Analytic Solutions to Schroedinger Equation

[YAHA]

Everyone knows that the above equation cannot be solved for anything, with the exception of some simple cases(infinite well, harmonic oscillator, hydrogenic atoms, etc).

Now, my question is whether we know that it is impossible to solve such differential equation for other potential energy functions or is it simply that we haven't discovered a way of solving it? In other words, is there some proof of the impossiblity of finding solutions for certain functions?

 

[Bill_K]

Soluble in terms of what? For the radial Schrödinger equation, for example, for any V(r), one can always define a new custom-made special function and/or a set of orthogonal polynomials in terms of which the solution can be exactly represented.