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generalized weak Weyl relation, time operator, canonical commutation relation, Hamiltonian, quantum dynamics, survival probability, decay in 
time, time-energy uncertainty relation, Schroedinger operator, Dirac operator, Fock space, second quantization, 
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\begin{document}

\title{\bf Generalized Weak Weyl Relation and Decay of
Quantum Dynamics}
\author{ Asao Arai\\
Department of Mathematics, Hokkaido University \\ 
Sapporo 060-0810, Japan \\
E-mail: arai@math.sci.hokudai.ac.jp} 


\maketitle 

\bigskip


\begin{abstract}
Let $H$ be a self-adjoint operator on a Hilbert space ${\cal H}$,
$T$ be a symmetric operator on ${\cal H}$ and $K(t)$ ($t\in \R$) be a
bounded self-adjoint operator on ${\cal H}$. 
We say that $(T,H,K)$ obeys the {\it generalized weak Weyl relation} (GWWR)
if $e^{-itH}D(T) \subset D(T)$ for all $t \in \R$ and
$Te^{-itH}\psi=e^{-itH}(T+K(t))\psi, \forall \psi \in D(T)$ (
$D(T)$ denotes the domain of $T$).
In the context of quantum mechanics where $H$ is the Hamiltonian
of a quantum system, we call $T$ a {\it generalized time opeartor} of $H$.
We first investigate, in an abstract framework,  mathematical structures and 
properties of triples $(T,H,K)$ obeying the GWWR.
These include the absolute continuity
of the spectrum of $H$ restricted to a closed subspace of ${\cal H}$,
an uncertainty relation between $H$ and $T$ (a 
\lq\lq{time-energy uncertainty relation}"), 
the decay property of transition probabilities $\left|\lang \psi,e^{-itH}\phi\rang
\right|^2$ as $|t| \to \infty$
for all vectors $\psi$ and $\phi$ 
in a subspace of ${\cal H}$. 
We describe methods to construct various examples of triples $(T,H,K)$ 
obeying the GWWR. 
In particular we show that there exist generalized time operators
of second quantization operators on Fock spaces (full Fock spaces,
boson Fock spaces, fermion Fock spaces) which may have
applications to quantum field models with interactions.
\end{abstract}

\noindent
{\it Keywords}:
generalized weak Weyl relation, time operator, canonical commutation relation, Hamiltonian, quantum dynamics, survival probability, decay in
time, time-energy uncertainty relation, Schroedinger operator, Dirac operator, Fock space, second quantization.

\tableofcontents

\section{Introduction}

In this  paper  we 
develop, in an abstract framework, 
an operator theory of a commutation relation, which
is a generalization of a variant  of the canonical commutation
relation (CCR) with one degree of freedom, and put a basis for 
applications
of the theory to
 quantum mechanics and quantum field theory. 

As is well-known, there is a stronger form
of the CCR's, called the Weyl relations
(e.g., \cite[pp.274--275]{RS1}). Schm\"udgen \cite{Schmu}
presented  and studied a weaker version  of 
the Weyl relation with one degree of freedom: 
Let $T$ be a symmetric  operator and $H$ be a
self-adjoint operator on a Hilbert space ${\cal H}$.
We say that $(T,H)$ obeys the {\it weak Weyl relation} (WWR) if
$e^{-itH}D(T)\subset D(T)$ for all $t \in \R$ (
$D(T)$ denotes the domain of $T$) and
$$
Te^{-itH}\psi=e^{-itH}(T+t)\psi, \quad \forall \psi \in D(T),\forall
t \in \R.
$$
This form of commutation relation was used to study
a time operator with application to survival probabilities 
in quantum dynamics \cite{Miya1,Miya2} (in \cite{Miya1}, 
the WWR is called the $T$-weak Weyl relation), where $H$
is taken to be the Hamiltonian of a quantum system. 
It was proven in \cite{Miya1} that, if
$(T,H)$ obeys the WWR, then $H$ has no point spectrum and
its spectrum is purely absolutely continuous \cite[Corollary 4.3, Theorem
4.4]{Miya1}. This kind class of $H$, however, is somewhat
restrictive. From this point of view, it would be natural
to investigate a  general version 
of  the WWR (if any) such that 
$H$ is not necessarily purely absolutely continuous.
This is one of the motivations of the present work.


The general version of the WWR we consider in the present paper
is defined as follows:

\begin{df}\label{GWWR}{\rm 
Let $T$ be a symmetric operator on a Hilbert space ${\cal H}$ , $H$ be a self-adjoint 
operator on ${\cal H}$ and
$K(t)$ ($t \in \R$) be 
a bounded self-adjoint operator on ${\cal H}$ with $D(K(t))={\cal H},\,
\forall t \in \R$.
We say that $(T,H,K)$ obeys
the {\it generalized weak Weyl relation} (GWWR) in ${\cal H}$ 
if  $e^{-itH} D(T)\subset 
D(T)$ for all $t \in \R$ and
\begin{equation}
Te^{-itH}\psi=e^{-itH}(T+K(t))\psi, \quad \forall \psi \in D(T), \, \forall t \in 
\R. \label{s-time}
\end{equation}
We call the operator-valued function $K$ the commutation
factor in the GWWR. Also we sometimes say that
$(T,H,K)$ is a representation of the GWWR.
}
\end{df}

Obviously the case $K(t)=t$ in the GWWR gives the WWR.
Triples $(T,H,K)$ obeying the GWWR 
are the main objects of the present paper.
As suggested above, in applications to quantum mechanics and
quantum field theory, 
we have in mind the case where
$H$ is the Hamiltonian of a quantum system. In this realization
of $H$, we call  $T$  a {\it generalized time operator}.
We show that the GWWR implies a \lq\lq{time-energy uncertainty
relation}" between $H$ and $T$. 
For physical discussions related to this aspect, see
\cite{Muga} and references therein.
Mathematically rigorous discussions for time-energy uncertainty
relations, which, however, do not use time operators, are given in \cite{PF}.
In the present paper we construct  generalized time operators for
Hamiltonians  in both relativistic and  nonrelativistic 
quantum mechanics including Dirac type operators as well as in
quantum field theory.


The outline of the present paper is as follows.
In Sections 2  we discuss fundamental  properties of
representations $(T,H,K)$  of  the GWWR.
In Section 3 we derive a decay property (in time $t \in \R$)
of transition probability amplitudes $\lang \psi, e^{-itH}\phi\rang$
for vectors $\psi,\phi$ in a suitable subspace. 
In Section 4, we establish a connection of the GWWR with
the generalized  weak CCR and prove a time-energy uncertainty 
relation.
Section 5 is concerned with properties of the point spectrum of
$T$. In Section 6 we develop functional calculus
for the GWWR. In a special  case where $K(t)=K_C(t):=tC$ with $C$ a bounded 
self-adjoint operator, we prove the absolute continuity of $H$
restricted to the closure of $\Ran (C)$, the range  of $C$.
In Section 7, we prove absence of minimum-uncertainty states
for each representation $(T,H,K_C)$ with $T$ being closed.
In Section 8, for representations $(T,H,K_C)$ of the GWWR,
we derive power laws for decays (in time)
of transition probability amplitudes as well as two-point 
correlation functions. Heat semi-groups $e^{-\beta H}$ ($\beta >0$) 
generated by $H$ (under the condition that $H$ is bounded from below)  are
also considered. In Section 9 we present an abstract version
of Wigner's time-energy uncertainty relation \cite{Wigner}.
In Section10, we show that there exists an structure
producing successively representations of the GWWR. In Section 11, 
we construct concrete classes of representations
of the GWWR, using partial differential
operators acting in $L^2(\R^d)$ ($d\in \N$). We also find
generalized time operators for an abstract Dirac operator.
 In the last section
we present a tensor representation of the GWWR and
construct generalized time operators for second quantization
operators on Fock spaces (full Fock spaces,
boson Fock spaces, fermion Fock spaces).
This puts a basis to investigations of quantum field models
with interactions.

\section{Fundamental Properties of the GWWR}

Throughout this section, we assume that
$(T,H,K)$ obeys the GWWR in a Hilbert space ${\cal H}$ (Definition \ref{GWWR}).


\subsection{Elementary facts}

\begin{pr}\label{Kzero}
For all $t \in \R$,
$e^{-itH}D(T)=D(T)$ and the operator
equality
\begin{equation}
Te^{-itH}=e^{-itH}(T+K(t)) \label{s-time1}
\end{equation}
holds. Moreover
\begin{equation}
K(0)=0. \label{K0}
\end{equation}
\end{pr}

{\it Proof} : Taking $-t$ as the $t$ in Definition
\ref{GWWR}, we have
$e^{itH}D(T) \subset D(T)$ for all $t \in \R$.
Hence  $D(T) \subset 
e^{-itH}D(T)\subset D(T)$ for all $t \in \R$, which implies
that $e^{-itH}D(T)=D(T)$ for all $t \in \R$.
By definition, we have
$e^{-itH}(T+K(t))\subset Te^{-itH}$ for all $t \in \R$.
On the other hand, the preceding result
implies that $D(Te^{-itH})=D(T)$ for all $t \in \R$.
Thus (\ref{s-time1}) follows.
Letting $t=0$ in (\ref{s-time1}), we have $K(0)=0$ on $D(T)$. Since $D(T)$ is dense
and $K(0)$ is bounded, we obtain (\ref{K0}).
\hfill \qed

\bigskip


\begin{pr}\label{barT} Let $\bar T$ be the closure of $T$. Then
$(\bar T,H,K)$ obeys the GWWR.
\end{pr}

{\it Proof} : For each $\psi \in D(\bar T)$,
there exists a sequence $\{\psi_n\}_{n=1}^{\infty}\subset
D(T)$ such that $\lim_{n\to \infty}\psi_n=\psi$ and 
$\lim_{n\to \infty}T\psi_n=\bar T\psi$.
We have $Te^{-itH}\psi_n=e^{-itH}(T+K(t))\psi_n$.
The right hand side (r.h.s.) strongly  converges to
$e^{-itH}(\bar T+K(t))\psi$ as $n\to \infty$.
We have $e^{-itH}\psi_n \to e^{-itH}\psi$ as $n\to \infty$.
Hence $e^{-itH}\psi \in  D(\bar T)$ and
$\bar Te^{-itH}\psi=
e^{-itH}(\bar T+K(t))\psi$. Thus $(\bar T,H,K)$ obeys
the GWWR.
\hfill \qed

\bigskip

For a linear operator $A$, we denote by $\sigma(A)$ (resp. $\sigma_{\rm p}
(A)$) 
the spectrum (resp. the point spectrum) of $A$.

\begin{cor} For all $t \in \R$, 
$\sigma(T+K(t))=\sigma(T)$ and
$\sigma_{\rm p}(T+K(t))=\sigma_{\rm p}(T)$, where
the multiplicity of each $\lambda\in \sigma_{\rm p}(T)$ 
is equal to that of $\lambda\in \sigma_{\rm p}(T+K(t))$.
\end{cor}

{\it Proof} : Operator equality (\ref{s-time1}) means the unitary equivalence
of $T+K(t)$ and $T$. Hence, by a general theorem, the desired 
results follow.\hfill \qed



\bigskip

\begin{df}{\rm 
We say that a linear operator $L$  on ${\cal H}$ strongly commutes
with $H$ if $e^{-itH}D(L)\subset D(L)$ for all $t \in \R$ and
$e^{-itH}L\subset Le^{-itH}$.}
\end{df}

\begin{rem}{\rm
In the same way as in the proof of Proposition \ref{Kzero}, one
can show that $L$ strongly commutes with $H$ if and only if 
operator equality $e^{-itH}L=Le^{-itH}$ holds for all $t\in \R$.
}
\end{rem}

\begin{pr}\label{TS} Let $S$ be a symmetric operator on ${\cal H}$ strongly commuting 
with $H$ 
such that
$D(S)\cap D(T)$ is dense (hence $T+S$ is a symmetric operator with
$D(T+S):=D(T)\cap D(S)$). 
Then $(T+S,H,K)$ obeys the GWWR.
\end{pr}

{\it Proof} : A simple calculation. \hfill \qed

\bigskip
 



We denote by ${\sf B}({\cal H})$ the Banach space of
all bounded linear operators on ${\cal H}$
with domains equal to ${\cal H}$.


\begin{pr}\label{reduc}  For all $t \in \R$,
\begin{equation}
e^{itH}K(-t)+K(t)e^{itH}=0. \label{KH}
\end{equation}
In particular
\begin{equation}
\sigma(K(t))=\sigma(-K(-t)), \quad \sigma_{\rm p}
(K(t))=\sigma_{\rm p}(-K(-t)), \quad 
\forall t \in \R. \label{specK}
\end{equation}
\end{pr}


{\it Proof} : Let $t \in \R$.
In general, for all $W \in {\sf B}({\cal H})$ and every
densely defined linear operator $A$ on ${\cal H}$,
we have $(WA)^*=A^*W^*$ (operator equality).
Hence, by taking the adjoint of (\ref{s-time1}), 
we have 
$e^{itH}T^*\subset (T^*+K(t))e^{itH}$ for all $t\in \R$. 
Hence, for all $\psi\in D(T)$,
$$
e^{itH}T\psi =(T+K(t))e^{itH}\psi
=e^{itH}(T+K(-t))\psi+K(t)e^{itH}\psi,
$$
where we have used (\ref{s-time1}) to obtain the second equality.
This implies that
$e^{itH}K(-t)\psi+K(t)e^{itH}\psi=0$ for all $t \in \R$. 
Since $D(T)$ is dense, we obtain (\ref{KH}).
Operator equality (\ref{KH}) implies the unitary equivalence
of $K(t)$ and $-K(-t)$. Hence (\ref{specK}) follows.
\hfill \qed

\bigskip



\subsection{Nonself-adjointness of generalized time operators}

In this subsection we prove the following theorem:

\begin{The}\label{nons-a}  
Assume that $K:\R \to {\sf B}({\cal H})$ is strongly differentiable on $\R$ 
and
let
\begin{equation}
K'(t):=\mbox{{\rm s-}}\frac{dK(t)}{dt},
\end{equation}
the strong derivative of $K$ in $t\in \R$.
Suppose that $K'(0)\not=0$,
$H$ is
semi-bounded (i.e.,  bounded
from below or bounded from above) and
\begin{equation}
K(t)T \subset TK(t) \label{KT}
\end{equation}
for all $t \in \R$.
Then $T$  is  not self-adjoint.
\end{The}


\begin{rem}{\rm In the simple case $K(t)=t$, the fact stated in 
Theorem \ref{nons-a}
has been pointed out
in \cite{Miya1}.
}
\end{rem}

We need some preliminary results.



\begin{pr} 
Suppose that  
$T$ is self-adjoint.
Then, for all $t\in \R$, $T+K(t)$ is self-adjoint and
\begin{equation}
e^{-isT}e^{-itH}=e^{-itH}e^{-is(T+K(t))}, \quad \forall s, \forall t 
\in \R. \label{s-time2}
\end{equation}
\end{pr}

{\it Proof} : The self-adjointness of $T+K(t)$ follows from a simple
application of the Kato-Rellich theorem, since
$K(t)$ is bounded and self-adjoint.
By (\ref{s-time1}), we have
\begin{equation}
e^{itH}Te^{-itH}=T+K(t) \label{s-time3}
\end{equation}
as operator equality. Hence, by the functional
calculus, we have for all $s,t\in \R$
$$
e^{itH}e^{-isT}e^{-itH}=e^{-is(T+K(t))},
$$
which is equivalent to (\ref{s-time2}).
\hfill \qed

\bigskip

\begin{df}{\rm We say that two self-adjoint operators
$A$ and $B$ on
a Hilbert space strongly commute
if their spectral measures commute : 
$E_A(J_1)E_B(J_2)=E_B(J_2)E_A(J_1)$ for all
Borel sets $J_1,J_2$ in $\R$, where
$E_A$ (resp. $E_B$) denotes the spectral measure of $A$
(resp. $B$).
}
\end{df}

For characterizations of the strong commutativity
of two self-adjoint operators, we refer the reader to
\cite[Theorem VIII.13]{RS1}.


\begin{lem}\label{stcom} Let $A$ and $B$ be self-adjoint operators
on a Hilbert space. Suppose that $B$ is bounded and
$BA \subset AB$. Then $A$ and $B$ strongly commute.
\end{lem}


{\it Proof} : The assumption implies that
$BD(A) \subset D(A)$ and $BA\psi=AB\psi,\ \forall \psi \in D(A)$.
This implies that, for all $n\in \N$ and
$\psi\in D(A)$, $B^n\psi \in D(A)$ and $B^nA\psi=AB^n\psi$. For $N\in \N$ and $t \in 
\R$, we set  $e_N:=\sum_{n=0}^N(it)^nB^n/n!$. let $\psi \in D(A)$. Then
$e_NA\psi=Ae_N\psi$. It is easy to see that, for all $\phi \in {\cal H}$,
 $e_N\phi \to
e^{itB}\phi$ as $N\to \infty$.  Hence $Ae_N\psi \to e^{itB}A\psi$ as
$N\to \infty$. Since $A$ is closed, it follows that
$e^{itB}\psi \in D(A)$ and $Ae^{itB}\psi=e^{itB}A\psi$.
In particular, $e^{itB}D(A) \subset D(A)$ for all $t \in \R$ 
and hence, in fact, $e^{itB}D(A)=D(A)$. Therefore opeartor equality
$e^{itB}Ae^{-itB}=A$ follows. By the functional calculus, we have
$e^{itB}e^{isA}e^{-itB}=e^{isA}, \, s,t\in \R$.
This implies the strong commutativity of $A$ and $B$ (apply \cite[Theorem 
VIII.13]{RS1}). \hfill \qed

\bigskip

\begin{lem}\label{prod}
Let $A$ and $B$ be strongly commuting self-adjoint operators
on a Hilbert space. Then 
$A+B$ is essentially self-adjoint and,
for all $t \in \R$,
\begin{equation}
e^{it\overline{(A+B)}}=
e^{itA}e^{itB}=e^{itB}e^{itA}. \label{seki}
\end{equation}
\end{lem}


{\it Proof} : The essential self-adjointness of $A+B$
follows from the two variable functional calculus (note that
$\overline{A+B}=\int_{\R^2}(\lambda+\mu)dE_{A,B}(\lambda,\mu)$, where
$E_{A,B}$ is the two dimensional
spectral measure such that $E_{A,B}(J_1\times J_2)=
E_A(J_1)E_B(J_2)$ for all Borel sets $J_1,J_2$ in $\R$). 
Formula (\ref{seki}) follows from the Trotter product
formula \cite[Theorem VIII.31]{RS1} and
\cite[Theorem VIII.13(c)]{RS1}. \hfill \qed

\bigskip




\noindent
{\bf Proof of Theorem \ref{nons-a}}

\medskip

Suppose that $T$
 were  self-adjoint.
Then it follows from (\ref{KT}) and Lemma \ref{stcom} that $K(t)$  and $T$ strongly 
commute. 
Hence, by (\ref{s-time2}) and Lemma \ref{prod}, we have for all $s\in \R,t\in 
\R\setminus
\{0\}$
\begin{equation}
\left(\frac{e^{-itH}-1}{t} \right)e^{-isT}\psi
=
e^{-isT}\left(\frac{e^{-itH}-1}t\right)
\psi-e^{-itH}e^{-isT}\left(\frac{e^{-isK(t)}-1}t\right)
\psi \label{HTK}
\end{equation}
for all $\psi \in{\cal H}$.  We can write
$$
\frac{e^{-isK(t)}-1}t=-is \frac {K(t)}{t}+M_s(t)
$$
with 
$$
M_s(t):=
\sum_{n=2}^{\infty}\frac{(-is)^n}{n!}\frac {K(t)}{t}K(t)^{n-1}
$$
in the operator norm topology. 
By the strong differentiability of $K$ with
$K(0)=0$ (Proposition \ref{Kzero}) and the principle of
uniform boundedness, we have for each $\delta >0$
$$
C_{\delta} :=\sup_{|t|<\delta}\|K(t)/t\| < \infty, \quad 
D_{\delta} :=\sup_{|t|<\delta}\|K(t)\|<\infty.
$$
Hence, for all $\psi\in{\cal H}$ and $0< |t| <\delta$,
$$
\|M_s(t)\psi\| \leq 
C_{\delta}\left(\sum_{n=2}^{\infty}\frac{|s|^n}{n!}D_{\delta}^{n-2}
\right)\|K(t)\psi\|.
$$ 
Using this estimate and the fact that $K(0)=0$,  we obtain
$\lim_{t\to 0}M_s(t)\psi=0$. 
Hecne
$$
\lim_{t\to 0}\frac{e^{-isK(t)}-1}{t}\psi=-isK'(0)\psi.
$$
Let $\psi\in D(H)$. Then, by (\ref{HTK}), 
$$
\lim_{t\to 0}
\left(\frac{e^{-itH}-1}{t} \right)e^{-isT}\psi
=
e^{-isT}(-iH)\psi
-e^{-isT}(-is)K'(0)\psi,
$$
which implies that $e^{-isT}\psi \in D(H)$ and
$$
(-iH)e^{-isT}\psi
=
e^{-isT}(-iH)\psi-e^{-isT}(-is)K'(0)\psi,
$$
i.e.,
$$
e^{isT}He^{-isT}\psi=H\psi-sK'(0)\psi.
$$
Hence
\begin{equation}
\lang e^{-isT}\psi, He^{-isT}\psi\rang=
\lang \psi,H\psi\rang-s\lang \psi,K'(0)\psi\rang. \label{HTK1}
\end{equation}
Since $K'(0)$ is a non-zero bounded self-adjoint  operator
by the present assumption,
there exists a vector $\eta \in D(H)$ such that $\|\eta\|=1$ and 
$\lang \eta, K'(0)\eta\rang\not=0$. Equation (\ref{HTK1})
implies that 
$\sup_{s\in \R} \lang e^{-isT}\eta, He^{-isT}\eta\rang=\infty$ and $
\inf_{s\in \R} \lang e^{-isT}\eta, He^{-isT}\eta\rang=-\infty$.
Hence $H$ is not semi-bounded (note that
 $\|e^{-isT}\eta\|=1$ for all $s\in \R$).
Thus we are led to a contradiction.
\hfill \qed

\bigskip



\subsection{Construction of triples obeying the GWWR 
in direct sums of Hilbert spaces}

Let ${\cal H}_1$  be a Hilbert space and
${\cal F}:={\cal H}\oplus {\cal H}_1$.
Let $(T_1, H_1, K_1)$ be a triple obeying
the GWWR in ${\cal H}_1$. We define
\begin{equation}
\widetilde H:=H\oplus H_1=\left(
\begin{array}{cc}
H & 0 \\
0 & H_1
\end{array}
\right).
\end{equation}



\begin{pr}\label{extpro1}  
Let $A$ be a  bounded linear operator from ${\cal H}$ to
${\cal H}_1$ with $D(A)={\cal H}$ and 
\begin{equation}
\widetilde T:=\left(
\begin{array}{cc}
T & A^*\\
A& T_1
\end{array}
\right), \quad
\widetilde K(t):=\left(
\begin{array}{cc}
K(t) &  e^{itH}A^*e^{-itH_1}-A^*\\
e^{itH_1}Ae^{-itH}-A& K_1(t)
\end{array}
\right).
\end{equation}
Then $(\widetilde T,\widetilde H,\widetilde K)$ obeys the GWWR in ${\cal F}$.
\end{pr}


{\it Proof} : By the functional calculus, we have
$e^{-it\widetilde H}=e^{-itH}\oplus e^{-itH_1}$ for all $t\in\R$. Then
direct computations yield the desired  result. \hfill \qed

\bigskip

Note that, in Proposition \ref{extpro1}, $\widetilde T$ is not diagonal
if $A\not=0$. 
This procedure of 
construction of a new triple obeying the GWWR  
obviously yields an algorithm to
obtain a triple obeying the GWWR in the $N$ direct sum
$\oplus_{n=1}^N{\cal H}_n$ of Hilbert spaces ${\cal H}_n$ 
($N\geq 2$), provided that, for each $n$, 
a triple $(T_n,H_n,K_n)$ obeying the GWWR
in ${\cal H}_n$ is given.


\subsection{Perturbations}

Let $V$ be a symmetric operator on ${\cal H}$ and assume that
\begin{equation}
H(V):=H+V
\end{equation}
is essentially self-adjoint. It is an interesting
problem to investigate if there exist a symmetric operator $T_V$ and
an operator-valued function $K_V:\R \to {\sf B}({\cal H})$
such that $(T_V,\overline{H(V)},K_V)$ obeys the GWWR.
Here we present only an abstract answer to this problem.


 


\begin{pr}\label{perturb} Assume that the following conditions 
(i)--(iii) hold:
\begin{list}{}{}
\item[{\rm (i)}] The operators $T,H$ and $K(t)$ ($t\in \R$) are reduced by
a closed subspace ${\cal M}$ of ${\cal H}$.
We denote their reduced part by $T_{\cal M}, H_{\cal M}$ and
$K_{\cal M}(t)$ respectively.
\item[{\rm (ii)}] The operator $\overline{H(V)}$ is reduced
by a closed subspace ${\cal N}$ of ${\cal H}$.
\item[{\rm (iii)}] There exists a unitary operator $U:{\cal M}\to {\cal N}$
such that $UH_{\cal M}U^{-1}=\overline{H(V)}_{\cal N}$.
\end{list}
Let
\begin{equation}
T_V:=(UT_{\cal M}U^{-1})\oplus 0, \quad K_V(t):=(UK_{\cal M}(t)U^{-1})
\oplus 0
\end{equation}
relative to the orthogonal decomposition ${\cal H}={\cal N}\oplus 
{\cal N}^{\perp}$. Then $(T_V, \overline{H(V)},K_V)$ obeys
the GWWR.
\end{pr}

{\it Proof} : It is obvious that $T_V$ is symmetric and
$K_V(t)$ is a bounded self-adjoint operator.
By direct computations, one sees that  $(T_V, \overline{H(V)},K_V)$ obeys
the GWWR. \hfill \qed

\bigskip


\begin{rem}{\rm A method to find the unitary operator $U$ in Proposition 
\ref{perturb} is to use the method of wave operators
with respect to the pair $(H,\overline{H(V)})$. In that case, 
$U$ would be one of the wave operators
$W_{\pm}:=\mbox{\rm s-}\lim_{t\to \pm\infty}e^{it\overline{H(V)}}Je^{-itH}P_{\rm 
ac}(H)$ 
(if they exist) (
$P_{\rm ac}(H)$ is the orthogonal projection
onto the absolutely continuous space of $H$ and $J$ is a linear operator),
${\cal M}=(\ker W_{\pm})^{\perp}$ and ${\cal N}=\overline{\Ran (W_{\pm})}
$ (e.g., \cite[\S 4.2]{kuroda}, \cite[p.34, Proposition 4]{RS3}).
 This method was taken
in \cite{Miya1,Miya2}
in the case where $H$ is the 1-dimensional Laplacian and
$V$ is a real-valued function on $\R$ (hence $H(V)$ is a one-dimensional
Schr\"odinger operator). In the present paper we do not develop this aspect.
}
\end{rem}


\section{Transition Probability  Amplitudes and the Point Spectra of  Hamiltonians}


If a self-adjoint operator $H$  on a Hilbert space 
represents the Hamiltonian of a quantum system,
then the transition probability amplitude
of an initial state $\psi \in {\cal H}$ with $\|\psi\|=1$
to a state $\phi \in {\cal H}$ with $\|\phi\|=1$ at time $t\in \R$ is given
by $\lang \phi, e^{-itH}\psi\rang$. The square
$\left|\lang \phi, e^{-itH}\psi\rang\right|^2$ of its modulus is called
the transition probability from $\psi$ at time $0$ to $\phi$ at time $t$. 
In particular $\left|\lang \psi, e^{-itH}\psi\rang\right|^2$
is referred to as the survival probability, at time $t$,  of the state $\psi$. 

Let $(T,H,K)$ be a triple obeying the GWWR in a Hilbert space ${\cal H}$.
The following proposition is concerned with upper bounds of
the modulus of a transition probabilty amplitude  in time $t$.

\begin{pr}\label{trans}
Suppose that there is a constant $\alpha >0$ such that the strong limit
\begin{equation}
L_{\alpha}:=\mbox{\rm s-}\lim_{t\to \infty}\frac {K(t)}{t^{\alpha}} \in
{\sf B}({\cal H})
\end{equation}
exists. Let $S$ be a symmetric operator strongly commuting
with $H$. Then, for all $\psi,\phi \in D(T)\cap D(S)$ and $t >0$,
\begin{equation}
\left|\lang \psi, e^{-itH}L_{\alpha}\phi\rang\right|
\leq \frac{\|(T+S)\psi\|\|\phi\|+\|\psi\|\|(T+S)\phi\|}{t^{\alpha}}
+\|\psi\|\left\|\left(L_{\alpha}-\frac{K(t)}{t^{\alpha}}\right)\phi\right\|.
\label{trp}
\end{equation}
\end{pr}


{\it Proof} : Let $L=L_{\alpha}$ and $L(t):=K(t)/t^{\alpha}$. Then
\begin{eqnarray*}
\left|\lang \psi, e^{-itH}L\phi\rang\right| &\leq &
\left|\lang \psi, e^{-itH}(L-L(t))\phi\rang\right|
+\left|\lang \psi, e^{-itH}L(t)\phi\rang\right| \\
&\leq & \|\psi\| \|(L-L(t))\phi\|+
\left|\lang \psi, e^{-itH}L(t)\phi\rang\right|.
\end{eqnarray*}
By Proposition \ref{TS}, we have
$e^{-itH}L(t)\phi=t^{-\alpha}[(T+S)e^{-itH}\phi-e^{-itH}(T+S)\phi]$.
Using this relation, we have
$$
\left|\lang \psi, e^{-itH}L(t)\phi\rang\right|
\leq \frac{\|(T+S)\psi\|\|\phi\|+\|\psi\|\|(T+S)\phi\|}{t^{\alpha}}.
$$
Hence (\ref{trp}) follows. \hfill \qed

\medskip


\begin{rem}{\rm Proposition \ref{trans} is a generalization
of  \cite[Theorem 4.1]{Miya1} where the special case $K(t)=t$ is considered.}
\end{rem}

The following corollary is a generalized version of 
\cite[Corollary 4.3]{Miya1}:

\begin{cor} Suppose that the assumption of Proposition \ref{trans}
holds. Then, for all $\psi,\phi \in {\cal H}$,
\begin{equation}
\lim_{t\to \infty}\lang \psi, e^{-itH}L_{\alpha}\phi \rang=0. 
\label{t-infty}
\end{equation}
\end{cor}

{\it Proof} : The proof is similar to that
of \cite[Corollary 4.3]{Miya1}.
By the polarization identity, we need only to prove
(\ref{t-infty}) with $\phi=\psi$. Since $D(T)$ is dense,
there exists a sequence $\psi_n \in D(T)$ such that
$\lim_{n\to \infty}\psi_n=\psi$. Then, in the same way as
in the proof of the preceding proposition, we have
\begin{eqnarray*}
\left|\lang \psi, e^{-itH}L_{\alpha}\psi\rang\right|
&\leq &\|\psi-\psi_n\|\|L_{\alpha}\psi\|+
\|\psi_n\| \|L_{\alpha}\| \|\psi-\psi_n\| \\
&& \quad +\left|\lang \psi_n,e^{-itH}L_{\alpha}\psi_n\rang\right|.
\end{eqnarray*}
By (\ref{trp}) with $\psi=\phi=\psi_n$ and $S=0$, we have
$\lim_{t\to \infty}\left|\lang \psi_n,e^{-itH}L_{\alpha}\psi_n\rang
\right|=0$.
Hence
$$
\limsup_{t\to \infty}\left|\lang \psi, e^{-itH}\psi\rang\right|
\leq \|\psi-\psi_n\|\|L_{\alpha}\psi\|+
\|\psi_n\| \|L_{\alpha}\| \|\psi-\psi_n\|.
$$
Then, taking $n\to \infty$, we obtain (\ref{t-infty}) with $\phi=\psi$.
\hfill \qed

\bigskip

This corollary implies an interesting structure of the point spectrum of
$H$: 

\begin{cor}\label{pspecH} Suppose that the assumption of Proposition \ref{trans}
holds. Then, for all $E\in \R$, 
$\ker(H-E)\subset \ker L_{\alpha}$. 
In particular, if $\ker L_{\alpha}=\{0\}$, then
$\sigma_{\rm p}(H)=\emptyset$.
\end{cor}

{\it Proof} : Let $\psi_E\in \ker(H-E)$. Then $e^{itH}\psi_E=e^{itE}\psi_E$.
Taking $\psi=\psi_E$ in (\ref{t-infty}), we obtain
$\lang \psi_E,L_{\alpha}\phi\rang=0$ for all $\phi \in {\cal H}$.
This implies that $L_{\alpha}\psi_E=0$, i.e., $\psi_E \in \ker L_{\alpha}$.
\hfill \qed

\bigskip

\begin{rem}{\rm Corollary \ref{pspecH} is a generalization
of \cite[Corollary 4.3]{Miya1} where
the case $K(t)=t$ is considered.
}
\end{rem} 


\section{Connection with the Generalized Weak CCR and
Time-Energy Uncertainty  Relations}



Let $A,B$ be symmetric operators on a Hilbert space 
${\cal H}$ and $C \in{\sf B}({\cal H})
$ be a self-adjoint operator. We say that
$(A,B,C)$ obeys the {\it generalized weak CCR} (GWCCR) if
\begin{equation}
\lang A\psi, B\phi\rang-\lang B\psi, A\phi\rang=\lang \psi,
iC\phi\rang, \quad \forall \psi,\phi \in D(A)\cap D(B).\label{gwCCR}
\end{equation}
The case $C=I$ (the identity on ${\cal H}$) is the usual
CCR with one degree of freedom in the sense of sesquilinear form.

For a symmetric operator $A$ on a Hilbert space, a constant $a\in \R$ and
a unit vector $\psi \in D(A)$, we define
\begin{equation}
(\Delta A)_{\psi}(a):=\|(A-a)\psi\| \label{delta},
\end{equation}
an {\it uncertainty of $A$ in the state vector}
$\psi$. The quantity $(\Delta A)_{\psi}(a)$ with  $a=\lang \psi,A\psi\rang$
is the usual uncertainty of $A$ in the state vector $\psi$.
We set
\begin{equation}
(\Delta A)_{\psi}:=(\Delta A)_{\psi}(\lang \psi, A\psi\rang).
\end{equation}
We also introduce
\begin{equation}
\delta_C:=\inf_{\psi \in (\ker C)^{\perp},\|\psi\|=1}|\lang \psi, C\psi\rang|.
\end{equation}

\begin{pr}\label{uncert} Suppose that
$(A,B,C)$ obeys the GWCCR. 
Then, for all  $\psi\in D(A)\cap D(B)\cap (\ker  C)^{\perp}$ with 
$\|\psi\|=1$ and all $a,b\in \R$,
\begin{equation}
(\Delta A)_{\psi}(a)(\Delta B)_{\psi}(b)
\geq \frac {\delta_{C}}2. \label{ABUR}
\end{equation}
\end{pr}

{\it Proof} : It is easy to see that
$(A-a, B-b, C)$ also obeys
the GWCCR. Hence, applying a general fact
on a non-negative, sesquilinear form on a space of linear operators
\cite[Lemma 1]{PF}, we have for all $\psi \in D(A)\cap D(B)$
$$
(\Delta A)_{\psi}(a)(\Delta B)_{\psi}(b) \geq \frac 12 |\lang\psi, C\psi\rang|.
$$
Thus (\ref{ABUR}) follows.
\hfill \qed




\bigskip


\begin{pr}\label{specB}  Suppose that
$(A,B,C)$ obeys the GWCCR with $C\geq 0$. 
Then, for all $\lambda \in \R$,
$\ker(B-\lambda) \cap  D(A)\subset \ker C$ and
$\ker(A-\lambda) \cap  D(B)\subset \ker C$.
\end{pr}

{\it Proof} : Let $\psi \in \ker(B-\lambda)\cap D(A)$.
Then, taking $\phi=\psi$ in (\ref{gwCCR}), we have
$\lang \psi, C\psi\rang=0$. Since $C$ is nonnegative,
it follows that $C\psi=0$, i.e., $\psi \in \ker C$.
\hfill \qed

\bigskip

The following proposition gives a connection
of the GWWR with the GWCCR:

\begin{pr}\label{sw} Let $(T,H,K)$ be a triple obeying
the GWWR in ${\cal H}$.
 Assume that $K$ is strongly differentiable on $\R$. 
Then $(T,H,K'(0))$ obeys the GWCCR:
\begin{equation}
\lang T\psi, H\phi\rang-\lang H\psi,T\phi\rang=\lang\psi, iK'(0)\phi\rang,
\quad \psi,\phi\in D(T)\cap D(H). \label{THCCR}
\end{equation}
\end{pr}

{\it Proof} : Let $\psi, \phi \in D(T)\cap D(H)$.
Then we have by (\ref{s-time})
\begin{equation}
\lang T\psi, e^{-itH}\phi\rang=\lang e^{itH}\psi, T\phi\rang
+\lang e^{itH}\psi, K(t)\phi\rang. \label{wst}
\end{equation}
It is well known that, for all $\eta \in D(H)$,
 $e^{isH}\eta$ is strongly differentiable with
$\mbox{s-}de^{isH}\eta/ds=iHe^{isH}\eta=ie^{isH}H\eta$.
Hence the both sides of (\ref{wst}) are differentiable
in $t$. Evaluating the derivatives at $t=0$ and using (\ref{K0}), we obtain  
(\ref{THCCR}).
\hfill \qed

\bigskip


Propositions \ref{sw} and \ref{uncert} yield the following
result:



\begin{cor} Suppose that  
the same assumption as in Proposition \ref{sw} holds.
Then, for all  $\psi\in D(T)\cap D(H)\cap (\ker K'(0))^{\perp}$ with 
$\|\psi\|=1$ and all $t, E\in \R$,
\begin{equation}
(\Delta T)_{\psi}(t)(\Delta H)_{\psi}(E)
\geq \frac {\delta_{K'(0)}}2. \label{THUR}
\end{equation}
\end{cor}

\bigskip

In applications to quantum theory, (\ref{THUR}) gives a time-energy
uncertainty relation if $H$ is the Hamiltonian  of a quantum system.


\section{The Point Spectra of  Generalized Time Operators}

For a linear opeartro $L$ on a Hilbert apce ${\cal H}$, we introduce 
a subset of ${\cal H}$:
\begin{equation}
N_0(L):=\{\psi \in D(L)| \lang \psi,L\psi\rang=0\}.
\end{equation}
It is obvious that $\ker L\subset N_0(L)$.

\begin{rem}{\rm If $L$ is a non-negative self-adjoint operator, then
$N_0(L)=\ker L$.
}
\end{rem}



\begin{pr}\label{p-spec}
 Assume that
$(T,H,K)$ obeys the GWWR and $K$ is strongly differentiable
on $\R$.  Then, for all $E \in \R$, 
\begin{equation}
\ker(T-E)\subset N_0(K'(0)). \label{PT1}
\end{equation}
\end{pr}

{\it Proof} :  
Let $\psi_0 \in \ker(T-E)$.
Then
$T\psi_0=E\psi_0$.
Taking the inner product of $\psi_0$ with 
the vector obtained from the operation of (\ref{s-time1}) to
$\psi_0$, we have $\lang \psi_0,e^{-itH}K(t)\psi_0\rang=0, \ t\in \R$. 
Dividing the both sides by $t\not=0$ 
and taking the limit
$t \to 0$,
we obtain  $\lang \psi_0,K'(0)\psi_0\rang=0$, where
we have used (\ref{K0}). Hence $\psi_0\in 
N_0(K'(0))$. 
Thus (\ref{PT1}) follows.
\hfill\qed
\bigskip



\begin{cor}\label{sTp} 
 Assume that
$(T,H,K)$ obeys the GWWR and $K$ is strongly differentiable
on $\R$. Then:
\begin{list}{}{}
\item[{\rm (i)}] If
$N_0(K'(0))=\{0\}$, then $\sigma_{\rm p}(T)=\emptyset$.
\item[{\rm (ii)}] If $K'(0)\geq 0$ or $K'(0) \leq 0$, then 
$\sigma_{\rm p}(T|[D(T)\cap(\ker K'(0))^{\perp}])=\emptyset$.
\end{list}
\end{cor}

\bigskip

\begin{rem}{\rm Corollary (\ref{sTp}) is a generalization
of \cite[Corollary 4.2]{Miya1} where
the case $K(t)=t$ is considered.
}
\end{rem} 

\begin{rem}{\rm It may be interesting to note
that the behavior of $K(t)$ at $t=0$ and $t=\infty$ is
respectively  related to
$\sigma_{\rm p}(T)$ (Corollary \ref{sTp}) 
and $\sigma_{\rm p}(H)$ (Corollary \ref{pspecH}).
}
\end{rem}



\section{Commutation Formulas and Absolute Continuity}

In this section we prove commutation relations
derived from the GWWR. Moreover,
in the special case where the commutation factor $K(t)$ is
of the form $tC$  with $C$ a bounded self-adjoint operator, 
we show that $H$ is reduced by $\overline{\Ran(C)}$
($\Ran(C)$ denotes the range of $C$) and
its reduced part  
is absolutely continuous.


\subsection{General cases}

For $p \geq 0$,
we introduce  a class of Borel measurable functions on $\R$:
\begin{equation}
L_p^1(\R):=\left\{F:\R \to \C, \mbox{Borel measurable}
\big| \int_{\R}|F(t)|(1+|t|^p)dt < \infty\right\}.
\end{equation}
It is easy to see that $L_p^1(\R)$ includes
the space ${\cal S}(\R)$ of rapidly decreasing
$C^{\infty}$-functions on $\R$.

We say that a Borel measurable 
function $f$ is in the set 
${\cal M}_p$ if it is the  Fourier transform of
an element $F_f \in L_p^1(\R)$:
\begin{equation}
f(\lambda)=\frac{1}{\sqrt{2\pi}}\int_{\R}F_f(t)e^{-it \lambda}dt, \quad
\lambda \in \R.
\end{equation}
Note that, for each $f \in {\cal M}_p$, $F_f$ is uniquely determined.
We have
\begin{equation}
{\cal  S}(\R) \subset {\cal M}_p.
\end{equation}
Moreover,
every element $f$ of ${\cal M}_p$ is bounded, $[p]$ times continuously
differentiable ($[p]$ denotes the largest integer
not exceeding $p$)  and, for  $j=1,\cdots,[p]$,  $d^jf/d\lambda^j$ is bounded. 



Let $H$ be a self-adjoint operator on a Hilbert space
${\cal H}$ and $S:\R \to {\sf B}({\cal H})$
be Borel measurable such that, for all $\psi \in {\cal H}$,
$$
\|S(t)\psi\| \leq  c(1+|t|^p)\|\psi\|, \quad t\in \R
$$ 
with constants $c>0$ and $p \geq 0$ independent of $\psi$.
Then, for all $\psi \in {\cal H}$ and $f \in {\cal M}_p$, 
the strong integral
\begin{equation}
f(H,S)\psi:=\frac 1{\sqrt{2\pi}}\int_{\R}F_f(t)e^{-it H}S(t)\psi dt
\end{equation}
exists and $f(H,S) \in {\sf B}({\cal H})$.





\begin{The}\label{FCAL}  Assume that  $(T,H,K)$ obeys
the  GWWR. Suppose that $K$ is strongly continuous and,
for all $\psi \in {\cal H}$,
\begin{equation}
\|K(t)\psi\| \leq c(1+|t|^p)\|\psi\|,
\end{equation}
where $c>0$ and $p \geq 0$ are constants 
independent of $\psi$.
Let $f \in {\cal M}_p$.
Then, for all $\psi \in D(\bar T)$ , we have $f(H)\psi \in D(\bar T)$ and
\begin{equation}
\bar T f(H)\psi=f(H)\bar T\psi+f(H,K)\psi, \label{TfH}
\end{equation}
where $f(H):=\int_{\R}f(\lambda)dE_H(\lambda)$.
\end{The}


{\it Proof} :  Let $ f\in {\cal M}_p$ and $\psi\in D(T^*),
\phi \in D(\bar T)$.
Then, by the functional calculus of the self-adjoint operator $H$,
we have
$$
\lang T^*\psi, f(H)\phi\rang=
\frac 1{\sqrt{2\pi}}\int_{\R}F_f(t)
\lang 
T^*\psi, e^{-it H}\phi\rang dt.
$$
Since $e^{-it H}\phi \in D(\bar T)=D(T^{**})$ by Proposition 
\ref{barT}, it follows that
$\lang 
T^*\psi, e^{-it H}\phi\rang=
\lang 
\psi, \bar Te^{-it H}\phi\rang$. 
Using the fact that $(\bar T,H,K)$ obeys the GWWR (Proposition 
\ref{barT}),
we  see that the r.h.s. is equal to
$\lang \psi, e^{-it H}\bar T\phi\rang +
\lang \psi, e^{-it H}K(t)\phi\rang$. 
Hence we obtain
$$
\lang T^*\psi, f(H)\phi\rang=\lang \psi, f(H)\bar T\phi\rang
+\frac 1{\sqrt{2\pi}}\int_{\R}F_f(t)
\lang \psi, e^{-it H}K(t)\phi\rang dt.
$$
Therefore 
$\lang T^*\psi, f(H)\phi\rang=\lang \psi, f(H)\bar T\phi+f(H,K)\phi\rang$.
Since $\psi \in D(T^*)$ is arbitray, it follows that
$f(H)\phi \in D(T^{**})=D(\bar T)$ and (\ref{TfH}) holds.
\hfill \qed

\bigskip


\subsection{A special case}

In this section 
we consider a special case of a triple $(T,H,K)$ obeying
the GWWR in a Hilbert space ${\cal H}$: We assume that $K$ is of the form
\begin{equation}
K_C(t):=tC, \quad t\in \R \label{KC}
\end{equation}
with $C$ being a bounded self-adjoint operator
on ${\cal H}$.
In this case a more detailed analysis is possible as shown below.

We set
\begin{eqnarray}
C_{\rm b}^1(\R)&:=&\{f \in C^1(\R)|\mbox{$f$ and $f'$ are bounded}\}, \\
C_{{\rm b},+}^1(\R)&:=&\{f \in C^1(\R)
|\mbox{for some $a \in \R$,
$\sup_{\lambda \geq a}|f(\lambda)| < \infty$ and} \nonumber\\
&& \qquad \hskip 1cm 
\mbox{$\sup_{\lambda \geq a}|f'(\lambda)| < \infty$}\}.
\end{eqnarray}

\begin{The}\label{fc1}  Let $C$ be a bounded self-adjoint operator
on ${\cal H}$ and
suppose that $(T,H,$   $K_C)$ obeys the GWWR.
\begin{list}{}{}
\item[{\rm (i)}] Let $f \in C^1_{\rm b}(\R)$. Then
$f(H)D(\bar T) \subset D(\bar T)$ and
\begin{equation}
\bar Tf(H)\psi-f(H)\bar T\psi=if'(H)C\psi \label{Tf(H)}
\end{equation}
for all $\psi \in D(\bar T)$.
\item[{\rm (ii)}] Suppose that $H$ is bounded from below.
Then, for all $f \in C_{{\rm b},+}^1(\R)$,
the same conclusion as that of part (i) holds.
In particular, for all $z \in \C$ with $\Re z > 0$,
$e^{-zH}D(\bar T)\subset D(\bar T)$ and, for all
$\psi \in D(\bar T)$
\begin{equation}
\bar Te^{-zH}\psi-e^{-zH}\bar T\psi=
-ize^{-zH}C\psi. \label{Texp}
\end{equation}
\end{list}
\end{The}

{\it Proof} : (i) We  first
 consider the case  where $f \in {\cal M}_1$. 
Then it is easy to see that 
$f(H,K_C)=if'(H)C$. By this fact and  Theorem \ref{FCAL}, (\ref{Tf(H)})
holds. 

We next consider the case where 
$f$ is an arbitrary element in the set
$C_0^1(\R):=\{f\in C^1(\R)
|\mbox{${\rm supp}f$ is compact}\}$, 
where ${\rm supp}f$ means the support of $f$.
For a function $g $ on $\R$, we set
$$
\|g\|_{\infty}:=\sup_{\lambda\in \R}|g(\lambda)|.
$$
Using the Friedrichs mollifier, we can find
a sequence $\{f_n\}_{n=1}^{\infty} \subset C_0^{\infty}
(\R)$  (the set of infinitely differentiable functions
on $\R$ with compact support) 
such that 
$\sup_{n\geq 1}\|f_n\|_{\infty} < \infty, 
\sup_{n\geq 1}\|f_n'\|_{\infty} < \infty$ and, for all
$\lambda \in \R$,  
$f_n(\lambda) \to f(\lambda),  \ 
f'_n(\lambda) \to f'(\lambda) $ 
as $n \to \infty$.
Then, by using the functional calculus for $H$,  we see that
$$
f_n(H) \to f(H), \quad f'_n(H) \to f'(H)
$$ 
strongly as $n \to \infty$.
By the fact that $C_0^{\infty}(\R) \subset {\cal M}_1$ and 
the preceding result, we have 
$$
\bar Tf_n(H)\psi=f_n(H)\bar T\psi+if'_n(H)C\psi
$$ 
for all $\psi \in D(\bar T)$. Hence $\bar Tf_n(H)\psi \to
f(H)\bar T\psi +if'(H)C\psi$ as $n\to \infty$.
Since $\bar T$ is closed, it follows that
$f(H)\psi \in D(\bar T)$ and (\ref{Tf(H)}) holds.


Finally we consider the case where $f \in C_{\rm b}^1(\R)$.
Let $\chi \in C_0^{\infty}(\R)$
such that $\chi(0)=1$ and set $f_n(\lambda):=\chi(\lambda/n)f(\lambda)$.
Then $f_n \in C_0^1(\R)$ and
$$
f_n'(\lambda)=\frac 1n \chi'\left(\frac{\lambda}n\right)f(\lambda)
+\chi\left(\frac{\lambda}n\right)f'(\lambda).
$$
It is easy to see that, for all $\lambda \in \R$, 
$f_n(\lambda) \to f(\lambda), f_n'(\lambda) \to f'(\lambda)$
as $n \to\infty$ and 
\begin{eqnarray*}
&&|f_n(\lambda)|
\leq \|\chi\|_{\infty}\|f\|_{\infty}, \quad
|f'_n(\lambda)| \leq \|\chi'\|_{\infty}\|f\|_{\infty}
+\|\chi\|_{\infty}\|f'\|_{\infty}.
\end{eqnarray*}
Hence, by the functional calculus for $H$, we obtain
that $f_n(H) \to f(H), f'_n(H) \to f'(H)$ strongly as 
$n \to \infty$. Then, in the same manner as in the preceding
paragraph, we obtain the desired conclusion.


(ii)  Let $H \geq -M$ with a constant $M \geq 0$
and $f \in C_{{\rm b},+}^1(\R)$.
Let $\chi$ be as above and define $f_n(\lambda):=\chi(\lambda/n)f(
\lambda)$. Then $f_n \in C_0^1(\R)$.
Hence, by part (i), 
\begin{equation}
\bar Tf_n(H)\psi=f_n(H)\bar T\psi+if_n'(H)C\psi 
\end{equation}
for all $\psi \in D(\bar T)$.
We have for all $\psi \in {\cal H}$
\begin{equation}
\|f_n(H)\psi-f(H)\psi\|^2
=\int_{[-M,\infty)}F_n(\lambda)d\|E_H(\lambda)\psi\|^2.
\label{Fn}
\end{equation}
with
$$
F_n(\lambda):=
\left| \chi\left(\frac {\lambda}n\right)
-1\right|^2|f(\lambda)|^2.
$$
We have
$$
|F_n(\lambda)|
\leq \left( \|\chi\|_{\infty}+1\right)^2
\left(\sup_{\lambda \geq -M}|f(\lambda)|\right)^2, 
\quad \lambda \in [-M,\infty).
$$
Hence, by the Lebesgue dominated convergence theorem,
the r.h.s. of (\ref{Fn}) converges to $0$ as $n \to \infty$.
Hence 
$f_n(H) \to f(H)$ strongly as $n \to \infty$.
Similarly we can show that
$f_n'(H) \to f'(H)$.
Thus, in the same way as in part (i), we obtain
(\ref{Tf(H)}).

If $\Re z > 0$ ($z \in \C$), then the function $f_z:\R \to \C$
defined by $f_z(\lambda):=e^{-z\lambda}$ is in $C_{{\rm b},+}^1
(\R)$. Hence, by applying the preceding
result, we obtain (\ref{Texp}).
\hfill \qed

\medskip

\begin{rem}{\rm 
Corollary \ref{fc1} is a slight generalization
of \cite[3.1 Proposition 1]{Schmu}.}
\end{rem}


\begin{pr}\label{HC} 
 Let $C$ be a bounded self-adjoint operator
on ${\cal H}$ and
suppose that $(T,H$,  $K_C)$ obeys the GWWR. Then
$H$ and $C$ strongly commute.
\end{pr}

{\it Proof} : We can apply Proposition \ref{reduc} to obtain
that $e^{-itH}Ce^{itH}=C,\ \forall t \in \R$.
This implies the strong commutativity of $H$ and $C$.
\hfill\qed 



\bigskip

\begin{lem} Let $A$ and $B$ be strongly commuting
self-adjoint operators on a Hilbert space ${\cal H}$.
Then $\overline{\Ran(B)}$ reduces $A$.
\end{lem}

{\it Proof} : Since we have the orthogonal decomposition
${\cal H}=\ker B\oplus \overline{\Ran(B)}$, it is sufficient
to prove that $\ker B$ reduces $A$. Let $P$ be the
orthogonal projection onto $\ker B$. Then we have
$P=E_B(\{0\})$. Hence, by the strong commutativity of
$A$ and $B$,  $e^{itA}P=Pe^{itA}$ for all $t \in \R$.
This implies that $PA \subset AP$. Thus $\ker B$
reduces $A$. \hfill \qed

\bigskip


This lemma and Proposition \ref{HC} imply
the following fact:

\begin{cor}\label{HC2}
 Under the same assumption as in Proposition \ref{HC},
$H$ is reduced by $\overline{Ran(C)}$.
\end{cor}

\bigskip




As in the case of \cite[3.2 Corollary 2]{Schmu}, we have
from Proposition \ref{fc1} and Corollary \ref{HC2} 
the following theorem. For a self-adjoint operator $H$, we set
$$
E_H(\lambda):=E_H((-\infty,\lambda]), \quad \lambda \in \R.
$$


\begin{The}\label{ABC}
Suppose that $(T,H,K_C)$ obeys
the GWWR.
Then $H$ is reduced by $\overline{\Ran(C)}$ and  the reduced part 
$H|\overline{{\rm Ran}(C)}$ is absolutely continuous. Moreover, 
for all $\psi,\phi \in D(\bar T)$, the Radon-Nikodym derivative
$d\lang \psi, E_H(\lambda)C\phi\rang/d\lambda$
is given by
\begin{equation}
\frac {d\lang \psi, E_H(\lambda)C\phi\rang}{d\lambda}
=i\left(\lang \bar T\psi, E_H(\lambda)\phi\rang
-\lang E_H(\lambda)\psi, \bar T\phi\rang\right). \label{RND}
\end{equation}
\end{The}

{\it Proof} : The reducibility of $H$ by $\overline{\Ran(C)}
$ has already been 
proved  in Corollary \ref{HC2}.
Let $f \in {\cal S}(\R)$ and  $\psi,\phi \in D(\bar T)$.
It is easy to see that ${\cal S}(\R) \subset C_{\rm b}^1(\R)$.
Hence, by Proposition \ref{fc1}, we have
\begin{equation}
\lang \bar T \psi, f(H)\phi\rang -
\lang \psi, f(H)\bar T\phi\rang=i\lang \psi, f'(H)C\phi\rang.
\label{fhc}
\end{equation}
Let $\mu(\lambda):=
\lang \psi, E_H(\lambda)C\phi\rang$ and
$\sigma(\lambda):=
\lang \bar T\psi, E_H(\lambda)\phi\rang
-\lang E_H(\lambda)\psi, \bar T\phi\rang$. Then, by
(\ref{fhc}) and 
the spectral theorem, we have
$$
\int_{\R} f(\lambda)d\sigma(\lambda)=
i\int_{\R}f'(\lambda)d\mu(\lambda).
$$
Applying  to the l.h.s.
the  integration by parts formula on the Stieltjes integral,
we have
\begin{equation}
-\int_{\R}f'(\lambda)\sigma(\lambda)d\lambda=
i\int_{\R}f'(\lambda)d\mu(\lambda).
\end{equation} \label{lm}
It is easy to see that
the functional $\Phi:{\cal S}(\R) \to \C$ defined by
$$
\Phi(g):=i\int_{\R}g(\lambda)d\mu(\lambda)+
\int_{\R}g(\lambda)\sigma(\lambda)d\lambda, \quad g \in {\cal S}(\R)
$$
is a tempered distribution on $\R$. Eq.(\ref{lm}) implies that
$\Phi(f')=0$ for all $f \in {\cal S}(\R)$.
Hence $\Phi(f)=\alpha \int_{\R}f(\lambda)d\lambda$
with $\alpha$ being a constant. This implies that
the measure $\mu$ is absolutely continuous and
its Radon-Nikodym derivative $d\mu/d\lambda$ is given by
\begin{equation}
\frac {d\mu(\lambda)}{d\lambda}=-i\alpha +i\sigma(\lambda). \label{RND1}
\end{equation}
By a limiting argument, the absolute continuity
of $\mu$ can be extended to that of
$\langle \psi, E_H(\cdot)C$ 
$\times\phi\rangle$ for all $\psi, \phi \in {\cal H}$.
Hence, in particular,  $H|\overline{{\rm Ran}(C)}$ 
is absolutely continuous.
It follows from (\ref{RND1}) that, for all $\lambda\in 
\R$ 
$$
|\alpha|\leq |\mu(\lambda+1)-\mu(\lambda)|
+\sup_{\lambda' \in (\lambda,\lambda+1]}|\sigma(\lambda')|.
$$ 
Noting the fact that $
\lim_{\lambda \to -\infty}\mu(\lambda)=0,
\lim_{\lambda \to -\infty}\sigma(\lambda)=0$, we obtain 
$\alpha=0$. Hence (\ref{RND}) follows.
\hfill \qed

\medskip





\section{Absence of Minimum-Uncertainty States}

Let $(A,B,C)$ be a triple obeying the GWCCR.
A vector $\psi_0
\in D(A)\cap D(B) \cap (\ker C)^{\perp}$ with $\|\psi_0\|=1$
which attains the equality $(\Delta A)_{\psi_0}(\Delta B)_{\psi_0}=
{\delta_C}/{2} >0 $ 
in the uncertainty relation (\ref{ABUR}) with $a=\lang \psi_0,A\psi_0\rang
$ and $b=\lang \psi_0,B\psi_0\rang$
is called a {\it minimum-unertainty state} for 
$(A,B,C)$.
In this section, we give
a sufficient condition
for 
a triple $(T,H,C)$ to have no
minimum-unertainty states.

\begin{The}\label{abs}{\rm (Absence of 
minimum-uncertainty state)} Suppose that $(T,H,K_C)$ obeys
the GWWR with $T$ being  closed.
Assume that $H$ is 
bounded from below and that $C \geq 0$ with
$\delta_C > 0$. 
Then there exist no  vectors $\psi_0
\in D(H)\cap D(T)\cap (\ker C)^{\perp}$ with $\|\psi_0\|=1$
such that 
\begin{equation}
(\Delta T)_{\psi_0}(\Delta H)_{\psi_0}=
\frac{\delta_C}{2} >0.   \label{mini}
\end{equation} 
\end{The}


\begin{rem}{\rm  This theorem is
an extension of \cite[Theorem 5.1]{Miya1}, where
the case $C=I$ is considered. A new point here is that
one does not need to assume the analytic continuation
property of the weak Weyl relation
(the GWWR with  $C=I$) as is done in \cite[Theorem 5.1]{Miya1}.
}
\end{rem}

To prove Theorem \ref{abs}, we need two lemmas.

\begin{lem}\label{abs-lem} Assume that $(A,B,C)$ obeys the  GWCCR with
$C \geq 0$.
Suppose that there exists a unit vector  $\psi 
\in D(A)\cap D(B)\cap (\ker C)^{\perp}$ such that
\begin{eqnarray*}
&&(A+aB+b)\psi=0,\\
&& \lang A\psi, B\psi\rang+\lang B\psi, A\psi\rang
-2\lang \psi,A\psi\rang \lang \psi, B\psi\rang=0,
\end{eqnarray*}
where $a$ and $b$ are complex constants.
Then $\Re a=0$ and $\Im a >0$.
\end{lem}

{\it Proof} :  In  the same way as in the proof
of \cite[Lemma 5.3]{Miya1}, we can show that
$i\lang \psi, C\psi\rang=2a(\Delta B)_{\psi}^2$. 
By Proposition \ref{specB},
 $\psi $ cannot be an eigenvector
of $B$. Hence $(\Delta B)_{\psi}\not=0$.
Thus $\Re a=0$ and
$\lang \psi, C\psi\rang=2(\Im a)(\Delta B)_{\psi}^2$.
The l.h.s. on the second equation 
is strictly positive,  since
$C\geq 0$ and $\psi \in (\ker C)^{\perp}$.
Hence $\Im a > 0$.
\hfill \qed

\medskip


For a self-adjoint operator $A$ on a Hilbert space,
we define
\begin{equation}
E_0(A):=\inf\sigma(A),
\end{equation}
called the {\it ground state energy}
of $A$, provided that $A$ is bounded from below.
If $E_0(A)$ is an eigenvalue of $A$, then
a non-zero vector in $\ker(A-E_0(A))$ is called
a {\it ground state} of $A$.

By the variational principle, one has
\begin{equation}
E_0(A)=\inf_{\psi \in D(A), \|\psi\|=1}\lang \psi, A\psi\rang.
\end{equation}
The following lemma is well known (e.g., \cite[Theorem 6.16]{Arai97}).

\begin{lem}\label{var}
Let $A$ be a self-adjoint operator on a Hilbert space
and bounded from below. Suppose that
there exists a unit vector $\psi_0\in D(A)$ such that
$\lang \psi_0,A\psi_0\rang=E_0(A)$.
Then $A\psi_0=E_0(A)\psi_0$, i.e., $\psi_0$ is a ground state.
\end{lem}



\noindent
{\bf Proof of Theorem \ref{abs}}

\medskip

It is sufficient to
prove Theorem \ref{abs} in the case where $H \geq 0$.
Suppose that there existed a unit vector $\psi_0
\in D(T)\cap D(H) \cap (\ker C)^{\perp}$ satisfying (\ref{mini}).
Set $
\hat T:=T-\lang \psi_0,T\psi_0\rang,
\hat H:=H-\lang \psi_0,H\psi_0\rang$.
Then, from the derivation of (\ref{ABUR})\cite[Lemma 1]{PF},
we have
\begin{eqnarray}
\|\hat T\psi_0\|\|\hat H\psi_0\|
&=&|\lang \hat T\psi_0, \hat H\psi_0\rang| 
= |\Im 
\lang \hat  T\psi_0, \hat H\psi_0\rang| \nonumber \\
&=& \frac 12\lang \psi_0, C\psi_0 \rang
= \frac 12 \delta_C. \label{THEQ}
\end{eqnarray}
The first equality is the equality 
in the Schwarz inequality for
$|\lang \hat T \psi_0, \hat H\psi_0\rang|$.
By this fact and the condition $\delta_C>0$,
there exists a constant $c \in \C, c\not=0$ such that
\begin{equation}
\hat T \psi_0+c \hat H\psi_0=0. \label{psi1}
\end{equation}
The second equality in (\ref{THEQ}) implies that
$\Re\lang \hat  T\psi_0, \hat H\psi_0\rang=0$.
Hence
\begin{equation}
\lang T\psi_0, H\psi_0\rang
+
\lang H\psi_0, T\psi_0\rang-2
\lang \psi_0,T\psi_0\rang
\lang \psi_0,H\psi_0\rang=0. \label{psi2}
\end{equation}
The forth equality  in (\ref{THEQ}) and Lemma \ref{var} imply that
\begin{equation}
C\psi_0=\delta_C\psi_0. \label{Cpsi}
\end{equation}
By (\ref{psi1}), (\ref{psi2}) and Lemma \ref{abs-lem},
we have $\Re c=0$ and $\Im c > 0$.
Hence, let $c=ia$ with $a > 0$. Then
(\ref{psi1}) implies that
\begin{equation}
T\psi_0+ia H\psi_0+b\psi_0=0, \label{psi3}
\end{equation}
where $b$ is a constant. Hence
$\lang \psi_0, T\psi_0\rang+ia \lang \psi_0,H\psi_0\rang +b=0$,
which implies that
$\Im b=-a\lang \psi_0,H\psi_0\rang < 0$, since $H \geq 0$ and
$H^{1/2}\psi_0\not=0$ (see (\ref{psi1})).
Let $z \in \C$ with $\Re z > 0$. Then, by
(\ref{Texp}), (\ref{psi3}) and (\ref{Cpsi}), we have
$$
Te^{-zH}\psi_0=e^{-zH}(-iaH\psi_0-b\psi_0)
-iz\delta_Ce^{-zH}\psi_0. 
$$
Since  $\Re (ib/\delta_C) > 0$, we can take $z=ib/\delta_C$
so that
$
Te^{-zH}\psi_0=-iae^{-zH}H\psi_0$. 
Hence
$$
\lang e^{-zH}\psi_0, Te^{-zH}\psi_0\rang
=-ia
\lang e^{-zH}\psi_0, He^{-zH}\psi_0\rang.
$$
The l.h.s. is real and the r.h.s. is pure imaginary.
Hence the both sides must vanish.
Hence $\lang e^{-zH}\psi_0, He^{-zH}\psi_0\rang=0$, which
implies that $H^{1/2}\psi_0=0$. 
But this is a contradiction.
\hfill \qed

\bigskip





\section{Power Decays of Transition Probability 
Amplitudes in Quantum Dynamics}

In Section 3 we  have derived an estimate
for transition probability amplitudes in time $t$.
In this section we consider a triple $(T,H,K_C)$
obeying the GWWR (discussed  in Section 6.2) and show that, for state vectors in 
\lq\lq{smaller}" subspaces,
 transition probability amplitudes  
decay in powers of $t$ as $|t| \to \infty$.
We apply the results to
two-point correlation functions
of Heisenberg operators. We also discuss decays of heat semi-groups
$e^{-\beta H}$ 
on $\beta >0$ the inverse of the absolute temperature.

Let $H$ be a self-adjoint  operator
on a Hilbert space ${\cal H}$ and $C\not=0$ be a bounded
self-adjoint opeartor on ${\cal H}$. We introduce
a set of generalized time  operators:
\begin{equation}
{\sf T}(H,C):=
\{T| \mbox{$(T,H,K_C)$ obeys the GWWR}\}.
\end{equation}
By Proposition \ref{TS}, if $T\in {\sf T}(H,C)$,
then $T+S \in {\sf T}(H,C)$ for all
symmetric operators $S$ on ${\cal H}$
strongly commuting with $H$ such that $D(T)\cap D(S)$ 
is dense in ${\cal H}$.



\subsection{A simple case}




\begin{The}\label{trans1} 
Let $T \in {\sf T}(H,C)$ and
$\psi,\phi \in D(T)$.
Then, for all $t\in \R\setminus\{0\}$,
\begin{equation}
\left|\lang \phi, e^{-itH}C\psi\rang\right|
\leq \frac 1{|t|} (\|T\phi\|\|\psi\|+\|\phi\|\|T\psi\|). \label{decay}
\end{equation}
\end{The}


{\it Proof} : In the present case, we have 
$L_{\alpha}=C$ with $\alpha=1$. Hence Proposition
\ref{trans} gives the desired result. \hfill \qed

\bigskip


\begin{rem}{\rm For vectors
$\phi,\psi \in D(T)$, we can define  a set of operators 
$$
{\sf T}_{\phi,\psi}(H,C)
:=\{T \in {\sf T}(H,C)| \phi, \psi \in D(T)\}
$$
and put
$$
c_{\phi,\psi}:=
\inf_{T \in {\sf T}_{\phi,\psi}(H,C)}
(\|T\phi\|\|\psi\|+\|\phi\|\|T\psi\|),
$$
then (\ref{decay}) implies that
\begin{equation}
\left|\lang \phi, e^{-itH}C\psi\rang\right|
\leq \frac {c_{\phi,\psi}}{|t|}. 
\end{equation}
}
\end{rem}


\begin{rem}{\rm  Let $T\in {\sf T}(H,C)$.
Then, for all $\psi \in D(T)$ with
$\|\psi\|=1$, 
 $T-\lang \psi, T\psi\rang$ is in the set ${\sf T}(H,C)$.
 Hence (\ref{decay}) implies that
\begin{equation}
\left|\lang \psi, e^{-itH}C\psi\rang\right|^2
\leq \frac {4(\Delta T)_{\psi}^2}{t^2}. \label{suvdecay}
\end{equation}
Hence Theorem \ref{trans1} gives a  generalization
of \cite[Theorem 4.1]{Miya1}.
}
\end{rem}






\bigskip

\subsection{Higher order dcays in
smaller subspaces}

As demonstrated in a concrete example
\cite[Proposition 3.2]{Miya1}, 
the modulus of a transition  probability amplitude
$\left|\lang\phi,e^{-itH}\psi\rang\right|$ 
may decay faster than $|t|^{-1}$ as $|t| \to \infty$ 
for a class of vectors $\phi$ and $\psi$.
In this subsection we investigate this aspect
in an abstract framework and show that
$\left|\lang\phi,e^{-itH}\psi\rang\right|$ 
decays faster than $|t|^{-1}$
for all  $\phi$ and $\psi$ 
in  smaller  subspaces.



\begin{lem}\label{TnH} Let $T \in {\sf T}(H,C)$.
Suppose that 
\begin{equation}
CT \subset TC. \label{CT}
\end{equation}
Then, for all $n \in \N$ and $t \in\R$,
\begin{equation}
e^{-itH}D(T^n)=D(T^n) \label{t-inv}
\end{equation}
\begin{equation}
T^ne^{-itH}=e^{-itH}(T+tC)^n. \label{Tn}
\end{equation}
\end{lem}



{\it Proof} :  By (\ref{s-time3}) 
we have the operator equality
$e^{itH}T^ne^{-itH}=(T+tC)^n$. 
Condition (\ref{CT}) implies that
$D((T+tC)^n)=D(T^n)$.
Hence \ref{t-inv}) follows and
(\ref{Tn}) holds.
\hfill \qed

\bigskip





\begin{The}\label{tn} Let $T \in {\sf T}(H,C)$.
Assume (\ref{CT}).
Let $n \in \N$ and $\psi,\phi \in D(T^n)$.
We define constants $d_k^T(\phi,\psi), k=1,\cdots,n$
by the following recursion relation:
\begin{eqnarray}
d_1^T(\phi,\psi)&:=& \|T\phi\|\|\psi\|+\|\phi\|\|T\psi\|, \\
d_n^T(\phi,\psi)&:=&
\|T^n\phi\|\|\psi\|+\|\phi\|\|T^n\psi\| 
 +\sum_{r=1}^{n-1}{}_nC_r
d_{n-r}^T\left(\phi,T^r\psi\right), \ 
 n \geq 2,
\end{eqnarray}
where ${}_nC_r:=
n!/[(n-r)!r!]$. 
Then, for all $t \in \R\setminus \{0\}$,
\begin{equation}
\left|\lang \phi, e^{-itH}C^n\psi\rang\right|
\leq \frac {d_n^T(\phi,\psi)}{|t|^n}. 
\label{dec1}
\end{equation}
\end{The}



{\it Proof} :  We prove (\ref{dec1}) by induction in $n$.
The case $n=1$ holds by Theorem \ref{trans1}.
Suppose that (\ref{dec1}) holds for $n=1,\cdots,m-1$ ($m \geq 2$).
Let $\psi,\phi \in D(T^{m})$.
Condition (\ref{CT})  implies that,
for all $k,r \in \N$,
$C^kT^r\subset T^rC^k$. 
By this fact and Lemma \ref{TnH}, we have
\begin{eqnarray*}
T^{m}e^{-itH}\psi&=&e^{-itH}(T+tC)^{m}\psi\\
&=&e^{-itH}T^{m}\psi+
\sum_{r=1}^{m-1}{}_mC_r
t^{m-r}e^{-itH}C^{m-r}T^{r}\psi
+t^{m}e^{-itH}C^m\psi.
\end{eqnarray*}
Hence
\begin{eqnarray*}
|t|^{m}\left|\lang \phi, e^{-itH}C^m\psi\rang\right|
&\leq&
\|T^{m}\phi\|\|\psi\|+ \|\phi\|\|T^{m}\psi\| \\
&&+\sum_{r=1}^{m-1}{}_mC_r
|t|^{m-r}\left|\lang \phi,e^{-itH}C^{m-r}T^r\psi\rang\right|.
\end{eqnarray*}
By the induction hypothesis, we have
$$
|t|^{m-r}\left|\lang \phi,e^{-itH}C^{m-r}T^r\psi\rang\right|
\leq  
d_{m-r}^T(\phi,T^r\psi).
$$
Hence
(\ref{dec1}) with $n=m$ follows.
\hfill \qed

\bigskip


Theorem \ref{tn} can be generalized. We need a lemma.



\begin{lem}\label{nT}
Let $T_1,\cdots, T_n \in {\sf T}(H,C)$ ($n\in \N$) such that
$CT_j\subset T_jC, \ j=1,\cdots,n$.
Then, for all $t \in \R$, 
\begin{equation}
e^{-itH}
D((T_1+tC)\cdots (T_n+tC))
\subset D(T_1\cdots T_n) \label{Dn1}
\end{equation}
and, for all $\psi \in 
D((T_1+tC)\cdots (T_n+tC))$,
\begin{equation}
T_1\cdots T_ne^{-itH}\psi=
e^{-itH}
(T_1+tC)\cdots (T_n+tC)\psi.\label{T-n}
\end{equation}
\end{lem}

{\it Proof} : We prove the assertion by induction in $n$.
The case $n=1$  obviously holds.
Suppose that, for an  $m\in\N$, (\ref{Dn1}) and (\ref{T-n})
hold. Let $\phi \in
D((T_1+tC)\cdots (T_{m+1}+tC))$. Then the vector $(T_{m+1}+tC)\phi$
is in 
$D((T_1+tC)\cdots (T_{m}+tC))$. Hence, by the induction hypothesis,
$e^{-itH}(T_{m+1}+tC)\phi$ is in 
$D(T_1\cdots T_m)$ and
$$
T_1\cdots T_m
e^{-itH}(T_{m+1}+tC)\phi=
e^{-itH}
(T_1+tC)\cdots (T_{m+1}+tC)\phi.
$$
On the other hand, $e^{-itH}\phi$ is in $D(T_{m+1})$
and $T_{m+1}e^{-itH}\phi=
e^{-itH}(T_{m+1}+tC)\phi$. Hence $T_{m+1}e^{-itH}\phi$ is in
$D(T_1\cdots T_m)$, i.e., $e^{-itH}\phi \in D(T_1\cdots T_{m+1})$, and
(\ref{T-n}) with $n=m+1$ holds.
Thus the assertion holds for $n=m+1$.
\hfill \qed

\bigskip



\begin{The}\label{ntop} 
Let $T,T_1,\cdots, T_n \in {\sf T}(H,C)$ such that
$CT\subset TC, CT_j\subset T_jC$,  $j=1,\cdots,n$.
Let $\phi \in  D(T_n\cdots T_1)\cap D(T^{n-1})$ and
$\psi \in \cap_{r=1}^{n-1}
\cap_{1 \leq i_1 < \cdots <i_r \leq n}D(T^{n-r}T_{i_1}\cdots T_{i_r})$.
For $k=1,\cdots, n$, we define a constant
\begin{eqnarray}
\delta_n^T(\phi,\psi;T_1,\cdots, T_n)
&:=&\|T_n\cdots T_1\phi\|\|\psi\|+\|\phi\| \|T_1\cdots T_n\psi\| \\
&& +\sum_{r=1}^{n-1}\sum_{1 \leq 
i_1<\cdots <i_r \leq n} d_{n-r}^{T}
(\phi,T_{i_1}\cdots T_{i_r}\psi).
\end{eqnarray}
Then,
for all $t \in \R\setminus \{0\}$,
\begin{equation}
\left|\lang \phi, e^{-itH}C^n\psi\rang\right|
\leq \frac {\delta_n^T(\phi,\psi;T_1,\cdots,T_n)}
{|t|^n}. \label{decay3}
\end{equation}
\end{The}

{\it Proof} :
By Lemma \ref{nT}, we have
\begin{eqnarray*}
T_1\cdots T_{n}e^{-itH}\psi
&=&e^{-itH}T_1\cdots T_{n}\psi\\
&& +\sum_{r=1}^{n-1} t^{n-r}
\sum_{1 \leq i_1 < \cdots <i_r \leq 
n}e^{-itH}C^{n-r}T_{i_1}\cdots T_{i_r}\psi
+t^{n}e^{-itH}C^n\psi.
\end{eqnarray*}
Hence
\begin{eqnarray*}
|t|^{n}\left|\lang \phi,e^{-itH}C^n\psi\rang\right|
&\leq &
\|T_{n}\cdots T_1\phi\|\|\psi\|+
\|\phi\|\|T_1\cdots T_{n}\psi\| \\
&&+
\sum_{r=1}^{n-1}
|t|^{n-r}
\sum_{1 \leq i_1 < \cdots <i_r \leq 
n}
\left|\lang \phi,e^{-itH}C^{n-r}T_{i_1}\cdots T_{i_r}\psi\rang\right.
\end{eqnarray*}
Note that $\phi,\,T_{i_1}\cdots T_{ir}\psi \in D(T^{n-r})$
for $r=1,\cdots,n-1$. Hence,
by Theorem \ref{tn}, we have
$$
|t|^{n-r}
\left|\lang \phi,e^{-itH}C^{n-r}T_{i_1}\cdots T_{i_r}\psi\rang\right|
\leq d_{n-r}^T(\phi,T_{i_1}\cdots T_{i_r}\psi).
$$
Thus (\ref{decay3}) follows.
\hfill \qed
\bigskip



Finally we discuss the case where condition (\ref{CT})
is not necessarily satisfied.
For $n \geq 2$ and $r=1,\cdots,n-1$, we introduce a set
\begin{equation}
{\sf J}_{n,r}:=\{j:=(j_1,\cdots, j_{r+1})
\in \{0,1\}^{r+1}|j_1+\cdots+j_{r+1}=n-r\}
\end{equation}
and, for each $j\in {\sf J}_{n,r}$, we define 
\begin{equation}
K_{n,r}^{(j)}:=T^{j_1}CT^{j_2}C\cdots CT^{j_r}CT^{j_{r+1}}.
\end{equation}
Let
\begin{eqnarray}
&&{\cal D}_n(T,C)\nonumber\\
&&:=\bigg\{\psi\in D(T^n)\cap 
\left(\cap_{r=1}^{n-1}\cap_{j\in {\sf J}_{n,r}}D(K_{n,r}^{(j)})\right)
\bigg| K_{n,r}^{(j)}\psi
\in {\rm Ran}(C^r|D(T^r)),\nonumber\\
&& \hskip 3cm j \in {\sf J}_{n,r},\,
 r=1,\cdots, n-1\bigg\} \quad (n\geq 2). \label{Dn}
\end{eqnarray}
We set ${\cal D}_1(T,C):=D(T)$.

\begin{rem}{\rm If (\ref{CT}) holds, then
${\cal D}_n(T,C)=D(T^n)$.
}
\end{rem}


\begin{The}\label{trans2} Let $T \in {\sf T}(H,C)$.
Then, for all $\phi
\in D(T^n)$ and 
$\psi \in {\cal D}_n(T,C)$ and $t \in \R\setminus\{0\}$,
\begin{equation}
|\lang \phi, e^{-itH}C^n\psi\rang|
\leq \frac {d_n(\phi,\psi)}{|t|^n}, \label{decay4}
\end{equation}
where $d_n(\phi,\psi) >0$ is a constant independent of $t$.
\end{The}

{\it Proof} : We prove (\ref{decay4}) by induction in $n$.
The case $n=1$ is just Theorem \ref{trans1}.
Suppose that (\ref{decay4}) holds with $n=1,\cdots,m-1$ ($m \geq 2$).
For all $\psi \in {\cal D}_m(T,C)$ we have
$$
T^me^{-itH}\psi
=e^{-itH}T^m\psi+t^me^{-itH}C^m\psi
+\sum_{r=1}^{m-1}t^r\sum_{j\in {\sf J}_{m,r}}e^{-itH}K_{m,r}^{(j)}\psi.
$$
Hence, for all $\phi \in D(T^m)$,
\begin{eqnarray*}
|t|^m\lang \phi, e^{-itH}C^m\psi\rang
&\leq &\|T^m\phi\| \|\psi\|
+\|\phi\| \|T^m\psi\|\\
&& +\sum_{r=1}^{m-1}
|t|^r
\sum_{j\in {\sf J}_{m,r}}
\left|\lang \phi, e^{-itH}K_{m,r}^{(j)}\psi\rang\right|.
\end{eqnarray*}
Since $K_{m,r}^{(j)}\psi \in {\rm Ran}(C^r|D(T^r))$, there is a vector
$\chi_j\in D(T^r)$ such that $K_{m,r}^{(j)}\psi=C^r\chi_j$.
By this fact and the induction hypothesis, we have
$|t|^r\left|\lang \phi, e^{-itH}K_{m,r}^{(j)}\psi\rang
\right|
\leq c_{m,r}$ 
with a constant $c_{m,r}$ independet of $t$.
Thus (\ref{decay4}) with $n=m$ follows.
\hfill \qed

\bigskip


\subsection{Correlation functions}

In this subsection, we show that 
the existence of generalized  time-operators gives
upper bounds for correlation functions
for a class of linear operators.
For a linear operator $A$ on ${\cal H}$ and a self-adjoint operator $H$
on ${\cal H}$, we define
\begin{equation}
A(t):=e^{itH}Ae^{-itH}, \quad t\in \R,
\end{equation}
the Heisenberg operator of $A$ with respect to
$H$. Let $B$ be a linear operator on ${\cal H}$.
Let
$$
\psi \in \cap_{t \in \R}[D(Ae^{-itH})\cap D(Be^{-itH})]
$$
with $\|\psi\|=1$.
Then
we can define
\begin{equation}
W(t,s;\psi):=\lang A(t)\psi, B(s)\psi\rang, \quad s,t\in \R.
\end{equation}
We call it the {\it two-point correlation function} of $A$ and $B$ 
with repect to the  vector $\psi$.



\begin{The}\label{corr} 
Let $T\in {\sf T}(H,C)$.
Suppose  that $\psi$ is an eigenvector of $H$
such that $A\psi \in D(T)$ and
$B\psi \in \Ran(C|D(T))$. 
Then, for all $t,s\in \R$ with $t\not=s$,
\begin{equation}
|W(t,s;\psi)|
\leq \frac {c_{A,B,T}}{|t-s|},
\label{decay5}
\end{equation}
where 
$$
c_{A,B,T}:=\inf_{\chi \in D(T), B\psi=C\chi}
\|TA\psi\|\|\chi\|+\|A\psi\|\|T\chi\|.
$$
\end{The}

{\it Proof} :  Let $E$ be the eigenvalue of $H$ with eigenvector
$\psi$. Then we have
\begin{equation}
W(t,s)=e^{i(t-s)E}
\lang A\psi, e^{-i(t-s)H}B\psi\rang. \label{Wrep}
\end{equation}
There exists a vector $\chi \in D(T)$ such that $B\psi=C\chi$.
Hence, applying Theorem \ref{trans1},
we obtain 
$$
|W(t,s;\psi)|\leq
\frac{\|TA\psi\|\|\chi\|+\|A\psi\|\|T\chi\|}{|t-s|}.
$$
Thus (\ref{decay5}) follows. \hfill \qed
\bigskip


Theorem \ref{corr}  can be strengthened:


\begin{The}\label{corr1} 
Let $T\in {\sf T}(H,C)$  with (\ref{CT}).
Suppose  that $\psi$ is an eigenvector of 
$H$  such that $\psi\in D(A)$ and
$A\psi \in D(T^n)$ and $B\psi \in {\rm Ran}(C^n|D(T^n))$. 
Then, for all $t,s\in \R$ with $t\not=s$,
\begin{equation}
|W(t,s)|
\leq \frac {c_{A,B,T}^{(n)}}{|t-s|^n},
\label{decay6}
\end{equation}
where 
$$
c_{A,B,T}^{(n)}
:=\inf_{\chi\in D(T^n), B\psi=C^n\chi}d_n^T(A\psi, \chi).
$$
\end{The}

{\it Proof} :  This follow from (\ref{Wrep}) and an application of
Theorem \ref{tn}.\hfill \qed

\bigskip


\subsection{Heat semi-groups}

In this subsection we assume the following:

\begin{list}{}{}
\item[{\bf Hypothesis (H)}]
The self-adjoint operator $H$ is bounded from below
and there exists a closed symmetric operator $T$
such that $T \in {\sf T}(H,C)$.
\end{list}
Then (\ref{Texp}) holds.
We set
\begin{equation}
\hat H:=H-E_0(H).
\end{equation}
We have for all $\beta \geq 0 $
$$
\|e^{-\beta\hat H}\|=1.
$$
Hence, in the same way as in Subsections 8.1 and 8.2,
we obtain the following results on 
the decay (in $\beta$) 
of the quantity $\lang \phi,e^{-\beta\hat H}C\psi\rang$.

\begin{The} Let $\phi,\psi\in D(T)$. Then, for all $\beta > 0$,
\begin{equation}
\left|\lang \phi,e^{-\beta\hat H}C\psi\rang\right|
\leq \frac 1{\beta} (\|T\phi\|\|\psi\|+\|\phi\|\|T\psi\|).
\end{equation}
\end{The}

\begin{The} Assume (\ref{CT}). Then, for all
$\phi,\psi \in D(T^n)$ and $\beta > 0$, 
\begin{equation}
\left|\lang \phi,e^{-\beta\hat H}C^n\psi\rang\right|
\leq \frac {d_n^T(\phi,\psi)}{\beta^n}.
\end{equation}
\end{The}

\begin{The} For all
$\phi\in D(T^n)$, $\psi \in {\cal D}_n(T,C)$ and $\beta > 0$,
\begin{equation}
\left|\lang \phi,e^{-\beta\hat H}C^n\psi\rang\right|
\leq \frac {c_n}{\beta^n},
\end{equation}
where $c_n$ is a constant independent of $\beta$.
\end{The}





\section{
Abstract Version of Wigner's  Time-Energy
Uncertainty Relation}

In this section we apply Theorems \ref{trans1} and \ref{trans2}
to establish 
an abstract version of Wigner's  time-energy
uncertainty relation
\cite{Wigner}.

Let $H$ be a self-adjoint operator on  a Hilbert space ${\cal H}$.
In the context of quantm mechanics where $H$ is interpreted
as the Hamiltonian of a quantum system,
the state vector at time $t \in \R$ is given by
\begin{equation}
\psi_(t):=e^{-itH}\psi
\end{equation}
with $\psi \in {\cal H}$ being the initial state.
Let $\phi_0 \in {\cal H}$ and suppose that
\begin{equation}
\int_{\R}t^2|\lang \phi_0, \psi(t)\rang|^2dt < \infty.
\label{finite1}
\end{equation}
Then 
\begin{equation}
N_0:=\int_{\R}|\lang \phi_0, \psi(t)\rang|^2dt 
\end{equation}
is finite and we can define
\begin{eqnarray}
&& \lang t\rang
:=\frac{\int_{\R}t|\lang \phi_0, \psi(t)\rang|^2dt }
{N_0}, \\
&&\Delta t:=\left(\frac{\int_{\R}
(t-\lang t\rang)^2
|\lang \phi_0, \psi(t)\rang|^2
dt }{N_0}\right)^{1/2}.
\end{eqnarray}
Physically $\lang t\rang$ may be interpreted as
the expectation value of the \lq\lq{arrival time}", i.e.,
the time $t$ at when
the state $\psi(t)$ \lq\lq{arrives}" at $\phi_0$.
In this interpretation, 
$\Delta t$ expresses the standard deviation
of the arrival time of the intial state $\psi$ to 
the state $\phi_0$.


We define
\begin{equation}
f_H(t;\phi_0,\psi)):=\left\{
\begin{array}{ll}
\lang \phi_0, H\psi(t)\rang \ ; \mbox{if $\psi \in D(H)$} \\
\lang H\phi_0,\psi(t)\rang \ ; \mbox{if $\phi_0 \in D(H)$} 
\end{array}
\right..
\end{equation}
We assume that 
$\phi_0 \in D(H)$ or $\psi \in D(H)$
and
\begin{equation}
\int_{\R}|f_H(t;\phi_0, \psi)|^2dt < \infty. \label{finite2}
\end{equation}
Then we can define
\begin{eqnarray}
&&\lang {\cal E}_H\rang:=
 \frac{\int_{\R}
\lang \psi(t),\phi_0\rang f_H(t;\phi_0,\psi)  dt}{N_0}, \\
&&\Delta {\cal E}_H:=
\left(\frac{\int_{\R}|f_H(t;\phi_0,\psi)|^2dt}{N_0}
-\lang {\cal E}_H\rang^2 \right)^{1/2}.
\end{eqnarray}


\begin{The}\label{Wig} 
Suppose that $\phi_0 \in D(H)$ or $\psi\in D(H)$ and that
(\ref{finite1}) and (\ref{finite2}) hold.
Then
\begin{equation}
\Delta t \cdot
\Delta {\cal E}_H \geq \frac{1}{2}. \label{teur}
\end{equation}
\end{The}

{\it Proof} : 
From the Schr\"odinger representation
$(\widehat E, \widehat t)$ of the CCR in $L^2(\R)$, where $\widehat t$ 
is the multiplication operator by the coordinate function $t\in \R$
and $\widehat E:=iD_t$ with $D_t$ being  the generalized differential
operator in the variable $t$, we have the standard uncertainty
relation
$$
\left\|\left(\widehat E-\lang u, \widehat Eu
\rang\right)u\right\|\cdot \left\|
\left(\widehat t-\lang u,\widehat tu\rang\right)u\right\| \geq \frac{1}{2}
$$
for all $u \in D(\widehat E)\cap D(\widehat t)$ with $\|u\|=1$.
We define  $f:\R \to \C$ by 
$f(t):=\lang \phi_0,\psi(t)\rang$. Then, by (\ref{finite1}) and
(\ref{finite2}),
$f $ is  in  $D(\widehat t)\cap D(\widehat E)$ with
$\widehat Ef(t)=f_H(t;\phi_0, \psi)$. 
Let $\widetilde f(t):= f(t)/\|f\|$. 
Then it is easy to see that
$$
\Delta {\cal E}_H=
\left\|\left(\widehat E-\lang \widetilde f, \widehat E
\widetilde f\rang\right)\widetilde f\right\|, \quad \Delta t:=
\left\|\left(
\widehat t-\lang \widetilde f,\widehat t \widetilde f\rang\right)\widetilde f
\right\|.
$$
Hence (\ref{teur}) follows.
\hfill \qed

\bigskip


\begin{The} Let $T \in {\sf T}(H,C)$.
Suppose that 
$\psi=C^2\chi$
with $\chi \in D(T^2)\cap D(TC)$
satisfying $TC\chi \in {\Ran}(C|D(T))$ and that
$\phi_0 \in D(T^2)\cap D(TH)$.
Then (\ref{teur}) holds.
\end{The}

{\it Proof} :  By the present assumption,  $\phi_0$ is in $D(H)$ with
$H\phi_0\in D(T)$ and $\psi=C(C\chi)$ with $C\chi \in D(T)$.
Hence we can apply Theorem \ref{trans1} to obtain
$$
\left|\lang H\phi_0, \psi(t)\rang\right|
\leq \frac 1{|t|}(\|TH\phi_0\|\|C\chi\|+\|H\phi_0\| \|TC\chi\|).
$$
Using this estimate, we see that
(\ref{finite2}) holds.
We have $\phi_0\in D(T^2)$ and $\chi \in {\cal D}_2(T,C)$, where
${\cal D}_n(T,C)$ is defined by (\ref{Dn}).
Hence, by Theorem \ref{trans2},
$\left|\lang \phi_0, \psi(t)\rang\right|
\leq {c}/{|t|^2}$
with $c > 0$ a constant. This implies that
(\ref{finite1}) holds.
Thus, by Theorem \ref{Wig}, we obtain the desired
result. \hfill \qed

\medskip

 
\section{Structure producing
successively triples obeying the GWWR}


Let $(Q,P,K_C)$ be a triple obeying the GWWR in a Hilbert space ${\cal H}$,
 i.e.,
\begin{equation}
Qe^{-itP}=e^{-itP}(Q+tC), \quad \forall t\in \R.
\end{equation}
(See Proposition \ref{Kzero}.)

\begin{lem}\label{ABTOP} \ \
\begin{list}{}{}
\item[{\rm (i)}] 
 For all $\psi \in D(\bar Q)$ and all $z \in \C$ with $\Re z> 0$,
we have $e^{-zP^2}\psi \in D(\bar Q)$ and
\begin{equation}
\bar Qe^{-zP^2}\psi
=e^{-zP^2}\bar Q\psi-2izPe^{-zP^2}C\psi. \label{QP1}
\end{equation}
\item[{\rm (ii)}] For all $\psi \in D(\bar Q)\cap D(P)$ 
and all $t\in \R$,
we have $e^{-itP^2}\psi \in D(\bar Q)\cap D(P)$ and
\begin{equation}
\bar Qe^{-itP^2}\psi
=e^{-itP^2}(\bar Q\psi+2tPC)\psi. \label{QP2}
\end{equation}
\end{list}
\end{lem}


{\it Proof} : (i) Let $\Re z >0$. Then
the function $:t \mapsto f(t):=e^{-zt^2}, t\in \R$
is in the class $C_{\rm b}^1(\R)$ and $f'(t)=-2zte^{-zt^2}$.
Hence, applying Theorem  \ref{fc1}-(i), we conclude that
$e^{-zP^2}D(\bar Q)\subset D(\bar Q)$ and
(\ref{QP1}) holds for all $\psi \in D(\bar Q)$. 

(ii) Let $\varepsilon >0$.
By part (i), we have for all $\phi\in D(Q^*)=D
\left(({\bar Q})^*\right)$ and $\psi \in D(\bar Q)
\cap D(P)$
$$
\lang Q^*\phi, e^{-(\varepsilon +it)P^2}\psi\rang
=\lang \phi, e^{-(\varepsilon+it)P^2}\bar Q\psi\rang
-\lang \phi, 2i(\varepsilon+it)Pe^{-(\varepsilon +it)P^2}C\psi\rang.
$$
Applying  Proposition \ref{HC} with $H=P$ and $T=Q$,
we see that $C\psi \in D(P)$. 
Hence, taking $\varepsilon \to 0$, we obtain
$$
\lang Q^*\phi, e^{-itP^2}\psi\rang
=\lang \phi, e^{-itP^2}\bar Q\psi\rang
+\lang \phi, 2tPe^{-itP^2}C\psi\rang.
$$
This implies that $e^{-itP^2}\psi \in D(Q^{**})=D(\bar Q)$ and
(\ref{QP2}) holds.\hfill \qed

\bigskip

In the rest of this section, 
we consider a simple  case such that
\begin{equation}
\ker C=\{0\}.\label{kerC}
\end{equation}
Then, by  Corollary \ref{pspecH} applied to $H=P$, $T=Q$ and $L_{\alpha}=C$,
we have 
$\sigma_{\rm p}(P)=\emptyset$. In particular,
$P$ is injective. Hence we can define
\begin{equation}
T(Q,P):=\frac 14 \left[\bar QP^{-1}+(\bar QP^{-1})^*\right],
\end{equation}
provided that $D(\bar QP^{-1})$ is dense in ${\cal H}$.

\begin{The}\label{ABTOP1} Assume (\ref{kerC}) and that 
$D(\bar QP^{-1})$ is dense in ${\cal H}$. Then, for all $t\in \R$,
$e^{-itP^2}D(T(Q,P)) \subset D(T(Q,P))$ and
\begin{equation}
T(Q,P)e^{-itP^2}\psi=
e^{-itP^2}(T(Q,P)+tC)\psi, \quad \psi \in D(T(Q,P)). \label{T-GWWR}
\end{equation}
\end{The}



{\it Proof} : Let $\psi \in D(\bar QP^{-1})$. Then
$P^{-1}\psi \in D(\bar Q)\cap D(P)$. Hence, by Lemma \ref{ABTOP}-(ii),
$e^{-itP^2}P^{-1}\psi \in D(\bar Q)\cap D(P)$ for all $t \in \R$ and
$$
\bar QP^{-1} e^{-itP^2}\psi=
e^{-itP^2}(\bar QP^{-1}+2tC)\psi,
$$ 
which implies that, for all $t\in \R$,
$e^{-itP^2}(\bar QP^{-1}+2tC) \subset
\bar QP^{-1}e^{-itP^2}$.  
Hence 
$$
e^{itP^2}(\bar QP^{-1})^* \subset
\left[(\bar QP^{-1})^*+2t C)e^{itP^2}\right]
$$ for all $t \in \R$.
This implies that
$
e^{-itP^2}
[(\bar QP^{-1})^*+2t C]
\subset (\bar QP^{-1})^*e^{-itP^2}$ for all $t \in \R$.
Thus the desired result follows. \hfill \qed

\bigskip

Theorem \ref{ABTOP1} shows 
that, if $D(\bar Q P^{-1})\cap
D((\bar QP^{-1})^*)$ is dense in addition to the assumption
there, then $(T(Q,P), P^2,K_C)$
obeys the GWWR. This structure is very interesting,
since, apart from domain problems,   it  produces
a series of triples obeying the GWWR. Indeed, let
\begin{eqnarray}
T_1(Q,P)&:=&T(Q,P), \\
T_n(Q,P)&:=&T(T_{n-1}(Q,P),P^{2^{n-1}}), \quad n\geq 2.
\end{eqnarray}
Then $(T_n(Q,P), P^{2^n},K_C)$ obeys the GWWR for all $n=1,\cdots,N$ ($N\in 
\N$), provided that
each $n=1,\cdots,N$, $T_n(Q,P)$ is symmetric.


\begin{ex}\label{ABO}{\rm A simple example of $(Q,P)$is given by 
the Schr\"odinger representation $(q, p)$ on $L^2(\R)$ of the CCR with
one degree of freedom, where $q$ is the multiplication
operator by the coordinate function $x\in \R$ and $p:=-iD_x$
($D_x$ is the generalized differential operator in the variable $x\in \R$).
It is easy to see that $(q, p,K_I)$ obeys the GWWR.
The operator $T(q, p)$ in this case is called
the {\it Aharonov-Bohm time operator} \cite{AB,Miya1,Miya2}.
Theorem \ref{ABTOP1} clarifies a general mathematical structure behind
this operator. A simple application of Theorem \ref{ABTOP1} to
this special case produces
generalized time operartors to $H=(-\Delta)^{2^{n-1}}$ ($n\in \N$).
}
\end{ex}

\bigskip


We can extend the theory presented above to
the case of finitely many degrees of freedom.
Let $Q_j$ ($j=1,\cdots,n, \, n\in \N$)be a symmetric operator
on ${\cal H}$ and $P_j$ be a self-adjoint operator
on ${\cal H}$ such that $P_j$  strongly commutes with
$Q_k$ and $P_k$ ($j,k=1,\cdots,n,
j\not=k$).
Suppose that each $(Q_j,P_j,K_C)$ obeys
the GWWR with $C$ satisfying (\ref{kerC}).
Then, as already shown, each $P_j$ is injective.
Suppose that $D(\bar Q_jP_j^{-1})$ is dense.
Then we can define 
\begin{equation}
T_j:=\frac 14[\bar Q_j P_j^{-1}+ (\bar Q_jP_j^{-1})^*], \quad j=1,\cdots,n.
\end{equation}
By the strong commutativity of $P_j$'s ($j=1,\cdots,n$), the
operator
\begin{equation}
H:=\sum_{j=1}^nP_j^2
\end{equation}
is self-adjoint and nonnegative.

\begin{The}\label{ABTOP2} Let the assumption stated  above on
$(Q_j,P_j)$ ($j=1,\cdots,n$) be satisfied. Then, for all $t \in \R$ and
$j=1,\cdots,n$,
$e^{-itH}D(T_j)) \subset D(T_j)$ and
\begin{equation}
T_je^{-itH}\psi=
e^{-itH}(T_j+tC)\psi, \quad \psi \in D(T_j). 
\end{equation}
\end{The}


{\it Proof} : By the strong commutativity of $Q_j$ with 
$P_k$ ($k\not=j$), we have
$Q_je^{-itH}=e^{-itH_j}Q_je^{-itP_j^2}$ on $D(P_j)$, where
$H_j:=\sum_{k\not=j}P_k^2$. Hence,
applying Theorem \ref{ABTOP1} with $Q=Q_j, P=P_j$ we obtain
the desired result.
\hfill \qed

\bigskip

In this case too, a remark  similar to the one after Theorem \ref{ABTOP1}
 is applicable.


\section{Generalized Time Operators of Partial Differential
Operators}

In this section we construct classes
of generalized time operators of partial differential
operators.

\subsection{Constructions from the Schr\"odinger
representation of the CCR with $d$ degrees of freedom}

Let ${\bf q}=(q_1,\cdots, q_d)$ and
${\bf p}=(p_1,\cdots, p_d)$ be the Schr\"odinger
representation of the CCR with $d$ degrees of freedom.
Namely, $q_j$ is the multiplication operator
by $x_j$, the $j$-th coordinate function ($x=(x_1,\cdots, x_d)\in \R^d$)
acting in $L^2(\R^d)$ and
$p_j=-iD_j$ on $L^2(\R^d)$ ($D_j$ is the generalized
partial differential operator in $x_j$).
The following properties are  well known:
\begin{list}{}{}
\item[(i)] $p_j$ and $p_l$ are strongly commuting self-adjoint
operators. 
\item[(ii)] For all $t \in \R$,
$$
q_je^{-itp_l}=e^{-itp_l}(q_j+\delta_{jl}t), \quad j,l=1,\cdots,d.
$$
\end{list}


We denote  by 
\begin{equation}
\hat \R^d:=\{k=(k_1,\cdots,k_d)
|k_j\in \R, j=1,\cdots,d\}
\end{equation}
the $d$-dimensional momentum space and
by $C^r(\hat \R^d)$ the set of 
$r$ times continuously differentiable
functions on $\hat \R^d$   



Let $F \in C^1(\R^d)$ be real-valued. Then
the operators
\begin{equation}
H_F:=F({\bf p})
\end{equation}
and
\begin{equation}
F_j^{(1)}:=(\partial_jF)({\bf p})
\end{equation}
are  self-adjoint.
Let $N_F^{(j)}$ be a closed subset of $\hat \R^d$ with Lebesgue measure
zero 
and $G\in C^1(\hat \R^d \setminus N_F^{(j)})$ such that
\begin{equation}
\sup_{k \in \hat \R^d\setminus N_F^{(j)}}|G(k)\partial_jF(k)|
< \infty. \label{unibd}
\end{equation}
Then
we can define a linear operator $T_{F,G}^{(j)}$
on $L^2(\R^d)$ by
\begin{equation}
T_{F,G}^{(j)}
:=G({\bf p})q_j+
q_jG({\bf p})^*, 
\quad
\end{equation}
with domain
\begin{equation}
D(T_{F,G}^{(j)}):={\cal F}^{-1}C_0^1(\hat \R^d\setminus N_F^{(j)}),
\end{equation}
where ${\cal F}:L^2(\R^d) \to L^2(\hat \R^d)$ is the
Fourier transform:
\begin{equation}
({\cal F}\psi)(k):=\frac {1}{(2\pi)^{d/2}}
\int_{\R^d}e^{-ikx}\psi(x)dx, \quad \psi \in L^2(\R^d)
\end{equation}
in the $L^2$-sense.
Note that, for all  $u \in C_0^1(\hat \R^d \setminus N_F^{(j)})$,
\begin{equation}
\left({\cal F} T_{F,G}^{(j)}{\cal F}^{-1}u\right)(k)
=G(k)i\frac{\partial u(k)}{\partial k_j}
+i\frac{\partial}{\partial k_j}[G(k)^*u(k)],  \quad
{\rm a.e.} k \in \hat \R^d \setminus N_F^{(j)}.
\end{equation}
It is easy to see that $T_{F,G}^{(j)}$ is symmetric [note that
$D(T_{F,G}^{(j)})$ is dense in $L^2(\R^d)$, 
since $C_0^{\infty}(\hat \R^d \setminus N_F^{(j)})$ is dense
in $L^2(\hat \R^d)$].



\begin{lem}\label{inv1} 
For all $t \in \R$ and $\psi \in D(T_{F,G}^{(j)})$, one has
$e^{-itH_F}\psi \in D(T_{F,G}^{(j)})$.
\end{lem}

{\it Proof} :  We have by the functional calculus
${\cal F}e^{-itH_F}{\cal F}^{-1}=e^{-itM_F}$, where
$M_F$ is the multiplication operator by the function $F$.
It is easy to see that, for all  $u \in C_0^1(\hat \R^d\setminus N_F^{(j)}$,
$e^{-itM_F}u \in C_0^1(\hat \R^d\setminus N_F^{(j)})$.
Thus the assertion follows.
\hfill \qed

\bigskip



For the functions $F$ and $G$ as above, the operator
\begin{equation}
C_{F,G}^{(j)}:=[G({\bf p})+G({\bf p})^*]F_j^{(1)}
\end{equation}
is bounded and self-adjoint.

\begin{pr}\label{HF1} Let $F$ and $G$  be as above.
Then
$(T_{F,G}^{(j)},H_F,K_{C_{F,G}^{(j)}})$ obeys the GWWR.
\end{pr}

{\it Proof} :  By using the Fourier transform, 
it is easy to see that
$$
q_je^{-itH_F}\psi=e^{-itH_F}q_j\psi+te^{-itH_F}F_j^{(1)}\psi
$$
for all $\psi \in D(q_j)\cap D(F_j^{(1)})$, which, together with
Lemma  \ref{inv1}, implies
that
\begin{eqnarray*}
&&G({\bf p})q_je^{-itH_F}\psi=e^{-itH_F}
G({\bf p})q_j\psi+te^{-itH_F}G({\bf p})F_j^{(1)}\psi, \\
&&q_jG({\bf p})^*e^{-itH_F}\psi=
e^{-itH_F}q_jG({\bf p})^*\psi+te^{-itH_F}F_j^{(1)}G({\bf p})^*\psi
\end{eqnarray*}
for all $\psi \in D(T_{F,G}^{(j)})$.
Adding these equations, we obtain
$$
T_{F,G}^{(j)}e^{-itH_F}\psi=
e^{-itH_F}T_{F,G}^{(j)}\psi+te^{-itH_F}C_{F,G}^{(j)}\psi.
$$
Thus the desired result follows.
\hfill \qed

\bigskip









\begin{ex}{\rm (The free Hamiltonian
of a nonrelativistic quantum particle)  Consider the case where
$F(k)=k^2/(2m)$ ($m > 0$ is a constant denoting
the mass of a quantum particle) and $G(k)=v(k)/\partial_jF(k)
=mv(k)k_j^{-1}$ with $v\in C^1(\hat \R^d)$ being bounded. Then 
\begin{eqnarray*}
&&H_F=H_{\rm NR}:=-\frac{\Delta}{2m}, \quad
T_{F,G}^{(j)}=T_v^{(j)}:=m \left(v({\bf p})p_j^{-1}q_j+q_jv({\bf p})^*p_j^{-
1}\right), \\
&& C_{F,G}^{(j)}=C_v:=v({\bf p})+v({\bf p})^*
\end{eqnarray*}
with $N_F^{(j)}=\{k\in \hat \R^d
| k_j=0\}$ (hence its $d$-dimensional Lebesgue measure is zero),
where
\begin{equation}
\Delta:=\sum_{j=1}^dD_j^2
\end{equation}
is  the $d$-dimensional generalized Laplacian on $L^2(\R^d)$.
Then $(T_v^{(j)},H_{\rm NR}, K_{C_v})$ obeys the GWWR.
The operator $T_v^{(j)}$ with  $d=1$ and $v=1/2$
 is just the Aharonov-
Bohm time operator mentioned in Example \ref{ABO}).
}
\end{ex}

\begin{ex}{\rm (The free Hamiltonian
of a relativistic quantum particle) Consider the case where 
$F(k)=\sqrt{k^2+m^2}$ with $m > 0$
(a constant). Let $G(k)=v(k)/\partial_jF(k)=v(k)\sqrt{k^2+m^2}k_j^{-1}$.
Then
\begin{eqnarray*}
&&H_F=H_{\rm rel}:=\sqrt{-\Delta+m^2}, \\
&&T_{F,G}^{(j)}=T_{\rm rel}^{(j)}:= 
v({\bf p})\sqrt{-\Delta+m^2}\,p_j^{-1}q_j+q_j
v({\bf p})^*\sqrt{-\Delta+m^2}\,p_j^{-1},\\
&& C_{F,G}^{(j)}=C_v
\end{eqnarray*}
with $N_F^{(j)}=\{k\in \hat \R^d
| k_j=0\}$.
Then $(T_{\rm rel}^{(j)}, H_{\rm rel},K_{C_v})$ obeys the GWWR.
}
\end{ex}



\begin{ex}{\rm
(Powers of the $d$-dimensional Laplacian) 
Let $\alpha >0$ be a constant and consider the case  where
$$
F(k)=F_{\alpha}(k):=|k|^{2\alpha}, \quad k \in \hat \R^d.
$$ 
In this case we have 
$$
H_{F_{\alpha}}=(-\Delta)^{\alpha}.
$$
Let $M_j:=\R^d \setminus \{k\in \hat \R^d| k_j=0\}$ and
$$
T_{j}^{(\alpha)}
:=\frac 1{2\alpha} \left\{ v({\bf p})(-\Delta)^{-\alpha+1}p_j^{-1}q_j
+q_jv({\bf p})^*(-\Delta)^{-\alpha+1}p_j^{-1}\right\},
$$
with $D(T_{j}^{(\alpha)}):={\cal F}^{-1}C_0^1(M_j)$. Then $(T_j^{(\alpha)},
H_{F_{\alpha}},K_{C_v})$ obeys the GWWR.
}
\end{ex}

\subsection{Abstract Dirac operators}


Let ${\cal K}$ be a Hibert space and
$A_j$ ($j=1,\cdots,d$) and $B$ be  bounded self-adjoint
operators satisfying the anticommutation relations
\begin{eqnarray*}
&&\{A_j,A_l\}=2\delta_{jl}, \quad \{A_j,B\}=0,\\
&& B^2=I, \quad j,l=1,\cdots,d,
\end{eqnarray*}
where $\{X,Y\}:=XY+YX$. 
Let $M$ be a  strictly positive, continuously differentiable
function on $\R^d$ such that $\partial_jM$ is bounded ($j=1,\cdots,d$)
and the set
$$
N_j:=\{k\in \hat \R^d|
k_j+M(k)(\partial_jM)(k)=0\}
$$
is closed with  Lebesuge measure zero.
Then
the operator 
\begin{equation}
p_j(M):=p_j+M({\bf p})(\partial_jM)({\bf p})
\end{equation}
is injective and ${\cal F}^{-1}C_0^{\infty}(\hat \R^d\setminus N_j)
\subset D(p_j(M)^{-1})$.
 
We define an operator of Dirac type
\begin{equation}
H_{\rm D}:=\sum_{j=1}^dA_j\otimes p_j +B\otimes M({\bf p})
\end{equation}
acting on ${\cal K}\otimes L^2(\R^d)$.  This is an abstract
Dirac operator. 

\begin{ex}{\rm The usual free Dirac operator \cite[p.7]{Thaller}
is given by taking $d=3, {\cal K}=\C^4, A_j=\alpha_j, M=m$ (a positive constant), 
$B=\beta$ with $\alpha_j$ and $\beta$ being $4 \times 4$-Hermitian matrices
satisfying $\{\alpha_j,\alpha_l\}=2\delta_{jl},\{\beta,\alpha_j\}=0,
\beta^2=I,\ j,l=1,2,3$.
}
\end{ex}

By a general theorem \cite[Theorem 4.3]{Arai93},
$H_{\rm D}$ is self-adjoint and
$$
H_{\rm D}^2=I\otimes (-\Delta+M({\bf p})^2).
$$
In what follows, under the natural identification
${\cal K}\otimes L^2(\R^d)=L^2(\R^d;{\cal K})$
(the Hilbert space of ${\cal K}$-valued
square integrable functions on $\R^d$), we write
$A_j\otimes p_j$ as $A_jp_j$. Let $v \in C^1(\hat \R^d)$ be bounded and
\begin{equation}
T_j:=v({\bf p})H_{\rm D}p_j(M)^{-1}q_j+q_jv({\bf p})^*H_{\rm D}
p_j(M)^{-1}
\end{equation}
with $D(T_j):={\cal K}\hat \otimes{\cal F}^{-1}C_0^{\infty}
(\hat \R^d\setminus N_j)$.

\begin{The} The triple $(T_j, H_{\rm D}, K_{C_v})$ obeys the GWWR.
\end{The}

{\it Proof} :  The self-adjoint operator $L:=(-\Delta+M({\bf p})^2)^{1/2}$
is strictly positive.  Hence it has a bounded inverse.
Let
$$
U:=\frac 1{\sqrt{2}}B(H_{\rm D}+BL)L^{-1/2}(L+M({\bf p})^{-1/2}.
$$
Then $U$ is unitary and
$$
UH_{\rm D}U^*=BL.
$$
(This is an abstract structure 
of the usual free Dirac operator,
see, \cite[\S 1.4]{Thaller}.)
We also have
$$
\hat T_j:=UT_jU^*=v({\bf p})BLp_j(M)^{-1}q_j+q_jv({\bf p})^*BL
p_j(M)^{-1}
$$
with $D(\hat T_j):=UD(T_j)$ (note that $UC_v=C_vU$).
Using the Fourier analysis, we can show that
$(\hat T_j, BL, K_{C_v})$ obeys the GWWR.
This implies that $(T_j,H_{\rm D},K_{C_v})$ obeys the GWWR.
\hfill \qed

\bigskip


\begin{rem}{\rm We can apply general results established
in Sections 2--10 to the Hamiltonians $H_F, H_{\rm NR}, H_{\rm rel}, 
H_{F_{\alpha}}$ and the Dirac operator $H_{\rm D}$.
In particular, we can derive decay estimates (in time $t$)
for transition probability amplitudes of states with these
Hamiltonians. But we do not write down them here.
}
\end{rem}


\section{Representations of the GWWR in Fock Spaces}

In this section we show that, given a triple 
obeying the GWWR in a Hilbert space ${\cal H}$, 
there exist triples obeying the GWWR in
Fock spaces (full Fock spaces, boson Fock spaces, fermion Fock spaces)
over ${\cal H}$.


\subsection{Tensor Representations of the GWWR }


Let  $n \in \N$ and
${\cal H}_j$ ($j=1,\cdots,n, n\in \N$)
be a Hilbert space. 
Suppose that  $(T_j,H_j,K_{C_j})$
obeys the GWWR in  ${\cal H}_j$ such that
$T_j$ is closed symmetric.
We consider  an operator
\begin{equation}
L:=\overline{\sum_{j=1}^n 
\underbrace{I \otimes \cdots \otimes I \otimes 
H_j \otimes I \otimes  
\cdots \otimes  I}_n
|\hat \otimes_{j=1}^nD(H_j)},
\end{equation}
where
$\hat \otimes$ denotes algebraic tensor product.
It is well known that $L$ is self-adjoint \cite[\S VIII.10]{RS1} and
\begin{equation}
e^{-itL}=e^{-itH_1}\otimes \cdots \otimes e^{-itH_n}, \quad t \in \R.
\label{unigr}
\end{equation}

Let
\begin{eqnarray}
\widetilde T_j&:=&
\underbrace{I \otimes \cdots \otimes I \otimes T_j  \otimes I \otimes  
\cdots \otimes  I}_n,\\
\widetilde C_j&:=&
\underbrace{I \otimes \cdots \otimes I \otimes C_j  \otimes I \otimes  
\cdots \otimes  I}_n.
\end{eqnarray}




\begin{pr}\label{tensor1} For all $j=1,\cdots, n$, 
$(\widetilde T_j, L, K_{\widetilde C_j})$ obeys the GWWR.
\end{pr}



{\bf Proof} :  Let $\Psi=\psi_1\otimes \cdots \otimes \psi_n$
with $\psi_j\in D(T_j), j=1,\cdots,n$.
Then, for all $t \in \R$,
$e^{-itL}\Psi=\otimes_{l=1}^n e^{-itH_l}\psi_l \in 
\hat \otimes_{l=1}^nD(T_l)$
and
\begin{eqnarray*}
\widetilde T_je^{-itL}\Psi
&=&
e^{-itH_1}\psi_1\otimes \cdots e^{-itH_{j-1}} 
\psi_{j-1}\otimes T_je^{-itH_j}\psi_j \otimes e^{-itH_{j+1}}\psi_{j+1}
\cdots \otimes \otimes e^{-itH_n}\psi_n \\
&=&
e^{-itH_1}\psi_1\otimes \cdots 
\otimes e^{-itH_j}(T_j+tC_j)\psi_j \otimes 
\cdots \otimes e^{-itH_n}\psi_n \\
&=&e^{-itL}(\widetilde T_j+t\widetilde C_j)\Psi.
\end{eqnarray*}
Since $\hat \otimes_{j=1}^nD(T_j)$ 
is a core of $\widetilde T_j$,
the last equality extends to all $\Psi
\in D(\widetilde T_j)$ with
$e^{-itL}\Psi \in D(\widetilde T_j)$.
Hence $(\widetilde T_j,L, K_{\widetilde C_j})$ obeys
the GWWR. 
\hfill \qed

\bigskip

\subsection{Constructions of triples obeying the GWWR in Fock Spaces}



Let ${\cal H}$ be a Hilbert space.
Then the full Fock space over ${\cal H}$ is defined by
\begin{equation}
{\cal F}({\cal H}):=\oplus_{n=0}^{\infty}
\otimes^n{\cal H}
\end{equation}
with $\otimes^0{\cal H}:=\C$ (see, e.g., \cite[\S 5.2]{BR}
or \cite[\S II.4, \S VIII.10]{RS1} for Fock space theory).
For a densely defined closed linear operator $A$ on ${\cal H}$
and $n\in\{0\}\cup \N$,
we define a linear operator $A_{\Sigma}^{(n)}$
on $\otimes^n{\cal H}$ as follows:
\begin{eqnarray}
A_{\Sigma}^{(0)}&:=&0, \\
A_{\Sigma}^{(n)}&:=&
\overline{\left(\sum_{j=1}^n
\underbrace{I \otimes \cdots \otimes I \otimes 
\stackrel{j{\rm-th}}{\stackrel{\smile}{A}} \otimes I \otimes  
\cdots \otimes I}_n\right)|\hat \otimes^nD(A) }
\end{eqnarray}
for $n\geq 1$. Then  the operator
\begin{equation}
d\Gamma(A):=\oplus_{n=0}^{\infty}A_{\Sigma}^{(n)},
\end{equation}
called
the second quantization of $A$, 
is
densely defined and closed.
If $A$ is self-adjoint, then so is $d\Gamma(A)$.

Let $H$ be a self-adjoint operator on ${\cal H}$,
$T$ be a closed symmetric operator on ${\cal H}$ and $C\in {\sf B}({\cal H})$
 be self-adjoint.
In what follows we take the  following hypothesis:
\begin{list}{}{}
\item[{\bf Hypothesis (I)}] 
The triple  $(T,H,K_C)$ obeys the  GWWR in ${\cal H}$. 
\end{list}

We define
\begin{equation}
T_j^{(n)}:=\left\{
\begin{array}{ll}
a_{n,j} & ; 0 \leq n < j \\
\underbrace{I \otimes \cdots \otimes I \otimes 
\stackrel{j{\rm-th}}{\stackrel{\smile}{T}} \otimes I \otimes   
\cdots \otimes I}_n & 
; n \geq j
\end{array}\right.,
\end{equation}
where $a_{n,j}$ is an arbitrary real constant.
For $j \in \N$, we define a linear operator $T_j$
on ${\cal F}({\cal H})$ by
\begin{equation}
T_j:=\oplus_{n=0}^{\infty}T_j^{(n)}.
\end{equation}
Then $T_j$ is  a closed symmetric operator.
Let
\begin{equation}
C_j^{(n)}:=\left\{
\begin{array}{ll}
0 & ; 0 \leq n < j\\
\underbrace{I \otimes \cdots \otimes I \otimes 
\stackrel{j{\rm-th}}{\stackrel{\smile}{C}} \otimes I \otimes   
\cdots \otimes I}_n & 
; n \geq j
\end{array}\right.
\end{equation}
and
\begin{equation}
C_j:=\oplus_{n=0}^{\infty}C_j^{(n)}.
\end{equation}
Then $C_j$ is a bounded self-adjoint operator on ${\cal F}({\cal H})$.




\begin{pr} Under Hypothesis (I),
for all $j \in\N$,  $(T_j, d\Gamma(H), 
K_{C_j})$ obeys
the GWWR.
\end{pr}

{\bf Proof} : It follows from Proposition \ref{tensor1} that, for all $n$, 
$(T_j^{(n)}, H_{\Sigma}^{(n)}, K_{C_j^{(n)}})$ obeys the GWWR. 
It is easy to see that this implies the desired result.
\hfill \qed


\bigskip

We next construct a triple obeying the GWWR
in ${\cal F}({\cal H})$ which is reduced by the
boson Fock space over ${\cal H}$
$$
{\cal F}_{\rm b}({\cal H}):=\oplus_{n=0}^{\infty}\otimes_{\rm s}^n{\cal H},
$$
where $\otimes_{\rm s}^n{\cal H}$ denotes the $n$-fold symemtric tensor product of 
${\cal H}$, and
 the
fermion Fock space over ${\cal H}$
$$
{\cal F}_{\rm f}({\cal H}):=\oplus_{n=0}^{\infty}\otimes_{\rm as}^n{\cal H},
$$
where $\otimes_{\rm as}^n{\cal H}$ denotes the $n$-fold antisymemtric tensor product 
of ${\cal H}$.
The operator
\begin{equation}
\hat N:=d\Gamma(I)
\end{equation}
on ${\cal F}({\cal H})$
is called the number operator.
The vector
\begin{equation}
\Omega:=\{1,0,0,\cdots\}\in {\cal F}({\cal H})
\end{equation}
is called the {\it Fock vacuum}.
We denote by $P_0$ is the orthogonal
projection
onto the one-dimensional subspace
$\{z\Omega|z\in \C\}$.
We set
\begin{equation}
Q_0:=I-P_0.
\end{equation}
Let
\begin{equation}
S:=Q_0\hat N^{-1/2}d\Gamma(C)\hat N^{-1/2}Q_0
\end{equation}

\begin{lem} The operator
$S$ is bounded and self-adjoint with $\|S\| \leq \|C\|$.
\end{lem}


{\it Proof} : 
It is easy to see that,
for all $\Psi \in D(S)$,
 $(S\Psi)^{(0)}=0$ and, for $n \geq 1$,
$$
(S\Psi)^{(n)}=\frac {\sum_{j=1}^n \widetilde C_j\Psi^{(n)}}{n}.
$$
Hence
$\|(S\Psi)^{(n)}\| \leq \|C\|\|\Psi^{(n)}\|$, which implies that
$\|S\Psi\|^2\leq \|C\|^2\|\Psi\|^2$.
Hence $S$ is bounded with $\|S\| \leq \|C\|$ and $D(S)={\cal F}({\cal H})$.
It is easy to see that $S$ is symmetric.
\hfill \qed

\bigskip



Let
\begin{equation}
{\cal D}_T:=\oplus_{n=0}^{\infty}\hat\otimes^nD(T)
\end{equation}
and
\begin{equation}
\tau(T):=\left(Q_0\hat N^{-1/2}d\Gamma(T)\hat N^{-1/2}Q_0
\right)|{\cal D}_T.
\end{equation}
Then $\tau(T)$ is symmetric.

\begin{The}\label{tau1} Assume Hypothesis (I). Then 
$(\tau(T),
d\Gamma(H), K_{S})$ obeys the GWWR.
\end{The}

{\bf Proof} :  For all $\Psi \in {\cal D}_T$, 
$\hat N^{-1/2}Q_0\Psi \in {\cal D}_T$, since
$$
(\hat N^{-1/2}Q_0\Psi)^{(0)}=0,\quad
(\hat N^{-1/2}Q_0\Psi)^{(n)}=
n^{-1/2}\Psi^{(n)}, \quad  n \geq 1.
$$
Using this property  and the easily proven fact
that $e^{-itH}\hat N^{-1/2}Q_0=
\hat N^{-1/2}Q_0e^{-itH}$, we have by direct computations
$$
d\Gamma(T)\hat N^{-1/2}Q_0e^{-itd\Gamma(H)}\Psi=
e^{-itd\Gamma(H)}d\Gamma(T)\hat N^{-1/2}Q_0\Psi
+te^{-itH}d\Gamma(C)\hat N^{-1/2}Q_0\Psi.
$$
 Operating $Q_0\hat N^{-1/2}$ to
the both sides, we obtain
$$
\tau(T)e^{-itH}\Psi=e^{-itH}\tau(T)\Psi+te^{-itH} S\Psi,
$$
which implies the desired result. \hfill \qed

\bigskip


\begin{rem}{\rm Applying the results in Section 8, we can derive
decay estimates for $|\lang \Phi,e^{-itd\Gamma(H)}\Psi\rang|$ ($t \in \R$)
for vectors $\Phi,\Psi$ in suitable subspaces. But we do not write down
them here. The same applies to other results implied by
the abstract theory.
}
\end{rem}



It is easy to see that $d\Gamma(H)$, $\tau(T)$ and $S$
are reduced by ${\cal F}_{\#}({\cal H})$ ($\#={\rm b, f}$).
Hence Theorem \ref{tau1}
holds for the reduced parts of them too.

It is more interesting  and important to
construct generalized time operators of perturbed Hamiltoinians
of the form $d\Gamma(H)+V$ 
on the boson Fock space ${\cal F}_{\rm b}({\cal H})$ 
or the fermion Fock space ${\cal F}_{\rm f}({\cal H})$
with $V$ a  symmetric operator. For this purpose, the method
given in Section 2.4 can be applied.
But in the present  paper we leave this problem for future study.

 

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\end{document}
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