Give a mathematician a situation which is the least bit ill-defined – he will first of all make it well defined. Perhaps appropriately, but perhaps also inappropriately. The hydrogen atom illustrates this process nicely. The physicist asks: “What are the eigenfunctions of such-and-such a differential operator?” The mathematician replies: “The question as put is not well defined. First you must specify the linear space in which you wish to operate, then the precise domain of the operator as a subspace. Carrying all this out in the simplest way, we find the following result…” Whereupon the physicist may answer, much to the mathematician's chagrin: “Incidentally, I am not so much interested in the operator you have just analyzed as in the following operator, which has four or five additional small terms – how different is the analysis of this modified problem?”¹⁾

To make the usual Schrödinger problem $i\frac{d}{dt}\psi=(-\Delta+V)\psi$ well-defined we have to answer:

1. for which potentials $V$ is the Hamiltonian self-adjoint and
2. what is the domain of such a Hamiltonian?

In the free case $V=0$ we have $D(H)=H^2(\mathbb{R^n})$ or for bounded space domains and zero boundary conditions $D(H)=H^2(\Omega)\cap H^1_0(\Omega)$.