|
|
|
Hermitizable, isospectral complex second-order differential operators |
Mu-Fa CHEN1,2,3( ), Jin-Yu LI2 |
1. Research Institute of Mathematical Science, Jiangsu Normal University, Xuzhou 221116, China
2. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3. Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China |
|
|
|
|
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
|
|
Corresponding Author(s):
Mu-Fa CHEN
|
|
Issue Date: 19 November 2020
|
|
| 1 |
F Bagarello, R Passante C Trapani. Non-Hermitian Hamiltonians in Quantum Physics. Springer Proc Phys, Vol 184. Heidelberg: Springer, 2016
https://doi.org/10.1007/978-3-319-31356-6
|
| 2 |
C M Bender, S Boettcher. Real spectra in non-Hermitian Hamiltonians having P T symmetry. Phys Rev Lett, 1998, 80: 5243–5246
https://doi.org/10.1103/PhysRevLett.80.5243
|
| 3 |
M F Chen. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004
https://doi.org/10.1142/5513
|
| 4 |
M F Chen. Criteria for discrete spectrum of 1D operators. Commun Math Stat, 2014, 2: 279–309
https://doi.org/10.1007/s40304-014-0041-y
|
| 5 |
M F Chen. Hermitizable, isospectral complex matrices or differential operators. Front Math China, 2018, 13(6): 1267–1311
https://doi.org/10.1007/s11464-018-0716-x
|
| 6 |
M F Chen. On spectrum of Hermitizable tridiagonal matrices. Front Math China, 2020, 15(2): 285–303
https://doi.org/10.1007/s11464-020-0832-2
|
| 7 |
M F Chen, X Zhang. Isospectral operators. Commun Math Stat, 2014, 2: 17–32
https://doi.org/10.1007/s40304-014-0028-8
|
| 8 |
P Dorey, C Dunning, R Tateo. Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics. J Phys A: Math Gen, 2001, 34: 5679–5704
https://doi.org/10.1088/0305-4470/34/28/305
|
| 9 |
J Y Li. Hermitizability of complex elliptic operators. Master’s thesis at Beijing Normal University, 2020 (in Chinese)
|
| 10 |
N Moiseyev. Non-Hermitian Quantum Mechanics. Cambridge: Cambridge Univ Press, 2011
|
| 11 |
A Mostafazadeh. 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 |
A D Polyanin, V F Zaitsev. Handbook of Exact Solutions for Ordinary Differential Equations. 2nd ed. Boca Raton: Chapman & Hall/CRC, 2003
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
