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Nonlinear Sciences > Exactly Solvable and Integrable Systems
Title: Screwon spectral statistics and dispersion relation in the quantum Rajeev-Ranken model
(Submitted on 20 Dec 2023 (v1), last revised 26 Apr 2024 (this version, v2))
Abstract: The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves (screwons) of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model based on a quadratic Hamiltonian on a nilpotent/Euclidean Poisson algebra is Liouville integrable. Upon adopting canonical variables in a slightly extended phase space, the model was interpreted as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. Here, we examine the spectral statistics and dispersion relation of quantized screwons via numerical diagonalization validated by variational and perturbative approximations. We also derive a semiclassical estimate for the cumulative level distribution which compares favorably with the one from numerical diagonalization. The spectrum shows level crossings typical of an integrable system. The $i^{\rm th}$ unfolded nearest neighbor spacings are found to follow Poisson statistics for small $i$. Nonoverlapping spacing ratios also indicate that successive spectral gaps are independently distributed. After displaying universal linear behavior over energy windows of short lengths, the spectral rigidity saturates at a length and value that scales with the square-root of energy. For strong coupling $\lambda$ and intermediate $k$, we argue that reduced screwon energies can depend only on the product $\lambda k$. Numerically, we find power law dependences on $\lambda$ and $k$ with an approximately common exponent $2/3$ provided the angular momentum quantum number $l$ is small compared to the number of nodes $n$ in the radial wavefunction. On the other hand, for the ground state $n = l = 0$, the common exponent becomes 1.
Submission history
From: G. S. Krishnaswami [view email][v1] Wed, 20 Dec 2023 15:41:13 GMT (564kb,D)
[v2] Fri, 26 Apr 2024 11:13:30 GMT (566kb,D)
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