Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures
Abstract
:1. Introduction
2. Generalized Oscillators
- (I)
- Harmonic oscillator: Making and , the operator (1) is reduced to the well known Hamiltonian of the harmonic oscillator:
- (II)
- Hermitian configuration: For , and such that , with the complex conjugate of , the operator is Hermitian. The harmonic oscillator Hamiltonian (2) is properly included in this class.
- (III)
- Global non-Hermitian configuration: In general, for arbitrary complex-valued functions , , and , the operator (1) is non-Hermitian. The two cases mentioned above are therefore relevant subclasses of this configuration.
- (IV)
- Non-Hermitian configuration: A subset of the global non-Hermitian class is characterized by real coefficients , , and . This includes the harmonic oscillator as well as a subset of the Hermitian classes.
- Then, (4) is written in the self-adjoint form:Clearly, there is a one-to-one correspondence between the sets and . Thus, by fixing the parameters in , one can determine the parameters in , and vice versa.
- Non-Hermitian configuration: For real coefficients , , and , Equations (4)–(7) yield such that . The non-Hermiticity is due to the real-valued functions and v, which may be cancelled by making . Noticeably, the latter case is consistent with the Hermitian configuration mentioned in the previous item after making . For and , the appropriate transformation shows that, providing , the self-adjoint operator coincides with .
2.1. Space of Solutions
2.2. Time-Dependent Model with Non-Hermiticity
3. Applications
3.1. Classical Picture
- Time-dependent mass. For , we may introduce the variable to getWe can immediately identify that the homogeneous part of (35) coincides with the Bessel differential equations [55]. In this case, we use the solutions
3.2. Quantum Picture
- For the constant mass case, we use the homogeneous solutions and their respective Wronskian as
3.3. Hermitian Conjugate and Bi-Orthogonality
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Point Transformations
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Zelaya, K.; Rosas-Ortiz, O. Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures. Quantum Rep. 2021, 3, 458-472. https://doi.org/10.3390/quantum3030030
Zelaya K, Rosas-Ortiz O. Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures. Quantum Reports. 2021; 3(3):458-472. https://doi.org/10.3390/quantum3030030
Chicago/Turabian StyleZelaya, Kevin, and Oscar Rosas-Ortiz. 2021. "Exact Solutions for Time-Dependent Non-Hermitian Oscillators: Classical and Quantum Pictures" Quantum Reports 3, no. 3: 458-472. https://doi.org/10.3390/quantum3030030