Landis' conjecture and related estimates

Abstract: In the late 60's, E.M. Landis conjectured that if $\Delta u+Vu=0$ in

$\R^n$ with $\|V\|_{L^{\infty}(\R^n)}\le 1$ and $\|u\|_{L^{\infty}(\R^n)}\le C_0$ satisfying $|u(x)|\le C\exp(-C|x|^{1+})$, then $u\equiv 0$. Landis' conjecture was disproved by Meshkov who constructed such $V$ and nontrivial $u$ satisfying $|u(x)|\le C\exp(-C|x|^{\frac 43})$. He also showed that if $|u(x)|\le C\exp(-C|x|^{\frac 43+})$, then $u\equiv 0$. A quantitative form of Meshkov's result was derived by Bourgain and Kenig in their resolution of Anderson localization for the Bernoulli model in higher dimensions. It should be noted that both $V$ and $u$ constructed by Meshkov are \emph{complex-valued} functions. It remains an open question whether Landis' conjecture is true for real-valued $V$ and $u$. In this talk I would like to discuss some recent joint works with C. Kenig, L. Silvestre, and Davey on Landis' conjecture and related questions in two dimensions.