¹Qinggong College, Hebei United University, Tangshan, China
²College of Mathematics, Inner Mongolia University for Nationalities, Tong Liao, China
³Department of Mathematics and Physics, Shijiazhuang TieDao University, Shijiazhuang, China

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Abstract—This paper discusses the second-order matrix eigenvalue problem by means of the nonlinearization of the Lax pairs, then the author gives the Bargmann and Neumann constraints of this problem. The relation between the potentials and the eigenfunction is set up based on these constraints. By means of the nonlinearization of the Lax pairs, the author found these systems of the eigenvalue problem can be equal to the Hamilton canonical system in real symplectic space. In the end, the infinite-dimensions Dynamical systems can be transformed into the finitedimensions Hamilton canonical systems in the symplectic space. As well, this paper obtains the representations of the solutions for the evolution equations.

Index Terms—Eigenvalue problem, integrable system, evolution equation, lax representation.

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Cite: Shujuan Yuan, Shuhong Wang, Wei Liu, Xiaohong Liu, Li Li, "Bargmann and Neumann System of the Second- Order Matrix Eigenvalue Problem," Journal of Computers vol. 7, no. 9, pp. 2216-2223, 2012.