In clssical mechanics it doesn't really seem to matter which reference point you use for your potential. All that matters is the difference in potential between two points. Does the same hold true for the potential you plug into the Schrödinger equation? It doesn't seem so since plugging in V=-10J or V=0 should certainly provide different solutions. But the difference between the two potentials is merely just your choife of reference point, and that seems weird.