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It is unclear for me what processes are exactly in place during the absorption-reemission process of a fluorescing photon.
I am thinking about the case when the absorption and emission spectrum overlap: Can a low(-ish, ie a photon with the largest wavelength allowed) energy photon lead to the emission of a high-energy one.
In other words, does the emission spectrum vary depending on the excitation wavelength?
Conservation of energy cannot be violated - so you cannot have a single interaction in which a photon of energy $E$ results in the emission of a photon of energy $E + \Delta E$ unless there is another source of energy.
It is conceivable that an atom in an already excited state could result in such a phenomenon; and of course there is two-photon interaction (two photons absorbed, one emitted at a higher energy).
If you have overlapping absorption and emission spectra, it is likely that there is indeed a range of atomic energy states involved. I can imagine (but have no solid data) that this would allow what you describe.
Usually a higher-energy photon hits a molecule, excites it, and in some time afterwards (from picoseconds to days) the molecule re-emits another photon with lower energy. Using two monochromators, you can obtain two-dimensional fluorescence spectra, identifying that some spectral features depend on the excitation wavelength, while others do not.
In some cases, the material may emit a higher-energy photon than the incident one, picking the energy difference from the thermal vibrations; this is the case e.g. of weak anti-Stokes lines in Raman scattering spectra.
Aside from this, some molecules may be optically excited, and the re-radiation of a visible photon must be triggered by an infrared photon. This is widely used in the infrared-laser visualisation screens.
