Dimensional reduction for a system of 2D anyons

Anyons with a statistical phase parameter \(\alpha \in (0,2)\) are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave guide. More specifically, we study the 2D magnetic gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the y-direction than in the x-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the x-direction is given by the impenetrable Tonks–Girardeau Bose gas, which has no dependency on \(\alpha \), and no trace left of the long-range interactions of the 2D model.