Some separation criteria and inequalities associated with linear second order differential operators (Q2712730)

The authors consider the symmetric second-order differential expression \(M(y)= -(py')+ qy\) defined on \(I= (a,\infty)\), \(a>-\infty\) and \(M_w(y)= W^{-1}M(y)\) in \(L^2\)-space with weight \(W\). \(M\) is assumed to satisfy the so-called minimal properties. The paper gives several new separation criteria (with left endpoint regular and right endpoint \(\infty\)). The first section gives the general idea of separation, its definition and the connection between separation and inequality. Further three significant results from this area which are representative of the type of theorems that have been obtained are stated. In the next section on ``The separation criteria'' a few theorems related to the main problem are enunciated along with relevant notes. These theorems are proved in the last section. Though the method of proof follows that of \textit{W. N. Everitt} and \textit{M. Gierts} [Proc. Lond. Math. Soc., III. Ser. 28, 352-372 (1974; Zbl 0278.34009)], there are some new criteria which are independent of the results of Everitt and Giertz.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00028].