Spectral invariance of Gaussian Schell-model beams

It is well known that in general the spectrum of a beam that is generated by a partially coherent source will change on propagation. Here we derive necessary and sufficient conditions under which the often-used Gaussian Schell-model sources can produce beams whose normalized spectrum is invariant everywhere, or is invariant just along the beam axis. These sources are not necessarily quasi-homogeneous or obeying the scaling law.