On the uniqueness of variable coefficient Schrödinger equations. (English) Zbl 1563.35153

Federico, Serena; Li, Zongyuan; Yu, Xueying

Commun. Contemp. Math. 27, No. 3, Article ID 2450016, 45 p. (2025).

Summary: 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 L. Escauriaza et al. [Commun. Partial Differ. Equations 31, No. 12, 1811–1823 (2006; Zbl 1124.35068); J. Eur. Math. Soc. (JEMS) 10, No. 4, 883–907 (2008; Zbl 1158.35018); Duke Math. J. 155, No. 1, 163–187 (2010; Zbl 1220.35008)].

© 2025 World Scientific Publishing Company

MSC:

35B60	Continuation and prolongation of solutions to PDEs
35Q41	Time-dependent Schrödinger equations and Dirac equations
35B45	A priori estimates in context of PDEs
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Keywords:

Schrödinger equations with variable coefficients; unique continuation; Carleman inequality; logarithmic convexity

Citations:

Zbl 1124.35068; Zbl 1158.35018; Zbl 1220.35008

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