Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, RR Liouville integrable. Here, we interpret as novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. quantum has two dimensionless parameters. Upon separating variables Schrödinger equation, find that radial equation four-term recurrence relation. It type [0, 1, 6 ] and lies beyond ellipsoidal Lamé Heun equations Ince’s classification. At strong coupling λ, energies highly excited states are shown depend on scaling variable λk. energy spectrum at weak its dependence double-scaling limit obtained. semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition expressed terms elliptic integrals. Numerical inversion enables us establish ( λk) 2/3 dispersion relation for energetic quantized “screwons” moderate coupling. We also suggest mapping between our one Zinn-Justin Jentschura could facilitate resurgent WKB expansion levels. In another direction, show motion can be viewed Euler step-3 nilpotent Lie algebra. use canonical uncover infinite dimensional reducible unitary representation this algebra, which then decomposed using Casimir operators.