Abstract
The first aim of the paper is to study the Hermitizability of second-order 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.
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Acknowledgements
Thanks are given to the referees for their careful reading of an earlier version of the paper. 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., Li, JY. Hermitizable, isospectral complex second-order differential operators. Front. Math. China 15, 867–889 (2020). https://doi.org/10.1007/s11464-020-0859-4
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DOI: https://doi.org/10.1007/s11464-020-0859-4