  Using sum rules and a new dipole-free sum-over-states expression, we
calculate the fundamental limits of the dispersion of the real and imaginary
parts of all electronic nonlinear-optical susceptibilities. As such, these
general results can be used to study any nonlinear optical phenomena at any
wavelength, making it possible to push both applications and our understanding
of such processes to the limits. These results reveal the ultimate constraints
imposed by nature on our ability to control and use light.
