Quantum Rajeev-Ranken model as an anharmonic oscillator

TL;DR

This paper analyzes the quantum and classical properties of the Rajeev-Ranken model, revealing its structure as an anharmonic oscillator, deriving energy spectra, and exploring its algebraic and integrable features.

Contribution

It introduces a novel interpretation of the RR model as a 3D anharmonic oscillator and provides detailed quantum spectra and algebraic structures, including a reducible nilpotent Lie algebra representation.

Findings

Energy spectrum depends on the scaling variable λk.

Dispersion relation for highly energetic states scales as (λk)^{2/3}.

Radial equation is a complex recurrence relation beyond classical special functions.

Abstract

The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schr\"odinger equation, we find that the radial equation has a four-term recurrence relation. It is of type $[0,1,1_6]$ and lies beyond the ellipsoidal Lam\'e and Heun equations in Ince's classification. At strong coupling $\lambda$, the energies of highly excited states are shown to depend on the scaling variable $\lambda k$. The energy spectrum at weak coupling and its dependence on wavenumber $k$ in a double-scaling strong coupling limit are obtained. The semi-classical WKB quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a $(\lambda k)^{2/3}$ dispersion relation for highly energetic quantized 'screwons' at moderate and strong coupling. We also suggest a mapping between our radial equation and one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.