Separation of the two dimensional Grushin operator by the disconjugacy property

In this work we have introduced a new proof of the separation of the Grushin differential operator of the form G{[}u] = -(partial derivative(2)u/partial derivative x(2) + x(4)partial derivative(2)u/4 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 2C(1)(Omega), by the disconjugacy property. We show that certain properties of positive solutions of the disconjugate second order differential expression G{[}u] imply the separation of minimal and maximal operators determined by G i.e. the property that G{[}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, see {[}W.N. Everitt and M. Giertz, Some properties of the domains of certain differential operators, Proc. London Math. Soc. 23 (1971), pp. 301-324]; a property leading to a new proof and generalizing of a 1971 separation criterion due to Everitt and Giertz. A final result of this article shows that the disconjugacy of G - lambda q(2) for some lambda > 0 implies the separation of G. This work is a generalizing of our work in {[}H.A. Atia and R. A. Mahmoud, Separation of two dimensional Laplace operator by disconjugacy property, Panamerican Math. J. 20(2) (2009), pp. 93-103].