臺灣博碩士論文加值系統

考慮一正則 Sturm-Liouville 方程式定義在一有限區間 [0,1] 上 (p(x)
y')'+(.lambda. w(x)-q(x))y=0 伴隨分離邊值條件。對於一正則 Sturm-
Liouville 系統伴隨 Dirichlet 邊值條件且當位能 q 為正值，
Ashbaugh 和 Benguria (1993) 給出了多種固有值比例的最佳上界。對於
一般的正則 Sturm-Liouville 系統，關於固有值我們給出了多種的估計
。尤其是對於 Neumann 邊值條件，其上界的估計是非常敏銳的。在本論
文中，我們利用一些基礎的微積分的辦法證明一三角學不等式。修飾的
Prufer 替代法和比較定理是我們證明的主要技巧。同時，第一固有值的
正負號我們也是必須探討的。利用修飾 Prufer 替代法及比較定理，我們
證明第一個固有值為正值，在大部份的分離邊值條件下。總而言之，這篇
論文可說是修飾 Prufer 替代法及比較定理的應用。

We consider the regular Sturm-Liouville equation on [0,1] (p(x)
y')'+(.lambda. w(x)-q(x))y=0 together with separated boundary
conditions. In 1993, Ashbaugh and Benguria [2] gave various
optimal bounds of eigenvalue ratios for the Sturm- Liouville
system with Dirichlet boundary conditions. when q .gdsim. 0.
For the general regular Sturm-Liouville system, we prove the
various estimates of eigenvalue ratios under the same
assumption. For the Neumann boundary conditions, the upper
bound is a sharp estimate. The modified Prufer substitution and
the Comparison Theorem are the key techniques in the proof. A
trigonometric inequality given in [1] is also found to be
useful. In this thesis, we give an alternatively proof with
elementary methods. We also give an elementary proof of an
trigonometric inequality were given in [1]. The sign of
.lambda. of the index 1 need to be discussed too. Using the
modified Prufer substitution and the Comparison Theorem, we
prove that .lambda.1 .relbo. 0 for most of the for most of the
separated boundary conditions. In conclusion, this thesis can
be viewed as an application of the methods of the modified
Prufer substitution and the Comparison Theorem to eigenvalue
problems.