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. We show that the spectral properties of $\mathrm{ H}_{ X,\alpha}$ like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators $\mathrm{ H}_{ X,\alpha}$ to be self-adjoint, lower-semibounded, and discrete in the case $d_*=0$. The operators with $\delta'$-type interactions are investigated too. The obtained results demonstrate that in the case $d_*=0$, as distinguished from the case $d_*>0$, the spectral properties of the operators with $\delta$ and $\delta'$-type interactions are substantially different.
Categories: math.SP math-ph math.MP