On the boundary conditions for products of Sturm-Liouville differential operators (Q2783577)

The author considers the Sturm-Liouville differential expressions of the form NEWLINE\[NEWLINE \tau[y]=w^{-1}\left[-(py')'+qy\right] NEWLINE\]NEWLINE on the interval \((a,b)\), \(-\infty\leq a<b\leq\infty\), with real-valued functions \(p^{-1},q,w\in L^1_{\text{loc}}(a,b)\), \(w(x)>0\), and their product, too. He recalls some information about the deficiency indices and the notions of limit-point and limit-circle case, and discusses possible forms of boundary conditions of selfadjoint extensions. More precisely, it is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained by e.g. \textit{A. M. Krall} and \textit{A. Zettl} [Differ. Integral Equ. 1, No. 4, 423-432 (1988; Zbl 0723.34023)], and in several other quoted papers.