Cold Collision Frequency Shift of an 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. We apply the method to analyzing spectral data for magnetically trapped hydrogen atoms and determine the value of the $1S-2S$ scattering length.