### Abstract

Under suitable assumptions on the potentials V and a, we prove that if u ∈ C ([0,1],H^{1}) is a solution of the linear Schrödinger equation (i∂_{t} + Δ_{x})u = Vu + a · ∇_{x}u on ℝ^{d} × (0,1) and if u ≡ 0 in { x > R} × {0,1} for some R ≥ 0, then u ≡ 0 in ℝ^{d} × [0,1]. As a consequence, we obtain uniqueness properties of solutions of nonlinear Schrödinger equations of the form (i∂_{t} + Δ_{x})u = G(x, t, u, ū, ∇_{x}u, ∇_{x}ū) on ℝ^{d} × (0,1), where G is a suitable nonlinear term. The main ingredient in our proof is a Carleman inequality of the form ∥e^{βφλ(x1)} ν ∥_{Lx2Lt2} + ∥ e^{βφλ(x1)} ∇_{x} ν ∥_{Bx∞,2Lt2} ≤ C̄ ∥ e^{βφλ(x1)} (i∂_{t} + Δ_{x}) ν ∥ _{Bx1,2Lt2} for any ν ∈ C(ℝ : H^{1}) with ν(., t) ≡ 0 for t ∉ [0,1]. In this inequality, B_{x}^{∞,2} and B_{x}^{1,2} are Banach spaces of functions on ℝ^{d}, and e^{βφλ(x1)} is a suitable weight.

Original language | English (US) |
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Pages (from-to) | 90-136 |

Number of pages | 47 |

Journal | Journal of Functional Analysis |

Volume | 232 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2006 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis

### Keywords

- Carleman inequalities
- Local smoothing
- Parametrices
- Uniqueness of solutions

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## Cite this

*Journal of Functional Analysis*,

*232*(1), 90-136. https://doi.org/10.1016/j.jfa.2005.06.005