We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form H = A † A + αA 2 + βA † 2, where α, β are real constants, with α ̸ = β, and A † and A are generalized creation and annihilation operators. Thus these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation ρ, mapping the non-Hermitian Hamiltonian H to a Hermitian one h, is also obtained. It is shown that H and h share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is PT symmetric. PACS: 03.65-w
We analyze a class of non-Hermitian quadratic Hamiltonians, which are of the form H = A † A + αA 2 + βA † 2, where α, β are real constants, with α ̸ = β, and A † and A are generalized creation and annihilation operators. Thus these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation ρ, mapping the non-Hermitian Hamiltonian H to a Hermitian one h, is also obtained. It is shown that H and h share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen-Morse I and the hyperbolic Rosen-Morse II type potentials are obtained. We also study the case when the non-Hermitian Hamiltonian is PT symmetric. PACS: 03.65-w
We analyse a class of non-Hermitian quadratic Hamiltonians, which are of the form , where α, β are real constants, with α ≠ β, and and are generalized creation and annihilation operators respectively. Thus, these Hamiltonians may be classified as generalized Swanson models. It is shown that the eigenenergies are real for a certain range of values of the parameters. A similarity transformation ρ, mapping the non-Hermitian Hamiltonian H to a Hermitian one h , is also obtained. It is shown that H and h share identical energies. As explicit examples, the solutions of a couple of models based on the trigonometric Rosen–Morse I and the hyperbolic Rosen–Morse II type potentials are obtained. We also study the case where the non-Hermitian Hamiltonian is symmetric.
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.
Searching for non-Hermitian, -symmetric Hamiltonians [1] with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first order differential operators. Because the Hamiltonian is pseudo-Hermitian, we have obtained the Hermitian equivalent of which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened and hyperbolic Rosen-Morse II potentials. Using convenient sinilarity transformations, we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics [2].
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 class of non-Hermitian quadratic su (2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realizations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified, the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalized Swanson Hamiltonian ( Assis and Fring 2009 J. Phys. A: Math. Theor . 42 015203 ) are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su (2) system, every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su (2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite-dimensional representations for the su (2) Hamiltonian, as opposed to the su (1, 1) studied before, allows for comparison with explicit diagonalization results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalization of the problem, establishing a simple relation between both approaches.
PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+21)+α(b^2−(b^†)2) where ω and α are real constants, b^ and b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H is pseudo-Hermitian, we have obtained the Hermitian equivalent of H which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. H is called pseudo-Hermitian, if there exists a Hermitian and invertible operator η satisfying H†=ηHη−1. For the Hermitian Hamiltonian h, one can write h=ρHρ−1 where ρ=η is unitary. Using this ρ we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρ is obtained for each potential case.