Time optimal observability for Grushin Schrödinger equation

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