Schrödinger operators with δ′-interactions and the Krein-Stieltjes string

Abstract

We investigate one dimensional symmetric Schrödinger operator H X, β with δ′-interactions of strength β = “β n ” = 1 ∞ ⊂ ℝ on a discrete set X = “x n ” = 1 ∞ ⊂ [0, b), b ≤ +∞ (x n ↑ b). We consider H X, β as an extension of the minimal operator H min:= −d 2/dx 2⌈W 0 2.2 (ℝ\X) and study its spectral properties in the frame-work of the extension theory by using the technique of boundary triplets and the corresponding Weyl functions. The construction of a boundary triplet for H min * is given in the case d *:= infn ∈ ℕ\x n − x n − 1\ = 0. We show that spectral properties like self-adjointness, lower semiboundedness, nonnegativity, and discreteness of the spectrum of the operator H X, β correlate with the corresponding properties of a certain Jacobi matrix. In the case β n > 0, n ∈ ℕ, these matrices form a subclass of Jacobi matrices generated by the Krein-Stieltjes strings. The connection discovered enables us to obtain simple conditions for the operator H X, β to be self-adjoint, lower semibounded and discrete. These conditions depend significantly not only on β but also on X. Moreover, as distinct from the case d * > 0, the spectral properties of Hamiltonians with δ- and δ′-interactions in the case d * = 0 substantially differ.