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

Abstract

Spectral properties of 1-D Schrödinger operators H X , α : = − d 2 d x 2 + ∑ x n ∈ X α n δ ( 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 | x n − x k | > 0 . Our paper is devoted to the case d ∗ = 0 . We consider H X , 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.