In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted in order to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behaviour of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the d − 1 -dimensional sphere whose vector potential reflects the behaviour of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.

This article is devoted to analyzing control properties for the heat equation with singular potential −µ/|x| 2 arising at the boundary of a smooth domain Ω ⊂ R N , N ≥ 1. This problem was firstly studied by Vancostenoble and Zuazua [19] and then generalized by Ervedoza [10] in the context of interior singularity. Roughly speaking, these results showed that for any value of parameters µ ≤ µ(N ) := (N − 2) 2 /4, the corresponding parabolic system can be controlled to zero with the control distributed in any open subset of the domain. The critical value µ(N ) stands for the best constant in the Hardy inequality with interior singularity.When considering the case of boundary singularity a better critical Hardy constant is obtained, namely µN := N 2 /4.In this article we extend the previous results in [19], [10], to the case of boundary singularity. More precisely, we show that for any µ ≤ µN , we can lead the system to zero state using a distributed control in any open subset.We emphasize that our results cannot be obtained straightforwardly from the previous works [19], [10].

The aim of this paper is two folded. Firstly, we study the validity of the Pohozaev-type identity for the Schrödinger operatorin the situation where the origin is located on the boundary of a smooth domain Ω ⊂ R N , N ≥ 1. The problem we address is very much related to optimal Hardy-Poincaré 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 Schrödinger equations corresponding to the singular operator A λ . 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.