This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete...This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete spectrum of the matrices is presented,with a slight and technical restriction.The problem is well known,but from the author’s knowledge,it has been largely opened for quite a long time.It is important in various application,in quantum mechanics for instance.The main tool to solve the problem is the isospectral technique developed a few years ago.Two alternative constructions of the isospectral operator are presented;they are helpful in theoretical analysis and in numerical computations,respectively.Some illustrated examples are included.展开更多
The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which o...The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.展开更多
The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties ar...The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.展开更多
Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computi...Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.展开更多
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11771046),the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete spectrum of the matrices is presented,with a slight and technical restriction.The problem is well known,but from the author’s knowledge,it has been largely opened for quite a long time.It is important in various application,in quantum mechanics for instance.The main tool to solve the problem is the isospectral technique developed a few years ago.Two alternative constructions of the isospectral operator are presented;they are helpful in theoretical analysis and in numerical computations,respectively.Some illustrated examples are included.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.12090011,11771046,11771188,11771189)the National Key R&D Program of China(No.2020YFA0712900)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171162)the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.
基金supported in part by the National Natural Science Foundation of China(Grant No.11771046)the project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.
基金supported in part the National Natural Science Foundation of China (Grant No. 11771046)the Project from the Ministry of Education in Chinathe Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Based on a series of recent papers, a powerful algorithm is reformulated for computing the maximal eigenpair of self-adjoint complex tridiagonal matrices. In parallel, the same problem in a particular case for computing the sub-maximal eigenpair is also introduced. The key ideas for each critical improvement are explained. To illustrate the present algorithm and compare it with the related algorithms, more than 10 examples are included.
