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arXiv:0806.2982

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

Omar Mustafa, S. Habib Mazharimousavi

Published Jun 18, 2008• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0806.2982

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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.
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.
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.
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.
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)
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian PT-symmetric part, 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 non-Hermitian (not necessarily PT-symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Non-Hermitian oscillator Hamiltonian and su(1,1)