1D nonnegative Schrodinger operators with point interactions

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 ${delta(x-y)}_{yin Y}, ;{delta'(x-y)}_{yin Y}, {delta(x-y),;delta'(x-y)}_{yin Y}$ form Riesz bases in the corresponding closed linear spans in the Sobolev spaces $W_2^{-1}(mathbb R )$ and $W_2^{-2}(mathbb R )$. 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 $A =-frac{ d^2}{ dx^2},$ $mathop{rm dom} (A)=W^2_2(mathbb R )$ to the linear manifolds $mathop{rm dom} (A_0)=left{ fin W_2^2(mathbb{R})colon f(y)=0,; yin Y right}$, $mathop{rm dom} (A')={ gin W_2^2(mathbb{R})colon g'(y)=0,; yin Y },$ and $mathop{rm dom} (H_0)=left{fin W_2^2(mathbb{R})colon f(y)=0,;f'(y)=0,; yin Y right}$, respectively. Using the divergence forms, the basic nonnegative boundary triplets for $A^*_0$, $A'^*$, and $H^*_0$ are constructed.