在這篇論文，我研究在(0,π)上定義的Sturm-Liouville方程式的Dirichlet邊界特徵值比率λ_n/λ_m之最佳化估計。在2005年，Horvath和Kiss[10]證明了當勢函數q≥0而且是一個single-well函數時，即有λ_n/λ_m≤(n/m)^2。並且這是一個最佳化上界估計，等號成立的充分必要條件是q=0。之前，Ashbaugh和Benguria[2]已證明過若q≥0，則λ_n/λ_1≤n^2，並提出有關λ_n/λ_m上界的猜想。以上結果證實了這個猜想。這裡我首先簡化證明Horvath和Kiss[10]的結果。我使用一種修飾過的Prufer代換y(x)=r(x)sin(ωθ(x))，y'(x)=r(x)ωcos(ωθ(x))，這裡的ω=√λ。這個修飾Prufer代換似乎比Horvath和Kiss[10]提出的代換更有效，因為它能簡化證明。此外我的方法可能被推廣至一維p-Laplacian特徵值問題。我研究方程式 -[(y')^(p-1)]'=(p-1)(λ-q)y^(p-1) 當p>1的Dirichlet問題，此中f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f。此論文的主定理是：假設q(x)≥0且q是一個在定義域(0,π_p)上的single-well函數，則特徵值比率滿足λ_n/λ_m≤(n/m)^p，而且這也是一個最佳化上界估計。

In this thesis, we study the optimal estimate of eigenvalue ratios λ_n/λ_m of the
Sturm-Liouville equation with Dirichlet boundary conditions on (0, π). In 2005, Horvath and Kiss [10] showed that λ_n/λ_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that λ_n/λ_1≤n^2 when q ≥ 0.
Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(ωθ(x)), y'(x)=r(x)ωcos(ωθ(x)), where ω =
√λ. This modified phase seems to be more effective than the phases φ and ψ that
Horvath and Kiss [10] used. Furthermore our approach can be generalized to study
the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet
problem of the equation -[(y')^(p-1)]'=(p-1)(λ-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies λ_n/λ_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, π_p). Again this is an optimal upper estimate.

1 Introduction 4
2 Eigenvalue ratios for the Sturm-Liouville operator 11
3 Eigenvalue ratios for the p-Laplacian 20
Bibliography 31

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