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1–D Schrödinger Operators with Local Interactions on a Discrete Set

By Aleksey Kostenko and Mark Malamud


Spectral properties of 1-D Schrödinger operators HX,α: = − d2 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 (n) 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 ne

Topics: Schrödinger operator, local point interaction, self-adjointness, lower semiboundedness, discreteness
Year: 2009
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