A Resolution to Landis’ Conjecture in the Plane

In the late 1960s, E.M. Landis made the following conjecture:  If u and V are bounded functions, and u is a solution to the Schrodinger equation in n-dimension Euclidean space that decays at a super-linear exponential rate, then u must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded, complex-valued functions u and V that solve the Schrodinger equation in the plane and satisfy a much faster rate of decay. The examples of Meshkov were accompanied by qualitative unique continuation estimates for solutions in any dimension. Meshkov's estimates were quantified in 2005 by J. Bourgain and C. Kenig. These results, and the generalizations that followed, have led to a fairly complete understanding of these unique continuation properties in the complex-valued setting.

In the real-valued setting, Landis' conjecture remains open. We will discuss a recent result of A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov that resolves the real-valued version of Landis' conjecture in the plane.