Perturbation theory of KMS states

TL;DR

This paper extends a recent perturbation formula for equilibrium states to a broader class of KMS states within general $W^*$-dynamical systems, enhancing the theoretical understanding of quantum statistical mechanics.

Contribution

It generalizes Hastings' perturbation formula to KMS states in $W^*$-dynamical systems, broadening its applicability.

Findings

Extended perturbation formula to KMS states

Applicable to general $W^*$-dynamical systems

Provides new tools for quantum statistical mechanics

Abstract

We extend the new perturbation formula of equilibrium states by Hastings to KMS states of general $W^*$-dynamical systems.

Equations

$$\frac{\mathrm{Tr}(e^{-\beta H}A)}{\mathrm{Tr}(e^{-\beta H})}.$$

$$E_V^{\tau }\left(\frac{i\beta }{2}\right)=\sum _{n\ge 0}{\left(-\frac{\beta }{2}\right)}^n{\int }_{0\le s_n\le \cdots \le s_1\le 1}d{s}_1\cdots d{s}_n{\tau }_{\frac{i\beta }{2}{s}_n}(V)\cdots {\tau }_{\frac{i\beta }{2}{s}_1}(V)$$

$$e^{-\beta (H+V)}=E_V^{\tau }\left(\frac{i\beta }{2}\right)e^{-\beta H}{\left(E_V^{\tau }\left(\frac{i\beta }{2}\right)\right)}^{\ast }.$$

$$e^{-\beta (H+V)}=\mathcal{O}(V)e^{-\beta H}{\left(\mathcal{O}(V)\right)}^{\ast }.$$

$$f(t)=\{\begin{array}{ll}\sum _{n=0}^{\infty }\frac{2}{n+\frac{1}{2}}{e}^{-2\pi \left(n+\frac{1}{2}\right)|t|}=-2\log \left(\tanh \frac{\pi |t|}{2}\right),& (t\ne 0)\\ 0& (t=0)\end{array}$$

$$\Phi (V;u)={\int }_{\mathbb{R}}f\left(t\right){\sigma }_t^{uV}\left(V\right)dt,$$

$$\theta (V)=\sum _{n=0}^{\infty }{\int }_0^{1}d{u}_1{\int }_0^{u_1}d{u}_2\cdots {\int }_0^{u_{n-1}}d{u}_n\Phi \left(V;u_1\right)\dots \Phi \left(V;u_n\right),$$

$${\omega }_V(A)=(\Omega ,\theta (V{)}^{\ast }A\theta (V)\Omega ),A\in \mathfrak{M}.$$

$$\frac{d}{ds}{\Omega }_s=\frac{1}{2}{\int }_0^{1}du{\sigma }_{-\frac{iu}{2}}^{sV}(V){\Omega }_s.$$

$$e^{z\left(L+sV\right)}\Omega =E_{sV}(-iz)\Omega ,$$

$$e^{z\left(\frac{L+sV}{2}\right)}{e}^{\left(1-z\right)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega =E_{sV}\left(-\frac{iz}{2}\right)E_{\tilde{s}V}{\left(\frac{i\bar{z}}{2}\right)}^{\ast }{e}^{\frac{L+\tilde{s}V}{2}}\Omega .$$

$$\frac{d}{dz}\left(e^{z\left(\frac{L+sV}{2}\right)}{e}^{\left(1-z\right)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)=\frac{s-\tilde{s}}{2}{e}^{z\left(\frac{L+sV}{2}\right)}V{e}^{\left(1-z\right)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega$$

$$D_s:=\bigcup _{N\in \mathbb{N}}{E}_{[-N,N]}\mathcal{H},$$

$$\left(\xi ,\frac{d}{dz}{e}^{z\left(\frac{L+sV}{2}\right)}{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)=\frac{d}{dz}\left(e^{\bar{z}\left(\frac{L+sV}{2}\right)}\xi ,e^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)$$

$$=\left(\left(\frac{L+sV}{2}\right)e^{\bar{z}\left(\frac{L+sV}{2}\right)}\xi ,e^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)+\left(e^{\bar{z}\left(\frac{L+sV}{2}\right)}\xi ,-\left(\frac{L+\tilde{s}V}{2}\right)e^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)$$

$$=\frac{s-\tilde{s}}{2}\left(e^{\bar{z}\left(\frac{L+sV}{2}\right)}\xi ,V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right).$$

$$V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \in D\left(e^{z\left(\frac{L+sV}{2}\right)}\right)$$

$$e^{z\left(\frac{L+sV}{2}\right)}V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega$$

$$=E_{sV}^{\sigma }\left(-\frac{iz}{2}\right){\sigma }_{-\frac{iz}{2}}(V){\left(E_{sV}^{\sigma }\left(-\frac{\overline{iz}}{2}\right)\right)}^{\ast }{E}_{sV}^{\sigma }\left(-\frac{iz}{2}\right){\left(E_{\tilde{s}V}^{\sigma }\left(-\frac{\overline{iz}}{2}\right)\right)}^{\ast }{e}^{\left(\frac{L+\tilde{s}V}{2}\right)}\Omega .$$

$$z\mapsto \left(e^{\bar{z}\left(\frac{L+sV}{2}\right)}\xi ,V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)$$

$$z\mapsto \left(\xi ,E_{sV}^{\sigma }\left(-\frac{iz}{2}\right){\sigma }_{-\frac{iz}{2}}(V){\left(E_{sV}^{\sigma }\left(-\frac{\overline{iz}}{2}\right)\right)}^{\ast }{E}_{sV}^{\sigma }\left(-\frac{iz}{2}\right){\left(E_{\tilde{s}V}^{\sigma }\left(-\frac{\overline{iz}}{2}\right)\right)}^{\ast }{e}^{\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right).$$

$$\left(e^{\overline{it}\left(\frac{L+sV}{2}\right)}\xi ,V{e}^{\left(1-it\right)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \right)$$

$$=\left(\xi ,e^{it\left(\frac{L+sV}{2}\right)}V{e}^{-it\left(\frac{L+\tilde{s}V}{2}\right)}{e}^{\frac{L+\tilde{s}V}{2}}\Omega \right)$$

$$=\left(\xi ,{\sigma }_{\frac{t}{2}}^{sV}(V)e^{it\left(\frac{L+sV}{2}\right)}{e}^{-it\frac{L}{2}}{\left(e^{it\left(\frac{L+\tilde{s}V}{2}\right)}{e}^{-it\frac{L}{2}}\right)}^{\ast }{e}^{\frac{L+\tilde{s}V}{2}}\Omega \right)$$

$$=\left(\xi ,E_{sV}^{\sigma }\left(\frac{t}{2}\right){\sigma }_{\frac{t}{2}}(V){\left(E_{sV}^{\sigma }\left(\frac{t}{2}\right)\right)}^{\ast }{E}_{sV}^{\sigma }\left(\frac{t}{2}\right){\left(E_{\tilde{s}V}^{\sigma }\left(\frac{t}{2}\right)\right)}^{\ast }{e}^{\frac{L+\tilde{s}V}{2}}\Omega \right).$$

$$\underset{s\to \tilde{s}}{\lim }{e}^{z\left(\frac{L+sV}{2}\right)}V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega =e^{z\left(\frac{L+\tilde{s}V}{2}\right)}V{e}^{(1-z)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega .$$

$$\mathbb{R}\ni u\mapsto e^{u\left(\frac{L+sV}{2}\right)}{e}^{(1-u)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega \in \mathcal{H}$$

$$\frac{d}{du}{e}^{u\left(\frac{L+sV}{2}\right)}{e}^{(1-u)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega =\left(\frac{s-\tilde{s}}{2}\right)e^{u\left(\frac{L+sV}{2}\right)}V{e}^{(1-u)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega ,$$

$${\Omega }_s-{\Omega }_{\tilde{s}}$$

$$={\int }_0^{1}du\frac{d}{du}{e}^{u\left(\frac{L+sV}{2}\right)}{e}^{(1-u)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega$$

$$=\frac{s-\tilde{s}}{2}{\int }_0^{1}du{e}^{u\left(\frac{L+sV}{2}\right)}V{e}^{(1-u)\left(\frac{L+\tilde{s}V}{2}\right)}\Omega$$

$$\frac{d}{ds}{\Omega }_s$$

$$=\frac{1}{2}{\int }_0^{1}du{\sigma }_{-\frac{iu}{2}}^{sV}(V){\Omega }_s.$$

$$\frac{d}{ds}{\omega }_s\left(A\right)={\int }_0^{1}du\left({\Omega }_s,{\sigma }_{iu}^{sV}\left(V\right)A{\Omega }_s\right)$$

$$\frac{d}{ds}{\omega }_s(A)$$

Full text

1. Introduction

Perturbation theory of equilibrium states is a basic problem in quantum statistical physics which has been studied from old time. For a finite system with Hamiltonian $H$, equilibrium state with inverse temperature $\beta$ is defined by the Gibbs state:

[TABLE]

If we perturb $H$ by $V$, $e^{-\beta H}$ is replaced by $e^{-\beta(H+V)}$. Using the Duhamel formula, $e^{-\beta(H+V)}$ can be obtained from $e^{-\beta H}$ by multiplying

[TABLE]

and its adjoint from left and right respectively. Here $\tau$ is the dynamics given by $H$, i.e., $\tau_{t}(A):=e^{itH}Ae^{-itH}$. Namely, we have

[TABLE]

In the thermodynamic limit, this $E_{V}^{\tau}\left(\frac{i\beta}{2}\right)$ can diverge. In spite of that, an analogous representation exists in infinite systems as well, thanks to Araki [A]. (See Theorem 9.) Corresponding to the possible divergence of $E_{V}^{\tau}\left(\frac{i\beta}{2}\right)$, the representation by Araki is written in terms of an unbounded operator.

In [H], Hastings introduced a new representation of $e^{-\beta(H+V)}$ with $E_{V}^{\tau}\left(\frac{i\beta}{2}\right)$ replaced by some bounded operator $\mathcal{O}(V)$:

[TABLE]

The main result of this paper is the extension of Hasting’s result to $W^{*}$-dynamical systems. For the notations and known facts about $W^{*}$-dynamical systems used in this paper, see Appendix A and Appendix B. Let $\mathcal{H}$ be a Hilbert space, $\mathfrak{M}\subset B(\mathcal{H})$ a von Neumann algebra. We denote the set of self-adjoint elements in $\mathfrak{M}$ by $\mathfrak{M}_{\rm sa}$. The predual of $\mathfrak{M}$ is denoted by $\mathfrak{M}_{*}$. Let $\Omega\in\mathcal{H}$ be a cyclic and separating unit vector for $\mathfrak{M}$. Define the functional $\omega\in\mathfrak{M}_{*}$ by $\omega(A)=(\Omega,\ A\Omega)$ for all $A\in\mathfrak{M}$. Let $\Delta$ be the modular operator and $J$ the modular conjugation associated to $(\mathfrak{M},\ \Omega)$. Let $\sigma$ be the modular automorphism group, and $\sigma^{Q}$ its perturbation by $Q\in{\mathfrak{M}}_{\mathrm{s}a}$. Define Liouvillean $L$ of $\sigma$ by $L=\log\Delta$. (See Appendix B.) For $V\in\mathfrak{M}_{sa}$, define $\Omega_{V}=e^{\frac{L+V}{2}}\Omega$ and the functional $\omega_{V}\in\mathfrak{M}_{*}$ by $\omega_{V}(A)=\left(\Omega_{V},\ A\Omega_{V}\right)$ for all $A\in\mathfrak{M}$. In [A], it was proven that $\frac{\omega_{V}}{\omega_{V}(1)}$ is a $(\sigma^{V},-1)$-KMS state. The explicit expansion of $\omega_{V}$ is given, in terms of unbounded operator, the Liouvillean $L$. This corresponds to the expansion (3). (See Appendix Theorem 9 (e).)

Our main Theorem is an alternative representation of $\omega_{V}$, in terms of bounded operators, which corresponds to the expression by Hastings (4).

Define the real function $f$ on $\mathbb{R}$ by

[TABLE]

Then, $f\in L^{1}(\mathbb{R})$. Throughout this paper we fix this $f$. Properties of $f$ are collected in Appendix C.

Define the maps $\Phi:\mathfrak{M}_{sa}\times\mathbb{R}\to\mathfrak{M}_{sa}$ by

[TABLE]

for all $V\in\mathfrak{M}_{\mathrm{s}a}$ and $u\in\mathbb{R}$. The right hand side integral is defined in the $\sigma$-weak topology [BR1]. Furthermore, $\Phi(V;u)$ is continuous in the norm topology with respect to $u\in{\mathbb{R}}$. (See Lemma 7.) Properties of $\Phi(V;u)$ we use in this paper are shown in Section 4. Define $\theta:\mathfrak{M}_{sa}\to\mathfrak{M}$ by

[TABLE]

for all $V\in\mathfrak{M}_{\mathrm{s}a}$. The integral and summation is in the norm topology. Here is our main Theorem.

Theorem 1**.**

For any $V\in{\mathfrak{M}}_{\mathrm{s}a}$, we have

[TABLE]

2. Proof of Theorem 1 for analytic $V$

In this section, we show Theorem 1 for analytic $V$.We will use facts in Appendix B. For $s\in\mathbb{R}$, define the vector $\Omega_{s}:=\Omega_{sV}=e^{\frac{L+sV}{2}}\Omega$ and the functional $\omega_{s}\in\mathfrak{M}_{*}$ by $\omega_{s}(A):=\omega_{sV}(A)=(\Omega_{s},\ A\Omega_{s})$ for all $A\in\mathfrak{M}$. Especially, we have $\Omega_{0}=\Omega$ and $\omega_{0}=\omega$. For $s\in\mathbb{R}$, the $W^{*}$-dynamics $\sigma^{sV}$ satisfies $\sigma_{t}^{sV}(A):=e^{it(L+sV)}Ae^{-it(L+sV)}$ for all $A\in\mathfrak{M}$ and $t\in\mathbb{R}$. Define $\Delta_{\Omega_{s}}=e^{(L+sV-sJVJ)}$ which is the modular operator of the pair $(\mathfrak{M},\Omega_{s})$. By Theorem 9(d), $\Omega_{s}$ is continuous in the norm topology with respect to $s\in\mathbb{R}$.

Lemma 2**.**

Let $V\in\mathfrak{M}$ be a $\sigma$-entire analytic and self-adjoint element. Then $\Omega_{s}$ is differentiable with respect to $s\in\mathbb{R}$ in the norm topology, and we have

[TABLE]

Proof.

First we recall several equalities which can be shown using identity theorem: We have $\Omega\in D\left(e^{z(L+sV)}\right)$ and

[TABLE]

for all $z\in{\mathbb{C}}$ and $s\in{\mathbb{R}}$. We also have $\Omega\in D\left(e^{z\frac{L+sV}{2}}e^{\left(1-z\right)\frac{L+\tilde{s}V}{2}}\right)$ for all $z\in\mathbb{C}$ and $s$, $\tilde{s}\in\mathbb{R}$,

[TABLE]

The proofs are analogous to that of (15), (2) below that we omit them.

Since the right-hand side of (11) is analytic in norm, the left-hand side is analytic too. Therefore, it is differentiable with respect to $z$ in norm. We claim that $\Omega\in D\left(e^{z\frac{L+sV}{2}}Ve^{\left(1-z\right)\frac{L+\tilde{s}V}{2}}\right)$ and the derivative is given by

[TABLE]

for all $z\in\mathbb{C}$ and $s$, $\tilde{s}\in\mathbb{R}$.

To prove this, define the subspace $D_{s}\subset\mathcal{H}$ for $s\in\mathbb{R}$, by

[TABLE]

where $\{E_{\lambda}\}_{\lambda\in\mathbb{R}}$ is the projection-valued measure such that $\int_{\mathbb{R}}dE_{\lambda}\lambda=L+sV$. Then, $D_{s}$ is a core for $e^{z(L+sV)}$ for all $z\in\mathbb{C}$.

For $\xi\in D_{s}$, we have

[TABLE]

We claim

[TABLE]

and

[TABLE]

From (2) and (15), we obtain (12).

To prove (15) and (2), let $\xi$ be an arbitrary element of $D_{s}$ and consider the following two complex functions on $\mathbb{C}$ :

[TABLE]

and

[TABLE]

These two functions coinide on $i\mathbb{R}$ since for $t\in\mathbb{R}$ we have

[TABLE]

The second equality follows by (76) and the third equality follows due to (75), (77). Since these two functions are analytic (recall (A) and (10) ), they coincide on $\mathbb{C}$. Since $D_{s}$ is a core for $e^{\bar{z}\left(\frac{L+sV}{2}\right)}$, we obtain the claim (15) and (2). Hence we have completed the proof of (12).

Finally, we show (9). To prove this, first note that from (2) and the expansion of $E_{sV}^{\sigma}(z)$ (79), we obtain the following continuity in the norm:

[TABLE]

For $s,\tilde{s}\in\mathbb{R}$, since the map

[TABLE]

is differentiable in the norm topology and satisfies

[TABLE]

(from (12)), we have

[TABLE]

Therefore from the continuity (20) and (2), using the Lebesgue’s dominated convergence theorem, $\Omega_{s}$ is differentiable with respect to $s$ in the norm topology and we have

[TABLE]

∎

Lemma 3**.**

Let $V\in\mathfrak{M}$ be a self-adjoint $\sigma$-entire analytic element, and $A\in\mathfrak{M}$. Then $\omega_{s}(A)$ is differentiable with respect to $s$ and

[TABLE]

for all $s\in\mathbb{R}$.

Proof.

By Lemma 2, $\omega_{s}(A)$ is differentiable and we have

[TABLE]

We obtain

[TABLE]

since we have

[TABLE]

Therefore we have

[TABLE]

∎

Lemma 4**.**

Let $V\in\mathfrak{M}$ be a $\sigma$-entire analytic and self-adjoint element, and $A\in\mathfrak{M}$. Let $f\in L^{1}(\mathbb{R})$ be the function defined in (5). Then for all $s\in\mathbb{R}$, we have

[TABLE]

where

[TABLE]

Proof.

Let $G$ be a function defined on $\mathbb{R}_{>0}$ by

[TABLE]

Define a function $F$ on $\mathbb{R}$ by

[TABLE]

It follows that $F(x)=G\left(e^{x}\right)$ for all $x\in\mathbb{R}$. Therefore, from (87) in Appendix C, we have

[TABLE]

for all $x\in\mathbb{R}$. We now show (30) for any $\sigma$-entire analytic element $A$. We have

[TABLE]

We have $A\Omega_{s}\in D(\Delta_{\Omega_{s}})$ due to the $\sigma$-entire analyticity of $A$.

We have

[TABLE]

Therefore we have

[TABLE]

We repeatedly used the self-adjointness of $\Phi(V;s)$.

To extend (30) to general $A\in{\mathfrak{M}}$, we just need to notice that both sides of (30) are continuous in $A$ with respect to the strong-topology and recall that ${\mathfrak{M}}_{\sigma}$ is strong-dense in $\mathfrak{M}$. ∎

Due to Lemma 7, we can define the map $\theta:\mathfrak{M}_{sa}\times\mathbb{R}\to\mathfrak{M}_{sa}$ by

[TABLE]

for all $V\in\mathfrak{M}_{sa}$ and $s\in\mathbb{R}$.

We now prove the statement of Theorem 1 for $\sigma$-entire analytic $V$.

Lemma 5**.**

Let $V\in\mathfrak{M}$ be a $\sigma$-entire analytic and self-adjoint element. Then we have

[TABLE]

Proof.

For each $s\in{\mathbb{R}}$, we define the map $\mathcal{L}_{s}:\mathfrak{M}\rightarrow\mathfrak{M}$ by

[TABLE]

By Lemma 3 and Lemma 4, for any $A\in{\mathfrak{M}}$, we have

[TABLE]

Next, define the state $\phi_{s}\in\mathfrak{M}_{*}$ by

[TABLE]

We now show that $\phi_{s}$ satisfies the same differential equation as $\omega_{s}$:

[TABLE]

By Lemma 7, the map $\mathbb{R}\ni s\mapsto\Phi(V;s)\in\mathfrak{M}$ is continuous in the norm topology. Therefore $\theta(V;s)$ is differentiable in the norm topology and we have

[TABLE]

Since $\Phi(V;s)$ is self-adjoint, we obtain (43):

[TABLE]

As $\phi_{s}$ and $\omega_{s}$ satisfies the same differential equation and $\omega_{0}=\phi_{0}$, we obtain $\omega_{s}=\phi_{s}$ for all $s\in\mathbb{R}$. ∎

3. Proof of Theorem 3

In order to extend the result to general $V$, we need the following continuity of $\theta(V;s)\Omega$.

Lemma 6**.**

Let $\{V_{m}\}_{m\in\mathbb{N}}\subset\mathfrak{M}_{sa}$ be a sequence such that $V_{m}\rightarrow V\in\mathfrak{M}_{\mathrm{s}a}$ strongly as $m\to\infty$. Then we have $\|\theta(V_{m};s)\Omega-\theta(V;s)\Omega\|\to 0$.

Proof.

From the uniform boundedness principle, we have $c:=\sup_{m}\left\|V_{m}\right\|<\infty$. By Lemma 7, $\Phi(V_{m};s)$ converges to $\Phi(V;s)$ strongly. From this convergence and the uniform boundedness

[TABLE]

we have

[TABLE]

We also have

[TABLE]

for any $u_{1},\ldots,u_{n}\in\mathbb{R}$ and $n\in\mathbb{N}$. From the definition of $\theta(V;s)$, we have

[TABLE]

From (47) and (48), applying Lebesgue dominated convergence Theorem, we obtain

[TABLE]

∎

Proof of Theorem 1.

Note that $\theta(V)=\theta(V;1)$. There exists a sequence $\{V_{m}\}\subset\mathfrak{M}_{sa}$ such that $V_{m}$ is $\sigma$-entirely analytic for all $m\in\mathbb{N}$ and that $V_{m}\to V$ strongly. Let $A\in\mathfrak{M}$. By Lemma 6, we have

[TABLE]

as $m\to\infty$.

By Theorem 9(d) , $\omega_{V_{m}}$ converges to $\omega_{V}$ in the norm topology. By Lemma 5 we have

[TABLE]

for all $m\in\mathbb{N}$. Therefore we conclude that

[TABLE]

∎

4. Properties of $\Phi(V;s)$

In this section, we collect some properties of $\Phi$ used in Section 2 and Section3.

Lemma 7**.**

1. (1)

For all $V\in\mathfrak{M}_{\mathrm{s}a}$ and $s\in\mathbb{R}$, we have

[TABLE] 2. (2)

For all $V\in\mathfrak{M}_{\mathrm{s}a}$ the map defined by $\mathbb{R}\ni s\mapsto\Phi(V;s)\in\mathfrak{M}$ is norm-continuous. 3. (3)

For any $s\in{\mathbb{R}}$ and sequence $V_{N}$ in $\mathfrak{M}_{\mathrm{s}a}$ such that $V_{N}\to V\in{\mathfrak{M}}_{sa}$ in the strong operator topology, we have $\Phi(V_{N};s)\to\Phi(V;s)$ in the strong operator topology.

Proof.

For each $V\in{\mathfrak{M}}_{\mathrm{s}a}$, $t\in\mathbb{R}$ and $n=0,1,\ldots$, set

[TABLE]

Here the integral is defined in $\sigma$-weak topology. Note that the map $\mathbb{R}\ni t\mapsto{\mathcal{T}}_{n}(V;t)\in{\mathfrak{M}}$ is norm-continuous. Furthermore, its norm is bounded

[TABLE]

We have

[TABLE]

where the summation converges in norm. From (75), we have

[TABLE]

where the summation in the last term converges in the norm topology.

For each $\varepsilon>0$, we fix $M_{\varepsilon}>0$ such that

[TABLE]

Now let us prove the claim of the Lemma. The first statement of the Lemma is trivial. Let us prove the second statement of the Lemma. For all $V\in{\mathfrak{M}}_{sa}$, $\xi\in{\mathcal{H}}$, $s,\tilde{s}\in\mathbb{R}$ and $\varepsilon>0$, using (57), (60), we have

[TABLE]

Therefore we have

[TABLE]

for any $\varepsilon>0$, $s,\tilde{s}\in{\mathbb{R}}$ and $V\in{\mathfrak{M}}_{\mathrm{s}a}$. As $\tilde{s}\to s$, the first term converges to [math] by Lebesgue dominated convergence theorem and we obtain

[TABLE]

for any $\varepsilon>0$, $s\in{\mathbb{R}}$ and $V\in{\mathfrak{M}}_{\mathrm{s}a}$. Therefore, we have

[TABLE]

for any $s\in\mathbb{R}$ and $V\in{\mathfrak{M}}_{\mathrm{s}a}$. This proves the second statement of Lemma.

To prove the third statement, let $V_{N}$ in $\mathfrak{M}_{\mathrm{s}a}$ be a sequence such that $V_{N}\to V\in{\mathfrak{M}}_{sa}$ in the strong operator topology. By the uniform boundedness principle, we have $c:=\sup_{N}\left\|V_{N}\right\|<\infty$. Fix arbitrary $\varepsilon>0$, $\xi\in\mathcal{H}$, and $s\in\mathbb{R}$. We have

[TABLE]

for all $N\in\mathbb{N}$. Note that

[TABLE]

From Lemma 8, the integrand on the last line converges to [math] as $N\to\infty$, point wise. On the other hand, they are bounded by $\|V\|^{n+m+1}+c^{n+m+1}$. Therefore, from Lebesgue dominated convergence Theorem, we see that

[TABLE]

From (4), we also have

[TABLE]

for any $|t|\leq M_{\varepsilon}$. From (67) and (68), applying the Lebesgue dominated convergence Theorem, we have

[TABLE]

Substituting this to (4), we obtain

[TABLE]

for any $\varepsilon>0$, $\xi\in\mathcal{H}$, and $s\in\mathbb{R}$. As $\varepsilon$ is arbitrary, we have

[TABLE]

for any $\xi\in\mathcal{H}$, and $s\in\mathbb{R}$. This proves the third statement. ∎

Appendix A $W^{*}$-dynamical systems

We collect known facts on $W^{*}$-dynamical systems. See [BR2] and [DJP] and references therein for more information. Let $\mathcal{H}$ be a Hilbert space and $\mathfrak{N}\subset B(\mathcal{H})$ a von Neumann algebra. We denote by $\mathfrak{N}_{*}$ the predual of $\mathfrak{N}$. We also use $\mathfrak{N}_{\mathrm{s}a}$ the set of all selfadjoint elements in $\mathfrak{N}$. Let ${\mathrm{Aut}}(\mathfrak{N})$ denote the group of all $*$-automorphisms of the von Neumann algebra $\mathfrak{N}$. A family $\{\tau_{t}\}_{t\in\mathbb{R}}\subset{\mathrm{Aut}}(\mathfrak{N})$ is called $W^{*}$-dynamics if it is satisfied that for all $t$, $s\in\mathbb{R}$, $\tau_{s}\circ\tau_{t}=\tau_{s+t}$, $\tau_{0}=1$ and $\tau_{t}(A)$ converges to $A$ in the $\sigma$-weak topology as $t\to 0$. As $\tau_{t}$ is an automorphism, for any $A\in\mathfrak{N}$, ${\mathbb{R}}\ni t\mapsto\tau_{t}(A)\in{\mathfrak{N}}$ is continuous with respect to the $\sigma$-strong topology. The pair $(\mathfrak{N},\tau)$ is called a $W^{*}$-dynamical system. For an entire $\tau$-analytic $A\in\mathfrak{N}$, we denote by $\tau_{z}(A)$, for $z\in{\mathbb{C}}$ the analytic continuation of ${\mathbb{R}}\ni t\mapsto\tau_{t}(A)\in\mathfrak{N}$. We denote the set of all $\tau$-entire analytic elements in $\mathfrak{N}$ by $\mathfrak{N}_{\tau}$. Then $\mathfrak{N}_{\tau}$ is a $*$-subalgebra in $\mathfrak{N}$, which is dense in $\mathfrak{N}$ with respect to the strong topology. Let $\beta\in\mathbb{R}$. A normal state $\omega$ on $\mathfrak{N}$ is called a ($\tau$, $\beta$)-KMS state if for all $A\in\mathfrak{N}$ and $B\in\mathfrak{N}_{\tau}$,

[TABLE]

Any ($\tau$,$\beta$)-KMS state is $\tau$-invariant.

Finally, we consider the perturbation theory of the dynamics. Let $(\mathfrak{N},\tau)$ be a $W^{*}$-dynamical system and $Q\in\mathfrak{N}_{sa}$. For $t\in\mathbb{R}$, we define $\tau_{t}^{Q}$ by

[TABLE]

for all $A\in\mathfrak{N}$. The integral is defined in the $\sigma$-weak topology and the summation is in the norm topology.

Then $\tau^{Q}$ is a $W^{*}$-dynamics. For all $t\in\mathbb{R}$, define $E_{Q}^{\tau}(t)\in\mathfrak{N}$ by

[TABLE]

The integral is defined in the $\sigma$-weak topology and the summation is in the norm topology. This $E_{Q}^{\tau}(t)$ is a unitary element of $\mathfrak{N}$ and

[TABLE]

for all $t\in\mathbb{R}$ and $A\in\mathfrak{N}$. Suppose that there exists a self-adjoint operator $L$ on $\mathcal{H}$ satisfying $\tau_{t}(A)=e^{itL}Ae^{-itL}$ for all $t\in\mathbb{R}$ and $A\in\mathfrak{M}$. Then, for all $t\in\mathbb{R}$, $A\in\mathfrak{M}$ and $Q\in\mathfrak{M}_{sa}$,

[TABLE]

and

[TABLE]

If $A,Q$ are entire-analytic for $\tau$, then $A$ is $\tau^{Q}$-entire analytic and ${\mathbb{R}}\ni t\mapsto E^{\tau}_{Q}(t)\in{\mathfrak{N}}$ has an entire analytic continuation. We have the expansion

[TABLE]

and

[TABLE]

for all $z\in\mathbb{C}$ respectively. The integration and the summation converges in the norm topology. ([DJP])

Appendix B Tomita-Takesaki Theory

We collect known facts Tomita-Takesaki Theory. See [BR2] and [DJP] and references therein for more information. Let $\mathcal{H}$ be a Hilbert space, $\mathfrak{M}\subset B(\mathcal{H})$ a von Neumann algebra, and $\Omega\in\mathcal{H}$ a cyclic and separating vector for $\mathfrak{M}$. Define the operator $S_{\Omega}^{0}$ on $\mathcal{H}$ with the domain $\mathfrak{M}\Omega$ by

[TABLE]

Then $S_{\Omega}^{0}$ is anti-linear and closable. Let $S_{\Omega}$ be its closure. Let $S_{\Omega}=J_{\Omega}\Delta^{\frac{1}{2}}_{\Omega}$ be the polar decomposition of $S_{\Omega}$. The operators $\Delta_{\Omega}$ and $J_{\Omega}$ are called respectively the modular operator and the modular conjugation for $(\mathfrak{M},\Omega)$. The modular operator $\Delta_{\Omega}$ is non-singular. The modular operator and the modular conjugation satisfy $J_{\Omega}\Delta_{\Omega}J_{\Omega}=\Delta_{\Omega}^{-1}$, $\Delta^{z}_{\Omega}\Omega=\Omega$ for all $z\in\mathbb{C}$ and $\log\Delta_{\Omega}\Omega=0$. Furthermore, $J_{\Omega}$ interchange $\mathfrak{M}$ and $\mathfrak{M}^{\prime}$, i.e., $J_{\Omega}\mathfrak{M}J_{\Omega}=\mathfrak{M}^{{}^{\prime}}$. From the fact that $\Delta_{\Omega}^{it}\mathfrak{M}\Delta_{\Omega}^{-it}=\mathfrak{M}$ all $t\in\mathbb{R}$, we can define a $*$-automorphism $\sigma_{t}$ on $\mathfrak{M}$ by $\sigma_{t}(A)=\Delta_{\Omega}^{it}A\Delta_{\Omega}^{-it}$, $A\in{\mathfrak{M}}$. This $W^{*}$-dynamics $\sigma$ is called the modular automorphism associated to $\Omega$. We set $L_{\Omega}=\log\Delta_{\Omega}$ and call it the Liouvillean of $\sigma$. One can check that the normalized state of $\omega$ is a $(\sigma,\ -1)$-KMS state.

The set $\mathfrak{M}_{\sigma}\Omega$ is a core for $S_{\Omega}$ and $\Delta_{\Omega}^{\frac{1}{2}}$.(Proposition 2.5.22 of [BR1].) We use the following lemma.

Lemma 8**.**

Let $Q\in\mathfrak{M}$, $n\in\mathbb{N}$ and $t_{1},\ldots,t_{n}\in\mathbb{R}$ . Let $\{Q_{m}\}_{m\in\mathbb{N}}\subset\mathfrak{M}$ satisfy that $Q_{m}\to Q$ and $Q_{m}^{*}\to Q^{*}$ strongly as $m\to\infty$. Then

[TABLE]

as $m\to\infty$, for any $\xi\in\mathcal{H}$.

The following theorem is shown in [A].

Theorem 9**.**

*Let $Q\in\mathfrak{M}_{sa}$. Then $\Omega\in D\left(e^{\frac{L_{\Omega}+Q}{2}}\right)$.

Define the vectors $\Omega_{Q}:={e^{\frac{L_{\Omega}+Q}{2}}\Omega}$, and the functional $\omega_{Q}$ on $\mathfrak{M}$ by*

[TABLE]

for all $A\in\mathfrak{M}$. Then the following conditions are satisfied;

1. (a)

$\Omega_{Q}$* is cyclic and separating for $\mathfrak{M}$;* 2. (b)

The normalized state of $\omega_{Q}$ is a ($\sigma^{Q}$, $-1$)-KMS state; 3. (c)

By (a), we can define the modular operator $\Delta_{\Omega_{Q}}$ and the modular conjugation $J_{\Omega_{Q}}$ for the pair $(\mathfrak{M},\ \Omega_{Q})$. Then $J_{\Omega}=J_{\Omega_{Q}}$,

[TABLE]

and

[TABLE] 4. (d)

Assume that a sequence $\{Q_{n}\}_{n\in\mathbb{N}}\subset\mathfrak{M}_{sa}$ converges to $Q\in\mathfrak{M}_{\mathrm{s}a}$ strongly. Then $\Omega_{Q_{n}}\to\Omega_{Q}$ and $\omega_{Q_{n}}\to\omega_{Q}$ in the norm topology; 5. (e)

For any $0\leq s_{n}\leq\cdots\leq s_{1}\leq 1$, $\Omega$ is in the domain of $e^{\frac{1}{2}s_{n}L}Q\cdot e^{\frac{1}{2}(-s_{n}+s_{n-1})L}Q\cdots e^{\frac{1}{2}(-s_{2}+s_{1})L}Q$ and $\Omega_{Q}$ has an expansion

[TABLE]

Due to (c) of this theorem, let us denote $J$ the modular conjugation if there is no danger of confusion.

Appendix C Properties of $f$

In this section we consider some properties of $f$. Clearly $f$ is in $L^{1}(\mathbb{R})$ by Fubini’s Theorem. This $f$ and $F$ in (33) are Fourier transform of each other.

Lemma 10**.**

For $f$ defined in (5) and $F$ defined in (33), we have

[TABLE]

for all $t\in\mathbb{R}\setminus\{0\}$ and

[TABLE]

for all $x\in\mathbb{R}$.

Proof.

In order to prove this we consider the analytic continuation of $F$ (defined in (33)) to $\mathbb{C}\setminus\left(2\pi i\left(\mathbb{Z}+\frac{1}{2}\right)\right)$ :

[TABLE]

The claim (86) for $t<0$ follows applying the residue theorem to $F(z)e^{-izt}$ along the path in the following picture.

\mathrm{R}e\{z\}$$\mathrm{I}m\{z\}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$$\gamma_{4}$$-R$$R$$2\pi in

Each pole $z=2\pi i\left(n+\frac{1}{2}\right)$ of $F(z)e^{-itz}$ is a simple pole of $F(z)e^{-izt}$, and

[TABLE]

From the residue Theorem, we obtain

[TABLE]

The integral along $\gamma_{3}$ goes to [math] as $n\to\infty$. The integral along $\gamma_{2},\gamma_{4}$ goes to [math] as $R\to\infty$. Hence we obtain (86) for all $t<0$.

To show that (86) for $t>0$, we note

[TABLE]

The final term converges to $f(-t)=f(t)$ as $R\to\infty$ due to $-t<0$. Hence we obtain (86) for all $t\neq 0$.

Finally we show (87). Note that $F|_{\mathbb{R}}\in L^{2}(\mathbb{R})$. Let $\hat{F}\in L^{2}(\mathbb{R})$ be the Fourier transform of $F|_{\mathbb{R}}$. For $R>0$, we define the function $\phi_{R}$ on $\mathbb{R}$ by

[TABLE]

for all $t\in\mathbb{R}$. Then we have $||\phi_{R}-\hat{F}||_{2}\to 0$ as $R\to\infty$ by Theorem 9.13 of [R]. Therefore there exists a sequence $\{n_{k}\}_{k\in\mathbb{N}}$ such that $\phi_{n_{k}}(t)\to\hat{F}(t)$ as $k\to\infty$ a.e. $t$. Since we have shown

[TABLE]

for all $t\in\mathbb{R}\setminus\{0\}$, it follows that $f(t)=\hat{F}(t)$ a.e. $t$. Hence $\hat{F}=f$ belongs to $L^{1}(\mathbb{R})$. By Theorem 9.14 of [R], we have

[TABLE]

a.e. $x$. The left-hand side is continuous. The right-hand side is continuous too due to $f\in L^{1}(\mathbb{R})$. Therefore the equality holds for all $x\in\mathbb{R}$.

∎