1-D Schrödinger operators with local point interactions on a discrete set

Spectral properties of 1-D Schrödinger operators H_X,α:=-d2/dx2+∑_xn∈Xα_nδ(x-x_n) with local point interactions on a discrete set X={x_n}_n=1^∞ are well studied when d*:=inf_{n,k∈N{double-struck}}|x_n-x_k|>0. Our paper is devoted to the case d*=0. We consider HX,α 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 H_X,α 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 H_X,α to be self-adjoint, lower semibounded, and discrete in the case d_*=0.The operators with δ′-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 δ- and δ′-type interactions are substantially different.