%\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} 

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