The periodic Schrödinger operators with potentials in the Morrey class

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36 Scopus citations

Abstract

We consider the periodic Schrödinger operator - Δ + V(x) in ℝd, d ≥ 3 with potential V in the Morrey class. Let Ω be a periodic cell for V. We show that, for p ∈ ((d - 1)/2,d/2], there exists a positive constant ε depending only on the shape of Ω, p and d such that, if lim sup sup r2/r→0 xεΩ {1/ B(x,r) ∫B(xr) V(y) pdy}1/p <ε, then the spectrum of - Δ + V is purely absolutely continuous. We obtain this result as a consequence of certain weighted L2 Sobolev inequalities on the d-torus. It improves an early result by the author for potentials in Ld/2 or weak-Ld/2 space.

Original languageEnglish
Pages (from-to)314-345
Number of pages32
JournalJournal of Functional Analysis
Volume193
Issue number2
DOIs
StatePublished - Aug 20 2002

Bibliographical note

Funding Information:
1Research supported in part by the NSF Grant DMS-9732894.

Keywords

  • Absolute continuous spectrum
  • Periodic potential
  • Schrödinger operator
  • Weighted uniform Sobolev inequalities

ASJC Scopus subject areas

  • Analysis

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