Self-adjoint abstract differential operators

Let $H$ be an abstract separable Hilbert space. We will consider the Hilbert space $H_1$ whose elements are functions $f(x)$ with domain $H$ and we will also consider the set of self-adjoint operators $Q(x)$ in $H$ of the form $Q(x)=A+B(x)$. In this formula $A\ge E$, $B(x)\ge0$, and the operator $B(x)$ is bounded for all $x$. An operator $L_0$ is defined on the set of finite, infinitely differentiable (in the strong sense) functions $y(x)\inH_1$ according to the formula: $L_0y=-y''+Q(x)y$ $(-\infty<x<\infty)$. It is proved that the closure of the operator $L_0$ is a self-adjoint operator in $H_1$ under the given assumptions.