Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator

Abstract

We carry out an exact quantization of a PT symmetric (reversible) Liénard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schrödinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.

10 pages, 1 figure, Fast Track Communication

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• Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
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• Exploring Branched Hamiltonians for a Class of Nonlinear Systems
• On the symmetries of a nonlinear non-polynomial oscillator
• Generalized Uncertainty principle and momentum-dependent effective mass Schrodinger equation
• On the linearization of isochronous centre of a modified Emden equation with linear external forcing
• Liénard Type Nonlinear Oscillators and Quantum Solvability
• A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
• Two-dimensional isochronous nonstandard Hamiltonian systems
• Generalized Liénard systems and isochronous connections
• Quantization of nonlinear non-Hamiltonian systems