Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields

  title={Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields},
  author={J. Br{\"u}ning and V. Geyler and Konstantin Pankrashkin},
  journal={Communications in Mathematical Physics},
  • J. Brüning, V. Geyler, Konstantin Pankrashkin
  • Published 2006
  • Mathematics, Physics
  • Communications in Mathematical Physics
  • We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band… CONTINUE READING
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