Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator

Details Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator

2011-01-05

arxiv.org/abs/1101.1040v1

Journal of Physics A 44 (2011) 062001 (Fast Track Communication)

Physics

Quantum Physics

Accepted in Journal of Physics A. Comments are welcome

Scientific paper

The generalized Swanson Hamiltonian $H_{GS} = w (\tilde{a}\tilde{a}^\dag+ 1/2) + \alpha \tilde{a}^2 + \beta \tilde{a}^{\dag^2}$ with $\tilde{a} = A(x)d/dx + B(x)$, 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 $[\tilde{a},\tilde{a}^\dag]=$ constant. However, the main objective of this paper is to show that though the commutator of $\tilde{a}$ and $\tilde{a}^\dag$ 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 $A(x)$ as the inverse square root of the mass function.

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