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

We investigate one dimensional symmetric Schrödinger operator H _X,β with δ'-interactions of strength β = {β_n}^∞_n=1⊂ℝ on a discrete set X = {x_n}^∞_n=1⊂[0, b),b ≤ +∞ (x_n↑ b). We consider H_{X, β} as an extension of the minimal operator H_min:= -d²/dx ² φ W^2.2₀ (RD\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^* is given in the case d _*:= inf_{n ∈ ND}|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 ∈ ND, 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.