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--- hamburgminicourse2017:classes_of_static_potentials [2017/04/02 15:33] – created markus
+++ hamburgminicourse2017:classes_of_static_potentials [2017/04/04 09:27] (current) – markus
@@ Line 7: / Line 7: @@
   - 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 $D(H)=H^2(\Omega)\cap H^1_0(\Omega)$.
+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.
+ >> [[Time-dependent potentials]]