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 $D(H)=H^2(\Omega)\cap H^1_0(\Omega)$. The full answer will be given by the Kato–Rellich theorem and potentials that are $\Delta$-bounded.

Definition. Let $A,B$ be densely defined operators, then $B$ is called $A$-bounded if:

1. $D(A)\subseteq D(B)$ and
2. there are $a,b>0$ such that for all $\varphi \in D(A)$ it holds $\|B\varphi\| \leq a\|A\varphi\| + b\|\varphi\|$.

The smallest such constant $a$ is called the relative bound.

Theorem (Kato–Rellich). For $A$ self-adjoint, $B$ symmetric and $A$-bounded with relative bound $<1$ we have $A+B$ self-adjoint on $D(A)$.

With tools of Fourier analysis one shows that for $A=-\Delta$ this indeed holds for $B=v \in L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, which includes the usual singular Coulomb potential, and it also holds for finite sums $B=V=\sum_i v(x_i) + \sum_{i<j} w(x_i-x_j)$ with $v,w$ like before. This class of potentials $V$ will be called the set of Kato perturbations $K$ and forms a Banach space with appropriate norm $\|\cdot\|_K$. The following important estimate then holds: \[ \|V\psi\|_{L^2} \leq \|V\|_K \|\psi\|_{H^2}. \]

One could ask if the Kato perturbations are the maximal class for the Kato–Rellich theorem to hold, but this is not the case. An almost maximal class is given by the so-called Stummel class potentials that are closely related to the Kato class (again something different than the Kato perturbations) of the form version of the Kato–Rellich theorem, called KLMN theorem.

» Time-dependent potentials