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.
Similar content being viewed by others
References
Bagarello F, Passante R Trapani C. Non-Hermitian Hamiltonians in Quantum Physics. Springer Proc Phys, Vol 184. Heidelberg: Springer, 2016
Bender C M, Boettcher S. Real spectra in non-Hermitian Hamiltonians having \(\mathscr{P}\mathscr{T}\) symmetry. Phys Rev Lett, 1998, 80: 5243–5246
Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004
Chen M F. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309
Chen M F. Hermitizable, isospectral complex matrices or differential operators. Front Math China, 2018, 13(6): 1267–1311
Chen M F. On spectrum of Hermitizable tridiagonal matrices. Front Math China, 2020, 15(2): 285–303
Chen M F, Zhang X. Isospectral operators. Commun Math Stat, 2014, 2: 17–32
Dorey P, Dunning C, Tateo R. Spectral equivalences, Bethe ansatz equations, and reality properties in \(\mathscr{P}\mathscr{T}\)-symmetric quantum mechanics. J Phys A: Math Gen, 2001, 34: 5679–5704
Li J Y. Hermitizability of complex elliptic operators. Master’s thesis at Beijing Normal University, 2020 (in Chinese)
Moiseyev N. Non-Hermitian Quantum Mechanics. Cambridge: Cambridge Univ Press, 2011
Mostafazadeh A. Physics of Spectral Singularities. In: Kielanowski P, Bieliavsky P, Odzijewicz A, Schlichenmaier M, Voronov T, eds. Geometric Methods in Physics. Trends Math. Cham: Springer, 2015, 145–165
Polyanin A D, Zaitsev V F. Handbook of Exact Solutions for Ordinary Differential Equations. 2nd ed. Boca Raton: Chapman & Hall/CRC, 2003
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-020-0859-4