Landis' conjecture and related estimates
Speaker 
Prof. JennNan Wang, National Taiwan University, Taiwan


When 
Sep 13, 2016
from 02:00 PM to 03:00 PM 
Where  LH 006 
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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 $u(x)\le C\exp(Cx^{1+})$, then $u\equiv 0$. Landis' conjecture was disproved by Meshkov who constructed such $V$ and nontrivial $u$ satisfying $u(x)\le C\exp(Cx^{\frac 43})$. He also showed that if $u(x)\le C\exp(Cx^{\frac 43+})$, 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{complexvalued} functions. It remains an open question whether Landis' conjecture is true for realvalued $V$ 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.