Although the equivalence theorem (ET) in the potential scattering theory has been proposed for a long time, its analysis is always confined to the idealized case where the incident field is a spatially coherent plane wave, which limits its practical applications. Here by exploiting Laplace's method for double integrals and the so-called beam condition, we generalize the ET in the potential scattering theory to partially coherent beams for the first time to the best of our knowledge. We present the analytical condition that two scattered fields, produced by Gaussian Schell-model beams on scattering from Gaussian Schell-model media, may have the same normalized spectral densities in the far zone. We find that the condition contains three implications, each corresponding to a statement of an ET for the spectral density in a scattering scenario, which exposes the concept of a previously unreported triad of ETs for the spectral density of partially coherent beams on scattering. Our results contribute to improving reconstruction accuracy when resolving the inverse scattering problem in practical situations, where the light field utilized to illuminate an unknown scatterer is a partially coherent beam rather than a plane wave.