Quasi-Hermitian quantum mechanics in phase space

@article{Curtright2007QuasiHermitianQM,
  title={Quasi-Hermitian quantum mechanics in phase space},
  author={Thomas L. Curtright and Andrzej Veitia},
  journal={Journal of Mathematical Physics},
  year={2007},
  volume={48},
  pages={102112-102112}
}
We investigate quasi-Hermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phase-space metrics, and explore the relationships between them. 
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References

SHOWING 1-10 OF 29 REFERENCES
Moyal products -- a new perspective on quasi-hermitian quantum mechanics
The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for
Pseudo-Hermiticity, symmetry, and the metric operator
The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction with the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and
On the regular Hilbert space representation of a Moyal quantization
It is shown that in the regular viz. phase space representation of a Moyal quantization real polynomials in the phase space variables become essentially self‐adjoint operators, and that functions in
Introduction to 𝒫𝒯-symmetric quantum theory
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability
Biorthogonal quantum systems
Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially
Isospectral Hamiltonians from Moyal products
Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more
...
...