There is a nice preprint Isosbestic Points: Theory and Applications by M. Greger, M. Kollar, and Dieter Vollhardt.
They show just how ubiquitous isosbestic [crossing] points are in the spectra of strongly correlated systems (both materials and models).
Previously I posted that isosbestic points were A signature of a "two fluid" picture for a strongly correlated system. Look at the post for the background and to see the pictures which clearly show what isosbestic points are.
The preprint gives a detailed mathematical analysis to show why the crossing points can exist for such large parameter variations.
It is still not completely clear to me physically why these points are so robust in such complex quantum many-body systems.
I feel that there is a profound physical question that might be emphasized and discussed more. To what extent does the existence of an isosbestic point justify a "two fluid" model for a specific system? e.g. ideas promoted by David Pines and collaborators in these two papers:
The two fluid description of the Kondo lattice.
Universal behaviour and the two-component character of magnetically underdoped cuprate superconductors
Such two fluid pictures seem to involve a lot of "dephasing" of different degrees of freedom. For example, one a Hubbard model one is sort of splitting up electronic degrees of freedom into a localised spin and an itinerant charge.
I thank Ben Powell for bringing the preprint to my attention.
Aside: Ard Louis has an interesting looking paper on how in classical fluids isosbestic points in the structure factor can be used to extract particle interaction potentials.