Degenerate elliptic operators in one dimension

Abstract

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\ddot{I}†=-(c\ddot{I}{†}^{\prime }{)}^{\prime }$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on $\Ri\backslash\{0\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\hat{I}½=\hat{I}{½}_{+}\vee \hat{I}{½}_{-}$ where $\hat{I}{½}_{\pm }(x)=\pm {\int }_{\pm x}^{\pm 1}{c}^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\hat{I}½\notin L_2(0,1)$ and a unique submarkovian extension if and only if $\hat{I}½\notin L_{\infty }(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty )$ and $L_2(-\infty ,0)$ invariant. In addition we prove that for a general non-negative $ c\in W^{1,\infty}_{\rm loc}(\Ri)$ the corresponding operator $H$ has a unique submarkovian extension.

28 pages