# 1D nonnegative Schrodinger operators with point interactions

Author Yu. G. Kovalev
yury_kovalev@ukr.net, yury.kovalev.lugansk@gmail.com
Department of Mathematical Analysis, East Ukrainian National University

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.
Keywords point interaction; Riesz basis; boundary triplet; the Friedrichs extension; the Krein extension
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Pages 150-163
Volume 39
Issue 2
Year 2013
Journal Matematychni Studii
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