Separation of the two dimensional Laplace operator by the disconjugacy property

Abstract

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.