Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results

The aim of this paper is two folded. Firstly, we study the validity of a Pohozaev-type identity for the Schrödinger operatorAλ:=−Δ−λ|x|2,λ∈R, in the situation where the origin is located on the boundary of a smooth domain Ω⊂RNΩ⊂RN, N⩾1N⩾1, showing some applications to semi-linear elliptic equations. The problem we address is very much related to optimal Hardy–Poincaré inequalities with boundary singularities which have been investigated in the recent past in various papers. In view of that, the proper functional framework is described and explained. Secondly, we use the Pohozaev identity to derive the method of multipliers and we apply it to study the exact boundary controllability for the wave and Schrödinger equations corresponding to the singular operator AλAλ. In particular, this complements and extends well-known results by Vancostenoble and Zuazua (2009) [38], who discussed the same issue in the case of interior singularity.