Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy

M. Burak Erdoǧan, Michael Goldberg, William R. Green

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate L 1(ℝ4) → L (ℝ4) dispersive estimates for the Schrödinger operator H = - Δ +V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator F t satisfying {pipe}F t{pipe}L1→L∞ ≲ 1/log t for t > 2 such that (Formula presented.) We also show that the operator F t = 0 if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.

Original languageEnglish (US)
Pages (from-to)1936-1964
Number of pages29
JournalCommunications in Partial Differential Equations
Volume39
Issue number10
DOIs
StatePublished - Oct 2014

Keywords

  • Dispersive estimate
  • Resonance
  • Schrödinger operator
  • Wave equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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