Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

M. Burak Erdoǧan, Wilhelm Schlag

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate boundedness of the evolution eitH in the sense of L2(ℝ3) → L2(ℝ3) as well as L1(ℝ3) → L(ℝ 3) for the non-selfadjoint operator H = [V2 -Δ+μ-V1Δ - μ +V1-V2 where μ > 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), see A1)-A4) below, but without imposing any restrictions on the edges ±μ of the essential spectrum. Our goal is to develop an "axiomatic approach," which frees the linear theory from any nonlinear context in which it may have arisen.

Original languageEnglish (US)
Pages (from-to)199-249
Number of pages51
JournalJournal d'Analyse Mathematique
Volume99
DOIs
StatePublished - 2006

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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