On quantitative uniqueness for elliptic equations

We address the question of quantitative uniqueness for the equation \(\Delta u=V u\) with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u corresponding to potential functions V such that u vanishes of order \(\mathrm{const} \Vert V\Vert _{L^\infty }^{2/3}\). The example also shows sharpness of recently obtained bounds in the case of a parabolic equation \(u_t-\Delta u= V u\).