Sturm-Liouville系統的特徵值差距(gap)和比率(ratio)問題，一直是被研究的主題。在 Schrodinger算子方面，Lavine [9]考慮凸的勢函數。他利用變分法，證明了Dirichlet特徵值差距的最佳下界。在1999年，黃明傑老師 [5] 採用此法，研究vibrating string的問題，對於非負凹密度函數證明了，Dirichlet特徵值比率的最佳下界。在這篇論文中，我們延續這個主題，利用不同的方法得到一般化的結果。
在過程中，Ashbaugh和Benguria的文章有重要的影響。他們考慮非負勢函數的Schrodinger算子利用修正的Prufer替代法和比較理論，證明Dirichlet特徵值比率有最佳化上界。而我們考慮非正勢函數，採用相同方法，得到對稱的結果。
這個結果，對我們的主題有很大的幫助。透過Liouville替代法和逼近理論， 非負凹密度函數的string方程可被轉換成非正勢函數的Schrodinger算子，因此從而推廣黃明傑老師的結果。

The eigenvalue gaps and eigenvalue ratios of the Sturm-Liouville systems have been studied in many papers. Recently, Lavine proved an optimal lower estimate of first eigenvalue gaps for Schrodinger operators with convex potentials. His method uses a variational approach with detailed analysis on different integrals. In 1999, (M.J.) Huang adopted his method to study eigenvalue ratios of vibrating strings. He proved an optimal lower estimate of first eigenvalue ratios with nonnegative densities. In this thesis, we want to generalize the above optimal estimate.
The work of Ashbaugh and Benguria helps in attaining our objective. They introduced an approach involving a modified Prufer substitution and a comparison theorem to study the upper bounds of Dirichlet eigenvalue ratios for Schrodinger
operators with nonnegative potentials. It is interesting to see that the counterpart of their result is also valid.
By Liouville substitution and an approximation theorem, the vibrating strings with concave and positive densities can be transformed to a Schrodinger operator with nonpositive potentials. Thus we have the generalization of Huang's result.

Contents
1 Introduction 6
2 Preliminaries 10
3 Schrodinger operators 12
4 Vibrating string problems 16

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