On the uniqueness of variable coefficient Schrödinger equations

In this paper, we prove unique continuation properties for linear variable coefficient Schrodinger equations with bounded real potentials. Under certain smallness conditions on the leading coefficients, we prove that solutions decaying faster than any cubic exponential rate at two different times must be identically zero. Assuming a transversally anisotropic type condition, we recover the sharp Gaussian (quadratic exponential) rate in the series of works by Escauriaza-Kenig-Ponce-Vega [On uniqueness properties of solutions of Schrodinger equations, Comm. Partial Differential Equations 31(10-12) (2006) 1811-1823; Hardy's uncertainty principle, convexity and Schrodinger evolutions, J. Eur. Math. Soc. (JEMS) 10(4) (2008) 883-907; The sharp Hardy uncertainty principle for Schrodinger evolutions, Duke Math. J. 155(1) (2010) 163-187].