  The superconductivity in very thin rings is suppressed by quantum phase
slips. As a result the amplitude of the persistent current oscillations with
flux becomes exponentially small, and their shape changes from sawtooth to a
sinusoidal one. We reduce the problem of low-energy properties of a
superconducting nanoring to that of a quantum particle in a sinusoidal
potential and show that the dependence of the current on the flux belongs to a
one-parameter family of functions obtained by solving the respective
Schrodinger equation with twisted boundary conditions.
