How can theories about 1D or 2D systems be generalized for 3D systems?

Question

I was watching a lecture video from MITx $^\dagger$ by professor Barton Zwiebach. He proved a pretty cool theorem "every attractive 1-dimensional potential has a bound state"; however, that only holds in 1-dimension, and the theorem doesn't hold in 3-dimension.

Then I was left wondering:

How do we generate any results only valid in 1D/2D to our 3D world (I mean, after all, our world is 3D) ? Or how this theorem predicts anything in our real world such that we can test it experimentally?

I am a total newbie in experiment, if this question sounds silly to you, I am sorry!

I was thinking if an electron being constraint on a very thin plate would work. But I am not so sure, say, what about a curved - but super thin - plate? Does it counts 1D? or 3D?

$\dagger$: The course's website here. Which requires you have a EdX ID(free) to view the material.

Related: Why the statement “there exist at least one bound state for negative/attractive potential” doesn't hold for 3D case? — Qmechanic, Apr 04 '19 at 16:06
I don't really understand what you're asking here: A theorem about 1D doesn't predict anything about the real world. Why do you think it does? — ACuriousMind, Apr 04 '19 at 16:47
@ACuriousMind I wonder if there is any experimental means to test this theorem, after all, physics is experimental science. (furthermore, I was thinking if we can achieve that experiment with help of (constraint) nearly infinite potential wall .) — Shing, Apr 04 '19 at 16:56
at 1st that idea sounds quite reason (at least to me), but later I get confused: considering a "curved" but yet super thin (say ~ electron size) plate, does the theorem still work or not? — Shing, Apr 04 '19 at 16:57
Navier–Stokes existence and smoothness, which is a Millennium Prize problem, has been solved in 2-dimensions since the 1960's. A 3-D result hasn't yet been reported despite its publicity. — Nat, Apr 05 '19 at 08:57
Maybe I'm misunderstanding, but could quantum wells or quantum wires be similar to what you are looking for? — DoeJohn, Apr 05 '19 at 10:12

How can theories about 1D or 2D systems be generalized for 3D systems?

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