Sum rule for the optical spectrum of a trapped gas

We develop an exact sum rule that relates the spectral shift of a trapped gas undergoing cold collisions to measurable quantities of the system. The method demonstrates the dependence of the cold collision frequency shift on the quantum degeneracy of the gas and facilitates extracting scattering lengths from the data. To illustrate the method, we consider optical shift for the Thomas-Fermi model of Bose-Einstein condensate in a harmonic optical trap.