Progress of Landis' conjecture in the plane
Time and place
2 PM on Monday, February 23rd, 2015; SH 378
Blair Davey (University of Minnesota)
Abstract
In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\R^n$ that decays like $|u(x)| <= c exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \R^2 \to \C$ that solve $\Delta u = V u$ in $\R^2$ and satisfy $|u(x)| <= c exp(- C |x|^{4/3})$. The result of Meshkov was accompanied by quantitative unique continuation estimates for $\R^n$. In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates. These results, and the generalizations that followed, have led to a fairly complete understanding the complex-valued setting. However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting. We will discuss very recent progress towards resolving the real-valued version of Landis' conjecture in the plane.
