Stationary states of a triangular prism I need to find the wavefunctions of the stationary states of a 3d square potential well with its boundaries defined by a triangular prism - like the one illustrated on the wikipedia page:
https://en.wikipedia.org/wiki/Triangular_prism 

The potential well (viewed in 1-d cross section) is a simple square potential well, and can be either finite (0 outside, -V inside) or infinite (0 inside, ∞ outside), both would be reasonable approximations for my purposes.   
I.e. the potential is something like this in cross-section, but its full 3-D shape is that of the triangular prism:

[Any solution for a close approximation of this geometry may also be helpful (for example, if the problem is easier to solve for a prism with a Reuleaux triangular cross-section instead of equilateral, or for a potential well described by a continuous function or something, it may be close enough).]   
Because of the reduced symmetry compared to the textbook cylindrical or spherical cases, I am not sure how to approach this.  
Is anyone able to point me in the direction of a solution? Many thanks!
 A: I'm unsure about the case with a finite well depth, but if the walls are infinitely hard this problem can be solved exactly. The solution is detailed in the papers


*

*Particle in an equilateral triangle: Exact solution of a nonseparable problem. Wai-Kee Li and S. M. Blinder. J. Chem. Educ. 64 no. 2, p. 130 (1987)

*Solution of the Schrödinger equation for a particle in an equilateral triangle. Wai‐Kee Li and S. M. Blinder. J. Math. Phys. 26, 2784 (1985)
Other papers with relevant solutions are here, here and here.
The loss of continuous rotational symmetry means that you do need to fully solve a two-dimensional PDE, but the discrete symmetry does help, in that the solutions are required to carry representations of the $D_3$ symmetry group. This means that there are strict relations between the values of the eigenfunctions at the different edges, and those can be exploited to 'stitch' together multiple copies of the domain to make up a translationally-invariant region,

and you therefore expect the solutions to be plane-wave exponentials in that expanded region, which then project back down to sums of exponentials on the inside of the triangle.

I'm not sure to what extent those methods carry over to the finite-depth version of the well, this paper uses numerical diagonalization to solve the problem, and googling for "triangular quantum dot" (probably the most helpful starting point) doesn't immediately yield anything promising, and neither of those bode very well for the existence of closed solutions. (Ditto for their absence in this review.) Since you state that the infinite-walls problem is OK for your purposes, I'd encourage you to just stick to that.
