﻿ 对角无穷维哈密顿算子点谱关于虚轴的对称性 Imaginary Axis Symmetry of the Point Spectrum of the Diagonal Infinite Dimensional Hamiltonian Operators

Vol.04 No.04(2015), Article ID:16153,6 pages
10.12677/AAM.2015.44038

Imaginary Axis Symmetry of the Point Spectrum of the Diagonal Infinite Dimensional Hamiltonian Operators

Lijun Yan, Angran Liu

School of Mathematical Sciences, Inner Mongolia University, Hohhot Inner Mongolia

Received: Sep. 26th, 2015; accepted: Oct. 9th, 2015; published: Oct. 14th, 2015

ABSTRACT

In this article, the point spectrum of infinite dimension of Hamilton operators is divided into four parts, getting the sufficient and necessary condition about symmetry of each part of the point spectrum. Using structural characteristics of spectrum of infinite dimension of Hamilton operators, then the symmetry axis of the point spectrum is characterized by using the residual spectrum of internal elements. In the end, some examples are constructed to illustrate the effectiveness of criterion.

Keywords:Infinite Dimensional Hamilton Operator, Point Spectrum, Residual Spectrum

1. 引言

2. 预备知识

;

;

;

;

;

;

;

;

1) 若，则;

2) 若，则;

1);

2);

3);

4);

3. 主要结果及证明

.

1);

2);

1)时，关于虚轴对称当且仅当

2)时，关于虚轴对称当且仅当

3)时，关于虚轴对称当且仅当

4)时，是关于虚轴对称的。

.

1) 当时，有，则关于虚轴对称

2) 当时，有，则关于虚轴对称

3) 当时，有关于虚轴对称

4)时，是平凡的，所以是关于虚轴对称的。

1)时，关于虚轴对称

2)时，关于虚轴对称

3)时，关于虚轴对称

4)时，是关于虚轴对称。

1)时，关于虚轴对称

2)时，关于虚轴对称

3)时，关于虚轴对称

4)时，是平凡的，所以关于虚轴对称。

4. 举例与应用

，则无穷维Hamilton算子

，考虑无穷维Hamilton算子

，则关于虚轴对称。

Imaginary Axis Symmetry of the Point Spectrum of the Diagonal Infinite Dimensional Hamiltonian Operators[J]. 应用数学进展, 2015, 04(04): 307-312. http://dx.doi.org/10.12677/AAM.2015.44038

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