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--- hamburgminicourse2017:classes_of_static_potentials [2017/04/02 15:44] – markus
+++ hamburgminicourse2017:classes_of_static_potentials [2017/04/04 09:27] (current) – markus
@@ Line 16: / Line 16: @@
 **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 also 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:
+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]]