On the discrete spectrum of ordinary differential operators in weighted function spaces (Q1898781)

Criteria are given which ensure that all self-adjoint realizations in a weighted space \(L_{2, r}\) of the expression \(A\) defined by \[ Ay= \sum^n_{k= 0} (- 1)^k(a_k(x) y^{(k)})^{(k)},\qquad x\in (0, \infty), \] are bounded below and have discrete spectra. These criteria are given in terms of the mean values of the functions \(a_0\), \(a^{- 1}_0\) and \(r^2\) on intervals \([x, x+ d(x)]\) which cover the semi-axis \((0, \infty)\). The main theorem proved includes and extends a number of known results.