数理解析研究所講究録
923 巻 1995 年 68-86

68

An Inverse Problem for the Rotating Wave

Approximation on a
By

廣川真男

-Algebra$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$(Masao HIROKAWA)

Advanced Research Laboratory, Hitachi Ltd., Hatoyama, Saitama 350-03, JapanI. INTRODUCTION.

A long-time behavior of the canonical correlation function as an infinite volume
limit interests us. In this paper, we would like to apply Arai’s results [5] concerning
long-time behavior of two-point functions to a class of canonical correlation functionsof position operators as infinite volume limit. In [5], Arai argued long-time behaviorof two-point functions of position operators for some models of a quantum harmonic
oscillator interacting with bosons.
We consider a quantum harmonic oscillator in thermal equilibrium with any

system in certain classes of bosons with infinitely many degrees of freedom in a finite
volume $V>0$ . Our models include photons in a laser interacting with oscillation

caused by a heat bath, which can be observed when the laser passes in the heatbath, and are photons in a laser interacting with oscillation caused by phonons on

the surface of a material, which can be observed when we irradiate the weak laseron the surface.

When a two-point function (or canonical correlation function)$R^{V}(t_{1}, t_{2})$ofthe position or momentum operator of the harmonic oscillator is given by anobservation, we take an infinite volume limit, $Varrow\infty$ , for$R_{\beta}^{\infty}(t_{1,2}t) \equiv\lim_{Varrow\infty}R^{V}(t1, t_{2})$havior ofLet$R_{\beta}^{\infty}(t1, t2)$

$O_{bs}$$R^{V}(t_{1},t_{2})$, and getunder suitable conditions. And we argue long-time be-.

be the position operator$q\mathrm{d}\mathrm{e}\mathrm{f}=(a+a^{+})/\sqrt{2\omega}$, momentum operator

69, or their smeared operators$p^{\mathrm{d}\mathrm{e}\mathrm{f}}=i\sqrt{\omega}(a^{+}-a)/\sqrt{2}$$O_{bs}^{S}m$, where$a$and$a^{+}$

are the an-nihilation and creation operators of the quantum harmonic oscillator, respectively,
and $\omega>0$ denotes the original frequen.cy of the quantum harmonic oscillator. In
this paper, we let that$R^{V}(t_{1}, t_{2})$$\omega$

is equal to 1, i.e., $\omega=1$ , for the sake of simplicity.

is the observed two-point function of

product. In our system to be considered, for$H_{a,b}^{V}$, given by the Bogoliubov scalar

there exists a Hamiltonian$R^{V}(t_{1}, t_{2})$which governs our system and is described by the annihilation operator , the$a$creation operator$b_{k}$$O_{bs}$$a^{+}$of the quantum harmonic oscillator; and annihilation operators, creation operators$e^{-\beta H_{a,b}^{V}}$$b_{k}^{+}$of bosons, i.e.,$H_{a,b}^{V}=^{\mathrm{e}}H^{V}\mathrm{d}\mathrm{f}(a,$is a trace class operator (where$\beta$Of course, the operator form oftr$b_{k},$$b_{k}+;k\in N)$, such that

denotes the inverse temperature and we

set the Planck constant $k=1$ ), furthermore,$R^{V}(t_{1,2}t)= \frac{1}{\beta \mathrm{t}\mathrm{r}(e^{-}\beta H_{a,b}^{V})}\mathrm{d}\mathrm{e}\mathrm{f}\int^{\beta}0d\lambda$$a^{+};$$R^{V}(t_{1}, t_{2})$is defined by$(e-(\beta-\lambda)H_{a,b}e,bOiH_{a}t_{1}bse-a,be,be^{i}a,bH_{a}VViHt_{1}-V\lambda VOH^{V}t2-iH_{a,b}bs)eVt_{2}$$H_{a,b}^{V}$is unknown, so there is a possibility that

$H_{a,b}^{V}$

is non-quadratic.
For applying Arai’s result [5, Theorem 1.3] to our case, we first solve the following

inverse problem: In terms of$x_{k}(k\in N)$$R^{V}(t_{1}, t_{2})$only, determine positive frequencies$x_{0}$

andof the quantum harmonic oscillator and scalar bosons, respectively; andcoupling constants $y_{k}\in C(k\in N)$ ; appearing in the Hamiltonian of the rotating
wave approximation (RWA),$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}(x, y)^{\mathrm{d}\mathrm{e}}=x0a^{+_{a+}} \sum_{k=1}\mathrm{f}\sum_{k=}\infty x_{k}b_{kk}^{+_{b}}+\infty 1(ykabk++\overline{yk}b_{k}+_{a})$$x^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}(x_{0}, X1, X2, \cdots)$,$y^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}(y1, y_{2}, \cdots)$,,(where means the complex conjugate of $c\in C$ ), and determine constants$\overline{c}$in terms of$R^{V}(t_{1}, t_{2})$$R^{V}(t_{1}, t_{2})$$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}(x, y)$levels of$R^{V}(t_{1}, t_{2})=\mathrm{a}$$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{\mathrm{v}}(x, y)\}=\{\mathrm{p}_{\mathrm{o}\mathrm{s}\mathrm{i}}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$representation using

poles of$W^{V}(t_{1}, t_{2})$,$c_{2}\in C$

recovers the originalin the following sense:$\{\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}$(1.1)

only such that the Hamiltonian$c_{1},$$\int_{0}^{\infty}dte^{it}R\mathcal{Z}(Vt)\}$

,

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where
$W^{V}(t_{1,2}t)\mathrm{d}=(\mathrm{e}\mathrm{f}\Omega_{0} ,.

e^{iH_{\mathrm{R}\mathrm{w}\mathrm{A}(}^{V}}Ox,y)t_{1}iH(x,y)t1iH_{\mathrm{R}}\mathrm{w}bse^{-}\mathrm{R}\mathrm{w}\mathrm{A}eVV\mathrm{A}(x,y)t_{2}obS’ 0e^{-i}\mathrm{W}\mathrm{A}y)t_{2}\Omega H_{\mathrm{R}}(xV)_{a,b}$,

which is the vacuum expectation of, and$e^{iH_{\mathrm{R}}^{V}}\mathrm{w}\mathrm{A}(x,y)t_{1}obs)e^{-iH_{\mathrm{R}\mathrm{W}}^{V}}\mathrm{A}(x,y)t_{1}e\mathrm{W}\mathrm{A}(x,ytiH_{\mathrm{R}}^{V}2ObSe^{-i}H^{V}\mathrm{R}\mathrm{w}\mathrm{A}(x,y)t_{2}$ner product of the Fock spacestate of

$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}(x, y)$$\mathcal{F}_{a,b}$. Moreover$\Omega_{0}$$(\cdot, \cdot)_{a,b}$

is a natural in-is the Fock vacuum and the ground.

Indeed there are some negative criticisms against RWA [14, \S V. ] and there$\mathrm{D}$exists the independent-oscillator model which is more useful in physics than RWA$[14,26]$ ,but we venture to use RWA in so far as our purpose of investigating thelong-time behavior. Why do we represent$R^{V}(t_{1}, t_{2})$by using RWA? Because it isnothing but easy to argue an infinite volume limit for the Hamiltonian of RWA inmathematics, and RWA is established in mathematics by [4,5,23]. So there is apossibility that we can investigate the long-time behavior of infinite volume limitof$R^{V}(t_{1}, t_{2})$$R^{V}(t_{1}, t_{2})$through infinite volume limit ofusing$W^{V}(t_{1}, t_{2})$infinite volume limit$W^{V}(t_{1}, t_{2})$in representation (1.1) of. Actually, what is better, the long-time behavior of the$W(t_{1}, t_{2})$of$W^{V}(t_{1}, t_{2})$of the position operator is investigatedexactly by Arai in [5].

An answer for this inverse problem for RWA is given by Theorem 2.1 in thispaper. By representation (1.1), we can consider an infinite volume$R^{V}(t_{1}, t_{2})$$R_{\beta}^{\infty}(t1, t2)$

offor the position operator , through the right side of (1.1). Then, we havea representation of$q$$R_{\beta}^{\infty}(t_{1}, t_{2})$by using$W(t_{1}, t_{2})$. And, applying Arai’s results in [5]to the representation, we consider the long-time behavior of

the position operator$q$$R_{\beta}^{\infty}(t)\equiv R_{\beta}^{\infty}(\mathrm{O}, t)$forin Theorem 2.3 of this paper.

II. STATEMENT OF MAIN RESULTS.

In this section, in order to introduce canonical correlation functions defined by
the Bogoliubov scalar product, and explain our main results, we first set up a general

71

framework. For a while, we fix a finite volume $V>0$ .
We give a complex Hilbert space $l^{2}(N)$ by$\{(C_{1}, C_{2}, \cdots)|c_{k}\in C,$

$f$

by

$k\in N,$ $\Sigma_{k=1}^{\infty}|c_{k}|^{2}<\infty\}$, i.e.,$(f\mathrm{o}, f_{1}, f2, \cdots)$$f=(f\mathrm{o}, f_{1}, f2, \cdots)$$l^{2}(N)=\mathrm{d}\mathrm{e}\mathrm{f}$. For each $f\in C\oplus l^{2}(N)$ , we denote, where $f_{0}\in C$ and $(f_{1}, f_{2}, \cdots)\in l^{2}(N)$ .

An inner product $( , )_{l^{2}}$ of $C\oplus l^{2}(N)$ is given by$C\oplus l^{2}(N))$, where$\overline{c}$$(f,g)_{l^{2}} \mathrm{d}\mathrm{e}\mathrm{f}=\sum^{\infty}k=0\overline{fk}gk(f,g\in$denotes the complex conjugate of $c\in C$ . We denote thesymmetric Fock space over $C\oplus l^{2}(N)$ by$\mathcal{F}_{S}(C\oplus l^{2}(N))$$\mathcal{F}_{S}(C\oplus l^{2}(N))\mathrm{d}\mathrm{e}\mathrm{f}=\oplus_{n=0}^{\infty s_{n}(c}\oplus l^{2}(N))^{n}$, which is defined by, where $S_{n}(C\oplus l^{2}(N))^{n}$ is the n-foldsymmetric tensor product of $C\oplus l^{2}(N)$ for each $n\in N$ and(see [32, p.53, Example 2]), and$S_{n}(C\oplus l^{2}(N))^{n}$for each

The operators

$a$

and

$S_{n}$$S_{0}(C\oplus l^{2}(N))0\mathrm{d}=^{\mathrm{e}}C\mathrm{f}$denotes an orthogonal projection onto$n\in N^{*}=\mathrm{d}\mathrm{e}\mathrm{f}\{0,1, \cdots\}$$a^{+}\mathrm{p}\mathrm{h}\mathrm{y}_{\mathrm{S}}\mathrm{i}\mathrm{c}\mathrm{a}11\mathrm{y}$(see again [32, p.53, Example 2]).

denote the annihilation and creation operatorsof the quantum harmonic oscillator, respectively, and likewise, operators$b_{k}$

and$b_{k}^{+}$$(k\in N)$ are the annihilation and creation operators of the bosons with infinitely

many degrees of freedom.

We consider a quantum harmonic oscillator in thermal equilibrium with a systemof bosons with infinitely many degrees of freedom in the finite volume. So, we givea state space for our system by a symmetric Fock space,denoted by simply$\mathcal{F}_{a,b}$

the inner product offor convenience, i.e.,$\mathcal{F}_{a,b}$$\mathcal{F}_{S}(C\oplus l^{2}(N))$$\mathcal{F}_{a,b}=\mathrm{d}\mathrm{e}\mathrm{f}\mathcal{F}s(C\oplus l^{2}(N))$, which is. And we denote

by $( , )_{a,b}$ .

For our system, there exists a Hamiltonian $H_{a,b}^{V}=H^{V}(a, a^{+}; bk, b_{k}^{+}, k\in N)$
whose form is unknown. So$H_{a,b}^{V}$

may be non-quadratic, but must be realized as aself-adjoint operator acting in the Fock space
the thermal equilibrium quantum system,$\mathcal{F}_{a,b}$. Since we are now considering

is a self-adjoint operator acting in

, and

(H)

where
$H_{a,b}^{1’}$$H_{a,b}^{V}$$\mathcal{F}_{a,b}$$\beta$$e^{-\tau H_{a,b}}V$

is a trace class operator on$\mathcal{F}_{a,b}$

for every$\tau\in(0, \beta]$,

is the inverse temperature. This condition implies that the spectra ofare purely discrete and the eigenvectors$\{\varphi_{n}|n\in N^{*}\}$

of$H_{a,b}^{\mathrm{v}}$form a complete

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orthonormal system ofof$(n\in N^{*})$$H_{a,b}^{V}$$\lambda_{n+1}\leq\cdots\nearrow\infty$, where$\mathcal{F}_{a,b}$$N^{*}\mathrm{d}\mathrm{e}\mathrm{f}=\{0,1, \cdots\}$in such a way that

$H_{a,b}^{V}\varphi_{n}=\lambda_{n}\varphi_{n}$$H_{a,b}^{V}$by$\{\varphi_{n}|n\in N^{*}\}$$\mathrm{D}_{a,b}$

denote the linear hull of a set$\mathrm{B}(\mathrm{D}_{a,b}, \mathcal{F}_{a},b)$Every element

$A$

in

, i.e.,$S$$\mathrm{D}_{a,b}=^{\mathrm{e}\mathrm{f}}\mathrm{d}$by L.

$\mathrm{h}.[S]$

to

, and
$\mathrm{D}_{a,b}$$A^{*}\lceil \mathrm{D}_{a,b}$$A^{+}$$\mathrm{x}_{c}(H_{a,b})V$, which is[22-24]. We denote the linear

L.h. $[\{\varphi_{n}|n\in N^{*}\}]$ . From here on, we$\mathrm{D}_{a,b}$

is dense in$\mathcal{F}_{a,b}$

the space of bounded linear operators from$\mathrm{B}(\mathrm{D}_{a,b}, \mathcal{F}_{a},b)$by$\mathcal{F}_{a,b}$. Obviously. Further,$\mathrm{D}_{a,b}$

has a unique extension to an element in

the space of bounded linear operators on$A^{-}$$0<\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{n}\leq$, we can construct a Liouville space

a set of adequate quantum operators acting in

we denote byand$\lambda_{n}$.

For the Hamiltonian

hull of. We count the eigenvalues$\mathcal{F}_{a,b}$to$\mathrm{B}(\mathcal{F}_{a,b})$. We denote the extension of, which means that the domain of operator$A^{*}$$\mathcal{F}_{a,b}$

$A$

.
,

by

is restricted.

In this paper, we consider the restricted position and momentum operators as$q^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}(a+a^{+})/\sqrt{2}\lceil \mathrm{D}_{a,b}$

and

We first define a classoperators$A$$p^{\mathrm{d}}=^{\mathrm{e}\mathrm{f}}i(a^{+}-a)/\sqrt{2}\lceil \mathrm{D}_{a,b}$$\mathrm{T}(H_{a,b}^{V})$of quantum operators, which is a set of quantum

satisfying the following conditions:(T.1) the domain of each operator is equal tooperator of each operator includes$\mathrm{D}(B)$$\mathrm{D}_{a,b}$(i.e.,

denotes the domain of each operator(T.2) for all

furthermore,

respectively.

$\tau$

in$(0, \beta]$

operators

with the Hilbert-Schmidt norm$||$$||_{2}$, and the domain of the adjoint$\mathrm{D}(A)=\mathrm{D}_{a,b}$

$B$

and

$\mathrm{D}(A^{*})\supset \mathrm{D}_{a,b}$, where);$e^{-\tau H^{\mathrm{v}_{A}}}a,b$$(e^{-\tau H_{a}^{\mathrm{v}_{b}}},A)-\mathrm{a}\mathrm{n}\mathrm{d}(Ae^{-\tau H_{a}^{\mathrm{v}_{b}}},)^{-}$$\mathrm{D}_{a,b}$

and$Ae^{-\tau H_{a,b}^{V}}$are in$\mathrm{B}(\mathrm{D}_{a,b}, \mathcal{F}_{a},b)$

are Hilbert-Schmidt operators on,$\mathcal{F}_{a,b}$.

We must now turn our attention to the unboundedness of operators because

it is known that limits on the precision of the measurement of observables forbounded operators (e.g., fermion) and unbounded operators (e.g., boson) are different [7,12,35]. For unbounded operators, the problem of their domains is delicate, so

we provide condition (T.1). Condition (T.2) addresses convergency with respect

73

to the Bogoliubov scalar product $[23,24]$ , [8, p. 96]. We note here thatis$\mathrm{T}(H_{a,b}^{V})$a linear space. We can then introduce the Bogoliubov (Kubo-Mori) scalar product$<$;$>_{H_{a,b}^{V}}$as$<A;B>_{H_{a,b}^{v^{\mathrm{d}}=\frac{1}{\beta Z(\beta)}\int_{0}(e^{-(\beta\lambda)}A}}\mathrm{e}\mathrm{f}\beta d\lambda \mathrm{t}\mathrm{r}(-H_{a,b}V*)^{-(e^{-\lambda H}}a,bVB)^{-})$

where$Z(\beta)\mathrm{d}\mathrm{e}\mathrm{f}=\mathrm{t}\mathrm{r}(e^{-\beta H_{a,b}^{V}})$

product of$\mathrm{T}(H_{a,b}^{V})$$<A;A>_{H_{a,b}}^{1/_{V}}2$. It can be easily proven that$<$,

$A,$ $B\in \mathrm{T}(H_{a,b}^{V})$;$>_{H_{a,b}^{V}}$(see, [23]). The inner product introduces a norm:. We can therefore obtain a partial -algebra$*$

Hilbert space which is the completion of
The definition of the$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$is an inner$||A||_{H^{V}}a,b\mathrm{d}\mathrm{e}\mathrm{f}=$$\mathrm{x}_{c}(H_{a,b})V$

defined by awith respect to the norm$\mathrm{T}(H_{a,b}^{V})$

,

$||$

$||_{H_{a,b}^{V}}$.

-algebra with a unit appears in [1-3,11,27]. We also

note here that an element in$\mathrm{x}_{c}(H_{a,b})V$

is not always an operator acting in$\mathcal{F}_{a,b}$.

It is noteworthy that Naudts et al. attempted to argue in general about linear
response theory on the Hilbert space which is constructed by a completion of a von
Neumann algebra with KMS-state [30]. Similarly, we deal with Mori’s theory instatistical physics on$\mathrm{x}_{c}(H_{a,b})V$, which was constructed by the completion concerning

the Bogoliubov scalar product.
Because we consider a system governed by the Hamiltonian
$a,$

all

$a^{+},$$b_{k}$, and $b_{k}^{+}(k\in N^{*})$ , on condition that$\tau\in(0, \beta]$, the condition that(O)

constructed by

is a trace class operator for$e^{-\tau H_{a,b}}V$

is natural assumption. Thus, in$O_{bs}\in \mathrm{T}(H_{a,b}^{V})$this paper we assume that the position operator$\mathrm{T}(H_{a,b}^{V})$$H_{a,b}^{V}$$O_{bs}$belongs to a dense subspacein the Liouville space.

For every $V>0,$$O_{bs}\in \mathrm{T}(H_{a,b}^{V})$.

REMARK 2.1. There are several examples satisfying condition (O). For instance,
there is an example that$q\in \mathrm{T}(H_{a,b}^{V})$, even if$H_{a,b}^{V}$

In order to introduce the Heisenberg operator
here the Liouville operator$\mathcal{L}_{a,b}^{V}$

is non-quadratic Hamiltonian.

$O_{bs}(t)$of the observable, we define

determined by the Hamiltonian$H_{a,b}^{V}$.

74

We can define, for adequate operators$A\in \mathrm{T}(H_{a}^{\tau_{b}^{\gamma}},)$$\mathcal{L}_{a,b}^{V}$

then contains a dense subspacesatisfying that:$\mathrm{D}_{a,b}arrow \mathrm{D}_{a,b}$

,

(2.1)

are in$AH_{a,b}^{V}\lceil \mathrm{D}_{a,b}$$A^{+}H_{a,b^{X}}^{V}$

are in$D_{a,b}$for all$\mathrm{D}(\mathcal{L}_{a,b}^{V})$byofof all elements$\mathrm{T}(H_{a,b}^{V})$$\mathrm{D}_{a,b}$$\mathcal{L}_{a,b}^{V}$; furthermore,

$x$

in

$\mathrm{D}_{a,b}$. Ac-[ $23$ , Lemmas 3.7 and 3.8]. More ex-$\mathcal{L}_{a,b}^{V}$

is as follows: We first define linear operators$\mathcal{L}_{a,b}^{V}$$m,$ $n\in N^{*}$$\mathrm{D}(\Phi_{m,n})^{\mathrm{d}}=^{\mathrm{f}}\mathrm{e}\mathrm{D}_{a,b}$, andis a core for$D_{a,b}$

act and easier definition of$\Phi_{m,n}$and$H_{a,b}^{V}A$$Ax,$ $A^{+}x,$ $H_{a}^{V},bAx,$ $H_{a}^{V},A^{+}Xb’ AHxa,bV$

tually, the subspace, the Liouville operator[23, Lemma 3.8]. The domain$\mathcal{L}_{a,b}^{V}A\mathrm{d}\mathrm{e}\mathrm{f}=[H_{a,b}^{V}, A]=H_{a,b}^{V}A-AH_{a,b}^{V}$

the Liouville operator$A$

defined by

,

$\{$

$\Phi_{m,n}x=\beta 1/2z(\mathrm{d}\mathrm{e}\mathrm{f}1/\beta)2W_{m}^{1}/,2n(\varphi_{n}, x)_{a,b}\varphi_{m}$,$x\in \mathrm{D}_{a,b;}m,$ $n\in N^{*}$

,

where
$\frac{\lambda_{n}-\lambda_{m}}{e^{-\beta\lambda_{m}}-e^{-\beta\lambda}n}$if$\lambda_{m}\neq\lambda_{n}$,$\beta^{-1}e^{\beta\lambda_{m}}$if$\lambda_{m}=\lambda_{n}$.

$W_{m,n}=^{\mathrm{e}\mathrm{f}}\mathrm{d}\{$It must be noted that
$W_{m,n}>0$

(2.2)

,

$m,$ $n\in N^{*}$,$m,$ $n\in N^{*}$

.

$\{$

$\iota\prime \mathrm{T}/_{m,n}=W_{n,m}r$

Then$\{\Phi_{m,n}\}_{m,n}=0,1,\cdots$,

is a complete orthonormal basis of$\mathcal{L}_{a,b}^{V}\Phi_{m},n=(\lambda_{m}-\lambda)n\Phi_{m,n}$(2.3)$\Phi_{m,n}^{+}=\Phi_{n,m}$

So, since it is clear that$\mathcal{L}_{a,b}^{V}$

self-adjoint operator acting in,is symmetric in$\mathrm{x}_{c}(H_{a,b})V$

For every,$m,$ $n\in N^{*}$.

$\mathrm{x}_{c}(H_{a,b})V,$$\mathcal{L}_{a,b}^{V}$$[\mathrm{f}^{\Phi_{m,n}}\}m,n=0,1,\cdots]$$[\{\Phi_{m,n}\}_{m,n=0},1,\cdots]\subset D_{a,b}\subset \mathrm{D}(\mathcal{L}_{a,b}^{V})$$A\in \mathrm{x}_{c}(H_{a,b})V$$m,$ $n\in N^{*}$

can be extended to a

, so$D_{a,b}$is included in
is a core for, we set$A(t)^{\mathrm{d}\mathrm{f}}=^{\mathrm{e}}e^{i}Ac_{a,b}\iota rt$, and, which is denoted by the same symbol. And

it is easy to show that the linear hull L.h.
clear that L.h.

,$\mathrm{x}_{c}(H_{a,b})V$,$R^{\mathrm{v}\mathrm{v}}(t)^{\mathrm{d}\mathrm{e}\mathrm{f}}=R(0, t)\equiv<O_{bs}(0);^{o_{bS}(t})>_{H_{a,b}^{V}}$$D_{a,b}$$\mathcal{L}_{a,b}^{V}$.

. It is75

REMARK 2.2. The time evolution $A(t)$ coincides with the Heisenberg picture$e^{iH_{a,b}}{}^{t}Ae^{-i}VVHa,bt$

for every operator

$A$

in

$D_{a,b}$and $t\in R$ [ $23$ , Proposition 3.13].

We can prove that
(2.4)$\sigma(\mathcal{L}_{a,b}^{V})=\overline{\{\lambda_{m}-\lambda|nN^{*}m,n\in\}}^{\mathrm{c}}1\mathrm{o}\mathrm{S}\mathrm{u}\mathrm{r}\mathrm{e}$In order to obtain suitable data of $R^{V}(t)$ for reconstruction, we note the following
fact: There exist non-negative constants $A_{m,n}(m, n\in N^{*})$ such that
(2.5)

$R^{V}(t)= \sum_{m,n=0}Ae^{it()}\infty m,n\lambda m-\lambda_{n}$(2.6)$0 \leq\sum_{m,n=0}^{\infty}A_{m,n}<\infty$,.

We define here a function $[R^{V}](Z)$ ( $z\in C$ with ${\rm Im} z>0$ ) by the Fourier-Laplacetransform as$[R^{V}](z)= \mathrm{d}\mathrm{e}\mathrm{f}\int_{0}^{\infty}dteR^{V}it\mathcal{Z}(t)$.

We denote the set of all positive poles of $[R^{V}](Z)$ bynegative poles of $[R^{V}](Z)$ by$\mathrm{P}_{-}^{R}$$\mathrm{P}_{+}^{R}$, and the set of all.

We here assume that
(A.O)
Moreover,$\mathrm{P}_{+}^{R}=\{\epsilon_{p}|p=0,1, \cdots\}$$\mathrm{P}_{-}^{R}=\{\eta_{p}|p=0,1, \cdots\}$

with$\inf_{p=0,1},\cdots(\mathcal{E}_{p+1^{-}}\epsilon)p>0$.

with $\inf_{p=0,1},\cdots(\eta_{p}-\eta_{p+1})>0$ .

When if condition (A.O) does not hold, we consider smeared observables given
in Definition 2.1 below.
By (2.5) and (A.O), it is clear that, for any$N^{*};$$\epsilon_{p}$andand $m^{-}(p),$ $n^{-(p)}\in N^{*}$ such that

(2.7)$\epsilon_{p}=-(\lambda_{m(p)}+-\lambda+(p))n$(2.8)$\eta_{p}=-(\lambda-m^{-}(p)\lambda)n^{-}(p)$’’$\eta_{p}$there exist $m^{+}(p),$ $n^{+}(p)\in$

76

and for the zero$0$

there exist $m^{0}(p),$ $n^{0}(p)\in N^{*}$ such that

(2.9)

$0=-(\lambda \mathrm{o}(\mathrm{p})-m\lambda 0)n(p)$ ,For every $z\in C$ with $z\neq 0$ andsuch that $|z-c|\leq|z-c^{;}|$ for all$z\not\in \mathrm{P}_{\pm}^{R}$, there exists a point$c’\in\{\mathrm{o}\}\cup \mathrm{P}_{+}^{R}\cup \mathrm{P}_{-^{\mathrm{b}}\mathrm{y}}R$$c\in\{0\}\cup \mathrm{P}_{+}^{R}\cup \mathrm{P}_{-}^{R}$

assumption (A.O). So, we have

if ${\rm Im} z>0$ . Thus, by applying Lebesgue’s dominated$| \int_{0}^{\infty}dte^{-i}et_{C}’itz|\leq|z-c|-1$

convergence theorem to (2.5) and (2.6), we note that
1$[R^{V}](z)=i \sum_{p=0}^{\infty}$(2.10)$z-\epsilon_{p}$$+i \sum_{p=0}^{\infty}$

1$z-\eta_{p}$$+i \sum_{p=0}^{\infty}\frac{1}{z}$ .

. Thus, by applying Weier-And, for$z\not\in\{0\}\cup \mathrm{P}_{+}^{R}\cup \mathrm{P}_{-}^{R},$strass’-test to (2.10), it is evident that $[R](z)$ can be extended into a meromorphic$\mathrm{M}$$| \frac{d}{dz}(\frac{1}{z-c’})|\leq|z-c|-2$function on the complex plain with singularities only at points in$\{0\}\cup \mathrm{P}_{+}^{R}\cup \mathrm{P}_{-}^{R}$by(A.0) and (2.10).

When condition (A.O) does not hold, we consider the following smeared observable. We can expand$O_{bs}$as$O_{b_{S}}= \sum_{m,n}<\Phi$ $m,n$in$\mathrm{x}_{\mathrm{c}}(H_{a,b})V$;$O_{b_{S}}>_{H_{a,b}^{V\Phi_{m}}},n$.

DEFINITION 2.1. By an observation, select$V>0$ . Then, for$O_{bs}$$\epsilon_{p}$, we define smeared observable

and$O_{b}^{sm}S$$\eta_{p}$bywith (A.O) for every

77$O_{bs}^{S}m\overline{=}O_{bs}^{sm}(\epsilon, \eta_{p}p;p\in N^{*})$$\mathrm{d}\mathrm{e}\mathrm{f}=\sum_{p=0}^{\infty}(_{\lambda_{m(p)}-}+\lambda_{n}’+\langle \mathrm{p})=-6ppm+(p)\sum_{\mathrm{p}n()}+,<\Phi+(p),n+m();O_{bs}>H_{a}^{V},)b\Phi_{m^{+}\langle p),n}+(p)$$+ \sum_{p=0}^{\infty}$
We call$O_{bs}^{sm}$$O_{bs}^{sm}$smeared position operator, which denotes

smeared momentum operator, which denotes

Of course, any smeared observable$O_{bs}^{S}m$$p^{sm}$$q^{sm}$$p\in N^{*}$$A_{m,n}$.

, if $O_{bs}=q$ . And we call

, if $O_{bs}=p$ .

satisfies condition (A.O).

The following fact is derived from (2.10) by Weierstrass’
that proper summations of$\Phi_{m}-(p),n-(p)$are determined in terms of$\mathrm{M}$-test, which tells us

only: For each$R^{\mathrm{v}}(t)$,,(2.11)$\lim_{zarrow\epsilon p}\frac{1}{i}(z-\epsilon_{p})[R^{V}](z)=\lambda_{m(p}+m+(p),,l+()-\lambda n\sum_{- ,+(p)^{=}\epsilon p}A_{m}+p),(p),n+(p)$(2.12),$\lim_{zarrow\eta_{p}}\frac{1}{i}(z-\eta_{p})[R^{V}](z)=$$\sum_{\prime,\lambda_{m^{-}}-()\lambda-()=-}A-mp-(\mathrm{p}),n-(np\mathrm{p})\eta_{p}m(p),n-(p)$.

(2.13)$\lim_{zarrow 0}\frac{1}{i}z[R^{V}](z)=\sum_{(p)^{=0}}\lambda_{m^{0_{(p)}}}-\lambda_{n}m0(p),n0_{(,0}p),A(p),n0m^{0}(p)$DEFINITION 2.2. For each $n\in N^{*}$ , we say that $d^{n}R^{V}(t)/dt^{n}$ is computable if$\sum_{p=0}^{\infty}(\lim_{zarrow\epsilon_{p}}\frac{1}{i}(z-\epsilon_{p})[R^{V}](z))\mathcal{E}_{p}^{n}<\infty$,

and$\sum_{p=0}^{\infty}(\lim_{zarrow\eta P}\frac{1}{i}(Z-\eta_{p})[R^{V}](z))(-\eta_{p})^{n}<\infty$

REMARK 2.3.
(1) If$q\in \mathrm{D}((\mathcal{L}_{a,b}^{V})n/2)$) for some $n\in N$ , then $d^{n}R^{V}(t)/dt^{n}$ is computable.

.

78(2) If $d^{n}R^{V}(t)/dt^{n}$ is computable, then we have for $n\in N$$\frac{d^{n}R^{V}(t)}{dt^{n}}$$=(-i)^{n} \sum_{=p0}^{\infty}\{(_{zarrow\epsilon_{p}}\lim\frac{1}{i}(Z-\epsilon_{\mathrm{P}})[R^{\iota}]/(_{Z}))\epsilon_{p}n-e+it\epsilon_{p}(_{zarrow}\lim_{\eta_{p}}\frac{1}{i}(Z-\eta_{p})[R^{V}](z))\eta_{p}ne-it\eta p\}$which is the meaning of “ computable.”(3) If $d^{2}R^{V}(t)/dt2$ is computable, then we have$O_{bs}\in \mathrm{D}(\mathcal{L}_{a,b}^{V})$since we assumed(A.0).
(4) There is an example of non quasi-free Hamiltonian satisfying$q\in \mathrm{D}(\mathcal{L}_{a,b}^{V})$.

Here we remember (2.6), $(2.11)-(2.13)$ , and note if $d^{2}R(t)/dt^{2}$ is computable,

converges by (2.11). So, we can define a con-$\Sigma_{p=}^{\infty}0\epsilon_{p^{\Sigma A+}}m2\lambda+(p)-m+(p),n+(p),m+(\lambda_{n}+(p)^{=}-\epsilon_{p}p),n(p)$stant(2.14)$\omega_{0}$by$\omega 0^{\mathrm{d}\mathrm{f}}=^{\mathrm{e}}\frac{\sum_{p=0}^{\infty}6_{p}(zarrow\epsilon Z\lim\frac{1}{i}(P-\in \mathrm{P})[R^{V}](z)\mathrm{I}}{\sum_{p=0}^{\infty}(_{z\epsilon}\lim_{arrow p}\frac{1}{i}(_{Z-}\epsilon_{p})[R^{V}](z)\mathrm{I}}$.

We furthermore define a function $D^{V}(z)$ by(2.15)$D^{V}(z)=^{\mathrm{f}} \mathrm{d}\mathrm{e}(\frac{2}{\omega_{0}(R^{V}(0)-R_{0)}^{V}}\sum_{p=0}^{\infty}(\lim_{zarrow\epsilon_{p}}\frac{1}{i}(z-\epsilon_{p})[R^{V}](z))\frac{\epsilon_{p}}{z-\epsilon_{p}})^{-1}$$R_{0}^{V}=-\mathrm{e}i\mathrm{d}\mathrm{f}zarrow 0;\mathrm{I}\mathrm{l}\mathrm{i}\mathrm{m}z\mathrm{m}>0^{Z}[R^{V}](_{Z)}$.

REMARK 2.4. We here note that the constantsalso determined by $[R^{V}](z)$ only.

,$R_{0}^{V}$$\omega_{0},$, and the function $D^{V}(z)$79Now, we can state one of our main theorems:

THEOREM 2.1. Under (H) and (O),$R^{V}(t_{1}, t_{2})$$suppo\mathit{8}e$that we get a two-point function, defined by the Bogoliubov scalar product, with (A.O) from an experimen-

tal observation satisfying the conditions that $d^{2}R^{V}(t)/dt^{2}$ is computable. Then thefunction$D^{V}(z)$can be extended to a meromorphic function on the complex plane,

and the set $\{\omega_{k}|k\in N\}$ of all zero points of $D^{V}(z)-D^{V}(0)$ except $z=0$ is counted
in such a way that$\omega_{k}\in(\epsilon_{k-1}, \epsilon_{k})$And the total Hamiltonian of$RW\mathrm{A}$,$k\in N$.

is given by$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}= \mathrm{d}\mathrm{e}\mathrm{f}\omega 0aa+\sum+bkkk+\omega\sum_{=k=11}+_{b}\rho k(\infty k\infty a^{+_{b+}}kb_{k}+_{a})$where$\rho_{k}=\mathrm{d}\mathrm{e}\mathrm{f}(\omega_{0}\omega_{k}/(D^{V})’(\omega_{k}))^{1/2}$

alized as a positive$\mathcal{F}_{a,b}$,$(i.e.)H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}\geq 0)$$k\in N$ ,,$(D^{V})’(z)\equiv dD^{V}(z)/dZ$.

$H_{\mathrm{R}\mathrm{W}\mathrm{A}}^{V}$

is re-

self-adjoint operator acting in the Fock spacesuch that$\sigma(H_{\mathrm{R}\mathrm{W}}^{V}\mathrm{A})=\{\epsilon_{0}n_{0}+\cdots+\epsilon_{N}n_{N}|n_{0}, \cdots , n_{N}\in N^{*}, N\in N^{*}\}$

whereis equal to the set of all positive poles of $[R^{V}](z)$ , which is the$\{\epsilon_{p}|p\in N^{*}\}$

set of all zero points of $D^{V}(z).$ Furthermoref$e^{iH_{\mathrm{R}\mathrm{w}\mathrm{A}}^{VV}}tobSe^{-}iH\mathrm{R}\mathrm{W}\mathrm{A}t$$O_{bs}(t)$is reconstructed as$t_{1},$$t_{2}\in R$$O_{\mathrm{R}\mathrm{W}\mathrm{A}}(t)=^{\mathrm{f}}\mathrm{d}\mathrm{e}$such that$R^{V}(t_{1}, t_{2})=2(R^{V}(0)-R^{V}0){\rm Re} W^{V}(t1, t2)+R_{0}^{V}$for every,,.

From now on, we consider the case that the observable is given by the position
operator, i.e., $O_{bs}=q$ . We define a set(2.16)$\Gamma_{V}$$\Gamma_{V}=\mathrm{d}\mathrm{e}\mathrm{f}\{k|k=\frac{2\pi n}{V},$of lattice points by$n=0,$ $\pm 1,$ $\pm 2,$$\cdots\}$.

80

Here we assume the following technical conditions for existence of an infinite

volume limit:(A.1)$\omega_{0}arrow\omega_{\beta,0}^{\infty}>0$

as $Varrow\infty$ .

(A.2) There exist a non-negative, continuously differentiable function

and real-valued continuous function$\rho_{\beta}(k)$$\omega_{\beta}(k)$,in $L^{2}(R)$ , which satisfy the following

conditions;$\omega_{\beta}(k’)<\omega_{\beta}(k)$there are$\mathrm{o}\mathrm{n}\mathrm{e}_{-}\mathrm{t}_{0}$andfor $0\leq k’<k$ ,one maps,$\delta_{1}$and$\delta_{2}$for $k\in R$ ,$\omega_{\beta}(-k)=\omega_{\beta}(k)$: $N\equiv\{1,2, \cdots\}arrow Z\equiv\{0, \pm 1, \pm 2, \cdots\}$ suchthat
(2.17)$\omega_{n}=\omega_{\beta}(\frac{2\pi\delta_{1}(n)}{V})$,,$\rho_{n}=\rho_{\beta}(\frac{2\pi\delta_{2}(n)}{V})/\sqrt{V}$and(2.18)$m \equiv\inf_{-\infty<k<\infty}\omega_{\beta}(k)>0$,$\int_{-\infty}^{\infty}\frac{\rho_{\beta}(k)^{2}}{\omega_{\beta}(k)}dk<\infty$

We define for every $V>0$ and $t\in R$ a function$R_{1}^{V}(t)$.

by$R_{1}^{V}(t)=^{\mathrm{f}} \sum_{p=}^{\infty}\mathrm{d}\mathrm{e}(_{zarrow}\lim_{\epsilon_{p}}\frac{1}{i}(Z-0\epsilon_{p})[R^{V}](_{Z}))e^{-it\epsilon_{p}}$,where $\{\epsilon_{p}|p=0,1, \cdots\}$ is the set of all positive poles of $[R^{V}](z)$ , which was ap-

peared in condition (A.O). And we set$[R_{1}^{V}](z)= \mathrm{d}\mathrm{e}\mathrm{f}\int_{0}^{\infty}dteR^{V}itz1(t)$,$z\in C^{+}\equiv\{\zeta\in C|{\rm Im}(>0\}$.

Then, we have

LEMMA

2.2.

If

$[R_{\beta,1}^{\infty}](z) \equiv\lim_{Varrow\infty}[R_{1}^{V}](z)$$R_{\beta}^{\infty}(0) \equiv\lim_{Varrow\infty}R^{\iota^{\gamma}}(0)$also exists.

and$R_{\beta,0}^{\infty} \equiv\lim_{Varrow\infty}R_{0}V$

exist,

then81

Here, for using Theorem 2.1 and Lemma 2.3, we assume that
(A.3) For every $V>0,$ $d^{2}R^{V}(t)/dt2$ is computable. And

and$R_{\beta,0}^{\infty} \equiv\lim_{Varrow\infty}R_{0}V$$R_{\beta}^{\infty}(0) \equiv\lim_{Varrow\infty}R^{V}(0)$exist.

We define a function$D_{\mathrm{R}\mathrm{w}\mathrm{A}}^{\beta}(z)$by$D_{\mathrm{R}\mathrm{W}\mathrm{A}}^{\beta}(z)^{\mathrm{d}}= \mathrm{e}\mathrm{f}(\frac{R_{\beta}^{\infty}(0)-R^{\infty}\beta)0}{2})\cross\frac{1}{i[R_{\beta,1}^{\infty}](_{Z})}$

It is clear that there exists the inverse function.

such that$\varphi_{\beta}(x)$$\varphi\beta(x)$is differ-entiable and monotone increasing in $(m, \infty)$ with$\lim_{x\downarrow m}\varphi\beta(X)=0$,,$\varphi_{\beta}’(x)=(\omega_{\beta}’(\varphi_{\beta}(x)))^{-1}$$x>m$ .

For using Arai’s results in [5], we assume a little more assumptions:(A.4)$\epsilon>0,x\geq\sup_{m}|\int_{-\infty}\infty\frac{\rho_{\beta}(k)^{2}}{(_{X-i\epsilon})-\omega_{\beta}(k)}|<\infty$(A.5) There exists a constanttion$\varphi_{\beta}’(x)\rho_{\beta}(\varphi\beta(X))^{2}$,$\theta(\beta)$$\lim_{\epsilon l0}I_{\beta}^{(0})(x-i\epsilon)=I_{\beta}(0)(x)$for all sufficiently large(2.20)$(0,2\pi)$,$x\geq m$$|z|(z\in \mathrm{D}_{m,\theta}^{\beta})$,$A_{m}^{(0)}(\beta)\neq 0$

and$\mathrm{p}_{0}(\beta;m)\geq 0$$I_{\beta}^{(0)}(Z)$onto the domain$|I_{\beta}^{(0)}(z)|\leq \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}|Z|-\mathrm{q}0(\beta)$with a constant$\mathrm{q}_{0}(\beta)\geq 0$

,

,

,

$0< \epsilon<0x\geq\inf_{\epsilon;m}|D_{\mathrm{R}\mathrm{W}\mathrm{A}}^{\beta}(x-i\epsilon)-2i\pi I_{\beta}((0)X-i\epsilon)|>0$

for all sufficiently small$\epsilon_{0}>0$.

such that the func-

with the following properties:$z arrow 0;z\in\lim_{\theta}\mathrm{D}_{m}^{\beta},\frac{I_{\beta}^{(0)}(m+z)}{z^{\mathrm{p}\mathrm{o}()}\beta;m}=A^{(}0)(m\beta)$with constant(2.21)$\in$has an analytic continuation$\mathrm{D}_{m,\theta}^{\beta \mathrm{d}}=^{\mathrm{e}\mathrm{f}}\{z\in C|{\rm Re} z>m, -\theta(\beta)<\arg z<0\}$(2.19)$\inf_{\epsilon>0,x\geq m}|D\beta(x-i\epsilon)\mathrm{R}\mathrm{W}\mathrm{A}|<\infty$.

So by using Arai’s result [5, Theorem 1.3], we obtain the following theorem:

82

THEOREM 2.3. Let $O_{bs}=q$ . There existsLet$R_{\beta}^{\infty}(t_{1}, t_{2}) \equiv\lim_{Varrow\infty}R^{V}(t_{1}, t_{2})$$B_{m}^{(0)}(\beta)=\mathrm{d}\mathrm{e}\mathrm{f}(D_{\mathrm{R}\mathrm{W}\mathrm{A}}^{\beta}(m)-2i\pi\delta_{0},\mathrm{P}\mathrm{o}(m)A_{m}(0))D_{\mathrm{R}\mathrm{W}}^{\beta}(\mathrm{A})m$(a) If$R_{\beta,0}^{\infty}\neq 0$(b) If$R_{\beta,0}^{\infty}=0$, then$\lim_{tarrow\infty}R_{\beta}^{\infty}(t)=R_{\beta,0}^{\infty}$

, and

.

$R_{\beta}^{\infty}(t)\equiv R_{\beta}^{\infty}(0, t)$.

.

, then$R_{\beta}^{\infty}(t)=R_{\beta,1}^{\infty}(t)+R_{\beta,1}^{\infty}(-t)$,,$R_{\beta,1}^{\infty}(t)\sim tarrow\infty$$\omega_{\beta,0}^{\infty}(R_{\beta}^{\infty}(\mathrm{o})-R_{\beta}^{\infty},)0\frac{A_{m}^{(0)}(\beta)e^{-}(\mathrm{p}_{\mathrm{o}(;}\beta m)+1)/2\Gamma i\pi(\mathrm{P}\mathrm{o}(\beta\cdot m)+1)}{B_{m}^{(0)}(\beta)}$$\cross e^{-im}tt-(\mathrm{p}\mathrm{o}(\beta;m)+1)$

REMARK 2.5. Concerning part (a), if the condition that

$R_{\beta,0}^{\infty}\neq 0$

occurs, maybe

it will be the case when there are infinitely many elements in the thermal states forevery $V>0$ such that the elements are not orthogonal to just like the superfluidity$q$

at $T=0$ . Here the thermal states is a physical notion given by(i.e.,$\mathcal{L}_{a,b}^{V}(\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{S})=\{0\}$$\overline{\mathrm{L}.\mathrm{h}.[\{\Phi n,n\}_{n}=0,1,\cdots]}$) in thermo field dynamics (e.g. [22]).

83

REFERENCES[1] J. -P. Antoine and W. Karwowski,$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$-algebra of closed linear operators

in Hilbert space, Publ. RIMS, Kyoto Univ., 21 (1985), 205-236.

Add.

$/\mathrm{E}\mathrm{r}\mathrm{r}$.

ibid. 22 (1986), 507-511.
[2] J. -P. Antoine and F. Mathot,$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$-algebra of closed linear operators andtheir commutants I. General structure, Ann. Inst. H. Poincare’, 46 (1987), 299-

324.
[3] J. -P. Antoine, A. Inoue and C. Trapani,$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$-algebra of closable operators

I. The basic theory and the abelian case, Publ. RIMS, Kyoto Univ., 26 (1990),

359-395.
[4] A. Arai, Spectral analysis of a quantum harmonic oscillator coupled to infinite

many scalar bosons, J. Math. Anal. Appl. 140 (1989), 270-288.
[5] A. Arai, Long-time behavior of two-point functions of a quantum harmonic

oscillator interacting with bosons, J. Math. Phys. 30 (1989), 1277-1288.
[6] A. Arai, Path integral representation of the index of K\"ahler-Dirac operators on

an infinite dimensional manifold, J. Func. Anal. 82 (1989), 330-369.
[7] H. Araki and M. Yanase, Measurement of quantum mechanical operators, Phys.

Rev. 120 (1960), 622-626.
[8] 0. Bratteli and D. W. Robinson, Operator algebras and Quantum Statistical

Mechanics II, (Springer-Verlag, New York, 1981)[9] I. Braun, Irreversible behavior of a quantum harmonic oscillator coupled to aheat bath, Physica, 129A (1985), 262-301.
[10] H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev.

83 (1951), 34-40.

84[11] G. Epifanio and C. Trapani,$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}*$-algebra of matrics and operators, J. Math.

Phys. 29 (1988), 536-540.
[12] H. Ezawa, Problem of Quantum Control and Measurement, 3-8, in QuantumControl and Measurement edited by H. Ezawa and Y. Murayama, (NorthHolland, Amsterdam, 1993).
[13] H. Ezawa and A. Arai, Quantum fields theory and statistical mechanics, Nihon$\mathrm{H}\mathrm{y}_{\overline{\mathrm{O}}\mathrm{r}}\mathrm{o}\mathrm{n}$Sha, Tokyo, 1988 (in Japanese).

[14] G. W. Ford, J. T. Lewis and R. F. O’Connell Quantum Langevin equation,

Phys. Rev. A, 37 (1988), 4419-4428.
[15] X. L. Li, G. W. Ford and R. F. O’Connell, Correlation in the Langevin theoryof Brownian motion, Am. J. Phys. 61 (1993), 924-929.
[16] K. Lindenberg and J. West, Statistical properties of quantum systems: The

linear oscillator, Phys. Rev. A, 30 (1984), 568-582.
[17] W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York,1973)[18] D. Forster, Hydrodynamic fluctuations,broken symmetry and correlation func-

tions, (Benjamin, 1975)[19] J. Glimm, Singular perturbations of selfadjoint operators, Comm. Pure Appl.
Math., 22 (1969), 401-414.
[20] J. Glimm and A. Jaffe, A$\lambda(\phi^{4})_{2}$quantum field theory without cutoffs I. Phys.

Rev. 176 (1968), 1945-1951.
[21] J. Glimm and A. Jaffe, The$\lambda(\phi^{4})_{2}$quantum field theory without cutoffs II. The

field operators and the approximate vacuum, Ann. Math. 91 (1970), 362-401.

85[22] M. Hirokawa, Rigorous construction of Liouville spaces and thermo field dy-

namics for bosonic systems in mathematics, Ann. Phys. (N.Y.)223 (1993),

1-36.
[23] M. Hirokawa, Mori’s memory kernel equation for a quantum harmonic oscillator

coupled to RWA-oscillator, Ann. Phys. (N.Y.)224 (1993), 301-341.
[24] M. Hirokawa, Mori’s memory kernel equation in equilibrium quantum systems

in finite volume, Ann. Phys. (N.Y.)229 (1994), 354-383./Errata, Ann. Phys.
(N.Y.)235 (1994), 240-241.
[25] M. Hirokawa, A mathematical relation between the potential of the rotating

wave approximation and an estimation of the fluctuation in Mori’s theory, Ann.
Phys. (N.Y.)234 (1994), 185-210.
[26] M. Hirokawa,

General Properties between Canonical Correlation andIndependent-Oscillator Model, (preprint) ARL Research Report No.95-001.
[27] S.S. Horuzhy, Introduction to algebraic quantum field theory, Kluwer Academic

Publishers, Dordrecht, Boston, London, 1990.
[28] H. Mori, Transport, collective motion and Brownian motion, Prog. Theo. Phys.

33 (1965), 423-455.
[29] H. Mori, A continued-fraction representation of the time-correlation functions,

Prog. Theo. Phys. 34 (1965), 399-416.
[30] J. Naudts, A. Verbeure and R. Weder, Linear response theory and the KMS

condition, Comm. Math. Phys. 44 (1975), 87-99.
[31] Y. Okabe, KMO-Langevin equation and fluctuation-dissipation theorem (I),

Hokkaido Mathematical Journal 15 (1986), 163-216.
[32] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol.I: Func-

tional Analysis, (Academic Press, New York, 1980)

86[33] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol.II:

Fourier Analysis, Self-Adjointness, (Academic Press, New York, 1975)[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol.IV:

Analysis of Operators, Self-Adjointness, (Academic Press, New York, 1978)[35] M. Ozawa, Dose a Conservation Law Limit Position Measurements? Phys. Rev.

Lett. 67 (1991), 1956-1959.
[36] B. Simon, Functional Integration and Quantum Mechanics, (Academic Press,

New York, 1979)[37] E. Turbowitz, The inverse problem for periodic potentials, Comm. Pure. Appl.

Math. 30 (1977), 321-337.