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 ሾ ݑሿ ൌെቆ ଶ ݑ ݔ ଶ ൅ ଶ ݑ ݕ ଶ ቇ ൅ ݍሺݔ,ݕሻݑሺݔ,ݕሻ ¹ሺߗሻ . 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 ሺݑሻ א ܮ ²ሺߗሻ ݍݑ א ܮ ²ሺߗሻ,ߗ א 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 െߣ ݍ ² for some ߣ൐0 � implies the separation of P.