A ug 2 00 9 1 – D Schrödinger operators with local interactions on a discrete set

Spectral properties of 1-D Schrödinger operators HX,α := − d 2 dx2 + ∑ xn∈X αnδ(x − xn) with local point interactions on a discrete set X = {xn}n=1 are well studied when d∗ := infn,k∈N |xn − xk| > 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 HX,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 HX,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.