  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.
