© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:0708.2403 · Journal of Physics A Mathematical and Theoretical

Metrics and isospectral partners for the most generic cubic {\cal PT} -symmetric non-Hermitian Hamiltonian

Paulo E G Assis, Andreas Fring

Published Jun 3, 2008• 30 citations

Abstract

We investigate properties of the most general -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten-parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic − x 4 -potential. In addition, it also reduces to various models not considered in the present context, namely the single-site lattice Reggeon model and a transformed version of the massive sextic ± x 6 -potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.

DOI: 10.1088/1751-8113/41/24/244001

Related papers

Swanson's non-Hermitian Hamiltonian and su(1,1): a way towards generalizations

C. Quesne

The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for Swanson’s PT-symmetric Hamiltonian, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized Swanson’s non-Hermitian (not necessarily PT-symmetric) Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Swanson’s non-Hermitian Hamiltonian and su(1,1)

•May 21, 2007•2 citations

Non-Hermitian PT-symmetric and Hermitian Hamiltonians' correspondence: Isospectrality and mass signature

Omar Mustafa, S. Habib Mazharimousavi

A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator $η$ are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.

arXiv:0806.2982•Jun 18, 2008•0 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.

arXiv:0804.4677•Journal of Physics A Mathematical and Theoretical•Nov 19, 2008•40 citations

Exact Metric Operators as the Ground State functions of the Hermitian Conjugates of a Class of Quasi-Hermitian Hamiltonians

Abouzeid M. Shalaby

We generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if $ψ$ is an eigen function of $H$, then $ϕ=ηψ$ is an eigen function of $H^{\dagger}$ with the same eigen value. Thus, knowing any pair $(ψ,ϕ)$ one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the $Q$ operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.

arXiv:0706.2509•Jun 17, 2007•0 citations

Pseudo supersymmetric partners for the generalized Swanson model

Anjana Sinha, Pinaki Roy

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., , where α, β are real constants, with α ≠ β, and and are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H − , and its partner H + , are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians H ± , related to their respective Hermitian counterparts h ± , through the same similarity transformation operator ρ: H ± = ρ −1 h ± ρ. The formalism is applied to the entire class of shape-invariant models.

arXiv:0806.4490•Journal of Physics A Mathematical and Theoretical•Jul 22, 2008•19 citations

A two-parameter family of non-Hermitian Hamiltonians with real spectrum

Ebru Ergün

We introduce a Hamiltonian H α, β depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H α, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT -symmetric. In the case when H α, β is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H α, β Hermitian. In a particular case we calculate the Hermitian counterpart of H α, β .

Journal of Physics A Mathematical and Theoretical•Oct 22, 2010•5 citations

We investigate properties of the most general -symmetric non-Hermitian Hamiltonian of cubic order in the annihilation and creation operators as a ten-parameter family. For various choices of the parameters we systematically construct an exact expression for a metric operator and an isospectral Hermitian counterpart in the same similarity class by exploiting the isomorphism between operator and Moyal products. We elaborate on the subtleties of this approach. For special choices of the ten parameters the Hamiltonian reduces to various models previously studied, such as to the complex cubic potential, the so-called Swanson Hamiltonian or the transformed version of the from below unbounded quartic − x 4 -potential. In addition, it also reduces to various models not considered in the present context, namely the single-site lattice Reggeon model and a transformed version of the massive sextic ± x 6 -potential, which plays an important role as a toy model to identify theories with vanishing cosmological constant.

The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for Swanson’s PT-symmetric Hamiltonian, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized Swanson’s non-Hermitian (not necessarily PT-symmetric) Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Swanson’s non-Hermitian Hamiltonian and su(1,1)

A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator

η

are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.

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 generalized a class of non-Hermitian Hamiltonians which introduced previously by us in such a way in which every member in the class is non-\textit{PT}-symmetric. For every member of the class, the ground state is a constant with zero energy eigen value. Instead of using an infinite set of coupled operator equations to calculate the metric operator we used a simple realization to obtain the class of closed form metric operators corresponding to the class of non-Hermitian and non-\textit{PT}-symmetric Hamiltonians introduced. The trick is that, if

ψ

is an eigen function of

H

, then

ϕ=ηψ

is an eigen function of

H^{\dagger}

with the same eigen value. Thus, knowing any pair

(ψ,ϕ)

one can deduce the form of the exact metric operator. We note that, the class of Hamiltonians generalized in this work has the form of that of imaginary magnetic field which can be absorbed by the quasi-gauge transformations represented by metric operators. Accordingly, it is expected that the

Q

operators will disappear for the whole members in the class in the path integral formulation. However, the detailed analysis of this issue will appear in another work.

New non Hermitian Hamiltonians are generated, as isospectral partners of the generalized Swanson model, viz., , where α, β are real constants, with α ≠ β, and and are generalized creation and annihilation operators. It is shown that the initial Hamiltonian H − , and its partner H + , are related by pseudo supersymmetry, and they share all the eigen energies except for the ground state. This pseudo supersymmetric extension enlarges the class of non Hermitian Hamiltonians H ± , related to their respective Hermitian counterparts h ± , through the same similarity transformation operator ρ: H ± = ρ −1 h ± ρ. The formalism is applied to the entire class of shape-invariant models.

We introduce a Hamiltonian H α, β depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H α, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT -symmetric. In the case when H α, β is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H α, β Hermitian. In a particular case we calculate the Hermitian counterpart of H α, β .