Time optimal observability for Grushin Schrödinger equation

Abstract: We consider two dimensional Grushin Schr\"odinger equation posed on a finite cylinder $\Omega=(-1,1)x\times \T_y$ with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time $T>0$ 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 $T\leq T_$.