奇异线性哈密顿算子自伴扩张的新描述

数学物理学报 ›› 2014, Vol. 34 ›› Issue (1): 157-170.

奇异线性哈密顿算子自伴扩张的新描述

郑召文|许艳丽

曲阜师范大学数学科学学院山东曲阜 273165

收稿日期:2012-10-08 修回日期:2013-09-06 出版日期:2014-02-25 发布日期:2014-02-25
基金资助:
国家自然科学基金(11171178, 11271225)和高校博士点专项科研基金(20103705110003)资助.

A New Description of Self-adjoint Domains of Singular Linear Hamiltonian Operators

ZHENG Zhao-Wen, XU Yan-Li

School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165

Received:2012-10-08 Revised:2013-09-06 Online:2014-02-25 Published:2014-02-25
Supported by:
国家自然科学基金(11171178, 11271225)和高校博士点专项科研基金(20103705110003)资助.

摘要/Abstract

摘要：

研究了具有一个奇异端点的线性哈密顿算子的自伴扩张的解析描述. 设最小哈密顿算子h 的亏指数为(d, d), 将 Im(h* y, y) 表示为秩为2d的二次型, 该文利用二次型的表示矩阵得到了最小哈密顿算子h 的自伴扩张域的一种新的完全描述.

关键词: 线性哈密顿算子, Hermite 矩阵, 自伴扩张, 亏指数

Abstract:

In this paper, the self-adjoint extension of the minimal Hamiltonian operator h with equal deficiency index (d, d) is considered. Since Im(h* y, y) can be represented as a quadratic form with rank being 2d, the new characterization of
the domains of self-adjoint extensions of h is obtained by the representation matrix.

Key words: Linear Hamiltonian operator, Hermitian matrix, Self-adjoint extension, Deficiency index

中图分类号:

34B05

郑召文, 许艳丽. 奇异线性哈密顿算子自伴扩张的新描述[J]. 数学物理学报, 2014, 34(1): 157-170.

ZHENG Zhao-Wen, XU Yan-Li. A New Description of Self-adjoint Domains of Singular Linear Hamiltonian Operators[J]. Acta mathematica scientia,Series A, 2014, 34(1): 157-170.

参考文献

[1] Everitt W N. Integrable-square solution of ordinary differential equations (III). Quart J Math, 1963, 14(2): 170--180
　
[2] Everitt W N, Zettl A. Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Archief voor Wiskunde, 1979, 27(3): 363--397

[3] Cao Zhijiang. On self-adjoint domains of 2nd order differential operators in limit-cirle case. Acta Math Sinica, New Series, 1985, 1(3): 175--180

[4] Krall A M. Hilbert Spaces, Boundary Value Problems and Orthogonal Polynomials. Basel, Boston, Berlin: Birkhauser-Verlag, 2002

[5] Naimark M A. Linear Differential Operators. New York: Ungar, 1968

[6] Shi Yuming. On the rank of the matrix radius of the limiting set for a singular linear Hamiltonian system. Linear
ALgebra and its Applications, 2004, 376: 109--123

[7] Sun Jiong, Wang Aiping, Anton Zettl. Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index. Journal of Functional Analysis, 2008, 255: 3229--3248

[8] Sun Jiong. On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices.
Acta Math Sinica, New Series, 1986, 2(2): 152--167

[9] Sun Huaqing, Shi Yuming. Self-adjoint extensions for singular linear Hamiltonian systems. Math Nachr, 2011, 284(5-6): 797--814

[10] Weidmann J. Linear Operators in Hilbert Spaces. Berlin, New York: Springer-Verlag, 1980

[11] Zheng Zhaowen, Chen Shaozhu. GKN theory for linear Hamiltonian systems. Applied Mathematics and Computation, 2006, 182: 1514--1527

[12] 曹之江. 常微分算子. 上海: 上海科学技术出版社, 1986

[13] 曹之江. 高阶极限圆型微分算子的自伴扩张. 数学学报, 1985, 28(2): 205--217

[14] 魏广生. 对称算子自伴域的一种新描述. 内蒙古大学学报, 1996, 27(3): 305--130

[15] 魏广生, 徐宗本. 亏指数为可数无穷的对称微分算子的自伴扩张. 数学进展, 2000, 29(3): 227--234