Separation of the two dimensional Laplace operator by the disconjugacy property

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.