On the number of negative eigenvalues for a Schrödinger operator with magnetic field

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Abstract

We consider the Schrödinger operator with magnetic field H = ∑nj=1 (1/i ∂/∂xj - aj)2 + V in ℝn. Under certain conditions on the magnetic field B = curl a, we generalize the Fefferman-Phong estimates (Bull. A. M. S. 9, 129-206 (1983)) on the number of negative eigenvalues for -Δ + V to the operator H. Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on B in particular are satisfied if the magnetic potentials aj(x) are polynomials.

Original languageEnglish
Pages (from-to)637-660
Number of pages24
JournalCommunications in Mathematical Physics
Volume182
Issue number3
DOIs
StatePublished - 1996

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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