Abstract
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.
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Acknowledgements
The author would like to acknowledge to the anonymous referees for their careful reading and corrections. Thanks are also given to Professors Hong-Kui Pang and Zhi-Gang Jia for their fruitful discussions, and also for their tests on the comparison of the algorithms given by (7) and Lemma 5, applied to Examples 6 and 11 by using MatLab. Up to N = 104; the outputs show that these algorithms are very close to each other, as those given in Example 11. 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.
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Chen, MF. On spectrum of Hermitizable tridiagonal matrices. Front. Math. China 15, 285–303 (2020). https://doi.org/10.1007/s11464-020-0832-2
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DOI: https://doi.org/10.1007/s11464-020-0832-2