Point interactions on the line and Riesz bases of δ -functions - Kovalev, Yu. G. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
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 $d := inf\bigl\{ | y′ y′ ′ | , y′ , y′ ′ \in Y, y′ ot = y′ ′ \bigr\}$. By using thisrelationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ δ (x y_j), y_j \in Y \bigr\} $ and their derivatives$\bigl\{ δ ′ (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 andKrein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the descriptionof all nonnegative selfadjoint extensions of the operator $A′$ are proposed.