Generalized Liénard systems and isochronous connections

Abstract

In this paper, we explore some classical and quantum aspects of the nonlinear Liénard equation $\ddot{x}+kx\dot{x}+{\omega }^{2}x+(k^2/9)x^3=0$, where $x=x(t)$ is a real variable and $k,\omega \in \mathbb{R}$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $\ddot{z}+J(z){\dot{z}}^2+F(z)\dot{z}+G(z)=0$, where $z=z(t)$ is a real variable and $\{J(z),F(z),G(z)\}$ are suitable functions to be specified. It can further be mapped to the harmonic oscillator by making use of a nonlocal transformation, establishing its isochronicity. Computations employing the Jacobi last multiplier reveal that the system exhibits a bi-Hamiltonian character, i.e., there are two distinct types of Hamiltonians describing the system. For each of these, we perform a canonical quantization in the momentum representation and explore the possibility of bound states. While one of the Hamiltonians is seen to exhibit an equispaced spectrum with an infinite tower of states, the other one exhibits branching but can be solved exactly in closed form for certain choices of the parameters.

arXiv commentv1: Comments are welcome; v2: To appear in IJTP; v3: This version includes the acknowledgments section which was missing in the previous version

References in corpus (19)

• Supersymmetry and Quantum Mechanics
• Quantum Time Crystals
• Exact solvability of potentials with spatially dependent effective masses
• A general scheme for the effective-mass Schrodinger equation and the generation of the associated potentials
• Hamiltonian for the zeros of the Riemann zeta function
• Ordering ambiguity revisited via position dependent mass pseudo-momentum operators
• Quantum particles trapped in a position-dependent mass barrier; a d-dimensional recipe
• Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems
• The quantum harmonic oscillator on the sphere and the hyperbolic plane
• Analytic results in the position-dependent mass Schrodinger problem
• Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator
• Remarks on the Solution of the Position Dependent Mass (PDM) Schrödinger Equation
• The Berry-Keating Hamiltonian and the Local Riemann Hypothesis
• Evolution profiles and functional equations
• Ladder operators for the Ben Daniel-Duke Hamiltonians and their SUSY partners
• Exploring Branched Hamiltonians for a Class of Nonlinear Systems
• So (2, 1) algebra, local Fermi velocity, and position-dependent mass Dirac equation
• A reappraisal of Lagrangians with non-quadratic velocity dependence and branched Hamiltonians
• Pseudo-Hermitian extensions of the harmonic and isotonic oscillators

Cited by in corpus (1)

• Quasi-harmonic spectra from branched Hamiltonians