Existence of solutions to the Efros self-consistency equation in all dimensions

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In this paper we develop in all dimensions an existence theory for the Efros energy-domain self-consistent equation governing the density of single-electron states. When the energy domain extends to infinity, we prove that solutions exist except in dimension one. When the energy domain is finite, we prove that solutions exist in all dimensions. The boundary condition near the Fermi level makes the integral diverge. Such a difficulty can be overcome by using a perturbed equation as an approximation and then passing to the zero perturbation limit.

Original languageEnglish (US)
Pages (from-to)853-861
Number of pages9
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2032
StatePublished - Apr 8 2003


  • Fixed-point theorem
  • Nonlinear integral equations
  • Perturbation

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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