Analytical non-separable solution for schrodinger equation

I am an undergraduate with the background of a first course in Quantum Mechanics. I want to find out if there exist non-unique solutions to Schrodinger equation. So I have to find potentials (Hamiltonians) for which there exist multiple solutions to a Schrodinger equation. For separable solutions, the problem is analogous to the classical problem of non-uniqueness in Newton's equation addressed by Abhishek Dhar in the paper http://dx.doi.org/10.1119/1.17411. But I would like to pursue the same for non-separable analytical solution to the Schrodinger equation. Can anyone suggest how to go about it or what resources could I use to attempt the problem?

I don't want to be too much precise, but the Schrödinger equation ($i \dot{\psi}= H\psi$ to avoid confusion) has at most an unique solution under very general assumptions on the Hamiltonian operator $H$, even if you see it as a liner equation in the more general setting of Banach spaces.
In particular, it is not a priori necessary that $H$ is self-adjoint for the solution to be unique (and anyways if it is, then it is unique).
• @illusion Mathematical assumptions that I am quite sure you do not want to see, for they are quite technical ;-) The main point is, however, that you have uniqueness for the solution of the Schrödinger Equation whenever the generator $H$ is self-adjoint (and also in other more general situations). And the self-adjointness of $H$ is a rather unavoidable assumption in standard QM. – yuggib Jun 23 '15 at 15:24