Time optimal observability for Grushin Schrödinger equation

Abstract

We consider the two-dimensional Grushin Schrödinger equation posed on a finite cylinder $\mathrm{\Omega }={(-1,1)}_x\times {\mathbb{𝕋}}_y$ with Dirichlet boundary condition. We obtain sharp observability by any horizontal strip, with the optimal time depending on the size of the strip. Consequently, we prove the exact controllability for the Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of a harmonic oscillator at $x=0$, we also show that the observability fails for any $T\le T_{\ast }$.