| Abstract |
Let Y be an infinite discrete set of points in R, satisfying
the condition
inf
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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