I would like to see if it's possible that neutrinos (with sufficiently slow velocities) could form bound states in a universe with matter (such as ours)

There is a cosmic neutrino background in the universe which is probably the coldest neutrinos we could find (meaning slowest speeds). Slower neutrinos would heat up by thermalizing to the background.

I have been told that their wave function is around 1cm in length at those temperatures. I would think there is simply too much other matter around for the neutrino to form some sort of bound gravitational state but I'm not sure.

However I've read that I could plug this into Schrödinger’s equation and replace the potential term with the gravitational potential and see what the bound states energy are (I'd have to guess at the mass of the neutrinos, but I could use some upper limit for your guess)

However, I'm studying physics by my own (as I come from another scientific field that is not very related with it) and I have a bit of trouble understanding Schrödinger's equation

I don't think this would be difficult, but could someone help me how to figure this out? How can I solve the equation for this? Which steps should I take?