====== Classes of static potentials ======

> 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?"[(Schwartz J, //The Pernicious Influence of Mathematics on Science//, in //Logic, Methodology and the Philosophy of Science//, p. 356-360 (Stanford University Press, 1962); also in //Discrete Thoughts: Essays in Mathematics, Science and Philosophy// (Springer, 1992))]

To make the usual Schrödinger problem $i\frac{d}{dt}\psi=(-\Delta+V)\psi$ well-defined we have to answer:
  - for which potentials $V$ is the Hamiltonian self-adjoint and
  - 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:
  - $D(A)\subseteq D(B)$ and
  - 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.

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