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I just started learning on path integral on my own.

It seems that the path integral method is not always able to be solved, depending on the potential. On the other hand, these potentials are solvable by Schrödinger method (i.e solving the Schrödinger equation).

My question is that which are the potentials that are unsolved yet, and why are these potential important? Is there any place I can read more about them as to why they are unsolvable although there are known solution by Schrödinger method.

Thank you.

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This question (v1) seems like a list question. – Qmechanic Dec 21 '15 at 17:08
To the best of my knowledge all the solvable potentials have also been evaluated with the path integral method, it's just a really, really tedious way of working these problems. If you are interested in some of it, look at non-holonomic space and time transformations. Kleinert's textbook "Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets" is the go-to resource, if I am not mistaken. Axel Pelster has also published copiously on the subject, I believe and the origins of this work go back further, but I forgot who the original authors were. – CuriousOne Dec 21 '15 at 17:19
The important potentials are the exactly solvable ones, as they are being used as starting point for a perturbative treatment of the others. It doesn't matter whether you do it via path integrals or via the Schroedinger equation. – Arnold Neumaier Dec 30 '15 at 11:37

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