• Carl M Bender

      Articles written in Pramana – Journal of Physics

    • Conduction bands in classical periodic potentials

      Tanwa Arpornthip Carl M Bender

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      The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time to perform the measurement. This paper considers the possibility that $\Delta E$, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behaviour of a classical particle in a one-dimensional periodic potential $V(x) = − \cos(x)$ is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviours: (i) The particle may hop from classically allowed site to nearest-neighbour classically allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunnelling processes or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunnelling. The classical conduction bands for this potential are determined numerically with high precision.

    • Compactons in $\mathcal{PT}$-symmetric generalized Korteweg–de Vries equations

      Carl M Bender Fred Cooper Avinash Khare Bogdan Mihaila Avadh Saxena

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      This paper considers the $\mathcal{PT}$-symmetric extensions of the equations examined by Cooper, Shepard and Sodano. From the scaling properties of the $\mathcal{PT}$-symmetric equations a general theorem relating the energy, momentum and velocity of any solitary-wave solution of the generalized KdV equation is derived. We also discuss the stability of the compacton solution as a function of the parameters affecting the nonlinearities.

    • Chaotic systems in complex phase space

      Carl M Bender Joshua Feinberg Daniel W Hook David J Weir

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      This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviours of these two $\mathcal{PT}$ -symmetric dynamical models in complex phase space exhibit strong qualitative similarities.

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