  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.
