摘要
三弦乐谱可由三对角复矩阵生成,人们听到的声音由其谱决定,自然要求此矩阵有实谱.如同量子力学,模型的描述是复算子,可观测量是实的.换言之,所述的三对角复矩阵应是关于某个测度的复内积空间上的自共轭算子.熟知,生灭Q矩阵可配称,自然就是自共轭的.我们将要介绍最新的一个代表性成果:对于相当广泛的自共轭三对角复矩阵,总可以构造出一个生灭Q矩阵,使得两者等谱(简单地说,两者有完全相同的特征值).这个问题浅显易懂,但我们曾在不同时期,从概率论、统计物理和计算数学三个不同的角度研究过,经历了漫长的求索岁月.本文是根据作者在"International Conference on Probability Theory and Its Applications"(湖南文理学院,2018年7月)的报告整理而成.共分三部分:1)生灭过程的新应用;2)从(非对角线元素非负的)实可配称矩阵到复可配称矩阵;3)此课题的来源(计算)及其判别准则的应用(统计物理及量子力学).
The Sanxian is a traditional Chinese three-stringed plucked instrument.Its music can be generated by tridiagonal complex matrices.The sound people hear is determined by its spectrum and naturally requires that the matrix has a real spectrum.As in quantum mechanics,the description of the model is a complex operator and the observable measurement is real.In other words,the tridiagonal complex matrix described is a self-adjoint operator on the complex inner product space with respect to a measure.It is well known that the birth-death Q matrix can be matched and naturally self-adjointed.We will introduce the latest representative results:for a fairly wide range of self-adjoint tridiagonal complex matrices,a birth-death Q matrix can always be constructed to make both isospectral (in simple words, both have the same eigenvalues).This problem is simple and easy to understand.But we have studied it from three different perspectives:probability theory,statistical physics and computational mathematics at different times,and have gone through a long time'of exploration. This article is based on the author's report on "International Conference on Probability Theory and Its Applications"(Hunan University of Arts and Science,2018/7).It consists of three parts:1)the new application of the birth-death processes;2)from the real matrices (with non negative off-diagonal element)to the complex matrices;3)the source of this topic (Computation)and the application of its criterion (Statistical Physics and Quantum Mechanics).
作者
陈木法
CHEN Mu-Fa(School of Mathematical Sciences,Beijing Normal University,Laboratory of Mathematics and Complex Systems (Beijing Normal University),Ministry of Education,Beijing,100875,China)
出处
《应用概率统计》
CSCD
北大核心
2018年第5期533-545,共13页
Chinese Journal of Applied Probability and Statistics
基金
国家自然科学基金项目(批准号:11771046)
教育部双一流大学建设项目
江苏省高校优势学科建设工程项目资助
关键词
复矩阵
可配称
复可配称
生灭矩阵
统计物理
量子力学
complex matrix
symmetrizable
Hermitizable
birth-death matrix
statistical physics
quantum mechanics
