A counterexample to dispersive estimates for Schrödinger operators in higher dimensions

M. Goldberg, M. Visan

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Abstract

In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class Cα, α < n−3 2 , for which the solutions to the linear Schrödinger equation in Rn −i∂tu =- Δu + V u, u(0) = f,fail to satisfy an evolution estimate of the formΙΙu(t, ·)ΙΙ ≤ C|t| − n /2 ΙΙu(0, ·)ΙΙ1. This contrasts with known results in dimensions n ≤ 3, where a pointwise decay condition on V is generally sufficient to imply dispersive bounds. The obstructions in our example are generated by an expression with scaling law |t| −n+ 3/2 +α, which becomes dominant in the time interval |t| « 1.
Original languageEnglish
Pages (from-to)211-238
JournalCommunications in Mathematical Physics
Volume266
Issue number1
DOIs
StatePublished - 2005

Keywords

  • math.AP
  • math-ph
  • math.MP
  • Inverse Fourier Transform
  • Riesz Potential
  • Modern Mathematical Physic
  • Dispersive Estimate
  • Schwartz Function

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