Eigenfunctions with double exponential rate of localization

We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\mathrm{\nabla }u)=\lambda u$, $\lambda >0,$ in the cylinder ${\mathbb{T}}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $\mathrm{\mid }u(x,y,t)\mathrm{\mid }\le C{e}^{-c{e}^{c\mathrm{\mid }t\mathrm{\mid }}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=\mathrm{\Delta }u+B\mathrm{\nabla }u$ 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.