On symmetric one-dimensional diffusions

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric linear diffusions. Let $(\mathcal{E},\mathcal{F})$ be a regular and local Dirichlet form on $L^2(I,m)$, where $I$ is an interval and $m$ is a fully supported Radon measure on $I$. We shall first present a complete representation for $(\mathcal{E},\mathcal{F})$, which shows that $(\mathcal{E},\mathcal{F})$ lives on at most countable disjoint `effective' intervals with corresponding scale function on each interval, and any point outside these intervals is a trap of the linear diffusion. Furthermore, we shall give a necessary and sufficient condition for $C_c^\infty(I)$ being a special standard core of $(\mathcal{E},\mathcal{F})$ and identify the closure of $C_c^\infty(I)$ in $(\mathcal{E},\mathcal{F})$ when $C_c^\infty(I)$ is contained but not necessarily dense in $\mathcal{F}$ relative to the $\mathcal{E}_1$-norm. This paper is partly motivated by a result of [Hamza, 1975], stated in [FOT, Theorem 3.1.6] and provides a different point of view to this theorem. To illustrate our results, many examples are provided.