Sufficient conditions for the self-adjointness of the Sturm–Liouville operator

Let $L$ be the minimal operator in $L_2(R^1)$ generated by the expression $ly=--y''+q(x)y$, $\operatorname{Im}q(x)\equiv0$, let $\Delta k$ ($k=\pm1,\pm2,\dots$) be a sequence of disjoint intervals going out to $\pm\infty$ for $k\to+\infty$, and let $\delta_k$ be the length $\Delta_k$. If $(ly,y)\ge-\gamma_k\|y\|^2$ on all smooth $y(x)$ with support in $\delta_k$, whereby $\gamma_k>0$,
$$\sum_{k=1}^\infty(\gamma_k+\delta_k^{-2})-1=\sum_{k=-\infty}^{-1}{(\gamma_k+\delta_k^{-2})-1=\infty},$$
then the operator $L$ is self-adjoint. This theorem generalizes criteria for the self-adjointness of $L$ obtained earlier by R. S. Ismagilov, A. Ya. Povzner, and D. B. Sears.