I excited a fused aromatic conjugated naphthalene-based system from 240-360 nm and the emission bank was continuously red shifted from 500-800 nm. The absorption maximum is around 360 nm.
You do not expect the emission wavelength to change with excitation wavelength. Though red-edge excitation shift is an exception. Does the quantum yield also changes? Please read about Kasha -Vavilov's rule. Before that, have you recorded excitation spectra? 500-800 nm emission looks like too extreme for a naphthalene based system. Why don't you show the spectra.
Dear Nathanael: I would suggest you to ensure that your compound is very pure before you try to find an explanation. Also ensure that light is not decomposing your molecule. This can be easily checked if you collect excitation spectrum with time. As an anecdote, when C.V. Raman was studying inelastic scattering (called Raman effect now) he had to thoroughly purify all his solvents, because nobody believed that scattering can change wavelengths.
A rather simple explanation of excitation wavelength dependence is the presence of more than one fluorophore (luminophore) in a large molecule. Kasha-Vavilov's rule may still individually hold true for that luminophore (and exceptions always exist).
All what has said above is right. In principle, if your change the excitation wavelength but keep exciting the same electronic level, the emission spectrum should not change if your compound is pure and your solvent impurity-free; you should simply see intensity changes reflecting the absorption spectrum and the lamp emission intensity.
A good way of trying to disentangle what is happening is to record
(i) blank spectra of the solvent for each excitation wavelength that you use; use preferably spectroscopy quality solvents.
(ii) excitation spectra for several wavelengths within the emission band; this may (a) help finding out if you have impurities generating spurious bands and (b) show you what is the best wavelength for exciting your sample. Note that excitation spectrum is the product of the absorption spectrum by the instrumental function, so that the most efficient wavelength for excitation may not correspond to the maximum in the absorption spectrum. If you correct the excitation spectrum for the instrumental function (mainly the emission spectrum of the lamp,the transmittance of the monochromator and the response of the detector), then both absorption and corrected excitation spectra should be identical.