Eigenvalue rations for the regulr Sturm-Liouville system,
Proc. Amer. Math. Soc., 124, 1427 { 1436.
[14] Law, C.-K. and Huang, Y. L. (1998) Eigenvalue rations and eigenvalue gaps of Sturm-Liouville
operators, Proc. Royal Soc. Edinburgh, 128, 337 { 347.