Point interactions on the line and Riesz bases of δ -functions

Abstract

We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$. Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$. By using this relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives $\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed.