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

TL;DR

This paper investigates boundary singularities in Schrödinger operators, establishing a Pohozaev identity, exploring Hardy inequalities, and applying these results to boundary controllability of wave and Schrödinger equations with boundary singularities.

Contribution

It introduces a Pohozaev identity for boundary singular Schrödinger operators and applies it to controllability problems, extending previous interior singularity results.

Findings

Pohozaev identity valid for boundary singularities

Extended controllability results for boundary singular Schrödinger and wave equations

Enhanced understanding of Hardy inequalities with boundary singularities

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.