| 用于周期有限带Dirac谱问题非线性化的迹公式 |
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| 引用本文： | 张广耀,朱慧倩.用于周期有限带Dirac谱问题非线性化的迹公式[J].数学研究及应用,2016,36(2):183-193. |
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| 作者姓名： | 张广耀 朱慧倩 |
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| 作者单位： | 湖州师范学院, 浙江 湖州 313000,湖州师范学院, 浙江 湖州 313000 |
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| 基金项目： | 国家自然科学基金(Grant No.61473332), 浙江省自然科学基金(Grant No.LQ14A010009), 湖州市自然科学基金(Grant No.2013YZ06). |
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| 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.
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| 关 键 词： | 迹公式 周期N-bands Dirac算子 非线性化 可积Hamilton系统 |
| 收稿时间： | 4/9/2015 12:00:00 AM |
| 修稿时间： | 2015/9/14 0:00:00 |
Trace Formulae for the Nonlinearization of Periodic Finite-Bands Dirac Spectral Problem |
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| Guangyao ZHANG and Huiqian ZHU.Trace Formulae for the Nonlinearization of Periodic Finite-Bands Dirac Spectral Problem[J].Journal of Mathematical Research with Applications,2016,36(2):183-193. |
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| Authors: | Guangyao ZHANG and Huiqian ZHU |
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| Institution: | School of Science, Huzhou University, Zhejiang 313000, P. R. China and School of Science, Huzhou University, Zhejiang 313000, P. R. China |
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| Abstract: | 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. |
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| Keywords: | trace formulae periodic $N$-bands Dirac operator nonlinearization integrable Hamiltonian system |
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