Separation and Disconjugacy

We show that certain properties of positive solutions of disconjugate second order differential expressions M [y] = −(py) + qy imply the separation of the minimal and maximal operators determined by M in L 2 (I a) where I a = [a, ∞), a > −∞, i.e., the property that M [y] ∈ L 2 (I a) ⇒ qy ∈ L 2 (I a). This result will allow the development of several new sufficient conditions for separation and various inequalities associated with separation. Some of these allow for rapidly oscillating q. It is shown in particular that expressions M with W KB solutions are separated, a property leading to a new proof and generalization of a 1971 separation criterion due to Everitt and Giertz. A final result shows that the disconjugacy of M − λq 2 for some λ > 0 implies the separation of M .