Hermitizable, isospectral complex second-order differential operators

Abstract

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.

Keywords

Hermitizable / isospectral / differential operators / non-Hermitian model / discrete spectrum

References

[1] Bagarello F, Passante RTrapani C. Non-Hermitian Hamiltonians in Quantum Physics. Springer Proc Phys, Vol 184. Heidelberg: Springer, 2016

[2] Bender C M, Boettcher S. Real spectra in non-Hermitian Hamiltonians having P T symmetry. Phys Rev Lett, 1998, 80: 5243–5246

[3] Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004

[4] Chen M F. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309

[5] Chen M F. Hermitizable, isospectral complex matrices or differential operators. Front Math China, 2018, 13(6): 1267–1311

[6] Chen M F. On spectrum of Hermitizable tridiagonal matrices. Front Math China, 2020, 15(2): 285–303

[7] Chen M F, Zhang X. Isospectral operators. Commun Math Stat, 2014, 2: 17–32

[8] Dorey P, Dunning C, Tateo R. Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. J Phys A: Math Gen, 2001, 34: 5679–5704

[9] Li J Y. Hermitizability of complex elliptic operators. Master’s thesis at Beijing Normal University, 2020 (in Chinese)

[10] Moiseyev N. Non-Hermitian Quantum Mechanics. Cambridge: Cambridge Univ Press, 2011

[11] 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

[12] Polyanin A D, Zaitsev V F. Handbook of Exact Solutions for Ordinary Differential Equations. 2nd ed. Boca Raton: Chapman & Hall/CRC, 2003

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