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Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators


We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice \(\mathbb {Z}\). These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in \(\mathbb {R}\). The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.

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The first author RAP thanks support by PNPD-CAPES scholarship of the Post-Graduate Program in Applied Mathematics of the IMECC-UNICAMP (under contract 88887.466594/2019-00). CRdO thanks the partial support by CNPq (Brazilian agency, under contract 303503/2018-1).

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Correspondence to Roberto A. Prado.

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Roberto A. Prado is supported by CAPES (Brazil).

César R. de Oliveira is partially supported by CNPq (Brazil)

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Prado, R.A., de Oliveira, C. & de Oliveira, E. Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators. Math Phys Anal Geom 24, 30 (2021).

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  • Discrete Dirac operators
  • Random potentials
  • Density of states
  • Lifshitz tails

Mathematics Subject Classification (2010)

  • Primary 47B80
  • 47A75
  • 82B44. Secondary 81Q15