這份論文探討在半無窮區間上的 Sturm-Liouville 問題。在這裡，如同 Fourier 展開式一樣，我們也會有一個包含譜函數rho的 Fourier 積分的較普遍的 Parseval 等式。並且，這個函數rho跟 Titchmarsh-Weyl 的 m 函數相關，而 m 函數正好給出了這個 Sturm-Liouville 問題的 L^2 解。譜可以看成是譜函數的非常數點。參照 Titchmarsh 的專書，我們探討由不同的位勢函數 q 的漸近行為所組合的譜的狀況，例如當q趨向無窮、0或是負無窮。

We give a report on the Sturm-Liouville problem defined on semi-infinite interval. Here as an extension of the Fourier expansion, we have a Parseval equality involving a Fourier integral with respect to a spectral function rho. This function rho is also related to Titchmarsh-Weyl m-function m(lambda) giving L2 solutions of the problem. The spectrum can be viewed as nonconstant points of the spectral function. Following Titchmarsh’s monograph, we shall investigate the nature of the spectrum associated with different asymptotic behaviors of the potential function q, namely, when q→∞, q→0 or q→-∞.

1 Introduction 1
2 Titchmarsh-Weyl m-functions and spectral functions 10
2.1 Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 m-function and the spectral function . . . . . . . . . . . . . . . . 18
2.3 Stieltjes Inversion Formula . . . . . . . . . . . . . . . . . . . . . . 25
3 The dependence of spectrum on the potential function 29
3.1 The case when q(x) ! 1 . . . . . . . . . . . . . . . . . . . . . . 30
3.2 The case when q(x) ! 0 . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 The case when q(x) ! −1 and R1 0 |q(x)|−12 dx is divergent . . . . 37
3.4 The case when q(x) ! −1 and R1 0 |q(x)|−12 dx is convergent . . . 46

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