In this article, we revisit a result by A. Koenig concerning the non-controllability of the half-heat equation posed on R, with a control domain that is an open set whose exterior contains an interval. The main novelty of the present article is to disprove the corresponding observability inequality by using as an initial condition a family of prolate spheroidal wave function (PSWF) translated in the Fourier space, associated to a parameter c that goes to ∞. The proof is essentially based on the dual nature of the PSWF together with direct computations, showing that the solution "does not spread out" too much during time. As a consequence, we obtain a new non-controllability result on the Grushin equation posed on R × R.