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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-24 04:20:39 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-04-24 04:20:39 +0200 |
commit | b3cf6a3ed2334719d5b7b047d5b5a6cbe4f14b30 (patch) | |
tree | 04f816cd10b076e460f58272979b297826feb0f2 /commutatortheorem.tex | |
parent | 0c4c33d879709ad8625d63267ae23a2ac0155ba4 (diff) | |
download | master-b3cf6a3ed2334719d5b7b047d5b5a6cbe4f14b30.tar.zst |
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diff --git a/commutatortheorem.tex b/commutatortheorem.tex new file mode 100644 index 0000000..9f6384f --- /dev/null +++ b/commutatortheorem.tex @@ -0,0 +1,21 @@ +%\chapter[Scales of Hilbert Spaces and Nelsons Commutator Theorem]{Scales of Hilbert Spaces\\ and Nelsons Commutator Theorem} +\chapter{Scales of Hilbert Spaces and~Nelsons~Commutator~Theorem} + +\begin{theorem}{Nelson Commutator Theorem}{nelson-commutator-theorem} + Let $N$ be a selfadjoint operator with $N \ge I$, + and let $q$ be a quadratic form with the same domain. + Suppose that there exist constants $c_1,c_2$ such that + \begin{align*} + \abs{q(\psi',\psi)} &\le c_1 \norm{N^{1/2}\psi} \norm{N^{1/2}\psi'} & \forall \psi,\psi' \in D(N^{1/2}) \\ + \abs{q(N\psi',\psi)-q(\psi',N\psi)} &\le c_2 \norm{N^{1/2}\psi} \norm{N^{1/2}\psi'} & \forall \psi,\psi' \in D(N^{3/2}) + \end{align*} + Then the operator $q_{\mathrm{op}}$ associated to $q$ is defined on the domain of $N$ + and is essentially selfadjoint on any core for $N$. +\end{theorem} + +\cite{ReedSimon2} +\cite{Nelson1972} + +\chapterbib + +%vim: syntax=mytex |