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arXiv:1101.1040 · Journal of Physics A Mathematical and Theoretical

Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator

Bikashkali Midya, P P Dube, Rajkumar Roychoudhury

Published Jan 17, 2011• 8 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/6/062001

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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.
The generalized Swanson Hamiltonian HGS=w(a~a~†+1/2)+αa~2+βa~†2H_{GS} = w (\tilde{a}\tilde{a}^\dag+ 1/2) + \alpha \tilde{a}^2 + \beta \tilde{a}^{\dag^2}HGS​=w(a~a~†+1/2)+αa~2+βa~†2 with a~=A(x)d/dx+B(x)\tilde{a} = A(x)d/dx + B(x)a~=A(x)d/dx+B(x), 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 [a~,a~†]=[\tilde{a},\tilde{a}^\dag]=[a~,a~†]= constant. However, the main objective of this paper is to show that though the commutator of a~\tilde{a}a~ and a~†\tilde{a}^\daga~† is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. Reason for this anomaly is discussed in the frame work of position dependent mass models by choosing A(x)A(x)A(x) as the inverse square root of the mass function.
A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator ηηη are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.
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.
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 study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
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