Cantor and Band Spectra for Periodic Quantum Graphs with Magnetic Fields
Abstract
We provide an exhaustive spectral analysis of the twodimensional 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 KronigPenney 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 structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 January 2007
 DOI:
 10.1007/s0022000600500
 arXiv:
 arXiv:mathph/0511057
 Bibcode:
 2007CMaPh.269...87B
 Keywords:

 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Spectral Theory
 EPrint:
 Misprints corrected