1D nonnegative Schrodinger operators with point interactions

Let $Y$ be an infinite discrete set of points in $dR$,satisfying the condition $inf{|y-y'|,; y,y'in Y, y' ey}>0.$ In the paper we prove that the systems${delta(x-y)}_{yin Y}, ;{delta'(x-y)}_{yin Y},{delta(x-y),;delta'(x-y)}_{yin Y}$ {form Riesz} bases in the corresponding closed linear spans in the Sobolev spaces $W_2^{-1}(dR)$ and $W_2^{-2}(dR)$. As an application, we prove the transversalness of the Friedrichs and Kreui n nonnegative selfadjoint extensions of the nonnegative symmetric operators $A_0$, $A'$, and $H_0$ defined {as restrictions} of the operator $A =-frac{ d^2}{ dx^2},$ $dom (A)=W^2_2(dR)${to} the linear manifolds $dom (A_0)=left{ finW_2^2(mathbb{R})colon f(y)=0,; yin Y ight}$, $dom(A')={ gin W_2^2(mathbb{R})colon g'(y)=0,; yin Y },$ and$dom (H_0)=left{fin W_2^2(mathbb{R})colonf(y)=0,;f'(y)=0,; yin Y ight}$, respectively. Using thedivergence forms, the basic nonnegative boundary triplets for$A^*_0$, $A'^*$, and $H^*_0$ are constructed.