$T\bar T$ deformation of Calogero-Sutherland model via dimensional reduction

We perform the dynamical change of coordinates to derive a generalization of the trace relation and apply it to the non-linear Schr\"odinger model. After that, we work out the dimensional reduction of the bilinear $T\bar T$ operator and obtain the new $T\bar T$-like deformation of the quantum mechanics of free non-relativistic fermions and interacting Calogero-Sutherland particles. The deformation modifies the spectrum of the Hamiltonian but does not alter its eigenfunctions. The deformed classical Lagrangian is also obtained. Finally, we study a particular deformation of the two-dimensional Yang-Mills theory that maps the gauge theory onto a system of $T\bar T$-perturbed fermions.