Non-Hermitian Hamiltonians of Lie algebraic type

Abstract

We analyse a class of non-Hermitian Hamiltonians, which can be expressed bilinearly in terms of generators of a sl(2,R)-Lie algebra or their isomorphic su(1,1)-counterparts. The Hamlitonians are prototypes for solvable models of Lie algebraic type. Demanding a real spectrum and the existence of a well defined metric, we systematically investigate the constraints these requirements impose on the coupling constants of the model and the parameters in the metric operator. We compute isospectral Hermitian counterparts for some of the original non-Hermitian Hamiltonian. Alternatively we employ a generalized Bogoliubov transformation, which allows to compute explicitly real energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.

27 pages

References in corpus (24)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
• Quantum Solitons in Affine Toda Field Theories
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Time evolution of non-Hermitian Hamiltonian systems
• $\mu -e$ conversion in nuclei within the CMSSM seesaw: universality versus non-universality
• Closed formula for the metric in the Hilbert space of a PT-symmetric model
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Delta-Function Potential with a Complex Coupling
• Semiclassical quantization of an N-particle Bose-Hubbard model
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Construction of a unique metric in quasi-Hermitian quantum mechanics: non-existence of the charge operator in a 2 x 2 matrix model
• Non-Hermitian oscillator Hamiltonian and su(1,1): a way towards generalizations
• A PT Symmetric QES Partner to the Khare Mandal Potential With Real Eigen Values
• Isospectral Hamiltonians from Moyal products
• Moyal products -- a new perspective on quasi-hermitian quantum mechanics
• Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• A complex periodic QES potential and exceptional points
• The S-matrix of the Faddeev-Reshetikhin Model, Diagonalizability and PT Symmetry
• sl(2,R) symmetry and solvable multiboson systems

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