Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L₂(R) with domain C^∞ _c and action H_φ = -(cφ^′)′ where c ε W^1,2 _loc(R) is a real function that is strictly positive on R\{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 ν = ν+ ∨ ν where ν±(x) = ± ∫^±1 _±x c^-1 then H has a unique self-adjoint extension if and only if ν ∉ L₂(0,1) and a unique submarkovian extension if and only if ν ∉ L_∞(0,1). In both cases, the corresponding semigroup leaves L₂(0,∞) and L₂(-∞,0) invariant. In addition, we prove that for a general non-negative c ε W^1,∞ _loc(R) the corresponding operator H has a unique submarkovian extension.