A one-to-one correspondence is known to exist between the spectra of the discrete states of the non-Hermitian Swanson-type Hamiltonian ) and an equivalent Hermitian Schrödinger Hamiltonian h , the two Hamiltonians being related through a similarity transformation. In this work, we consider the continuum states of h , and examine the nature of the corresponding states of H .
A one-to-one correspondence is known to exist between the spectra of the discrete states of the non-Hermitian Swanson-type Hamiltonian ) and an equivalent Hermitian Schrödinger Hamiltonian h , the two Hamiltonians being related through a similarity transformation. In this work, we consider the continuum states of h , and examine the nature of the corresponding states of H .
The generalized Swanson Hamiltonian with can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as constant. However, the main objective of this communication is to show that though the commutator of and is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. The reason for this anomaly is discussed in the framework of position-dependent mass models by choosing A ( x ) as the inverse square root of the mass function.
In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT-invariant Hamiltonian, with potential V = igx 3. In this case the corresponding h, which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P. 1
In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h , obtained from H by a similarity transformation, as pointed out by Mostafazadeh. In the first model, due to Swanson, h turns out to be just a scaled harmonic oscillator, which explains the form of its spectrum. However, the transformation is not unique, which also means that the observables of the original theory are not uniquely determined by H alone. The second model we consider is the original PT -invariant Hamiltonian, with potential V = i gx 3 . In this case the corresponding h , which we are only able to construct in perturbation theory, corresponds to a complicated velocity-dependent potential. We again explore the relationship between the canonical variables x and p and the observables X and P .
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.
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)