Trace Formulae for the Eigenfunctions of Periodic Finite-Bands Dirac Operator
Received:June 02, 2014  Revised:March 09, 2015
Key Words: trace formulae   eigenfunctions   periodic \textit{N}-bands Dirac operator  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11326131), the Natural Science Foundation of Zhejiang Province (Grant No.LQ14A010009) and the Natural Science Foundation of Huzhou City (Grant No.2013YZ06).
Author NameAffiliation
Zhen LI Department of Mathematics, Zhengzhou University, Henan 450001, P. R. China 
Guangyao ZHANG School of Science, Huzhou University, Zhejiang 313000, P. R. China 
Zhongyao WANG School of Science, Huzhou University, Zhejiang 313000, P. R. China 
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      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.
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