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 P{[}u] = -(partial derivative(2)u/partial derivative x(2) +partial derivative(2)u/partial derivative y(2)) + q(x, y)u(x, y) in the Hilbert space H = L-2(Omega); with potential q(x, y) is an element of C'(Omega). 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 P(u) is an element of L-2(Omega) double right arrow qu is an element of L-2(Omega), Omega is an element of R-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 P - lambda q(2) for some lambda > 0 implies the separation of P.