The answer is classical and negative. It is a particular instance of Thm X.11 of Reed-Simon [here][1].

Let $V(x)$ be a continous potential on $(0,+\infty)$. If
\begin{equation}
V(x) \geq \frac{3}{4x^2}
\end{equation}
Then $-\partial_x^2 +V$ it is essentially self-adjoint. On the other hand, if for some $\varepsilon >0$
\begin{equation}
0\leq V(x) \leq \left(\frac{3}{4} - \varepsilon\right)\frac{1}{x^2}
\end{equation}
Then $-\partial_x^2 +V$ is not essentially self-adjoint


  [1]: http://www.amazon.com/Fourier-Analysis-Self-Adjointness-Methods-Mathematical/dp/0125850026