Recent progress on the Landis conjecture

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Jeen-Nan Wang , National Taiwan University
Fine Hall 314

In the late 60's, E.M. Landis conjectured that if $\Delta u+Vu=0$ in $\mathbb{R}^n$ with $\|V\|_{L^{\infty}(\mathbb{R}^n)}\le 1$ and $\|u\|_{L^{\infty}(\mathbb{R}^n)}\le C_0$ 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 a recent joint work with Kenig and Silvestre on Landis' conjecture in two dimensions. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\|W\|_{L^{\infty}(\\mathbb{R}^2)}\le 1$ and $\|V\|_{L^{\infty}(\mathbb{R}^2)}\le 1$. Let $u$ be a real solution of $\Delta u-\nabla(Wu)-Vu=0$ in $\mathbb{R}^2$. Assume that $u(0)=1$ and $\|u\|_{L^{\infty}(\mathbb{R}^2)}\le\exp(C_0|z|)$. Then $u$ satisfies \$\underset{{|z_0|=R}}{\inf}\,\underset{|z-z_0|