Null controllability of the 1D heat equation with interior inverse square potential

This paper aims to answer an open problem posed by Morancey in 2015 concerning the null controllability of the heat equation on (−1,1) with an internal inverse square potential located at x=0. For the range of singularity under study, after having introduced a suitable self-adjoint extension that enables to transmit information from one side of the singularity to another, we prove null-controllability in arbitrary small time, firstly with an internal control supported in an arbitrary measurable set of positive measure, secondly with a boundary control acting on one side of the boundary. Our proof is mainly based on a precise spectral study of the singular operator together with some recent refinements of the Fattorini-Russell moment method. This in particular requires to use some fine (and sometimes new) properties of Bessel functions and their zeros.