Rigorous upper bound for the persistent current in systems with toroidal geometry

It is shown that the absolute value of the persistent current in a system with toroidal geometry is rigorously less than or equal to $e \hbar N /4 \pi m r_0^2$, where $N$ is the number of electrons, and $r_0^{-2} = \langle r_i^{-2}\rangle$ is the equilibrium average of the inverse of the square of the distance of an electron from an axis threading the torus. This result is valid in three and two dimensions for arbitrary interactions, impurity potentials, and magnetic fields.