We construct a real-valued solution to the eigenvalue problem -\text{div}(A\nabla u)=\lambda u, in the cylinder \mathbb{T}^2\times \mathbb{R} with a real, uniformly elliptic, and uniformly C^1 matrix A such that |u(x,y,t)|\leq C e^{-c e^{c|t|}} for some c,C>0. We also construct a complex-valued solution to the heat equation u_t=\Delta u + B \nabla u in a half-cylinder with continuous and uniformly bounded B, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed.
