Interference of Non-Hermiticity with Hermiticity at Exceptional Points
Abstract
:1. Introduction
2. Non-Hermitian and Hermitian Operators in Interaction
2.1. Motivation: The Access to EPs in Quantum Physics
2.2. Paradox of Non-Locality of Complex Delta-Function Interactions
3. Finite-Dimensional Toy-Model Hamiltonians
3.1. Partitioned Structure of the Relevant Hamiltonians
3.2. -Symmetry Requirement and Reparametrization
- [A] T elements of H lying on the main diagonal will form a real and equidistant sequence simulating the spectrum of the most common harmonic oscillator;
- [B] As long as a broad class of general matrices can be routinely tridiagonalized, we assume that all of our Hs are tridiagonal. Moreover, for the methodical reasons formulated in [49], their off-diagonal part was chosen to be antisymmetric;
4. Exceptional Points
4.1. Illustrative Example
4.2. Hamiltonians in the EP Limit
5. Hamiltonians in the Vicinity of EPs
5.1. The Perturbed Schrödinger Equation
5.2. Leading-Order Solution
5.3. Fine-Tuned anomaly in an Illustrative Example
6. Step-by-Step Hermitizations in the Generic Case
6.1. Elementary Generic Model
6.2. Linear Unfoldings at Arbitrary
6.3. Illustrative Model with
7. Summary
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Quantum Mechanics Using Non-Hermitian Operators
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… | − | |||||
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− | … | |||||
M | − | 1 | 2 | … | J | |
K | J | … | 1 | − |
t | 0 | 7 | 12 | 15 | 16 |
---|---|---|---|---|---|
− | 1 | 2 | 3 | 4 | |
EP(2K) | EP(8) | EP(6) | EP(4) | EP(2) | − |
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Znojil, M. Interference of Non-Hermiticity with Hermiticity at Exceptional Points. Mathematics 2022, 10, 3721. https://doi.org/10.3390/math10203721
Znojil M. Interference of Non-Hermiticity with Hermiticity at Exceptional Points. Mathematics. 2022; 10(20):3721. https://doi.org/10.3390/math10203721
Chicago/Turabian StyleZnojil, Miloslav. 2022. "Interference of Non-Hermiticity with Hermiticity at Exceptional Points" Mathematics 10, no. 20: 3721. https://doi.org/10.3390/math10203721