Essential self-adjointness criterion for one-dimensional graph Laplacians

Let $G=(V,c,E)$ be the one-dimensional weighted graph considered above, with edge conductances $c_{n,n-1}=a_n$ and energy Hilbert space $\mathscr{H}_{E}$. Let $\Delta$ denote the corresponding graph Laplacian regarded as an operator in $\mathscr{H}_{E}$. Essential self-adjointness criterion. The operator $\Delta$ is essentially self-adjoint in $\mathscr{H}_{E}$ if and only if

$$\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty.$$

If the complementary condition in equation holds, then the deficiency indices are $(1,1)$. This gives a precise limit-point/limit-circle criterion for the self-adjointness of the Laplacian on the weighted half-line; the parser supplies no evidence resolving the claim beyond the statement itself.