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We investigate one dimensional symmetric Schrödinger operator HX, β with δ'interactions of strength β = ⊂ on a discrete set X = ⊂ [0, b), b ≤ +∞ (xn ↑ b). We consider HX, β as an extension of the minimal operator Hmin := –d2/dx2 (\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 is given in the case d∗ := infn ∈ |xn – xn – 1| = 0. We show that spectral properties like self adjointness, lower semiboundedness, nonnegativity, and discreteness of the spectrum of the operator HX, β correlate with the corresponding properties of a certain Jacobi matrix. In the case βn > 0, 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 HX, 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.
Kostenko, A., Malamud, M.: Schrödinger Operators with δ'Interactions and the Krein–Stieltjes String.Doklady Mathematics, Vol. 81, no. 3, 2010, pp.342-347.