Content-Type: multipart/mixed; boundary="-------------0701312104872" This is a multi-part message in MIME format. ---------------0701312104872 Content-Type: text/plain; name="07-26.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-26.keywords" Spectrum; time operator; Hamiltonian; weak Weyl relation; quantum theory. ---------------0701312104872 Content-Type: application/x-tex; name="spectrum.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="spectrum.tex" \documentclass[12pt]{article} \topmargin=0cm \oddsidemargin=0cm \evensidemargin=0cm \textheight=23cm \textwidth=16cm \makeatletter \renewcommand{\theequation}{% \thesection.\arabic{equation}} \@addtoreset{equation}{section} \makeatother %%%%%MACROS%%%%%%%%%%%% \newtheorem{The}{\bf Theorem}[section] \newtheorem{df}[The]{\bf Definition} \newtheorem{lem}[The]{\bf Lemma} \newtheorem{prop}[The]{\bf Proposition} \newtheorem{cor}[The]{\bf Corollary} % \newtheorem{ex}{\bf Example}[section] \newtheorem{rem}{\bf Remark}[section] \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\qed}{\hbox{\rule{4pt}{7pt}}} \newcommand{\lang}{\left<} \newcommand{\rang}{\right>} \newcommand{\Ran}{{\rm Ran}} \renewcommand{\Im}{{\rm Im}\,} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsfonts} \begin{document} \title{\bf Spectrum of Time Operators} \author{ Asao Arai\thanks{ This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from the JSPS.}\\ 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 complex Hilbert space ${\cal H}$. A symmetric operator $T$ on ${\cal H}$ is called a time operator of $H$ if, for all $t\in \R$, $e^{-itH}D(T)\subset D(T)$ ($D(T)$ denotes the domain of $T$) and $Te^{-itH}\psi=e^{-itH}(T+t)\psi, \ \forall t\in \R, \forall \psi \in D(T)$. In this paper, spectral properties of $T$ are investigated. The following results are obtained: (i) If $H$ is bounded below, then $\sigma(T)$, the spectrum of $T$, is either $\C$ (the set of complex numbers) or $\{z\in \C| \Im z \geq 0\}$. (ii) If $H$ is bounded above, then $\sigma(T)$ is either $\C$ or $\{z\in \C| \Im z \leq 0\}$. (iii) If $H$ is bounded, then $\sigma(T)=\C$. The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified. Moreover spectral analysis is made on a generalized time operator. \end{abstract} \medskip \noindent {\it Keywords}: Spectrum; time operator; Hamiltonian; weak Weyl relation; quantum theory. \medskip \noindent Mathematics Subject Classification 2000: 81Q10, 47N50 \section{Introduction} In a paper \cite{Schmu}, Schm\"udgen studied a pair $(T,H)$ of a symmetric operator $T$ and a self-adjoint operator $H$ on a complex Hilbert space ${\cal H}$ (in the notation there, $T=P, H=-Q$) such that, for all $t\in \R$, $e^{-itH}D(T)\subset D(T)$ ($D(T)$ denotes the domain of $T$) and \begin{equation} Te^{-itH}\psi=e^{-itH}(T+t)\psi, \ \forall t\in \R, \forall \psi \in D(T).\label{WWR} \end{equation} This is a stronger version of representation of the canonical commutation relation (CCR) with one degree of freedom, since (\ref{WWR}) implies that \begin{equation} \lang T\phi, H\psi\rang-\lang H\phi, T\psi\rang =\lang \phi,i\psi\rang, \quad \psi,\phi \in D(T)\cap D(H), \end{equation} i.e., $T$ and $H$ satisfy the CCR in the sense of sesquilinear form on $D(H)\cap D(T)$ and hence, in particular, $TH-HT=i$ on $D(HT)\cap D(TH)$, the CCR in the original sense. We call (\ref{WWR}) the {\it weak Weyl relation} (WWR). About twenty years later, Miyamoto \cite{Miya1} used the WWR to introduce a proper concept of time operator in quantum mechanics. Namely a symmetric operator $T$ on ${\cal H}$ is called a {\it time operator} of $H$ if $(T,H)$ obeys the WWR (\ref{WWR}) (in \cite{Miya1}, (\ref{WWR}) is called the {\it $T$-weak Weyl relation}). We remark that, in this terminology, one has in mind the case where, in application to quantum mechanics, $H$ is the Hamiltonian of a quantum system. Some fundamental properties of the pair $(T,H)$ were investigated in \cite{Miya1}. The work of Miyamoto \cite{Miya1} was extended by the present author in a previous paper \cite{Ar05}, where a generalized version of the WWR (\ref{WWR}), called a {\it generalized weak Weyl relation}, is given and, in terms of it, a concept of {\it generalized time operator} is introduced. We remark that a time operator as well as a generalized one of a given self-adjoint operator $H$ is not unique \cite[Proposition 2.6, \S 11]{Ar05}. Physically the set of generalized time operators associated with a self-adjoint operator $H$ ( a Hamiltonian) can be regarded as a class of symmetric operators which play a role in controlling decays (in time) of survival probabilities as well as time-energy uncertainty relations \cite{Ar05,Miya1}. In this paper, we investigate spectral properties of (generalized) time operators. We first recall the definition of the spectrum of a linear operator $A$ on ${\cal H}$. The resolvent set of $A$, denoted $\rho(A)$, is defined to be the set of complex numbers $z$ satisfying the following three conditions: (i) $A-z$ is injective ; (ii) $\Ran(A-z)$, the range of $A-z$, is dense in ${\cal H}$ ; (iii) $(A-z)^{-1}$ with $D((A-z)^{-1})=\Ran(A-z)$ is bounded. Then the spectrum of $A$, denoted $\sigma(A)$, is defined by $\sigma(A):=\C\setminus \rho(A)$, where $\C$ is the set of complex numbers. It follows that, if $A$ is closable, then $\sigma(\bar A)=\sigma(A)$, where $\bar A$ is the closure of $A$, and $\Ran(\bar A-z)={\cal H}$ for all $z\in \rho(\bar A)=\rho(A)$. In particular, for all symmetric operators $S$ on ${\cal H}$, $\sigma(S)=\sigma(\bar S)$ and $\Ran(\bar S-z)={\cal H}$ for all $z\in \rho(\bar S)=\rho(S)$. One of the motivations for this work comes from the following fact: \begin{The}\label{th1-1}{\rm (\cite{Miya1}, \cite[Theorem 2.8]{Ar05})} If $H$ is a self-adjoint operator on ${\cal H}$ and semi-bounded (i.e., bounded below or bounded above), then every time operator $T$ of $H$ is not essentially self-adjoint . \end{The} This theorem combined with a general theorem \cite[Theorem X.1]{RS2} implies that, in the case where $H$ is semi-bounded, the spectrum $\sigma(T)$ of $T$ ($=\sigma(\overline{T})$) is one of the following three sets: \begin{list}{}{} \item{(i)} $\C$. \item{(ii)} $\overline {\Pi}_+$, the closure of the upper half-plane $\Pi_+:=\{z\in \C|\Im z >0\}$. \item{(iii)} $\overline{\Pi}_-$, the closure of the lower half-plane $\Pi_+:=\{z\in \C|\Im z <0\}$. \end{list} Then it is interesting to examine which one is realized, depending on properties of $H$. The outline of the present paper is as follows. In Section 2, we prove a theorem on the spectrum of time operators (Theorem \ref{th2-1}). In Section 3 we consider time operators on direct sums of Hilbert spaces. In Section 4, we identify the spectrum of concrete time operators, including the Aharonov-Bohm time operator \cite{AB} and time operators of a relativistic Schr\"odinger operator. In Section 5, we prove a theorem similar to Theorem \ref{th2-1} in the case where $T$ is a generalized time operator. \section{Main Result} In this section we prove the following theorem: \begin{The}\label{th2-1} Let $H$ be a self-adjoint operator on ${\cal H}$ and $T$ be a time operator of $H$. Then the following {\rm (i)---(iii)} hold: \begin{list}{}{} \item{{\rm (i)}} If $H$ is bounded below, then $\sigma(T)$ is either $\C$ or $\overline{\Pi}_+$. \item{{\rm (ii)}} If $H$ is bounded above, then $\sigma(T)$ is either $\C$ or $\overline{\Pi}_-$. \item{{\rm (iii)}} If $H$ is bounded, then $\sigma(T)=\C$. \end{list} \end{The} \begin{rem}\label{rem2-1}{\rm The time operator $T$ has no eigenvalues , i.e., the point spectrum $\sigma_{\rm p}(T)$ of $T$ is an empty set \cite[Corollary 4.2]{Miya1}}. \end{rem} \begin{rem}{\rm In the case where $\sigma(T)=\overline{\Pi}_+$ or $\overline{\Pi}_-$, $\overline{T}$ is maximally symmetric \cite[p.141]{RS2}.} \end{rem} Throughout the rest of this section, $T$ represents a time operator of $H$. The following lemma is a key fact to prove Theorem \ref{th2-1}. \begin{lem}\label{lem2-1} Suppose that $H$ is bounded below. Then, for all $\beta >0$, $e^{-\beta H}D(\overline{T})\subset D(\overline{T})$ and, for all $\psi\in D(\overline{T})$ \begin{equation} \overline{T}e^{-\beta H}\psi=e^{-\beta H}(\overline{T}-i\beta)\psi. \end{equation} \end{lem} {\it Proof}. Apply \cite[Theorem 6.2]{Ar05}. \hfill \qed \medskip We denote by $T^*$ the adjoint of $T$. \begin{lem}\label{lem2-2} Suppose that $H$ is bounded below. Then, for all $\beta >0$, $e^{-\beta H}D(T^*)\subset D(T^*)$ and, for all $\psi\in D(T^*)$ \begin{equation} T^*e^{-\beta H}\psi=e^{-\beta H}(T^*-i\beta)\psi. \end{equation} \end{lem} {\it Proof}. Lemma \ref{lem2-1} implies that $e^{-\beta H}(\overline{T}-i\beta)\subset \overline{T}e^{-\beta H}$. We have $(\overline{T})^*=T^*$. For each bounded linear operator $A$ on ${\cal H}$ with $D(A)={\cal H}$ and all densely defined linear operators $B$ on ${\cal H}$, $(AB)^*=B^*A^*$. Using these facts, one can show that $e^{-\beta H}T^*\subset (T^*+i\beta)e^{-\beta H}$. Thus the desired result follows. \hfill \qed \medskip \noindent {\bf Proof of Theorem \ref{th2-1}} (i) By the fact on the spectrum of $T$ mentioned after Theorem \ref{th1-1}, we need only to show that the case $\sigma(T)=\overline{\Pi}_-$ is excluded. For this purpose, suppose that $\sigma(T)=\overline{\Pi}_-$. Then $\Pi_+=\rho(T)=\rho(\overline{T})$. In general, we have for all $z\in \C\setminus\R$ the orthogonal decomposition \begin{equation} {\cal H}=\ker(T^*-z^*)\oplus \Ran(\overline{T}-z) \label{dcp} \end{equation} Applying this structure with $z=i\in \Pi_+$, we obtain $\ker (T^*+i)=\{0\}$. Since $T$ is not essentially self-adjoint by Theorem \ref{th1-1}, it follows that $\ker (T^*-i)\not=\{0\}$. Hence there exists a non-zero vector $\psi\in D(T^*)$ such that $T^*\psi=i\psi$. Then, by Lemma \ref{lem2-2}, $i(1-\beta)\in\sigma_{\rm p}(T^*)$. Since $\beta>0$ is arbitrary, we can take it to be $1< \beta$. Then $\gamma:=i(1-\beta)\in \Pi_-$. Taking $z=\gamma^*$ in (\ref{dcp}), we have the orthogonal decomposition $$ {\cal H}=\ker(T^*-\gamma)\oplus \Ran(\overline{T}-\gamma^*). $$ Hence $\Ran(\overline{T}-\gamma^*)$ is not dense in ${\cal H}$. Therefore $\gamma^*\in \sigma(\overline{T})=\sigma(T)$, i.e., $i(\beta-1) \in \sigma(T)$. But $i(\beta-1)\in \Pi_+$. This is a contradiction. Thus $\sigma(T)\not=\overline{\Pi}_-$. (ii) If $H$ is bounded above, then $\widehat H:=-H$ is bounded below. It is easy to see that $\widehat T:=-T$ is a time operator of $\widehat H$. Hence, by part (i), $\sigma(\widehat T)=\C$ or $\overline{\Pi}_+$. On the other hand, $\sigma(T)=\{-\lambda|\lambda \in \sigma(\widehat T)\}$, which implies that $\sigma(T)=\C$ or $\overline{\Pi}_-$. (iii) This follows from (i) and (ii). \hfill \qed \medskip In the next section we analyze the spectrum of nontrivial examples of time operators. Here we present only simple examples. \begin{ex}\label{ex2-1}{\rm We denote by $\hat r$ the multiplication operator on $L^2([0,\infty))$ by the variable $r\in [0,\infty)$: $(\hat r g)(r):=rg(r), \ {\rm a.e.}r\in [0,\infty), g\in D(\hat r)$. The operator $\hat r$ is self-adjoint and nonnegative. Let $p_0$ be an operator on $L^2([0,\infty))$ defined as follows: \begin{eqnarray} &&D(p_0):=C_0^{\infty}((0,\infty)), \\ && (p_0g)(r):=-ig'(r), \quad g\in D(p_0), \end{eqnarray} where, for an open set $\Omega\subset \R^n$ ($n\in \N$), $C_0^{\infty}(\Omega)$ denotes the set of infinitely differentiable functions on $\Omega$ with compact support in $\Omega$. Then it is easy to see that $-p_0$ is a time operator of $\hat r$ and that $$ \sigma(-p_0)=\overline{\Pi}_+. $$ Hence this is an example which illustrates one of the case of Theorem \ref{th2-1}-(i). } \end{ex} \begin{ex}\label{ex2-2}{\rm Let $L>0$ and $V_L:=(-L/2,L/2)\subset \R$. We denote by $\hat x_L$ the multiplication operator on $L^2(V_L)$ by the variable $x\in V_L$. Then $\hat x_L$ is a bounded self-adjoint operator. We define an operator $p_L$ as follows: \begin{eqnarray*} && D(p_L):=C_0^{\infty}(V_L),\\ && p_Lf:=-if', \quad f\in D(p_L). \end{eqnarray*} Then it is easy to see that $-p_L$ is a time operator of $\hat x_L$ and $$ \sigma(-p_L)=\C. $$ Hence this is an example which illustrates Theorem \ref{th2-1}-(iii). It should be remarked that $p_L$ has uncountably many self-adjoint extensions \cite[pp.257--259]{RS1}. } \end{ex} \section{Time Operators on Direct Sum Hilbert Spaces} In applications, time operators on direct sum Hilbert spaces may be useful. We briefly discuss this aspect here. Let $H_j$ ($j=1,2$) be a self-adjoint operator on a complex Hilbert space ${\cal H}_j$ which has a time operator $T_j$. Let \begin{equation} {\cal H}:={\cal H}_1\oplus {\cal H}_2. \end{equation} Then \begin{equation} T:=T_1\oplus T_2 \end{equation} is a time operator of $H_1\oplus H_2$ \cite[Proposition 2.14]{Ar05}. \begin{The}\label{th2-2} Let $H_j$, $T_j$ and $T$ be as above. Then: \begin{list}{}{} \item{{\rm (i)}} If $H_1$ is bounded below and $H_2$ is bounded above, then $\sigma(T)=\C$. \item{{\rm (ii)}} If one of $H_1$ and $H_2$ is bounded, then $\sigma(T)=\C$. \end{list} \end{The} {\it Proof}. (i) By Theorem \ref{th2-1}, $\sigma(T_1)=\C$ or $\overline{\Pi}_+$, and $\sigma(T_2)=\C$ or $\overline{\Pi}_-$. By a general theorem, we have $\sigma(T)=\sigma(T_1)\cup \sigma(T_2)$. Hence, in each case, we have $\sigma(T)=\C$. (ii) In this case, we can apply Theorem \ref{th2-1}-(iii) to conclude that one of $\sigma(T_1)$ and $\sigma(T_2)$ is equal to $\C$. Thus the desired result follows. \hfill \qed \medskip \begin{rem}{\rm In each case of Theorem \ref{th2-2}-(i) and (ii), $H_1\oplus H_2$ can be unbounded both above and below. } \end{rem} \begin{ex}{\rm Let $$ {\cal H}_L:=L^2([0,\infty))\oplus L^2(V_L), $$ $\hat r, p_0$ be as in Example \ref{ex2-1} and $\hat x_L, p_L$ be as in Example \ref{ex2-2}. Then $H_L:=\hat r\oplus \hat x_L$ on ${\cal H}_L$ is self-adjoint and bounded below (but unbounded above). Moreover $T_L:=(-p_0)\oplus (-p_L)$ is a time operator of $H_L$ and $\sigma(T_L)=\C$. Thus this example shows that the spectrum of a time operator of a self-adjoint operator which is bounded below, but unbounded above, can be equal to $\C$. } \end{ex} \section{Examples} \subsection{Time operators of the free Hamiltonian of a nonrelativistic particle } Let $\Delta$ be the $n$-dimensional generalized Laplacian acting in $L^2(\R_x^n)$ ($n\in \N$), where $\R_x^n:=\{x=(x_1,\cdots,x_n)|x_j\in \R,j=1,\cdots,n\}$, and \begin{equation} H_0:=-\frac {\Delta}{2m} \end{equation} with a constant $m>0$. In the context of quantum mechanics, $H_0$ represents the free Hamiltonian of a nonrelativistic particle with mass $m$ in the $n$-dimensional space $\R_x^n$. It is well known that $H_0$ is a nonnegative self-adjoint operator. We denote by $\hat x_j$ the multiplication operator on $L^2(\R_x^n)$ by the $j$-th variable $x_j\in \R_x^n$ and set \begin{equation} \hat p_j:=-iD_j \end{equation} with $D_j$ being the generalized partial differential operator in the variable $x_j$ on $L^2(\R_x^n)$. It is easy to see that $\hat x_j$ and $\hat p_j$ are injective. We introduce \begin{equation} \Omega_j:=\{k=(k_1,\cdots,k_n)\in \R_k^n| k_j\not=0\}, \quad j=1,\cdots,n. \end{equation} For a real-valued, Borel measurable function $G$ on $\R_k^n$ which is continuous on $\Omega_j$, we define a linear operator on $L^2(\R_x^n)$ by \begin{equation} G(\hat p):={\cal F}^{-1}G{\cal F},\label{G} \end{equation} where ${\cal F}:L^2(\R_x^n)\to L^2(\R_k^n)$ is the Fourier transform: \begin{equation} ({\cal F}f)(k):=\frac {1}{(2\pi)^{n/2}}\int_{\R^n_x}f(x)e^{-ikx}dx, \quad f\in L^2(\R_x^n),\,k=(k_1,\cdots,k_n) \in \R_k^n, \end{equation} in the $L^2$-sense, $\hat p:=(\hat p_1,\cdots, \hat p_n)$ and $G$ on the right hand side of (\ref{G}) represents the multiplication operator on $L^2(\R_k^n)$ by the function $G$. Since the Lebesgue measure of the set $\R_k^n\setminus\Omega_j$ is zero, it follows that $G(\hat p)$ is self-adjoint. For each $j=1,\cdots,n$, one can define a linear operator on $L^2(\R_x^n)$ by \begin{equation} T_{j}(G):=\frac m{2} \left(\hat x_j\hat p_j^{-1}+\hat p_j^{-1}\hat x_j\right)+G(\hat p) \end{equation} with domain \begin{equation} D(T_{j}(G)):={\cal F}^{-1}C_0^{\infty}(\Omega_j), \end{equation} where $C_0^{\infty}(\Omega_j)$ denotes the set of infinitely many differentiable functions on $\Omega_j$ with compact support in $\Omega_j$. It is easy to see that $T_j(G)$ is a symmetric operator on $L^2(\R_x^n)$. \begin{lem} The operator $T_j(G)$ is a time operator of $H_0$. \end{lem} {\it Proof}. We write $$ T_j(G)=T_j+G(\hat p) $$ with \begin{equation} T_j:=\frac m{2} \left(\hat x_j\hat p_j^{-1}+\hat p_j^{-1}\hat x_j\right). \end{equation} The operator $T_j$ is a time operator of $H_0$ (\cite{Miya1}, \cite[\S 10]{Ar05}). By using the Fourier transform, one can show that $e^{-itH_0}G(\hat p)\subset G(\hat p)e^{-itH_0}$ for all $t\in \R$. Hence, by applying \cite[Proposition 2.6]{Ar05}, $T_j(G)$ is a time operator of $H_0$. \hfill \qed \medskip \begin{rem}{\rm The operator $T_j$ is called the {\it Aharonov-Bohm time operator} \cite{AB}. Hence $T_j(G)$ is a perturbed Aharonov-Bohm time operator. } \end{rem} As for the spectrum of $T_j(G)$, we have the following theorem: \begin{The}\label{th3-1} $\sigma(T_j(G))=\overline{\Pi}_+,\ j=1,\cdots,n$. \end{The} {\it Proof}. By Theorem \ref{th2-1}-(i) and (\ref{dcp}), we need only to show that $\ker (T_j(G)^*-i) =\{0\}$. Let $f\in\ker (T_j(G)^*-i)$. Then $T_j(G)^*f=if$. This implies the equation $$ D_{k_j}\hat f(k)=\left(\frac{1}{2k_j}+\frac{k_j}{m}+ \frac {i}{m} k_jG(k)\right)\hat f(k) $$ in the sense of distribution on $\Omega_j$, where $\hat f:={\cal F}f$ and $D_{k_j}$ is the generalized partial differential operator in the variable $k_j$. Hence $$ \hat f(k)=c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n) \sqrt{|k_j|}e^{k_j^2/(2m)}e^{iG_j(k)/m}, \quad k\in \Omega_j $$ where $c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n)\in \C$ is independent of $k_j$ and $G_j$ is a differentiable function on $\Omega_j$ such that $\partial G_j(k)/\partial k_j=k_jG(k), \ k\in \Omega_j$. Since $\hat f$ is in $L^2(\R_k^n)$, it follows that $c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n)=0$ (a.e.). Hence $f=0$. Thus $\ker(T_j(G)^*-i)=\{0\}$. \hfill \qed \medskip \subsection{Time operators of the free Hamiltonian of a relativistic particle } A Hamiltonian of a free relativistic particle with mass $m\geq 0$ moving in $\R_x^n$ is given by \begin{equation} H_{\rm rel}:=\sqrt{-\Delta+m^2} \end{equation} acting in $L^2(\R_x^n)$. It is shown that the operator \begin{equation} T_j^{\rm rel}(G):=\sqrt{-\Delta+m^2}\,\hat p_j^{-1}\hat x_j+\hat x_j\sqrt{-\Delta+m^2}\,\hat p_j^{-1}+G(\hat p) \end{equation} with $D(T_j^{\rm rel}(G)):={\cal F}^{-1}C_0^{\infty}(\Omega_j)$ is a time operator of $H_{\rm rel}$ \cite[Example 11.4]{Ar05}. \begin{The}\label{th3-2} $\sigma(T_j^{\rm rel}(G))=\overline{\Pi}_+,\ j=1,\cdots,n$. \end{The} {\it Proof}. As in Theorem \ref{th3-1}, we need only to show that $\ker (T_j^{\rm rel}(G)^*-i) =\{0\}$. Let $f\in\ker (T_j^{\rm rel}(G)^*-i)$ and $\omega(k):=\sqrt{k^2+m^2}, \, k\in \R_k^n$. Then $$ D_{k_j}\hat f(k)=\frac{1}{2}\left(\frac{k_j}{\omega(k)}-\frac{k_j}{\omega(k)} \left(\frac{\partial}{\partial k_j}\frac{\omega(k)}{k_j}\right) -\frac{k_jG(k)}{\omega(k)i}\right)\hat f(k) $$ in the sense of distribution on $\Omega_j$. Hence $$ \hat f(k)=c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n) \sqrt{\frac{|k_j|}{\omega(k)}}e^{\omega(k)/2}e^{iF_j(k)/2}, \quad k\in \Omega_j $$ where $c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n)\in \C$ is independent of $k_j$ and $F_j$ is a differentiable function on $\Omega_j$ such that $\partial F_j(k)/\partial k_j=k_jG(k)/\omega(k), \ k\in \Omega_j$. Since $\hat f$ is in $L^2(\R_k^n)$, it follows that $c(k_1,\cdots, k_{j-1},k_{j+1},\cdots, k_n)=0$ (a.e.). Hence $f=0$. Thus $\ker(T_j^{\rm rel}(G)^*-i)=\{0\}$. \hfill \qed \medskip \section{A Class of Generalized Time Operators} In this section we consider spectral properties of a class of generalized time operators. Let $H$ be a self-adjoint operator on a complex Hilbert space ${\cal H}$ and $T$ be a symmetric operator on ${\cal H}$. We call the operator $T$ a {\it generalized time operator} of $H$ if $e^{-itH}D(T)\subset D(T)$ for all $t\in \R$ and there exists a bounded self-adjoint operator $C\not=0$ on ${\cal H}$ with $D(C)={\cal H}$ such that \begin{equation} Te^{-itH}\psi=e^{-itH}(T+tC)\psi, \quad \psi\in D(T).\label{GWWR} \end{equation} We call $C$ the {\it noncommutative factor} for $(H,T)$. The following facts are known: \begin{The}\label{KR} Let $T$ be a generalized time operator of $H$ with noncommutative factor $C$. \begin{list}{}{} \item{{\rm (i)}}{\rm (\cite[Theorem 2.8]{Ar05})} Let $H$ be semi-bounded and \begin{equation} CT\subset TC. \label{TC} \end{equation} Then $T$ is not essentially self-adjoint . \item{{\rm (ii)}}{\rm (\cite[Corollary 5.3-(ii)]{Ar05})} If $C\geq 0$ or $C\leq 0$, then $\sigma_{\rm p}(\overline{T}|[D(\overline{T})\cap (\ker C)^{\perp}])=\emptyset$. \item{{\rm (iii)}}{\rm (\cite[Theorem 6.2-(ii)]{Ar05})} Let $H$ be bounded below. Then, for all $\beta>0$, $e^{-\beta H}D(\overline{T})\subset D(\overline{T})$ and \begin{equation} \overline{T}e^{-\beta H}\psi-e^{-\beta H}\overline{T}\psi=-i\beta e^{-\beta H}C\psi, \quad \psi \in D(\overline{T}). \end{equation} \item{{\rm (iv)}}{\rm (\cite[Proposition 6.4, Corollary 6.6]{Ar05})} The operators $H$ and $C$ strongly commute (i.e., their spectral measures commute) and $H$ is reduced by $\overline{\Ran(C)}$. \end{list} \end{The} In what follows, $T$ is a generalized time operator of $H$ with noncommutative factor $C$ satisfying (\ref{TC}). \subsection{The case where $C$ has a non-zero eigenvalue} We first consider the case where $C$ has a non-zero eigenavlue $a \in \R\setminus\{0\}$, i.e., \begin{equation} {\cal K}_a:=\ker (C-a)\not=\{0\}. \end{equation} We have the orthogonal decomposition \begin{equation} {\cal H}={\cal K}_a\oplus {\cal K}_a^{\perp}. \label{orth} \end{equation} Relation (\ref{TC}) implies that \begin{equation} C\overline{T}\subset \overline{T}C. \end{equation} Then it follows that $\overline{T}$ is reduced by ${\cal K}_a$ and hence by ${\cal K}_a^{\perp}$. We denote the reduced part of $\overline{T}$ to ${\cal K}_a$ and ${\cal K}_a^{\perp}$ by $\overline{T}_a$ and $\overline{T}_a^{\perp}$ respectively. Hence we have \begin{equation} \overline{T}=\overline{T}_a\oplus \overline{T}_a^{\perp} \end{equation} relative to the orthogonal decomposition (\ref{orth}). Therefore \begin{equation} \sigma(T)=\sigma(\overline{T})=\sigma(\overline{T}_a)\cup \sigma(\overline{T}_a^{\perp}). \end{equation} By the strong commutativity of $H$ and $C$ (Theorem \ref{KR}-(iv)), $H$ also is reduced by ${\cal K}_a$. We denote the reduced part of $H$ to ${\cal K}_a$ by $H_a$. \begin{lem}\label{lem3-1} The operator $\overline{T}_a$ is a time operator of $H_a/a$. \end{lem} {\it Proof}. Let $\psi \in D(\overline{T}_a)=D(\overline{T})\cap {\cal K}_a$. Then, for all $t\in \R$, $e^{-itH_a}\psi=e^{-itH}\psi\in D(\overline{T})\cap {\cal K}_a =D(\overline{T}_a)$ and, by (\ref{GWWR}), $$ \overline{T}_ae^{-itH_a}\psi=e^{-itH_a}(\overline{T}_a+ta)\psi. $$ Thus the desired result follows. \hfill \qed \medskip \begin{The} Let $T$ be a generalized time operator of $H$ with noncommutative factor $C$ satisfying (\ref{TC}). Then the following {\rm (i)---(v)} hold: \begin{list}{}{} \item{{\rm (i)}} If $H_a$ is bounded below and $a>0$, then $\sigma(\overline{T}_a)$ is either $\C$ or $\overline{\Pi}_+$. \item{{\rm (ii)}} If $H_a$ is bounded above and $a<0$, then $\sigma(\overline{T}_a)$ is either $\C$ or $\overline{\Pi}_+$. \item{{\rm (iii)}} If $H_a$ is bounded above and $a>0$, then $\sigma(\overline{T}_a)$ is either $\C$ or $\overline{\Pi}_-$. \item{{\rm (iv)}} If $H_a$ is bounded below and $a<0$, then $\sigma(\overline{T}_a)=\C$ or $\overline{\Pi}_-$. \item{{\rm (v)}} If $H_a$ is bounded, then $\sigma(\overline{T}_a)=\C$. \end{list} \end{The} {\it Proof}. In (i) and (ii), $H_a/a$ is bounded below. Hence, Lemma \ref{lem3-1} and Theorem \ref{th2-1}-(i) yield the results stated in (i) and (ii). Similarly other cases follow. \hfill \qed \medskip \subsection{The case where $\Ran(C)$ is closed} We next consider the case where $\Ran(C)$ is closed. Then ${\cal H}$ is decomposed as \begin{equation} {\cal H}=\ker C\oplus \Ran(C). \end{equation} By the closed graph theorem, there exists a constant $M>0$ such that \begin{equation} \|C\psi\|\geq M\|\psi\|, \quad \psi \in (\ker C)^{\perp}=\Ran(C). \label{C1} \end{equation} The operators $T$ and $C$ are reduced by $\Ran(C)$. We denote by $\widetilde T$ and $\widetilde C$ the reduced part of $T$ and $C$ to $\Ran(C)$ respectively. The operator $\widetilde T$ is symmetric and $\widetilde C$ is a bounded self-adjoint operator on $\Ran(C)$ which is bijective with $\widetilde C^{-1}$ bounded. It follows from (\ref{TC}) that \begin{equation} \widetilde C\widetilde T\subset \widetilde T\widetilde C.\label{TC1} \end{equation} \begin{lem}\label{lem-TC} The operator \begin{equation} T_C:=\widetilde C^{-1}\widetilde T=\widetilde T\widetilde C^{-1}|D(\widetilde T) \end{equation} on $\Ran(C)$ is a symmetric operator. \end{lem} {\it Proof}. We have $D(T_C)=D(T)\cap \Ran(C)$. Hence $D(T_C)$ is dense in $\Ran(C)$. Relation (\ref{TC1}) implies that \begin{equation} \widetilde C^{-1}\widetilde T\subset \widetilde T\widetilde C^{-1}. \end{equation} Hence $T_C^*=\widetilde T^*\widetilde C^{-1}\supset \widetilde T\widetilde C^{-1}\supset T_C$. Thus the desired result follows. \hfill \qed \medskip By Theorem \ref{KR}-(iv), $H$ is reduced by $\Ran(C)$. We denote the reduced part of $H$ to $\Ran(C)$ by $\widetilde H$. \begin{The} The operator $T_C$ is a time operator of $\widetilde H$ and the following {\rm (i)---(iii)} hold: \begin{list}{}{} \item{{\rm (i)}} If $\widetilde H$ is bounded below, then $\sigma(T_C)$ is either $\C$ or $\overline{\Pi}_+$. \item{{\rm (ii)}} If $\widetilde H$ is bounded above, then $\sigma(T_C)$ is either $\C$ or $\overline{\Pi}_-$. \item{{\rm (iii)}} If $\widetilde H$ is bounded, then $\sigma(T_C)=\C$. \end{list} \end{The} {\it Proof} Since $D(T_C)=D(T)\cap \Ran(C)$, it follows that $e^{-it\widetilde H}D(T_C)\subset D(T_C)$ for all $t \in\R$. Using (\ref{GWWR}), one can see that $T_C$ satisfies $$ T_Ce^{-it\widetilde H}\psi=e^{-it\widetilde H}(T_C+t)\psi, \quad \psi \in D(T_C). $$ These facts and Lemma \ref{lem-TC} imply that $T_C$ is a time operator of $\widetilde H$. Then (i)---(iii) follow from application of Theorem \ref{th2-1}. \hfill \qed \medskip \begin{ex}{\rm A simple example is given by the case where $C\not=0$ is an orthogonal projection. Then $T_C=\widetilde T$. To construct such examples, see \cite[\S 11]{Ar05}. } \end{ex} \begin{thebibliography}{99} \bibitem{AB}Y. Aharonov and D. Bohm, Time in the quantum theory and the uncertainty relation for time and energy, {\it Phys. Rev.} {\bf 122}, 1649--1658(1961). \bibitem{Ar05}A. Arai, Generalized weak Weyl relation and decay of quantum dynamics, {\it Rev. Math. Phys.} {\bf 17} (2005), 1071--1109. \bibitem{Miya1}M.~Miyamoto, A generalized Weyl relation approach to the time operator and its connection to the survival probability, {\it J. Math. 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