Weak and strong extensions of ordinary differential operators

Let A be a second order differential operator with positive leading term defined on an interval J of R. In this paper we study conditions for the equality D₀(A) = D₁(A) to hold. Here D₀(A) and D₁(A) are the domains of the minimal and maximal extensions of A respectively. Under the general assumption that $\text{A(1)}\text{and}\text{A}{}^1\text{(1)}$ are bounded above it is proven that under certain conditions D₀(A) = D₁(A) if functions which are constant near the boundaries of J are in $\text{D}{}_0\text{(A) ∩ D}{}_0\text{(A}{}^1\text{)}$ whenever they are in $\text{D}{}_1\text{(A) ∩ D}{}_1\text{(A}{}^1\text{)}$. In particular if A is formally selfadjoint and 1 ?D₁(A) then D₁(A) = D₀(A) if and only if 1 ?D₀(A). When the measure of J is infinite at both ends D₀(A) is always equal to D₁(A). This fact is used to show that the leading term of A as well as its terminal coefficient can be chosen arbitrarily (although not independently of one another) in such a way that the equality D₀ = D₁ holds.