WIAS Preprint No. 1538, (2010)

Random block operators


  • Kirsch, Werner
  • Metzger, Bernd
  • Müller, Peter

2010 Mathematics Subject Classification

  • 82B44 82D55


  • Bogoliubov-de Gennes equations, random Schrödinger operators, integrated density of states, Wegner estimates




We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.

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