Real Description of Classical Hamiltonian Dynamics Generated by a Complex Potential

Abstract

Analytic continuation of the classical dynamics generated by a standard Hamiltonian, H = p^2/2m + v(x), into the complex plane yields a particular complex classical dynamical system. For an analytic potential v, we show that the resulting complex system admits a description in terms of the phase space R^4 equipped with an unconventional symplectic structure. This in turn allows for the construction of an equivalent real description that is based on the conventional symplectic structure on R^4, and establishes the equivalence of the complex extension of classical mechanics that is based on the above-mentioned analytic continuation with the conventional classical mechanics. The equivalent real Hamiltonian turns out to be twice the real part of H, while the imaginary part of H plays the role of an independent integral of motion ensuring the integrability of the system. The equivalent real description proposed here is the classical analog of the equivalent Hermitian description of unitary quantum systems defined by complex, typically PT-symmetric, potentials.

9 pages, slightly revised published version with updated references

References in corpus (12)

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Cited by in corpus (13)

• On complexified mechanics and coquaternions
• A Hamiltonian Formulation of the Pais-Uhlenbeck Oscillator that Yields a Stable and Unitary Quantum System
• Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report
• A return to observability near exceptional points in a schematic PT-symmetric model
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• Quantum and Classical Statistical Mechanics of a Class of non-Hermitian Hamiltonians
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• PT-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
• Integrable models from PT-symmetric deformations
• Metric operators for non-Hermitian quadratic su(2) Hamiltonians
• Crypto-Harmonic Oscillator in Higher Dimensions: Classical and Quantum Aspects
• Hidden Gauge Symmetry in Holomorphic Models
• A Differential Integrability Condition for Two-Dimensional Hamiltonian Systems