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Non-Hermitian Hamiltonians with real and complex eigenvalues: An sl(2,C) approach

Bijan Kumar Bagchi, C. Quesne

Potential algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the complex Lie algebra Ai.

arXiv:math-ph/0209031CRC Press eBooksNov 30, 20030 citations

sl(2, ) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues

Bijan Kumar Bagchi, C. Quesne

arXiv:math-ph/0008020Physics Letters ASep 1, 2000193 citations

Lie Algebraic Approach to Non-Hermitian Hamiltonians

Sanjib Meyur, S. Debnath, Rabindra Sarani

The Lie-algebraic approach has been used to obtain both real and complex eigenvalues of non-Hermitian Hamiltonians, corresponding to regular eigenfunctions.

Jan 1, 20090 citations

Exponential Type Complex and Non-Hermitian Potentials in PT-Symmetric Quantum Mechanics

Özlem Yeşiltaş, Mehmet Şimşek, R. Sever +1 more

Using the NU method [A F Nikiforov and V B Uvarov 1988 “Special Functions of Mathematical Physics” (Birkhauser, Basel)], we investigated the real eigenvalues of the complex and/or PT-symmetric, non-Hermitian and the exponential type systems, such as Pöschl–Teller and Morse potentials.

arXiv:hep-ph/0303014Physica ScriptaJan 1, 200352 citations

Non-Hermitian Hamiltonians of Lie algebraic type

Paulo E G Assis, Andreas Fring

We analyse a class of non-Hermitian Hamiltonians, which can be expressed in terms of bilinear combinations of generators in the -Lie algebra or their isomorphic su (1, 1)-counterparts. The Hamiltonians 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 Hamiltonians. Alternatively, we employ a generalized Bogoliubov transformation, which allows us to compute explicitly real-energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.We analyse a class of non-Hermitian Hamiltonians, which can be expressed in terms of bilinear combinations of generators in the -Lie algebra or their isomorphic su (1, 1)-counterparts. The Hamiltonians 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 Hamiltonians. Alternatively, we employ a generalized Bogoliubov transformation, which allows us to compute explicitly real-energy eigenvalue spectra for these type of Hamiltonians, together with their eigenstates. We compare the two approaches.

arXiv:0804.4677Journal of Physics A Mathematical and TheoreticalNov 19, 200840 citations

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

Carla Figueira de Morisson Faria, A. Fring

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.

arXiv:quant-ph/0609096Laser PhysicsApr 1, 200748 citations

Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework

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Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework

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