Time optimal observability for Grushin Schr\"odinger equation

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