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].
Federico, S., Li, Z., Yu, X. (2024). On the uniqueness of variable coefficient Schrödinger equations. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, Online first, Online first-Online first [10.1142/s0219199724500160].
On the uniqueness of variable coefficient Schrödinger equations
Federico, Serena;
2024
Abstract
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].File | Dimensione | Formato | |
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Federico-Li-Yu_Uniqueness of Variable Schrödinger.pdf
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