On the group-theoretical approach to energy quantization of a perturbed vortex ring: Spectrum calculating in the pipe-type domain

In this study, the problem of the energy spectrum of a quantum vortex loop moving in a thin long pipe is solved for the first time. We quantize this dynamic system using a new method, which leads to non-trivial results for circulation Γ and energy values E. It is shown that the spectrum has a quasi-continuous fractal structure. In the final form, we present the spectrum of the vortex loop in the form of a “Regge trajectory” E=E(Γ). The vortex quantization problem is considered outside of two-fluid hydrodynamics and other conventional approaches. We also discuss ways to improve the model, which could allow us to apply the results we have obtained to describe a quantum turbulent flow.