Uniqueness properties of solutions of Schrödinger equations

Alexandru D. Ionescu, Carlos E. Kenig

Research output: Contribution to journalArticle

14 Scopus citations

Abstract

Under suitable assumptions on the potentials V and a, we prove that if u ∈ C ([0,1],H1) is a solution of the linear Schrödinger equation (i∂t + Δx)u = Vu + a · ∇xu on ℝd × (0,1) and if u ≡ 0 in { x > R} × {0,1} for some R ≥ 0, then u ≡ 0 in ℝd × [0,1]. As a consequence, we obtain uniqueness properties of solutions of nonlinear Schrödinger equations of the form (i∂t + Δx)u = G(x, t, u, ū, ∇xu, ∇xū) on ℝd × (0,1), where G is a suitable nonlinear term. The main ingredient in our proof is a Carleman inequality of the form ∥eβφλ(x1) ν ∥Lx2Lt2 + ∥ eβφλ(x1)x ν ∥Bx∞,2Lt2 ≤ C̄ ∥ eβφλ(x1) (i∂t + Δx) ν ∥ Bx1,2Lt2 for any ν ∈ C(ℝ : H1) with ν(., t) ≡ 0 for t ∉ [0,1]. In this inequality, Bx∞,2 and Bx1,2 are Banach spaces of functions on ℝd, and eβφλ(x1) is a suitable weight.

Original languageEnglish (US)
Pages (from-to)90-136
Number of pages47
JournalJournal of Functional Analysis
Volume232
Issue number1
DOIs
StatePublished - Mar 1 2006
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

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

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