On the uniqueness of variable coefficient Schrödinger equations

Abstract

In this paper, we prove unique continuation properties for linear variable coefficient Schrödinger 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 Schrödinger equations, Comm. Partial Differential Equations 31(10–12) (2006) 1811–1823; Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS) 10(4) (2008) 883–907; The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 155(1) (2010) 163–187].

Carleman estimates for third order operators of KdV and non KdV-type and applications

“…In the last decades variable coefficient Schrödinger equations have attracted lots of attention. Smoothing and Strichartz estimates have been proved under different hypotheses (see [2, 11, 20-22, 29, 34, 37, 41] and references therein), nonlinear problems have been solved (see [11,20,21,25,29,34]), and uniqueness results have been proved (see [7,8,10,27] in the constant coefficient case, and [3,16] and references therein for variable coefficient cases). For KdV-type equations the investigation has not been pushed that far, possibly because of the unknown real analogue of this equation in dimensions higher than two, and also because variable coefficient third order equations can be much more challenging to study.…”