Spectral equivalences, Bethe ansatz equations, and reality properties in 𝒫𝒯-symmetric quantum mechanics

Dorey, Patrick; Dunning, Clare; Tateo, Roberto

doi:10.1088/0305-4470/34/28/305

Cited by 530 publications

(660 citation statements)

References 40 publications

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“…Here, P denotes the spatial reflection p → −p and x → −x, and T denotes the time reversal p → −p, x → x, and i → −i. Many PT -symmetric quantum-mechanical Hamiltonians have been studied in the recent literature [3,4,5,6]. However, the Hamiltonian (1) is special because when g = 0 the boundary conditions on the eigenfunctions may be imposed on the real-x axis, as opposed to the interior of a wedge in the complex-x plane, as we will now show: The quantization condition satisfied by the eigenfunctions requires that ψ n (x) must vanish exponentially in a pair of wedges in the complex-x plane.…”

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confidence: 99%

Semiclassical analysis of a complex quartic Hamiltonian

2006

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It is necessary to calculate the C operator for the non-Hermitian PT -symmetric Hamiltonian H = 1 2 p 2 + 1 2 µ 2 x 2 − λx 4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H = 1 2 p 2 + 1 2 µ 2 x 2 + igx 3 − λx 4 , whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter λ.

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confidence: 99%

Semiclassical analysis of a complex quartic Hamiltonian

2006

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“…The Schwinger model may be solved exactly, and exhibits a violation of unitarity, as was discussed in detail in [12]. Thus, it appears that there are problems with unitarity not only in (PT QED) 2,4 , but also in any quantum mechanical system. Analytic properties required by the probability considerations and the Euclidean postulate seem to be generically violated.…”

Section: Discussionmentioning

confidence: 92%

“…Theories [2] and proved in 2001 [3,4]. There remained the question of unitarity or probability conservation, and that was established in the following year [5].…”

Section: Introductionmentioning

confidence: 99%

PT -symmetric quantum electrodynamics and unitarity

Milton

Abalo

Parashar

et al. 2013

Phil. Trans. R. Soc. A.

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More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, . It was shown that, if is unbroken, energies were, in fact, positive, and unitarity was satisfied. Since quantum mechanics is quantum field theory in one dimension—time—it was natural to extend this idea to higher-dimensional field theory, and in fact an apparently viable version of -invariant quantum electrodynamics (QED) was proposed. However, it has proved difficult to establish that the unitarity of the scattering matrix, for example, the Källén spectral representation for the photon propagator, can be maintained in this theory. This has led to questions of whether, in fact, even quantum mechanical systems are consistent with probability conservation when Green’s functions are examined, since the latter have to possess physical requirements of analyticity. The status of QED will be reviewed in this paper, as well as the general issue of unitarity.

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“…The spectrum of H c is shown in Figure 1. It was shown in [DDT01] that the spectrum of H c is real and positive. Moreover, H c is closed and has compact resolvent [CGM80,Mez01] so the spectrum is also discrete.…”

Section: Non-selfadjoint Operators and Pseudospectramentioning

confidence: 99%

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

Dondl

Dorey

Rösler

2016

Appl Math Res Express

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We are concerned with the non-normal Schrödinger operatorThe spectrum of this operator is discrete and contained in the positive half plane. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense.By exploiting the fact that the semigroup e −tH is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum{z : |z − λ| < δ}.In particular, the unbounded part of the pseudospectrum escapes towards +∞ as ε decreases.Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.Mathematics Subject Classification (2010): 35P99, 47B44.

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Spectral equivalences, Bethe ansatz equations, and reality properties in 𝒫𝒯-symmetric quantum mechanics

Cited by 530 publications

References 40 publications

Semiclassical analysis of a complex quartic Hamiltonian

Semiclassical analysis of a complex quartic Hamiltonian

PT -symmetric quantum electrodynamics and unitarity

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

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