Singular Sturm-lioville Differential Formula Type Of Geometry Description

This paper introduces a new geometric frame to deal with the Sturm-Liouvilledifferential expressions L=-D~2+q,x∈[0,∞). The central concept of this frameis the argument function (?)(x). We obtain two basic equations concerning (?)(x),with which, we give geometric description of the concepts of limit-point/limit-circle,oscillatory/non-oscillatory and dis-conjugation of this type of expressions. This state-ment emphasizes the geometric characters of these concepts and so their connec-tions. And we deal with some criteria of limit-point/limit-circle and oscillatory/non-oscillatory classification, including the classical theorem of N. Levinson. We also pro-vide a natural explanation of the phenomenon that a typical limit-point criterion is ofinterval-type, and a simple method of constructing differential expressions with certainprescribed properties. The qualitative property of L, that how the asymptotic behav-ior of (?)'(x) at∞affects that of q(x), is also discussed.