Trace Formulae for the Eigenfunctions of Periodic Finite-Bands Dirac Operator

Abstract:

      In this paper, we deal with a Dirac operator with periodic and finite-bands potentials. Taking advantage of the commutativity of the monodromy operator and the Dirac operator, we define the Bloch functions and multiplicator curve. Further, we obtain the formulae of Dubrovin-Novikov's type, which illustrate the inherent relations between the Bloch functions and potentials. Finally, we get the trace formulae of eigenfunctions corresponding to the left end-points, right end-points and all end-points of the spectral bands by calculation of residues on the complex sphere, respectively.