Spectral properties of 1-D Schr\"odinger operators
$\mathrm{H}_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in
X}\alpha_n\delta(x-x_n)$ with local point interactions on a discrete set
$X=\{x_n\}_{n=1}^\infty$ are well studied when
$d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We
consider $\mathrm{H}_{X,\alpha}$ in the framework of extension theory of
symmetric operators by applying the technique of boundary triplets and the
corresponding Weyl functions.