On the uniqueness of variable coefficient Schrödinger equations

TL;DR

This work establishes global unique continuation for linear Schrödinger equations with variable coefficients and bounded real potentials. The authors develop a robust framework combining log-convexity of Gaussian-weighted energies, parabolic-dissipation regularization, and Carleman inequalities to derive lower bounds and contradict potential nonzero solutions under strong endpoint-decay hypotheses. Under a smallness condition on the leading coefficients, they prove uniqueness when endpoint decays are faster than any cubic exponential; under a transversally anisotropic coefficient structure they obtain the sharp Gaussian-decay regime, recovering the classical EKPV Gaussian-rate results in a variable-coefficient setting. The results bridge uncertainty-principle-inspired decay constraints with variable-coefficient dispersive dynamics, and they provide a roadmap for sharp unique continuation in anisotropic media with potential extensions to weaker asymptotic flatness.

Abstract

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 [14, 17, 18].

Theorems & Definitions (35)

• Theorem A: Theorem 1 in EKPV_Duke
• Theorem B: Hardy's uncertainty principle in Hardy
• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 3.1
• Corollary 3.2
• Remark 3.3
• Proposition 3.4
• Remark 3.5