用于周期有限带Dirac谱问题非线性化的迹公式

Trace Formulae for the Nonlinearization of Periodic Finite-Bands Dirac Spectral Problem

中文关键词:  迹公式  周期N-bands Dirac算子  非线性化  可积Hamilton系统

英文关键词:trace formulae  periodic $N$-bands Dirac operator  nonlinearization  integrable Hamiltonian system

中文摘要:

      本文研究具有周期有限带位势的Dirac算子,利用Dirac算子与单值算子的交换性,定义Bloch函数和乘子曲线,获得Dubrovin-Novikov型公式;进而通过复球面上的留数计算及规范变换,分别得到相应于谱带左端点、右端点以及双侧端点的特征函数的迹公式.作为应用,将Dirac谱问题非线性化得到在Liouville意义下完全可积的Hamilton系统.

英文摘要:

      This paper deals 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, which leads to the formula of Dubrovin-Novikov's type. Further, by calculation of residues on the complex sphere and via gauge transformation, we get the trace formulae of eigenfunctions corresponding to the left end-points and right end-points of the spectral bands, respectively. As an application, we obtain a completely integrable Hamiltonian system in Liouville sense through nonlinearization of the Dirac spectral problem.