臺灣博碩士論文加值系統

目前對於Sturm-Liouville算子在特徵值的差和特徵值的比之最佳估計上已是基本且重要的研究課題。近年來由Keller、Chern-Shen、Lavine、Huang和Horvath等數學所做的一系列工作中，顯示出在Dirichlet邊界條件下的Schrodinder算子的第一個特徵值差距和在Dirichlet邊界條件下的繩子震動方程式的第一個特徵值比是有對偶的關係。更進一步的，當位勢函數與密度函數被限制在某些集合中，這些問題能用變分法（不同於所有用一個參數去找的式子）配合一些基本分析來解決。
在這篇論文中，我們將對這些結果作一個報告。特別地，我們將證明三個對偶的定理。首先，當q是凸的（ρ 是凹的）時，則假如q是單坑函數和轉折點是π/2時 （ρ 是單峯函數和轉折點是π/2時）,則λ2-λ1≧3 （λ2/λ1≧4）。當q（ρ）是常數時，所有這些的下界是可以算出來的。最後當q是有界的（ρ是有界的）再加上一些條件，則λ2-λ1 在階梯位勢時達到最小值（ λ2/λ1在階梯密度時達到最小值）。我們將對這些結果用統一的方法一併處理。

The optimal estimates of the eigenvalue gaps and eigenvalue ratios for the Sturm-Liouville operators have been of fundamental importance. Recently a series of works by Keller [7],Chern-Shen [3], Lavine [8], Huang [4] and Horvath [6] show that the first eigenvalue gap of the Schrodinger operator under Dirichlet boundary condition and the first eigenvalue ratio（λ2/λ1）of the string equation under Dirichlet boundary condition are dual problems of each other. Furthermore the problems when the potential functions and density functions are restricted to certain classes of functions can all be solved by a variational calculus method (differentiating the whole equation with respect to a parameter t to find λn''(t)) together with some elementary analysis. In this thesis, we shall give a short survey of these result. In particular, we shall prove $3$ pairs of theorems. First when q is convex (ρ is concave), then λ2-λ1≧3 （λ2/λ1≧4）.If q is a single
well and its transition point is π/2 (ρ is a
single barrier and its transition point is π/2), then
λ2-λ1≧3（λ2/λ1≧4）.All these lower bounds are optimal when q(ρ) is constant. Finally when q is bounded (ρ is bounded), then λ2-λ1 is minimized by a step function (λ2/λ1
is minimized by a step function), after some additional
conditions. We shell give a unified treatment to the above
results.

1.Introduction
2.Preliminaries
3.Convex potentials and concave densities
4.Single-well potentials and single-barrier densities
5.The class of $L^{infty}$ potentials and densities
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