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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?

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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).

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  • $\begingroup$ What do you mean by very general assumptions? $\endgroup$ – illusion Jun 23 '15 at 15:14
  • $\begingroup$ @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. $\endgroup$ – yuggib Jun 23 '15 at 15:24

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