On the formula for the distribution of the eigenvalues of singular differential operators

In this note we construct exampLes of a function $q(x)$, which grows arbitrarily rapidly, and a function $q(x)$ ($c_1|x|^\alpha\le q(x)\le c_2|x|^\beta$, $\beta>\alpha>0$) such that for a Sturm–Liouville operator with the constructed potential functions $q(x)$, the classical formula for the number of eigenvalues of the operator that do not exceed $\lambda$ is not true.