Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L ₂(R) with domain ${C_c^infty({bf R})}Let H be the symmetric second-order differential operator on L ₂(R) with domain C_c^￥(R){C_c^infty({bf R})} and action Hj = -(c j^￠)^￠{Hvarphi=-(c,varphi^{prime})^{prime}} where c ? W^1,2_loc(R){ cin W^{1,2}_{rm loc}({bf R})} is a real function that is strictly positive on R{0}{{bf R}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 n = n₊ún_-{nu=nu_+veenu_-} where n_±(x)=±ò^±1_±x c^-1{nu_pm(x)=pmint^{pm 1}_{pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L₂(0,1){nunotin L_2(0,1)} and a unique submarkovian extension if and only if n ? L_￥(0,1){nunotin L_infty(0,1)}. In both cases, the corresponding semigroup leaves L ₂(0,∞) and L ₂(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W^1,￥_loc(R){ cin W^{1,infty}_{rm loc}({bf R})} the corresponding operator H has a unique submarkovian extension.