The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain

@article{Dobson2012TheSO,
  title={The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain},
  author={Matthew Dobson and Christoph Ortner and Alexander V. Shapeev},
  journal={Multiscale Model. Simul.},
  year={2012},
  volume={10},
  pages={744-765}
}
We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a real, positive spectrum. The spectrum is identical to that of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbor interactions. We construct an eigenbasis for the linearized QCF operator whose condition number is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent numerical observations [M… 
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