Sturmian theorems and positive resolvents

Abstract

1. Introduction.The classical Sturmian theorem of ordinary differential equations deals with functions u(x) and v(x) which are, respectively, solutions of differential equations(1) Um-^^j+tumO,(2) Mv=-l{J^+yv = 0.Under the assumption that "F is larger than M" (in the sense that a(x) ä a(x) > 0 and c(x) £ y(x)) one can infer information about all solutions of (2) from knowledge about a particular nontrivial solution of (1)-i.e. if u(xx) = u(x2)=0 then every solution of (2) has a zero in [xx, x2].These ideas have been generalized to second order elliptic equations by several authors ([l]-[4]) considering elliptic operators and also by Protter [5] and Swanson [6] considering the nonselfadjoint case.Given a proper relation among the coefficients of F and M and that Lu=0 has a nontrivial solution with nodal domain Ü, then it can be shown that every solution of Mv = 0 has a zero in Í2.While all the above proofs of this fact make essential use of some sort of ordering among elliptic operators, the nature of this ordering is never defined in operator-theoretic terms.The results of §2 below suggest that it is an order relationship between certain resolvents of the differential operators F and M which underlies the separation properties characteristic of Sturmian theorems.It will be shown that quite general operator equations in a Banach space 3S satisfy a type of Sturmian theorem if the operators' resolvents satisfy prescribed positivity requirements with respect to a cone SP.In order to apply this theory to differential operators, one must first establish the corresponding positivity properties for their resolvents.This is done in §3 for sufficiently regular nonselfadjoint second order elliptic operators, and the general theory of §2 is then applied in the proof of two Sturmian theorems and the establishment of criteria for certain Green's functions to be positive.