Schr\"{o}dinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results

Abstract

The aim of this paper is two folded. Firstly, we study the validity of the Pohozaev-type identity for the Schr\"{o}dinger operator $$A_\la:=-\D -\frac{\la}{|x|^2}, \q \la\in \rr,$$ in the situation where the origin is located on the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$. The problem we address is very much related to optimal Hardy-Poincar\'{e} inequality with boundary singularities which has been investigated in the recent past in various papers. In view of that, the proper functional framework is described and explained. Secondly, we apply the Pohozaev identity not only to study semi-linear elliptic equations but also to derive the method of multipliers in order to study the exact boundary controllability of the wave and Schr\"{o}dinger equations corresponding to the singular operator $A_\la$. In particular, this complements and extends well known results by Vanconstenoble and Zuazua [34], who discussed the same issue in the case of interior singularity.