Point Interactions in a Line and the Riesz Bases of δ-Functions

Abstract

We present a description of the relationship between the Sobolev spaces $$ {W}_2^1 $$ (ℝ) and $$ {W}_2^2 $$ (ℝ) and the Hilbert space l2. Let Y be a finite or countable set of points on ℝ and let d ≔ inf {|y′ − y′′|, y′, y′′ ∈ Y, y′ ≠ y′′}. By using this relationship, we prove that if d = 0, then the systems of δ –functions {δ(x − yj), yj ∈ Y} and their derivatives {δ′(x − yj), yj ∈ Y} do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $$ {W}_2^{-1} $$ (ℝ) and $$ {W}_2^{-2} $$ (ℝ) but, on the contrary, form the indicated bases in the case where d > 0. We also present the description of Friedrichs and Krein extensions and demonstrate their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative self-adjoint extensions of the operator A′ are presented.