In this paper we have studied the separation for the Laplace differential operator of the form , , in the Hilbert space ², with potential , ¹. We show that certain properties of positive solutions of the disconjugate second order differential expression P[u] imply the separation of minimal and maximal operators determined by P i.e, the property that ² ², ². A property leading to a new proof and generalization of a 1971 separation criterion due to Everitt and Giertz. This result will allow the development of several new sufficient conditions for separation and various inequalities associated with separation. A final result of this paper shows that the disconjugacy of ² for some 0 implies the separation of P.