I'm thinking about the possibility of a discontinuous solution of the Schrödinger equation and what is needed to get such an object.
It should be a condition to the potential, I think. If the potential is bounded everywhere a discontinuous wavefunction does not make sense because the second derivative will be singular and so the left side of the equation while the right side ist still bounded. So the only possibility for such solutions should be a singularity in the potential. Is this correct so far? Does there exit any potential which results in a discontinuous solution?
If I'm thinking about the particle in a box problem: Why isn't it possible that this solution becomes discontinuous at the boundary points?