quantum mechanics of fractal potential wells.

[kram1032]

Instead of potentially wondering how quantum-mechanics might apply to fractals, there is actual research on this topic, as found here:
P-adic QM (Wiki)
Note, there are lots of references to more information on this topic, like, for instance:
Michael Berry's publications, for instance the so called "Diffractals _(pdf)" - Waves that have encountered fractals and thus interfered in fractal ways on all scales.
I encourage searching through the references for further info beyond Wikipedia.

[jehovajah]

Righto, Kram. Will give it a shot!

[Alef]

Could it be created real 3D mandelbrot set on p-adic numbers? Or on adele ring?
Well, it seems not so, even I realy don't understand this concept an I'm not a matematician, but could there be some new fractals based on this?

[kram1032]

I'm fairly sure (but I'm not an expert either, obviously), that you gotta have a complete limit set description of the set...
Stuff like the usual cantor set, where you end up with a very predictable pattern for the numbers in the set and -Analogues of it, like the sierpinski carpets, sponges, etc.
I could easily be wrong though.