| Abstract |
Let $Y$ be an infinite discrete set of points in $\mathbb R $, satisfying
the condition
$\inf\{|y-y'|,\; y,y'\in Y, y'\ne y\}>0.$
In the paper we prove that the systems
$\{\delta(x-y)\}_{y\in Y}, \;\{\delta'(x-y)\}_{y\in Y},
\{\delta(x-y),\;\delta'(x-y)\}_{y\in Y}$
form Riesz bases in the corresponding closed linear spans in the Sobolev spaces
$W_2^{-1}(\mathbb R )$ and $W_2^{-2}(\mathbb R )$. As an application, we prove
the transversalness of the Friedrichs and Kre\u\i n nonnegative selfadjoint extensions of
the nonnegative symmetric operators $A_0$, $A'$, and $H_0$
defined as restrictions of the operator $A =-\frac{ d^2}{ dx^2},$ $\mathop{\rm dom} (A)=W^2_2(\mathbb R )$
to the linear manifolds $\mathop{\rm dom} (A_0)=\left\{ f\in W_2^2(\mathbb{R})\colon
f(y)=0,\; y\in Y \right\}$, $\mathop{\rm dom} (A')=\{ g\in
W_2^2(\mathbb{R})\colon g'(y)=0,\; y\in Y \},$ and $\mathop{\rm dom}
(H_0)=\left\{f\in W_2^2(\mathbb{R})\colon f(y)=0,\;f'(y)=0,\; y\in Y
\right\}$, respectively. Using the divergence forms, the basic
nonnegative boundary triplets for $A^*_0$, $A'^*$, and $H^*_0$ are
constructed. |
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