On the uniqueness of variable coefficient Schr\"odinger equations

TL;DR

This paper establishes unique continuation properties for variable coefficient Schrödinger equations, showing solutions with rapid decay at two times are trivial, and recovers sharp Gaussian decay rates under specific conditions.

Contribution

It introduces new unique continuation results for variable coefficient Schrödinger equations, including conditions for triviality and sharp decay rate recovery.

Findings

Solutions decaying faster than cubic exponential at two times are zero.

Under transversally anisotropic conditions, sharp Gaussian decay rates are achieved.

The results extend previous work to variable coefficient settings.

Abstract

We prove unique continuation properties for linear variable coefficient Schr\"odinger 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 [14, 17, 18].