A criterion for discrete spectrum of a class of differential operators. (Q1873641)

Let \(T\) be a self-adjoint differential operator in the Hilbert space \(L_r^2({\mathbb R}):=L^2({\mathbb R}, r^2\,dx)\) generated by the differential expression \[ ly := r^{-2}\sum\limits_{j=0}^n (-1)^j(\rho_j^2y^{(j)})^{(j)}, \qquad ry\in L^2(\mathbb R). \] Here, \(r\) and \(\rho_j \,\,(0\leq j \leq n)\) are real-valued measurable functions such that \[ \rho_j, \rho_j^{-1}, r \in L^2_{loc}({\mathbb R}), \qquad \rho_0 \not\in L^2(-\infty, 0). \] Moreover, the functions \(\rho_j\;(0\leq j \leq n)\) satisfy some (rather complicated) condition on their mutual behaviour. Adapting the approach of the paper by \textit{D.~E.~Edmunds} and \textit{J.~Sun} [Proc. R. Soc. Lond. A, 434(2), 643--656 (1991; Zbl 0773.47028)], the author obtains a criterion for discreteness of the spectrum of the operator~\(T\) and a criterion for uniform positivity of~\(T\). The paper does not contain any example of the applications of these criteria and therefore it is difficult to estimate their effectiveness.