151

数理解析研究所講究録
第2074巻 2018年 151-156

MULTI‐PARAMETER ASYMPTOTICS FOR TRUNCATEDWIENER‐HOPF OPERATORSALEXANDER V. SOBOLEV

1. INTRODUCTION

This note is devoted to the study of (bounded, self‐adjoint) operators of the form
(1.1)

W(a; $\alpha \Lambda$) :=$\chi$_{ $\alpha \Lambda$}Ta\mathcal{F}$\chi$_{ $\alpha \Lambda$}, $\alpha$>0,

on \mathrm{L}^{2}(\mathb {R}^{d}) , d\geq 1 , where $\chi$_{ $\Lambda$} is the indicator function of a set $\Lambda$\subset \mathbb{R}^{d} , and $\alpha \Lambda$=\{ $\alpha$ \mathrm{x}:\mathrm{x}\in$\Lambda$\} . The notation \mathcal{F} stands for the unitary Fourier transform in \mathrm{L}^{2}(\mathb {R}^{d}) . The real‐valued
function a , called symbol is assumed to be bounded and smooth. We call the operator(1.1) \mathrm{a} (truncated) Wiener‐Hopf operator. We are interested in the asymptotics of thetrace of the following operator difference(1.2)

D(a, $\alpha \Lambda$;f) :=$\chi$_{ $\alpha \Lambda$}f(W(a; $\alpha \Lambda$))$\chi$_{ $\alpha \Lambda$}-W(f\mathrm{o}a; $\alpha \Lambda$) ,as $\alpha$ \rightarrow \infty , with some suitably chosen functions f . The second operator on the right‐

hand side of (1.2) can be viewed as a regularizing term: it makes the operator (1.2) traceclass even if f(0) \neq 0 and $\Lambda$ is unbounded, under some extra mild conditions on $\Lambda$ andf . On the othcr hand, if f(0)=0, $\Lambda$ is bounded and the symbol a decays fast at infinity,
then both operators on the right‐hand side of (1.2) are casily shown to be trace class.
Asymptotic propertics of D(a, $\alpha \Lambda$;f) depend strongly on the smoothness of the symbola.
For the full asymptotic expansion of tr D(a, $\alpha \Lambda$;f) in powers of $\alpha$^{-1} with smoothsymbols a , smooth functions f and smooth bounded domains $\Lambda$ , we refer to A. Budylin‐V. Buslaev [1] and H. Widom [15]. The leading term of this expansion is of order $\alpha$^{d-1}.

For symbols a with jump discontinuities we only mention the papers by H. Landau‐1 ) and by A.V. Sobolev [8], [9] (for arbitrary
H. Widom [3], H. Widom [13] (for dd\geq 1) . Compared to the smooth case, the leading asymptotic term acquires an extra=\log ‐factor. For example, for the symbol a=$\chi$_{ $\Omega$} with a bounded piece‐wise smooth region$\Omega$ , the trace tr D($\chi$_{ $\Omega$}, $\alpha \Lambda$;f) is of order $\alpha$^{d-1}\log $\alpha$.
The mentioned asymptotic results find their applications in the study of large‐scale
behaviour of the spatially bipartite entanglement entropy of free fermions in thermal

equilibrium, see [2], [4], [5]. For this application the symbol is taken to be the Fermi
2010 Mathematics Subject Classification. Primary 47\mathrm{G}30, 35\mathrm{S}05 ; Secondary 45\mathrm{M}05, 47\mathrm{B}10, 47\mathrm{B}35.
Key words and phrases. Non‐smooth functions of Wiener‐Hopf operators, asymptotic trace formulas,entanglement entropy.

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symbola_{T}( $\xi$)=a_{T, $\mu$}( $\xi$) :=\displaystyle \frac{1}{1+\exp\frac{h( $\xi$)- $\mu$}{T} , $\xi$\in \mathb {R}^{d},(1.3)

where T>0 is the temperature, and $\mu$\in \mathbb{R} is the (fixed) chemical potential.

The real‐valucd free Hamiltonian h\in \mathrm{C}^{\infty} is assumed to satisfy h( $\xi$) \geq c| $\xi$|^{ $\beta$}, $\beta$>0 , as | $\xi$| \rightarrow\infty,
and is such that the level set \{ $\xi$\in \mathbb{R}^{d} : h( $\xi$)= $\mu$\} is a smooth surface with finitely many
conncctcd components. Thus the Fermi sea$\Omega$=\{ $\xi$\in \mathbb{R}^{d}:h( $\xi$)< $\mu$\}(1.4)

is a smooth bounded region. It is natural to define this symbol for T=0 as thc point‐
wise limit a_{0}( $\xi$)=$\chi$_{ $\Omega$}( $\xi$)=\displaystyle \lim_{T\rightarrow 0}a_{T}( $\xi$) . Since the symbol a_{0} is discontinuous, it is not
surprising that the nature of large‐scale entropy asymptotics is different for T=0 and
T>0 , see [4],[5].

Partly motivated by the above example, in this note we concentrate on the transition
from smooth to discontinuous symbols. The precise statements, proofs and detailed

discussions arc found in the paper [11]. In this note we illustrate the results of [11] byconsidering just one example of such a “transitional” symbol, the Fermi symbol (1.3).
The parameter $\mu$ is kept fixed, but the temperature T is allowed to vary simultaneouslywith the scaling parameter $\alpha$.

Acknowledgements. The author is grateful to the organizers for giving him an
opportunity to present his results at the Kato Centennial Conference, Scptember 2017.
2. THE RESULTS

2.1. Main results. For a function f : \mathbb{R}\rightar ow \mathbb{C} and any s_{1}, s_{2}\in \mathbb{R} define the integral(2.1)

U(s_{1}, s_{2};f)=\displaystyle \int_{0}^{1}\frac{f( 1-t)s_{1}+ts_{2})-[(1-t)f(s_{1})+tf(s_{2})]}{t(1-t)}dt.

It is clear that U(S_{1}, \mathcal{S}_{2;1)}=U(s_{1}, s_{2};t)=0 , for all s_{1}, s_{2}\in \mathbb{R} . This integral is finite for
any Hölder function f . For a smooth symbol a=a( $\xi$) , $\xi$\in \mathbb{R} , define(2.2)\displayst le\mathcal{B}(a;f):=\frac{1}8$\pi$^{2}\lim_{$\varepsilon$\rightarow0}\int_{|$\xi$_{1}-$\xi$_{2}|>$\varepsilon$}\frac{U(a$\xi$_{1}),a($\xi$_{2});f}{|$\xi$_{1}-$\xi$_{2}|^{2}d$\xi$_{1}d$\xi$_{2}.

If f is smooth, then this definition coincides with the standard double integral. The
principal value definition becomes necessary for functions f featuring in the theorems
below, see [10] for details.
1 , for smooth f and a we have tr D(a, \mathbb{R}_{+};f)
As shown in [14], in the case d\mathcal{B}(a;f) . For the multi‐dimensional case the asymptotic coefficient is defined as follows.
For a unit vector \mathrm{e}\in \mathbb{R}^{d}, d\geq 2 , introduce the hyperplane=$\Pi$_{\mathrm{e} :=\{ $\xi$\in \mathbb{R}^{d}:\mathrm{e}\cdot $\xi$=0\}.

=

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TRUNCATED \mathrm{V}\mathrm{f}\mathrm{l}\mathrm{E}\mathrm{N}\mathrm{E}\mathrm{R}-\mathrm{H}\mathrm{O}\mathrm{P} $\Gamma$ OPERATORS

Introduce the orthogonal coordinates $\xi$= (\mathring{ $\xi$}, t) such that \mathring{$\xi } \in $\Pi$_{\mathrm{e} and t \in \mathbb{R} . Then weset(2.3)\displaystyle \mathcal{B}_{d}(a;\partial $\Lambda$, f):=\frac{1}{(2 $\pi$)^{d-1} \int_{\partial $\Lambda$}A_{d}(a, \mathrm{n}_{\mathrm{x} ;f)dS_{\mathrm{x} , A_{d}(a, \displaystyle \mathrm{e};f):=\int_{$\Pi$_{\mathrm{e} \mathcal{B}(a (\mathring{ $\xi$}, \cdot); f)d\mathring{ $\xi$}.

As illustrated in Theorem 2.2 below, this coefficient describes the large‐scale behaviour
for smooth symbols. Here and henceforth we assume that

(2.4)\{$\Lambda$

is a region with finitely many connected components

such that the boundary \partial $\Lambda$ is a unionof bounded piece‐wise smooth surfaces.

Thus the integral (2.3) is well‐defined.

For discontinuous symbols we need a different asymptotic coefficient. Define the quan‐tity\displayst le\mathfrak{V}(\parti l$\Lambda$,\parti l$\Omega$)=\frac{1}(2$\pi$)^{d+1}\int_{\parti l$\Lambda$}\int_{\parti l$\Omega$}|\mathrm{n}_{$\xi$}\cdot\mathrm{n}_{\mathrm{x}|dS_{$\xi$}dS_{\mathrm{x},(2.5)

where \mathrm{n}_{\mathrm{x} , \mathrm{n}_{$\xi$} are the exterior unit normals to the surfaces \partial $\Lambda$ and \partial $\Omega$ at the points \mathrm{x}and $\xi$ respectively. Now we can describe the asymptotics of tr D(a_{T}, $\alpha \Lambda$;f) .
In the following theorems $\Omega$ is the Fermi sea defined in (1.4).

Theorem 2.1. [See [11]] Let d \geq 2 . Suppose that $\Lambda$ satisfies (2.4), and let X\{z_{1}, z_{2}, . . . , z_{N}\} \subset \mathbb{R}, N < \infty , be a collection of points on the real line. Suppose that=f\in \mathrm{C}^{2}(\mathbb{R}\backslash X) is a function such that in a neighbourhood of each point z\in X it satisfies

the bound|f^{(k)}(t)| \leq C_{k}|t-z|^{ $\gamma$-k}, k=0, 1, 2 ,(2.6)
with some $\gamma$>0.

Let a_{T} be as defined in (1.3), 0<T\leq T_{0;} with a fixed T_{0}>0 . Then(2.7)\displayt e\lim_{$\alph$^{T}\vec{T\geq}^{01} \displayst le\frac{1}$\alpha$^{d-1}\log\frac{1}T}tr D(a_{T}, $\alpha \Lambda$;f)=U(0,1;f)\mathfrak{V}(\partial $\Lambda$, \partial $\Omega$) ,and(2.8)$\alph$\displayte\vc{T}\leq1\lim_{$\alph$\infty} \displayst le\frac{1} $\alpha$^{d-1}\log$\alpha$} tr D(a_{T}, $\alpha \Lambda$;f)=U(0,1;f)\mathfrak{B}(\partial $\Lambda$, \partial $\Omega$) .

Note that both fomulas (2.7) and (2.8) require that T\rightarrow 0 . The next theorem treats

the case T=const,$\alpha$\rightarrow\infty.

Theorem 2.2. [See [7]] Suppose that the region $\Lambda$ and function f are as in Theorem
2.1. Then(2.9)\displaystyle\lim_{$\alpha$\rightar ow\infty}$\alpha$^{1-d} tr D(a_{T}, \mathrm{a} $\Lambda$;f)=\mathcal{B}_{d}(a_{T}, \partial $\Lambda$;f) ,154
A.V. SOBOLEV

for each T>0.

The formula (2.9) is proved in [7] for much more general smooth symbols. At thispoint one should recall that this formula was established first by H. Widom in [12] evenin the matrix case, but for smooth domains $\Lambda$ and smooth functions f . We emphasizethat the result of [7] (just as that of [11]) holds for non‐smooth functions f and piece‐wisesmooth $\Lambda$.

As we see from the next theorem, the asymptotic formulas in Theorems 2.1 and 2.2 arein agreement with each other. We show this by comparing the asymptotic coefficients in(2.7) and (2.8) with the one in (2.9).

Theorem 2.3. [See [11]] Suppose that the region $\Lambda$ and function f. are as in Theorem
2.1. Then\displaystyle\lim_{T\rightar ow0}\frac{1}{\log\frac{1}{T} \mathcal{B}_{d}(a_{T};\partial$\Lambda$,f)=U(0,1;f)\mathfrak{V}(\partial$\Lambda$,\partial$\Omega$) .

(2.10)

Thcrcfore thc formula (2.7) can be rewritten in the form (2.9), and can be viewed as
an extension of (2.9) to thc asymptotics in two parameters, $\alpha$ and T , as $\alpha$ \rightarrow \infty and$\alpha$ T\geq 1 . Note that (2.8) cannot bc rewritten in the same way.
2.2. Entropy: large‐scale behaviour. The regions $\Lambda$ and $\Omega$ are the same as before.
In order to study the entropy we use Theorems 2.1 and 2.2 with the $\gamma$ ‐Rényi entropy
function $\eta$_{$\gam a$} : \mathbb{R}\mapsto[0, \infty ), defined for all $\gamma$>0 as follows. If $\gamma$\neq 1 , then

(2.11)$\eta$_{ $\gamma$}(t):=\left\{ begin{ar y}{l \frac{1} -$\gam a$}\log[t^{$\gam a$}+(1-t)^{$\gam a$}]&\mathrm{f}\mathrm{o}\mathrm{}t\in(0,1) \ 0&\mathrm{f}\mathrm{o}\mathrm{}t\not\in(0,1) \end{ar y}\right.

and for $\gamma$=1 (the von Neumann case) it is defined as the limit(2.12)$\eta$_{1}(t) :=\displaystyle \lim_{ $\gamma$\rightar ow 1}$\eta$_{ $\gamma$}(t)=\left\{ begin{ar ay}{l} -t\log(t)-(1-t)\log(1-t)&\mathrm{f}\mathrm{o}\mathrm{r}t\in(0,1),\ 0&\mathrm{f}\mathrm{o}\mathrm{r}t\not\in(0,1). \end{ar ay}\right.

For $\gamma$\neq 1 the function $\eta$_{$\gamma$} satisfies condition (2.6) with $\gamma$ replaced with x=\displaystyle \min\{ $\gamma$ , 1 \},\{0 , 1 \} . The function $\eta$_{1} satisfies (2.6) with an arbitrary $\gamma$ \in (0,1) , andand with X=

the same sct X.

Various cntropies were studied in [4], [5] and [6]. For the sake of illustration we

discuss only the $\gamma$ ‐Rényi entanglement entropy (EE) with respect to the bipartition\mathbb{R}^{d}= $\Lambda$\cup(\mathbb{R}^{d}\backslash $\Lambda$) , as defined in [6, Section 10]:(2.13)\mathrm{H}_{ $\gamma$}(T, $\mu$; $\alpha \Lambda$) :=\mathrm{t}\mathrm{r}D(a_{T, $\mu$}, $\alpha \Lambda$;$\eta$_{ $\gamma$})+\mathrm{t}\mathrm{r}D(a_{T, $\mu$}, \mathbb{R}^{d}\backslash $\alpha \Lambda$;$\eta$_{ $\gamma$}) .

We are interested in the behaviour of this quantity when T\rightarrow 0 and $\alpha$\rightarrow\infty.
Theorem 2.4. Let d\geq 2 . The EE satisfies(2.14)$\tau$0_{1}\displayst le\lim_{$\alpha$\vec{T\geq} \frac{1}$\alpha$^{d-1}\log\frac{1}T}\mathrm{H}_{$\gam a$}(T, $\mu$; \alpha\Lambda$)=$\pi$^{2}\frac{1+$\gam a$}{3$\gam a$}\mathfrak{V}(\partial$\Lambda$,\partial$\Omega$),

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TRUNCATED WIENER‐HOPF OPERATORS

and$\alpha$\displayst le\vec{T}\leq1\lim_{$\alpha$\infty}\frac{1}$\alpha$^{d-1}\log$\alpha$}\mathrm{H}_{$\gam a$}(T, $\mu$; \alpha\Lambda$)=$\pi$^{2}\frac{1+$\gam a$}{3$\gam a$}\mathfrak{V}(\partial$\Lambda$,\partial$\Omega$)(2.15).

If T>0 is fixedf then\displaystyle \lim_{ $\alpha$\rightar ow\infty}$\alpha$^{1-d}\mathrm{H}_{ $\gamma$}(T, $\mu$; $\alpha \Lambda$)=2\mathcal{B}_{d}(a_{T, $\mu$}, \partial $\Lambda$;$\eta$_{ $\gamma$}) .

(2.16)

Proof. Formulas (2.14) and (2.15) follow from (2.7) and (2.8) respectively upon observing(cf. [4]) thatU(0,1;$\eta$_{$\gam a$})=\displaystyle\int_{0}^{1}\frac{$\eta$_{$\gam a$}(t)}{t(1-t)}dt=$\pi$^{2}\frac{1+$\gam a$}{6$\gam a$}.

The formula (2.16) is a direct consequence of (2.9).

\squareFor d=1 and $\alpha$ T\geq 1 Theorem 2.4 was proved in [6]. We also stress that the formula(2.15) agrees with the large‐scale asymptotics of the cntropy \mathrm{H}_{$\gam a$} for the zero temperaturecase, which were found in [4].
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A.V. SOBOLEV[13] H. Widom, On a Class of Integral Operators with Discontinuous Symbol. Toeplitz Centennial (TelAviv, 1981), Operator Theory: Adv. Appl., vol. 4, 477‐500, Birkhäuser, Basel‐Boston, Mass., 1982.
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DEPARTMENT OF MATHEMATICS, UNIVERSITY COLLEGE LONDON, GOWER STREET, LONDON,WCIE 6\mathrm{B}\mathrm{T} UK
E‐mail address:\mathrm{a}. sobolev@ucl.ac.uk