Extending PT symmetry from Heisenberg algebra to E2 algebra
arXiv:1009.3236 · doi:10.1007/s10773-010-0511-2
Abstract
The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.
8 pages, 7 figures
References in corpus (12)
- Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
- Making Sense of Non-Hermitian Hamiltonians
- Complex Extension of Quantum Mechanics
- PT-Symmetric Quantum Mechanics
- Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
- Introduction to PT-Symmetric Quantum Theory
- Supersymmetry and the spontaneous breakdown of PT symmetry
- The ODE/IM Correspondence
- Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
- Complex periodic potentials with real band spectra
- Scalar Quantum Field Theory with Cubic Interaction
- The energy spectrum of complex periodic potentials of the Kronig-Penney type
Cited by in corpus (8)
- PT-Symmetric Representations of Fermionic Algebras
- E2-quasi-exact solvability for non-Hermitian models
- PT phase transition in multidimensional quantum systems
- Spontaneous PT-symmetry breaking for systems of noncommutative Euclidean Lie algebraic type
- Non-Hermitian systems of Euclidean Lie algebraic type with real eigenvalue spectra
- WKB Analysis of PT-Symmetric Sturm-Liouville problems
- Stability boundaries of a Mathieu equation having PT symmetry
- Vector Models in PT Quantum Mechanics