text:
With the scattering function formalism we can now present the concrete choice of the abstract operators $L, A, N, D, C_n$, $n=0,1,2,3$, of \Autoref{sec:abstract virial}. Subsequently, we prove the assumptions of the abstract virial theorem which we use in order to get the concrete virial theorem \autoref{th:result virial}. It will be one main ingredient for the proof of the main result of this paper. We give its proof at the end of \Autoref{subsec:verification assumptions}.

\subsection{Scattering States}
\label{sec:scattering}
\input{sr_scattering}
\subsection{Setup for the Virial Theorems}
First, we describe the setting on the particle space $\H_\p$. We consider a dense subspace given by
\[
\Dps := \Vc^* \Dp \oplus \ran \Pd .
\]
Note that $\Dps$ is dense since $\Vc^* \Dp \subseteq \ran \Pc$ is dense in $\ran \Pc$.
Now, based on the definition of the generator of dilations,
\[
\Adil = \frac{1}{4} (\q \po + \po \q),
\]
we define on $\Dps$ the conjugation operator
\[
\Aps := \Vc^* \Adil  \Vc.
\]
It is clear that both $H_\p$ and $\Aps$ leave $\Dps$ invariant. Thus we can define $\ad^{(n)}_{\Aps}(H_\p)$ on $\Dps$ for all $n \in \IN$. Furthermore, the bounding operator is chosen as
\begin{align*}
\Lps &:= \Vc^* (\q^2  + \po^2) \Vc + \Id.
\end{align*}
Next, on the field space we set
\begin{align*}
A_\f &:= \dG(\i \partial_u), \\
\Lambda_\f &:= \dG(\mult{u} ^2 + 1).
\end{align*}
Now, we can define on the dense subspace of the complete space $\H$,
\begin{align*}
\Ds &= \Dps \otimes \Dps \otimes  \Df,
\end{align*}
the operators
\begin{align*}
\Ls &= \Lps \otimes \Id_\p  \otimes \Id_\f + \Id_\p \otimes \Lps \otimes \Id_\f + \Id_\p \otimes \Id_\p \otimes \Lambda_\f, \\
L &= L_\lambda, \\
N &= \Id_\p \otimes \Id_\p \otimes N_\f, \\
\As &= (\Aps \otimes \Id_\p - \Id_\p \otimes \Aps) \otimes \Id_\f + \Id_\p \otimes \Id_\p \otimes A_\f, \\
D &= \i [L_\lambda,\Id_\p \otimes \Id_\p \otimes N_\f].
\end{align*}
For symmetric operators $X,Y$ with a dense domain $\D_0$ we define multiple commutators by $\ad^{(0)}_Y(X) = X$ and $\ad^{(n+1)}_Y(X) = \i [\ad^{(n)}_Y(X),Y]$ in the form sense on $\D_0 \times \D_0$. If the form is bounded, we denote the corresponding bounded operator by the same symbol.
Now we set
\muun
{
\label{eq:cn_formula 2}
W_n &:= \ad^{(n)}_{\As}(\Phi(I))  \\
&=  \sum_{k=0}^n  \binom{n}{k}  \Phi \big(  (-\i \partial_u)^k \tau_\beta( \ad^{(n-k)}_{ \Aps }(G) ) \otimes \Id_\p  \nonumber \\ &\qquad -  (-\i \partial_u)^k  e^{-\beta u  /2}   \Id_\p  \otimes \tau_\beta( \ad^{(n-k)}_{\Aps} (\G^*) )  \big), \nonumber  \\
C_n &= \delta_{n,1} \Id_\p \otimes \Id_\p \otimes N_\f + \ad_{\Aps}^{(n)}(H_\p) \otimes \Id_\p + (-1)^n \Id_\p \otimes \ad_{\Aps}^{(n)}(H_\p) + \lambda W_n. \nonumber
}

Notice that the conditions in \autoref{lemma:gluing_integrable} for $m=3$ are satisfied due to \ref{assumption:I2}, \ref{assumption:I3}, which guarantees that the expressions in the field operators in \eqref{eq:cn_formula 2} are indeed in $L^2(\IR \times \IS^2, \L(\H_\p))$. Hence, it is relatively $\Id_\p \otimes \Id_\p \otimes N_\f^{1/2}$-bounded, thus, also $\Id_\p \otimes \Id_\p \otimes \Lambda_\f$-bounded.
We will see shortly, that $C_n$, $n=1,2,3$, are indeed essentially self-adjoint on $\D$ and we denote their self-adjoint extensions bei the same symbols.
Moreover, it will be shown below in \thref{th:total estimate steps} that $C_1$ is actually bounded from below. Thus, we can assign to $C_1$ a quadratic form $\qce$.

\subsection{Verification of the Assumptions of the Virial Theorems}
\label{subsec:verification assumptions}

In the given setting just described we can now start to prove the assumptions of the virial theorem \thref{thm:firstvirial}. Above all, we have to check the GJN condition for the different commutators. The most difficult part will be the discussion of interaction terms $W_n$, $n=1,2,3$. Here, the expressions in the field operators need to be sufficiently bounded. These bounds will be collected in the following proposition which is the main result of \Autoref{sec:estimates scattering}.
\begin{prop}
\label{th:G commutator conditions}
Let $\partial_u$ denote the derivative of a $\L(H_\p)$-valued function in the sense of the strong operator topology.
For all $n,m \in \{0,1,2,3\}$, $j \in \{1,2,3\}$, and for all $(u,\Sigma)$ the operators
\begin{enumerate}[label=(\arabic*)]
\item
\label{item:no derivative}
$\partial_u^m  \ad^{(n)}_{\Aps}(  \tau_\beta(G)(u,\Sigma)  )$,
\item
\label{item:qj derivative}
$\partial_u^m   \ad_{\Vc^* \q_j \Vc}(  \ad^{(n)}_{\Aps}(  \tau_\beta(G)(u,\Sigma)  ) )$,
\item
\label{item:pj derivative}
 $ \partial_u^m \ad_{\Vc^* \po_j \Vc}(  \ad^{(n)}_{\Aps}(  \tau_\beta(G)(u,\Sigma)   ) )$,
\end{enumerate}
and for all $n,m \in \{0,1\}$,
\begin{enumerate}
\item[(4)]
\label{item:pj additional}
 $ \partial_u^m  \Vc^* \po_j \Vc    \ad^{(n)}_{\Aps }(  \tau_\beta(G)(u,\Sigma)   )$,
\end{enumerate}
are well-defined, and the corresponding functions $\IR \times \IS^2 \rightarrow \L(H_\p)$ are continuous and in $L^2(\IR \times \IS^2, \L(H_\p))$.  Moreover there exists a constant $C$ independent of $\beta$ such that for $n,m \in \{0,1\}$,
\[
\nn{  \partial_u^m  \Vc^* \po_j \Vc    \ad^{(n)}_{\Aps }(  \tau_\beta(G)  ) }_{L^2(\IR \times \IS^2, \L(H_\p))} \leq C(1+ \beta^{-\frac{1}{2}}).
\]
\end{prop}

Having this we can now verify the necessary GJN conditions.
\begin{prop}
\label{th:gjn check}
The following triples are GJN:
\begin{enumerate}[label=(\arabic*)]
\item $(\Aps, \Lps, \Dps)$,
\item $(H_\p, \Lps, \Dps)$, \label{Hp Lps Dps}
\item $(L, \Ls, \D)$, \label{Ll essentially self-adjoint}
\item $(D, \Ls, \D)$,
\item $(C_i, \Ls, \D)$, $i \in \{1,2,3\}$.
\end{enumerate}
In particular, $L = L_\lambda$ is essentially self-adjoint on $\D$ for any $\lambda \in \IR$ due to \ref{Ll essentially self-adjoint}.
\end{prop}
\begin{proof}
\begin{enumerate}[label=(\arabic*)]
\item
Using that $(\Adil, \Lambda_\p, \Dp)$ is GJN, we have, for $\psi \in \Dps$,
\begin{align*}
\nn{ \Aps \psi} = \nn{\Vc^* \Adil \Vc \psi} = \nn{\Adil \Vc \psi} \leq C \nn{ \Lambda_\p  \Vc  \psi} = C \nn{ \Vc^* \Lambda_\p  \Vc  \psi},
\end{align*}
for some constant $C$, and
\begin{align*}
\pm \i &(\sc{\Aps \psi, \Lps \psi} - \sc{\Lps \psi, \Aps \psi})  \\\ &=  \pm \i \left(\sc{\Adil \Vc \psi, (\q^2 + \po^2) \Vc \psi} - \sc{(\q^2 + \po^2)\Vc \psi, \Adil\Vc \psi \psi}  \right)  \\
&\leq C \sc{\Vc \psi, (\q^2 + \po^2) \Vc \psi} \\
&\leq C \sc{\psi,  \Lps \psi}.
\end{align*}
\item
We have, with regard to the first GJN condition,
\begin{align*}
\nn{H_\p \psi} = \nn{ \Vc^* \q^2 \Vc \psi + H_\p \Pd \psi} \leq C \nn{\Vc^* (\q^2 + \po^2) \Vc \psi} + \sup_{\lambda \in \sigma_\d(H_\p)} \abs \lambda \nn{\psi},
\end{align*}
as $	\q^2$ is relatively bounded by $\po^2 + \q^2$. Furthermore,
\begin{align*}
&\sc{ H_\p \psi, \Lps \psi} - \sc{ \Lps \psi,   H_\p \psi} \\ &= \sc{\Vc^* \q^2 \Vc \psi, \Vc^* (\po^2 + \q^2) \Vc \psi} - \sc{\Vc^* (\po^2 + \q^2) \Vc \psi,  \Vc^* \q^2 \Vc \psi}
\\ &= \sc{\q^2 \Vc \psi, \po^2 \Vc \psi} - \sc{ \po^2  \Vc \psi,  \q^2 \Vc \psi}
\end{align*}
Using now that the commutator $[\po^2,\q^2]$ is obviously bounded in the operator sense by a multiple of $\po^2 + \q^2$, we get also the second GJN condition.

\item
As $H_\p$ is relatively bounded by $\Lps$ (by the previous argument), $L_0 = (H_\p \otimes \Id_\p - \Id_\p \otimes H_\p) \otimes \Id_\f$ is relatively bounded by $\Ls$.
Next, we know that the interaction terms $I^{(l)}, I^{(\r)}   \in L^2(\IR \times \IS^2)$. Hence $W$ is bounded by $\Id_\p \otimes \Id_\p \otimes \Nf^{1/2}$ and thus bounded by $\Id_\p \otimes \Id_\p \Lambda_\f^{1/2}$. Therefore, the first GJN condition is satisfied.

For the second GJN condition, as $(H_\p, \Lps, \Dps)$ is GJN, we get that, for $\psi \in \D$,
\[
\pm \i ( \sc{L_0 \psi, \Ls \psi} - \sc{ \Ls  \psi, L_0 \psi}) \leq C\sc{ \psi, (\Lps \otimes \Id_\p  + \Id_\p \otimes \Lps)  \otimes \Id_\f  \psi}
\]
for some constant $C$.

By \ref{assumption:I2}, \ref{assumption:I3} we know that $(u,\Sigma) \mapsto (u^2 + 1) I^{(\l)}(u,\Sigma)$ and $(u,\Sigma) \mapsto (u^2 + 1) I^{(\l)}(u,\Sigma)$  are in $L^2(\IR \times \IS^2, \L( \H_\p \otimes \H_\p))$. We have
\begin{align*}
&\abs{ \sc{ \Phi( I^{(\l)}) \psi ,  \Id_\p \otimes \Id_\p \otimes \Lambda_\f \psi } - \sc{  \Id_\p \otimes \Id_\p \otimes \Lambda_\f \psi , \Phi( I^{(\l)})  \psi } }
\\ &\qquad = 2^{-\frac{1}{2}} \abs{ \sc{\psi,(  a\left(( \mult{u} ^2 + 1) I^{(\l)}) - a^*(( \mult{u} ^2 + 1) I^{(\l)}) \right) \psi} }
\\ &\qquad\leq C \nn{\Id_\p \otimes \Id_\p \otimes \Nf^{1/2} \psi} \nn{\psi}
\\ &\qquad\leq C \sc{ \psi, \Ls \psi}
\end{align*}
for some constants $C$. The same thing can be shown for the commutator with $\Phi(I^{(\r)})$.

It remains to consider the commutator of $W$ with the $\Lps$ terms. One has to show that the weak commutators
\[
[\Phi(I^{(\l)} ), \Vc^* (\po^2 + \q^2) \Vc], \qquad [\Phi(I^{(\r)} ), \Vc^* (\po^2 + \q^2) \Vc]
\]
are weakly bounded by $\Lps$. This follows directly from \thref{th:G commutator conditions} as we can write in the weak sense on $\D$,
\[
[\Phi(I^{(\l)} ), \Vc^* \po^2 \Vc] = \sum_{j} [\Phi(I^{(\l)} ), \Vc^* \po_j \Vc]  \Vc^* \po_j \Vc +   \Vc^* \po_j \Vc [\Phi(I^{(\l)} ), \Vc^* \po_j \Vc]
\]
and analogously for $\Phi(I^{(\r)})$, and for the commutator with $\Vc^* \q^2 \Vc$.

\item
We have
\[
D =  \frac{\i}{\sqrt 2} ( a(I) - a^*(I) ).
\]
Thus, the proof works as the one of $L$.
\item
We first consider again the free part. We have on $\Dps$,
\begin{align*}
[H_\p, \Aps] =  [\Vc^* \q^2 \Vc, \Aps] =     \Vc^* [\q^2, A_\p] \Vc = \i \Vc^* \q^2 \Vc
\end{align*}
Repeating this procedure yields for $n \in \{1,2,3\}$,
\begin{align*}
\ad^{(n)}_{\Aps}(H_\p) =  (-1)^n  \Vc^* \q^2 \Vc.
\end{align*}
Then we can show as in the proof of $(H_\p, \Lps, \Dps)$ that also $(\ad^{(n)}_{\Aps}(H_\p), \Lps, \Dps)$, $n \in \{1,2,3\}$, is GJN and so is $(\ad^{(n)}_\As (L_0), \Ls, \D)$, $n=1,2,3$.

It remains to verify the GJN conditions for  $(W_n, \Ls, \D)$, $n=1,2,3$. Analogously to the proof of the GJN condition of $L$ we have to show that the expressions in the field operators are bounded, and also stay bounded if we commute with the square root of $\Lps$, that is, we have to show that
\begin{align*}
(u,\Sigma) &\mapsto  \partial_u^k \tau_\beta(\ad^{(n-k)}_{\Aps} (G))(u,\Sigma), \\
(u,\Sigma) &\mapsto  \partial_u^k [\tau_\beta(\ad^{(n-k)}_{\Aps} (G))(u,\Sigma), \Vc^* \q_j \Vc], \\
(u,\Sigma) &\mapsto  \partial_u^k [\tau_\beta(\ad^{(n-k)}_{\Aps} (G))(u,\Sigma), \Vc^* \po_j \Vc],
\end{align*}
$n=1,2,3, ~k = 0, \ldots,n$,  $j=1,2,3$,
are in $L^2(\IR \times \IS^2)$. As $\kappa$ has the right decay behavior, this follows directly from \thref{th:G commutator conditions} together with \thref{lemma:gluing_integrable}. \qedhere
\end{enumerate}
\end{proof}

Now we can prove the main result of this section, the concrete virial theorem of our setting.
\begin{thm}[Concrete virial theorem]
\label{th:result virial}
Let $C_1$ defined as below.
Assume that there exists $\psi \in \Def(L_\lambda)$ with $L_\lambda \psi = 0$. Then $\psi \in \Def(\qce)$ and $\qce(\psi) \leq 0$.
\end{thm}

\begin{proof}
We have seen in \thref{th:gjn check} that $L_\lambda, A, N, D, C_n$, $n=0,1,2,3$, satisfy the GJN condition on $\D$. Furthermore, it is shown there that also $(\Aps, \Lps, \Dps)$ is a GJN triple. Hence \thref{thm:invariance_gjn_domain} implies that, for all $t \in \IR$, $e^{ \i t \Aps}$ leaves $\Def(\Lambda_\p)$ invariant. Moreover, one computes on $\Df$,
\muu
{
\Lambda_\f e^{\i t A_\f} = e^{\i t A_\f} (\Lambda_\f + \dG(2\mult{u}t - t^2)).
}
Thus, for some fixed $t$ there is a constant $C$ such that $\nn{ \Lambda_\f e^{\i t A_\f} \psi} \leq C\nn{ \Lambda_\f \psi}$ for all $\psi \in \D$. As $\Df$ is a core for $\Lambda_\f$, we find $e^{\i t A_\f} \Def(\Lambda_\f) \subseteq \Def(\Lambda_\f)$. Thus, we conclude $\Def(\Lambda)$ is invariant under the unitary group associated to $A$. Finally, by definition of $A$ it is clear that $\Id_\p \otimes \Id_\p \otimes N_\f$ and $e^{\i t A}$, $t \in \IR$, commute in the stronge sense on $\D$.

Therefore, by \thref{thm:firstvirial} we find a sequence $(\psi_n)_{n\in \IN}$ in $\Def(C_1) \cap \Def(L_\lambda)$ such that $\lim_{n\to\infty} \psi_n = \psi$ and $\lim_{n\to\infty} \sc{\psi_n, C_1 \psi_n} = 0$. Now, as $\qce$ is closed, thus continuous from below, we obtain $\psi \in \Def(\qce)$ and
\[
\qce(\psi) \leq \lim_{n \to \infty} \qce(\psi_n) = 0. \qedhere
\]
\end{proof}


TITLE: Dense countable set in an infinite Baire and $T_1$ space has dense complement
QUESTION [2 upvotes]: I'm right now covering chapter 8 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problem.
If $X$ is an infinite, $T_1$ and Baire space with a subset $D$ dense and countable, then $D^c$ is dense and $D$ cannot be an intersection of countable open sets.
My approach:
Let $D = \{d_1,d_2,\dots\}$ If $X$ is T1, then every set $U_n = \{d_1,\dots,d_n\}$ is closed.
$$D^c = \bigcap_{i=1}^\infty U_n^c$$
Every $U_n^c$ is dense and open, then by Baire property $D^c$ is dense. I don't know if this is right, specially because I'm not sure if the $U_n^c$ are dense.
For the other part of the problem, the hint says we can take $D = \bigcap W_n$ with $W_n$ open sets and consider the sets $V_d = X-\{d\}$ and to prove that $V_d$ and $W_n$ are dense. I'm not sure why this is true or why this is sufficient to prove the statement.
Thanks in advance.

REPLY [4 votes]: As has been noted in the comments, the result is false as stated. In this case one can discover exactly what is going on by trying to complete the proof suggested by the hint and seeing where it runs into a problem.
Let $X$ be a $T_1$ space with a countable dense subset $D$. Suppose that there is a countable family $\mathscr{W}$ of open sets such that $D=\bigcap\mathscr{W}$. For each $d\in D$ let $V_d=X\setminus\{d\}$; $X$ is $T_1$, so each singleton $\{d\}$ is closed, and therefore each $V_d$ is open. Let $\mathscr{V}=\{V_d:d\in D\}$; then $\mathscr{V}$ is a countable collection of open sets, and $\bigcap\mathscr{V}=X\setminus D$. Thus, $\mathscr{W}\cup\mathscr{V}$ is a countable collection of open sets whose intersection is empty:
$$\bigcap(\mathscr{W}\cup\mathscr{V})=\bigcap\mathscr{W}\cap\bigcap\mathscr{V}=D\cap(X\setminus D)=\varnothing\;.$$
If $X$ is a Baire space, this implies that at least one member of $\mathscr{W}\cup\mathscr{V}$ must fail to be dense in $X$. Clearly each $W\in\mathscr{W}$ is dense in $X$: if $U$ is any non-empty open set, then
$$U\cap W\supseteq U\cap D\ne\varnothing\;.$$
Thus, if $X$ is a Baire space there must be at least one $d_0\in D$ such that $V_{d_0}$ is not dense in $X$. Then there is a non-empty open $U$ such that
$$\varnothing=U\cap V_{d_0}=U\cap(X\setminus\{d_0\})=U\setminus\{d_0\}\;,$$
so $U=\{d_0\}$, and $d_0$ is an isolated point of $X$.
If, on the other hand, $X$ has no isolated points, then each of the sets $V_d$ is dense in $X$, $\mathscr{W}\cup\mathscr{V}$ is a countable family of dense open sets whose intersection is empty, and $X$ is not a Baire space.
In short, the result in the exercise is true if and only if $X$ has no isolated points. This exception is not really surprising, since — as was also mentioned in the comments — a dense set in a space must contain every isolated point of that space.