Periodic orbits for classical particles having complex energy

Anderson, Alexander G.; Bender, Carl M.; Morone, Uriel

doi:10.1016/j.physleta.2011.07.051

Cited by 18 publications

(25 citation statements)

References 32 publications

(37 reference statements)

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“…and make contact with the model studied in Anderson et al [48]. As explained in Cavaglia et al [18], integrating (4.6) twice with respect to ζ , we obtain with integration constants κ 1 , κ 2 ∈ C. Identifying u → x and ζ → t for the travelling wave equation, together with the constraints…”

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confidence: 63%

PT -symmetric deformations of integrable models

Fring

2013

Phil. Trans. R. Soc. A.

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We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and non-linear integrable field equations of Korteweg-de-Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models we provide three alternative deformations: A complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real valued field equations of non linear integrable systems and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of KdV-type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.Comment: 21 pages, 3 figure

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confidence: 63%

PT -symmetric deformations of integrable models

Fring

2013

Phil. Trans. R. Soc. A.

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“…The quantum‐classical transitions in complex space for model systems, including the free Gaussian wave packet, the harmonic oscillator, and the Morse potential, have been analyzed using complex transition trajectories. In addition, complex classical trajectories have attracted considerable interest due to similarities between complex classical mechanics and quantum mechanics in the last decade . It has been shown that many quantum features can be reproduced in the context of complex classical mechanics.…”

Section: Discussionmentioning

confidence: 92%

Quantum‐classical transition equation with complex trajectories

Chou

2016

Int J of Quantum Chemistry

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A quantum‐classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex‐valued quantum potential term from the complex‐extended time‐dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum‐classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum‐classical transition of physical systems in complex space.

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“…While complex classical mechanics is a relatively new field there have already been many papers published on the properties of complex phase space . Most of these studies have focused on Hamiltonians of the form

H (x, p) = p^{2} + V (x)

.…”

Section: Introductionmentioning

confidence: 99%

Complex Classical Motion in Potentials with Poles and Turning Points

Bender

Hook

2014

Stud Appl Math

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Complex trajectories for Hamiltonians of the form H = p n + V (x) are studied. For n = 2 timereversal symmetry prevents trajectories from crossing. However, for n > 2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V (x) have the opposite effect -they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.

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Periodic orbits for classical particles having complex energy

Cited by 18 publications

References 32 publications

PT -symmetric deformations of integrable models

PT -symmetric deformations of integrable models

Quantum‐classical transition equation with complex trajectories

Complex Classical Motion in Potentials with Poles and Turning Points

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