Separation properties for Sturm–Liouville operators

Let $q(x)$ be a positive function given on the interval $I$ of the real axis; let $P$ be the minimal operator generated in $L_2(0,+\infty)$ by the differential expression $P[\cdot]=-\frac{d^2}{dx^2}+q(x)$; let $Q$ be the operator of multiplication by the function $q(x)$.
If $D_{P^*}\subset D_Q$, then $P[\cdot]$ is said to be separated. In this note the separation of $P[\cdot]$ is proved for some growth regularity conditions on the fonction $q(x)$, without assuming anything on its smoothness. One proves that if $D_{P^*}\subset D_S$, where $S$ is the multiplication operator by the function $s(x)$, satisfying some growth regularity condition, then $D_Q\subset D_S$.