For example, U-235 fission cross section looks like this:
(source: science20.com)
As I understand it, the resonances peaks correspond to discrete quantum states of the excited compound nucleus. As you go higher, the density of states is too high to be resolved and you get that continuum.
But at thermal energies (left part of the graphic), I don't really understand what's going on, since the available states should be low. Consequently. I expect the cross section to be low too.
Is it a tail of a resonance peak corresponding to low energy states? Is the 1/v behavior dominating the decline of that resonance peak?
I'm expanding the question a little, since I'm not satisfied with the answers. Here is what I believe should be happening (the example is done with the absorption of a neutron by Indium-115):
Left is before absorption, right is after. The orange level is not a level in the compound nucleus, so absorption would be diminished.
This also happens with Uranium-238, so the question is not about fission only.
Best Answer
This is because U-235 is fissile, that is you only have to deliver the neutron to the nucleus for the magic to happen. Unlike U-238 where just delivering it doesn't do the job, there you also have to impart the nucleus with the neutrons kinetic energy.
Once we know this, it becomes clear that for low energy neutrons, their de Broglie wavelength is very big. So the cross-section is effectively determined by the quantum size of the neutron, rather than any other dynamics, so roughly $\sigma \approx \pi \lambda_{dB}^2\approx \frac{1}{E}$