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. In the first part of the paper we consider a family {Sn}n∈N of abstract symmetric operators Sn and investigate the problem when a direct sum Π = ⊕n∈NΠn of boundary triplets Πn = {Hn,Γ 0 ,Γ (n) 1 } for S∗ n forms a boundary triplet for the operator S∗ = ⊕n∈NS∗ n. We completely solve this problem and present regularization procedures for constructing a new family {Π̃n}n∈N of triplets for S∗ n such that the direct sum Π̃ = ⊕n∈NΠ̃n forms a boundary triplet for S∗. Assuming xn+1 > xn, n ∈ N, we apply these results to the direct sum Hmin = ⊕n∈NHn of minimal symmetric operators Hn generated in L [xn, xn+1] by − d 2 dx2 . This enables us to 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.