Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications

Hua WANG^{1,2}, Alatancang^{1}, Jun-jie HUANG^{1}

1. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China;
2. College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, China

Abstract This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σ_{p}(H) = σ_{p}(A)∪σ_{p}^{1}(-A*). Using the characteristic of the set σ_{p}^{1}(-A*), we divide the point spectrum σ_{p}(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ_{p}^{1}(-A*) and one part of σ_{p}(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σ_{p}(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.

Abstract：
This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σ_{p}(H) = σ_{p}(A)∪σ_{p}^{1}(-A*). Using the characteristic of the set σ_{p}^{1}(-A*), we divide the point spectrum σ_{p}(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ_{p}^{1}(-A*) and one part of σ_{p}(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σ_{p}(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.

Supported by the National Natural Science Foundation of China (No. 11061019, 10962004, 11101200), the Chunhui Program of Ministry of Education of China (No. Z2009-1-01010), and the Natural Science Foundation of Inner Mongolia (No. 2010MS0110, 2009BS0101),and the Cultivation of Innovative Talent of ‘211 Project’ of Inner Mongolia University.

Cite this article:

.Symmetry of the Point Spectrum of Infinite Dimensional Hamiltonian Operators and Its Applications[J] Acta Mathematicae Applicatae Sinica, English Serie, 2012,V28(1): 149-156

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