[1]Maurizio Fagotti The authors contributed equally to this work.

The authors contributed equally to this work.

[1]Université Paris-Saclay, CNRS, LPTMS, Orsay, 91405, France

# Asymptotic behaviour of determinants through the expansion of the Moyal star
product

<EMAIL_ADDRESS>Vanja Marić
<EMAIL_ADDRESS>*

###### Abstract

We work out a generalization of the Szegö limit theorems on the determinant of
large matrices. We focus on matrices with nonzero leading principal minors and
elements that decay to zero exponentially fast with the distance from the main
diagonal, but we relax the constraint of the Toeplitz structure. We obtain an
expression for the asymptotic behaviour of the determinant written in terms of
the factors of a left and right Wiener-Hopf type factorization of an
appropriately defined symbol. For matrices with elements varying slowly along
the diagonals (e.g., in locally Toeplitz sequences), we propose to apply the
analogue of the semiclassical expansion of the Moyal star product in phase-
space quantum mechanics. This is a systematic method that provides
approximations up to any order in the typical scale of the inhomogeneity and
allows us to obtain explicit asymptotic formulas.

###### keywords:

determinant, Toeplitz, Moyal star product, Wiener-Hopf factorization

###### Contents

  1. 1 Introduction
    1. 1.1 Notations
    2. 1.2 Overview of the Szegö limit theorems
    3. 1.3 Statement of the problem and organization of the article
  2. 2 Main results
    1. 2.1 Readaptation of standard results
    2. 2.2 Asymptotic expansion
    3. 2.3 On block matrices
  3. 3 Limit behavior of determinants
    1. 3.1 A Borodin-Okounkov-Case-Geronimo type formula—proof of proposition 2.1
    2. 3.2 Szegö type limit theorems—proof of theorem 2.2
  4. 4 Matrices with slowly varying elements along the diagonals
    1. 4.1 Truncated Moyal product
    2. 4.2 Truncated Wiener-Hopf star factorisation
    3. 4.3 Truncated star functions
    4. 4.4 Asymptotic expansion—proof of theorem 2.3
  5. 5 Discussion
  6. A Basic results
  7. B Additional details

##  1 Introduction

Determinants of Toeplitz matrices are ubiquitous in physical problems with
translational symmetry: every time that a quantity can be associated with a
matrix whose indices are identified with the position, translational
invariance manifests itself in the matrix being (block-)Toeplitz. On the other
hand, when translational symmetry is broken, the Toeplitz structure breaks
down and only few cases can be addressed as powerfully and effectively (for
example, when the matrices exhibit particular Toeplitz+Hankel structures [1,
2, 3, 4, 5, 6]). More generally, moving from translationally
invariant—homogeneous—settings to inhomogeneous ones is like moving from
Toeplitz to generic matrices.

A very common physical situation is when the typical scale of the
inhomogeneity is controlled by a parameter, in the sense that the
inhomogeneity becomes milder and milder as the parameter is increased. Such
settings are often described by local density approximations or hydrodynamic
theories, which generally become effective in the limit of low inhomogeneity.
There are however systems in which (generalised) hydrodynamics turns out to be
an exact alternative description: we mention, for example, spin-$\frac{1}{2}$
chain systems that are dual to free fermions [7]. The absence of interactions
manifests itself in the applicability of the Wick’s theorem, which allows one
to fully characterize the system by means of a single matrix filled with the
two-point fermionic correlations. In the presence of translational symmetry,
the correlation matrix is a (block-)Laurent operator, and many quantities of
interest reduce to the calculation of determinants of (block-)Toeplitz
matrices; estimating the same quantities in the presence of inhomogeneity, on
the other hand, presents itself as a much more difficult problem. Here is
where generalised hydrodynamics takes the lead. Such a theory provides an
alternative exact description, which defines a phase-space formulation of
quantum mechanics where matrices are replaced by functions [8]. The latter are
nothing but the symbols of the matrices and live in a space where
multiplication is represented by the so-called Moyal star product [9].
Notwithstanding the phase-space formulation being completely equivalent to the
standard description in terms of the correlation matrix, the former is
superior when the symbol varies over large scales, as it allows for asymptotic
expansions.

This was the motivation underlying our interest in the behaviour of
determinants of matrices when the Toeplitz structure breaks down. In this
respect, we point out Tilli’s work [10] on locally Toeplitz sequences. He
considered indeed matrices that resemble Toeplitz ones when inspecting a
neighbourhood of elements, which is analogous to the low inhomogeity limit
mentioned above. Imposing a particular structure (later relaxed to some extent
by Serra-Capizzano and collaborators [11, 12]), Tilli obtained a result that
can be read as the analogue of the weak Szegö limit theorem. In this respect,
we also mention Refs [13, 14], which focussed on the generalisation of the
Szegö limit theorems to operators with almost periodic diagonals.

In this paper, the analogy with the theory of Toeplitz operators will be used
to develop a systematic method to approximate the determinant of a class of
matrices that arise in the study of physical systems with inhomogeneities. We
start by deriving a Borodin-Okounkov-Case–Geronimo type formula as well as
Szegö limit type theorems for a class of matrices with elements that decay to
zero exponentially fast with the distance from the main diagonal. We then
specialise the generalization to matrices with local smoothness properties
along the diagonals and exhibit asymptotic expressions for their determinants.
Finally, we comment on the natural generalizations to matrices with an
emergent block structure.

###  1.1 Notations

Even if the notations will be explained the first time they are used, the main
ones are collected here for easy reference.

  * •

$\mathbb{Z}$ denotes the set of all integers; $\mathbb{N}$ is the set of
natural numbers, and $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$.

  * •

$\mathbb{T}$ denotes the unit circle $\\{z\in\mathbb{C}:|z|=1\\}$.

  * •

$L^{p}$ and $l^{p}$ are the function and sequence spaces respectively.

  * •

$(a)_{k}$ ($k\in\mathbb{Z}$) are the Fourier coefficients of a function $a\in
L^{\infty}(\mathbb{T})$.

  * •

$\mathbb{R}/(2\pi\mathbb{Z})$ and $\mathbb{C}/(2\pi\mathbb{Z})$ denote the
quotient spaces of $\mathbb{R}$ and $\mathbb{C}$, respectively, by
$2\pi\mathbb{Z}$ in which $x\sim x+2\pi$ (we will use this notation to define
$2\pi$-periodic functions).

  * •

$g^{+}:(x,p)\mapsto\sum_{j=0}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij(p-q)}g(x,q)dq$,

$g^{-}:(x,p)\mapsto\sum_{j=-\infty}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij(p-q)}g(x,q)$,

$\tilde{g}=g^{+}-g^{-}$.

  * •

$\star$ denotes the Moyal (Weyl-Groenewold) star product.

  * •

$a^{-1_{\star}}$ is the star inverse of $a$.

  * •

$\log_{\star}a$ is the star logarithm of $a$.

  * •

$(a^{\overset{n}{\sim}})_{x}(z)=a_{n+1-x}(z^{-1})$,

$a^{\sim}=a^{\overset{0}{\sim}}$.

  * •

$\texttt{T}^{t}:a_{x}(z)\mapsto a_{x+t}(z)$.

  * •

$[A,B]$ denotes the commutator $AB-BA$.

  * •

$\\{a,b\\}_{M}$ denotes the Moyal brackets $-i(a\star b-b\star a)$.

  * •

$\\{a,b\\}$ denotes the Poisson brackets
$\partial_{1}a\partial_{2}b-\partial_{2}a\partial_{1}b$, where $a:(x,p)\mapsto
a(x,p)$ and $b:(x,p)\mapsto b(x,p)$.

###  1.2 Overview of the Szegö limit theorems

Here we review some results on Toeplitz determinants directly connected with
this work. More details can be found, e.g., in textbook [15].

Let $a$ be a function in $L^{\infty}(\mathbb{T})$. A Laurent operator $L(a)$
is a bounded operator on $\ell^{2}(\mathbb{Z})$ and is represented by the
doubly infinite matrix with elements $[L(a)]_{ij}=(a)_{i-j}$, where
$i,j\in\mathbb{Z}$. Such matrices provide the matrix representation of
multiplication operators on $L^{\infty}(\mathbb{T})$ with respect to the
orthonormal basis $\\{\frac{e^{inp}}{\sqrt{2\pi}}\\}_{n\in\mathbb{Z}}$ and, in
particular, $L(a)L(b)=L(ab)$ for any $a,b\in L^{\infty}(\mathbb{T})$. The
function $a(z)=\sum_{j\in\mathbb{Z}}(a)_{j}z^{j}$, with $z\in\mathbb{T}$, is
known as the symbol of $L(a)$.

Closely related to Laurent operators are Toeplitz operators. The Toeplitz
operator $T(a)$ induced by the symbol $a$ is the bounded operator on
$\ell^{2}(\mathbb{N})$ represented by the infinite matrix with elements
$[T(a)]_{ij}=(a)_{i-j}$, where $i,j\in\mathbb{N}$. Such matrices provide the
matrix representation of multiplication operators on the Hardy spaces in which
all the negative or all the positive Fourier coefficients vanish.

Toeplitz operators play a crucial role in the calculation of determinants of
large Toeplitz matrices $T_{n}(a)$, which are finite sections of Toeplitz
operators:

$T_{n}(a)=\begin{pmatrix}(a)_{0}&(a)_{-1}&(a)_{-2}&\ldots&(a)_{1-n}\\\
(a)_{1}&(a)_{0}&(a)_{-1}&\ldots&(a)_{2-n}\\\
(a)_{2}&(a)_{1}&(a)_{0}&\ldots&(a)_{3-n}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
(a)_{n-1}&(a)_{n-2}&(a)_{n-3}&\ldots&(a)_{0}\end{pmatrix}\,.$ (1.1)

One of the basic results on the large $n$ behaviours of such determinants is
referred to as the strong Szegö limit theorem:

######  Theorem 1.1 (Strong Szegö Limit Theorem).

Let $a\in L^{\infty}(\mathbb{T})$ satisfy

  * -

$\sum_{n\in\mathbb{Z}}|(a)_{n}|<\infty$

  * -

$\sum_{n\in\mathbb{Z}}(|n|+1)(a)_{n}^{2}<\infty$

  * -

$a(z)\neq 0$ for every $z\in\mathbb{T}$

  * -

$a$ has zero winding number with respect to the origin.

Then

$\lim_{n\rightarrow\infty}\frac{\det T_{n}(a)}{\exp(n(\log
a)_{0})}=e^{E(a)},\quad E(a):=\sum_{k=1}^{\infty}k(\log a)_{k}(\log
a)_{-k}\,.$ (1.2)

The theorem was proven initially by Szegö [16] under stronger conditions on
$a$ and was subsequently generalized by many authors (see e.g. Refs [17, 18]
for details and a historical account). A consequence of the theorem is

$\lim_{n\rightarrow\infty}\frac{\det T_{n}(a)}{\det T_{n-1}(a)}=\exp((\log
a)_{0})\,,$ (1.3)

which is referred to as the weak Szegö limit theorem and historically predates
the strong version.

These results have been generalised in different directions. Particularly
important in physics are the following two generalisations. First, Fisher and
Hartwig [19, 20] derived some formulas covering the case of a non-zero winding
number and conjectured a formula dealing with singular and vanishing symbols,
which has been proven afterwards (see e.g. Ref. [21] for a review). Second,
some of the results were generalised to block-Toeplitz matrices [22, 23, 24].
Historically, the development on Toeplitz determinants had a direct connection
with the investigations into the two-dimensional classical Ising model [18].
More recently, they have been applied in the calculation of two-point
correlation functions as well as Rényi/von Neumann entanglement entropies of
subsystems in translationally invariant quantum many-body systems both in and
out of equilibrium. Just to provide a list of recent works, largely incomplete
and manifestly biased but at least varied for applications and theorems
involved, we refer the reader to Refs [25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42]; we cannot however fail to mention also
applications in random matrix theory [43].

Nowadays, on the other hand, much attention is being paid to the behaviour of
systems prepared in traps or in inhomogeneous states [44]. In those situations
the quantities that, in homogeneous settings, could be described in terms of
determinants of (block-)Toeplitz matrices are now related to determinants of
matrices that could still exhibit some nice smoothness properties but that are
far from being (block-)Toeplitz. Can one predict their asymptotic behaviour
notwithstanding the breakdown of translational symmetry? We present a series
of results that go in the desired direction.

###  1.3 Statement of the problem and organization of the article

We are interested in the large $n$ behavior of determinants of $n\times n$
matrices $T_{n}$, with elements

$(T_{n})_{j,k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}a_{\frac{j+k}{2}}(e^{ip})e^{-ip(j-k)}dp\;,\quad
j,k=1,2,\ldots,n,$ (1.4)

where $a_{x}$, with $x\in\frac{1}{2}\mathbb{Z}$, is an integrable function
that determines the elements on the antidiagonal associated with $x$ as $a$
does in the Toeplitz matrix (1.1). Such a parametrisation does not constrain
the matrix elements for given $n$ but still provides a natural generalisation
of Toeplitz matrices; we keep the analogy alive by referring to $T_{n}$ as the
“star-Toeplitz” matrix associated with $a$ (we will be more precise in a
moment).

We can separate our results in two categories. First, we accommodate some
standard theorems on Toeplitz and Hankel operators to the less structured
operators we are interested in. Our underlying goal is to emphasize the
analogy with the theory of Toeplitz and Hankel operators, and indeed we will
not do more than adapting standard proofs to our situation. Proposition 2.1,
its (weak Szegö limit type) corollary, and theorem 2.2 will be the most
general results obtained, but such a generality is arguably accompanied by
impracticality: the problem of computing the determinant is simply moved to
the problem of working out two factorisations. The second category of results
deals with this latter problem. We restrict ourselves to matrices whose
elements are slowly varying along the diagonals. This allows us to work out
the factorisations and obtain the results stated in theorem 2.2, which we also
use to infer the asymptotics of determinants in locally Toeplitz sequences.

In this respect, Section 2 collects the main determinant formulas and its
first two subsections follow the aforementioned division in two categories.
Some definitions are required to read the statement of the theorems. We report
first those that are needed in both of the subsections. The remaining, more
specific, definitions will be provided when necessary. We point out that, even
if the first subsection of Section 2 provides an overview of the framework, it
is not preparatory to the second subsection, thus we encourage the reader most
interested in application-oriented results to read Section 2.2 first. Finally,
Section 2.3 sketches the minor changes needed to generalise the results to
matrices with an emergent block structure. The reader can find the lemmas and
proofs underlying the results reported in Section 2.1 and Section 2.2 in
Section 3 and Section 4, respectively.

##  2 Main results

Just as Laurent matrices represent Laurent operators through the Fourier
coefficients of their symbol, so we can use the same parametrisation as in
(1.4) to construct the following doubly-infinite matrix.

a

######  Definition 2.1.

Let $a_{x}$, for $x\in\frac{1}{2}\mathbb{Z}$, be functions in
$L^{\infty}(\mathbb{T})$ and denote by $a$ the function
$a:\frac{1}{2}\mathbb{Z}\times\mathbb{T}\to\mathbb{C}$, $(x,z)\mapsto
a_{x}(z)$. By $L(a)$ we denote the operator represented by the doubly infinite
matrix

$L_{j,k}(a)=\frac{1}{2\pi}\int_{-\pi}^{\pi}a_{\frac{j+k}{2}}(e^{ip})e^{-ip(j-k)}dp\;,\quad
j,k\in\mathbb{Z}\,.$ (2.1)

######  Definition 2.2.

For $\rho\in(0,1)$, we denote by $V_{\rho}$ the set of all $a$ with the
following properties:

  1. 1.

For every $x\in\frac{1}{2}\mathbb{Z}$, $a_{x}(z)$ has an analytic extension to
the annulus $\rho<|z|<\rho^{-1}$.

  2. 2.

$a_{x}(z)$ is uniformly bounded in $x$ for $\rho<|z|<\rho^{-1}$.

When we do not need to be specific, we say that a symbol is in
$V\equiv\cup_{\rho\in(0,1)}V_{\rho}$.

> ######  Remark 2.1.
>
> These properties provide a sufficient condition for the boundedness of the
> operator represented by (2.1). Namely, if $a\in V$ then $L(a)$ is a bounded
> linear operator on $l^{2}(\mathbb{Z})$ (lemma A.3 in Appendix A).

a

######  Definition 2.3 (Symbol).

The symbol of $L(a)$ is the equivalence class of all functions generating the
same $L(a)$

$a\sim
b\Leftrightarrow\\{(a_{\frac{n}{2}})_{2m-\frac{1-(-1)^{n}}{2}}=(b_{\frac{n}{2}})_{2m-\frac{1-(-1)^{n}}{2}},\quad\forall
m,n\in\mathbb{Z}\\}\,.$ (2.2)

In the following we will usually represent the symbol with a function
$f:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))\rightarrow\mathbb{C}$, and we
will write $a=[f]$ as a way to say

$(a_{\frac{n}{2}})_{2m-\frac{1-(-1)^{n}}{2}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\tfrac{n}{2},p)e^{-i(2m-\frac{1-(-1)^{n}}{2})p}dp\qquad\forall
n,m\in\mathbb{Z}\,.$ (2.3)

> ######  Remark 2.2.
>
> If $a=[f]$ and
> $f:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))\rightarrow\mathbb{C}$ is
> independent of its first argument, $L(a)$ is a Laurent operator and $T(a)$
> is a Toeplitz operator.

a

a

###  2.1 Readaptation of standard results

Similarly to the relation between Laurent matrices and operators on
$L^{\infty}(\mathbb{T})$, (2.1) provides the matrix representation of the
following star product:

######  Definition 2.4 (Moyal-Weyl–Groenewold product).

The Moyal star product $a\star b$ of two symbols $a,b$ is defined by the
double series

$(a\star
b)_{x}(e^{ip})=\frac{1}{(2\pi)^{2}}\sum_{m\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}a_{x+\frac{m}{2}}(e^{iq_{1}})b_{x-\frac{n}{2}}(e^{iq_{2}})e^{-i(nq_{1}+mq_{2})}d^{2}q\,.$
(2.4)

> ######  Remark 2.3.
>
> If $a,b\in V$ then the double series converges absolutely and the Moyal
> product is well defined. In particular, the following properties hold
> (Appendix A)
>
>   1. 1.
>
> $a\star b\in V$.
>
>   2. 2.
>
> $L(a\star b)=L(a)L(b)$.
>
>   3. 3.
>
> $(a\star b)\star c=a\star(b\star c)$.
>
>

a

######  Definition 2.5 (Star inverse).

A symbol $a$ is star invertible if there is $a^{-1_{\star}}$ such that

$a\star a^{-1_{\star}}=a^{-1_{\star}}\star a=1\,.$

We then say that $a^{-1_{\star}}$ is the star inverse of $a$.

######  Definition 2.6 (Star function).

Let $f(z)$ be analytic inside a closed simple curve $C$ of the complex plane
strictly surrounding the spectrum of $L(a)$; the star function $f_{\star}(a)$
reads

$f_{\star}(a)=\frac{1}{2\pi
i}\oint_{C}f(\zeta)(\zeta-a)^{-1_{\star}}\,\mathrm{d}\zeta\,.$ (2.5)

> ######  Remark 2.4.
>
> This definition is consistent with the definition of operator valued
> functions in the holomorphic functional calculus
>
> $f(L(a))=\frac{1}{2\pi
> i}\oint_{C}f(\zeta)(\zeta-L(a))^{-1}\,\mathrm{d}\zeta\,,$
>
> in the sense that $f(L(a))=L(f_{\star}(a))$.

> ######  Remark 2.5.
>
> If the curve $C$ in definition 2.6 can be chosen to be a circle centered at
> $0$, the star-function can also be defined through its series expansion
> around $0$
>
> $f_{\star}(a)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}a^{n_{\star}}$
>
> where $a^{n_{\star}}$ can be defined resursively as $a^{n_{\star}}=a\star
> a^{(n-1)_{\star}}$, $a^{0_{\star}}=1$.

a

######  Definition 2.7 (Star operators).

We call star-Toeplitz the operator represented by the semi-infinite matrix
$T(a)$ with elements

$T_{j,k}(a)=L_{j,k}[a],\quad j,k=1,2,3,\ldots$ (2.6)

We call star-Hankel the operator represented by the semi-infinite matrix
$H(a)$ with elements

$H_{j,k}(a)=\frac{1}{2\pi}\int_{-\pi}^{\pi}a_{\frac{j-k+1}{2}}(e^{ip})e^{-ip(j+k-1)}dp\;,\quad
j,k=1,2,3,\ldots$ (2.7)

By extension, we will refer to $L(a)$ as the star-Laurent operator induced by
$a$.

######  Definition 2.8 (Star-Toeplitz matrix).

The $n$-by-$n$ star-Toeplitz matrix $T_{n}(a)$ is the finite section (1.4) of
the star-Toeplitz operator $T(a)$.

######  Definition 2.9 (Reflection).

We define the reflected symbol $a^{\overset{n}{\sim}}$ as follows

$(a^{\overset{n}{\sim}})_{x}(z)=a_{n+1-x}(z^{-1})\,.$ (2.8)

and, for the sake of compactness, we omit the number when $n=0$ :
$a^{\sim}\equiv a^{\overset{0}{\sim}}$.

######  Definition 2.10 ($\pm$ symbols).

We say that a symbol $a$ is a $+$ symbol if
$m<0\Rightarrow(a_{\frac{n}{2}})_{2m+\frac{1-(-1)^{n}}{2}}=0$ for all
$n\in\mathbb{Z}$. Similarly, we say that $a$ is a $-$ symbol if
$m>0\Rightarrow(a_{\frac{n}{2}})_{2m-\frac{1-(-1)^{n}}{2}}=0$ for all
$n\in\mathbb{Z}$. Equivalently, $a$ is a $+$ or $-$ symbol if $L(a)$ is lower
or upper triangular, respectively. We will typically add a subscript $+$ (or
$-$) to a symbol if it is a $+$ (or $-$) symbol.

######  Definition 2.11 ($\pm$ functions).

If $g:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))$ is an integrable function
of the second argument at any fixed value of the first argument, we define the
$\pm$ functions $g^{\pm}:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))$ as
follows:

$\displaystyle g^{+}(x,p)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}g(x,q)e^{in(p-q)}dq$
(2.9) $\displaystyle g^{-}(x,p)$
$\displaystyle=\sum_{n=-\infty}^{-1}\frac{1}{2\pi}\int_{-\pi}^{\pi}g(x,q)e^{in(p-q)}dq\,.$

We also use $\tilde{g}$ to indicate $g^{+}-g^{-}$.

> ######  Remark 2.6.
>
> A $+$ symbol can be represented by a $+$ function, but it’s not always
> possible to represent a $-$ symbol by a $-$ function (the diagonal of the
> star Laurent matrix induced by a $-$ symbol is generally nonzero).

a

Finite matrices in which all the principal leading submatrices are invertible
exhibit the so-called LU decomposition, where L/U are lower/upper triangular
matrices. This notion is lifted to the operator level by introducing the
following Wiener-Hopf factorization.

a

######  Definition 2.12 (Wiener-Hopf star factorization with zero winding
number).

We say that a symbol $a\in V$ has zero star winding number if it can be
decomposed as follows

$a=a_{+}^{L}\star a_{-}^{L}=a_{-}^{R}\star a_{+}^{R}\,,$

where $a_{\pm}^{L/R}\in V$ are $\pm$ symbols for which
$\log_{\star}a_{\pm}^{L/R}$ exists and belongs to $V$. We call this
decomposition Wiener-Hopf star factorization.

> ######  Remark 2.7.
>
> Any finite section $T_{n}(a)$ of $L(a)$ is independent of $a_{x}$ with
> $x<1$, therefore the symbol $a$ can be redefined in such a way that
> $a^{\sim}=a$ without affecting the value of $\det T_{n}(a)$. The left
> factorisation is then readily obtained from the right factorisation (and
> vice versa) in that $a_{\pm}^{L}=(a^{R}_{\mp})^{\sim}$. In particular, the
> symmetry ensures the conditional existence of one factorisation given the
> existence of the other.

> ######  Remark 2.8.
>
> The Wiener-Hopf star factorization is not unique. It becomes unique (in the
> quotient space of symbols) if we adopt e.g. the convention
> $((a_{-}^{R/L})_{x})_{0}=1$ for every $x\in\mathbb{Z}$.

> ######  Remark 2.9.
>
> When $a=[f]$ and
> $f:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))\rightarrow\mathbb{C}$ can
> be chosen to be independent of its first argument (i.e., when the star
> Laurent matrix is a Laurent matrix), this definition reduces to the
> requirement that the symbol has zero winding number. The Wiener-Hopf star
> factorization is then the standard Wiener-Hopf factorization, an example of
> which is provided by $a_{\pm}^{L/R}=[e^{(\log f)^{\pm}}]$.

a

######  Definition 2.13.

For $n\in\mathbb{Z}$ we use the compact notation $z^{n}$ for the symbol
represented by the function that maps $(x,z)\mapsto z^{n}$.

a

The following Borodin-Okounkov-Case–Geronimo type formula holds:

######  Proposition 2.1.

For a symbol $a\in V$ with zero star winding number the operator
$T(a_{-}^{L}\star(a_{+}^{R})^{-1_{\star}})T((a_{-}^{R})^{-1_{\star}}\star
a_{+}^{L})$ is invertible and the determinant of the star-Toeplitz matrix
$T_{n}(a)$ satisfies the identity

$\det T_{n}(a)=\tfrac{\det\left(I-H\left(z^{-n}\star
a_{-}^{L}\star(a_{+}^{R})^{-1_{\star}}\right)H\left(((a_{-}^{R})^{-1_{\star}}\star
a_{+}^{L})^{\sim}\star
z^{-n}\right)\right)}{\det(T(a_{-}^{L}\star(a_{+}^{R})^{-1_{\star}})T((a_{-}^{R})^{-1_{\star}}\star
a_{+}^{L}))}\exp\Bigl{(}\sum_{j=1}^{n}\log\bigl{(}(a_{+,j}^{L})_{0}(a_{-,j}^{L})_{0}\bigr{)}\Bigr{)}\,.$
(2.10)

> ######  Remark 2.10.
>
> A similar expression is known for block-Toeplitz matrices, which suffer from
> the same lack of commutativity of the factors of the Wiener-Hopf
> factorisation—see, e.g., Ref. [45].

> ######  Remark 2.11.
>
> If $a\in V_{\rho}$, the numerator in the formula satisfies the bound
>
> $\det\left(I-H\left(z^{-n}\star
> a_{-}^{L}\star(a_{+}^{R})^{-1_{\star}}\right)H\left(((a_{-}^{R})^{-1_{\star}}\star
> a_{+}^{L})^{\sim}\star z^{-n}\right)\right)=1+O(\rho_{1}^{2n})$
>
> for any $\rho_{1}\in(\rho,1)$.

a

A weak Szegö limit type theorem readily follows:

######  Corollary 2.1.

From (2.10) it follows

$\lim_{n\rightarrow\infty}\frac{\det T_{n}(a)}{\det
T_{n-1}(a)(a_{+,n}^{L})_{0}(a_{-,n}^{L})_{0}}=1\,.$ (2.11)

> ######  Remark 2.12.
>
> There is an intuitive reason why this asymptotic behaviour is expressed only
> in terms of the factors of the left factorization: the formula compares the
> determinant of a star-Toeplitz matrix with the one of the submatrix obtained
> by removing the last row and column. A different result is obtained by
> comparing it with the submatrix in which the first row and column are
> removed. Specifically, since $\det T_{n}(a^{\overset{n}{\sim}})=\det
> T_{n}(a)$ and $\det T_{n-1}(a^{\overset{n}{\sim}})=\det
> T_{n-1}(\texttt{T}^{1}a)$, where T is the shift operator
> $\texttt{T}^{t}:a_{x}(z)\mapsto a_{x+t}(z)$, corollary 2.1 can be
> alternatively expressed as
>
> $\lim_{n\rightarrow\infty}\frac{\det T_{n}(a)}{\det
> T_{n-1}(\texttt{T}^{1}a)(a_{-,n}^{R})_{0}(a_{+,n}^{R})_{0}}=1\,.$ (2.12)

a

The denominator in (2.10) captures the leading correction to (2.11) and is the
main character of the star generalization of the strong Szegö limit theorem.
In the theory of Toeplitz and Hankel operators, such contribution is usually
reduced to the calculation of the trace of a commutator of Toeplitz operators.
In the generalisation that we are considering, a similar approach would
involve additional assumptions to ensure that some operators are trace
class111We have followed that route in a preliminary version of this work in
which we used the generalisation of the Helton-Howe-Pincus formula [46] worked
out by Ehrhardt in Ref. [47]. Star-Laurent matrices, however, live in a larger
space than Laurent ones, and this allows us to ease some of the complications
underlying operator algebra. This is where we depart somewhat from the
standard Toeplitz formalism.

For stating the result, we need two additional definitions.

######  Definition 2.14.

Given two symbols $c,d\in V$, $\Phi_{BCH}(c,d)$ stands for

$\Phi_{BCH}(c,d)=\left(\int_{0}^{1}\int_{0}^{1}e^{-is\mathcal{M}(c)}\psi_{1}(e^{i\mathcal{M}(c)}e^{it\mathcal{M}(d)})\,ds\,dt\right)d\\\
-\left(\int_{0}^{1}\int_{0}^{t}e^{is\mathcal{M}(c)}e^{i\mathcal{M}(d)}\psi_{1}(e^{-i\mathcal{M}(d)}e^{-it\mathcal{M}(c)})\,ds\,dt\right)c\,,$
(2.13)

where $\psi_{1}(x)=2x\frac{x\log x+1-x}{(x-1)^{2}}$ and
$\mathcal{M}(c)d=\\{c,d\\}_{M}$.

> ######  Remark 2.13.
>
> If $b_{\pm}\in V$ are $\pm$ symbols and
> $a=\exp_{\star}(b_{-})\star\exp_{\star}(b_{+})$, the Baker-Campbell-
> Hausdorff formula for $\log_{\star}a$ gives
>
>
> $\log_{\star}a-b_{-}-b_{+}=\frac{i}{4}\\{b_{-},\Phi_{BCH}(b_{-},b_{+})\\}_{M}-\frac{i}{4}\\{\Phi_{BCH}(-b_{+},-b_{-}),b_{+}\\}_{M}\,.$
> (2.14)

> ######  Remark 2.14.
>
> The first terms of the expansion of $\nu^{-1}\Phi_{BCH}(\nu c,\nu d)$ as
> $\nu\rightarrow 0$ read
>
> $\nu^{-1}\Phi_{BCH}(\nu c,\nu
> d)=d+\frac{1}{3}i\nu\\{c,d\\}_{M}+\frac{1}{12}\nu^{2}\\{\\{c,d\\}_{M},d\\}_{M}+\dots$
> (2.15)

a

######  Definition 2.15.

We denote by $V^{2}$ the subset of $V\times V$ for which, if $c,d\in V^{2}$
then the power series in $\nu$ of $\nu^{-1}\Phi_{BCH}(\nu c,\nu d)$ is
absolutely convergent to a symbol in $V$ (with respect to the metric induced
by the norm $\|L(\cdot)\|$) in an open interval including $\nu=1$.

a

######  Theorem 2.2.

Let $a\in V$ be a symbol with zero star winding number and
$b_{\pm}^{L/R}=\log_{\star}a_{\pm}^{L/R}$ be such that
$(b_{+}^{L},b_{-}^{L}),(b_{-}^{R},b_{+}^{R})\in V^{2}$. The following formula
holds

$\log\det
T_{n}(a)=\sum_{j=1}^{n}((\log_{\star}a)_{j})_{0}+\frac{E^{R}_{\frac{1}{2}}+E^{L}_{n+\frac{1}{2}}}{2}+O(\rho^{n})$
(2.16)

for some $0<\rho<1$, where

$E^{R/L}_{x}=\frac{1}{2}\sum_{m=1}^{\infty}\sum_{j=1}^{m}(b^{R/L}_{+,x+j-\frac{m+1}{2}})_{m}(d^{R/L}_{(-),{x+j-\frac{m+1}{2}}})_{-m}+(d^{R/L}_{(+),x+j-\frac{m+1}{2}})_{m}(b^{R/L}_{-,{x+j-\frac{m+1}{2}}})_{-m}$
(2.17) $\displaystyle d^{L}_{(-)}=$
$\displaystyle\Phi_{BCH}(b_{+}^{L},b_{-}^{L})$ $\displaystyle d^{R}_{(+)}=$
$\displaystyle\Phi_{BCH}(b_{-}^{R},b_{+}^{R})$ (2.18) $\displaystyle
d^{L}_{(+)}=$ $\displaystyle-\Phi_{BCH}(-b_{-}^{L},-b_{+}^{L})$ $\displaystyle
d^{R}_{(-)}=$ $\displaystyle-\Phi_{BCH}(-b_{+}^{R},-b_{-}^{R})\,.$

> ######  Remark 2.15.
>
> For a symbol inducing a Laurent operator the formula is reduced to (1.2)
> and, in particular, $E^{R}_{x}=E^{L}_{x}=E(a)$.

> ######  Remark 2.16.
>
> Since $b_{\pm}^{R/L},d_{\pm}^{R/L}\in V$, the dependence of $E_{x}$ on
> $a_{y}(z)$ is exponentially suppressed with $|y-x|$, making it clear that
> $E^{R}_{\frac{1}{2}}$ and $E^{L}_{n+\frac{1}{2}}$ are determined by the
> matrix elements close to the upper-left and bottom-right corners of the
> matrix.

###  2.2 Asymptotic expansion

Similarly to the standard block-Toeplitz case, the generality of theorem 2.2
is paid by the difficulty in computing the Wiener-Hopf star factorizations.
This problem is actually manageable when the symbol $a=[f]$ can be represented
by a function $(x,p)\mapsto f(x,p)$ whose typical scale of variation with
respect to $x$ becomes large. Equation (2.16) shows that, ignoring
exponentially small corrections $O(\rho^{n})$, $\log\det T_{n}(a)$ is
determined by two contributions: corner terms, which we denote by $C_{ul}$ and
$C_{br}$, do not change when the matrix is perturbed in some region far from
the upper-left or bottom-right corner of the matrix, respectively; the
remaining contributions are bulk terms, which we denote by $D$

$\log\det T_{n}(a)=D+\frac{C_{ul}+C_{br}}{2}+O(\rho^{n})\,.$ (2.19)

Our ultimate goal is to express $C_{up}$, $C_{br}$ and $D$ as functionals of
$g=\log f$ and its derivatives as follows

$\begin{gathered}C_{ul}=\int_{-\pi}^{\pi}\mathcal{C}_{-}\Bigl{(}g(\tfrac{1}{2},p),\partial
g(\tfrac{1}{2},p),\dots\Bigr{)}dp\qquad
C_{br}=\int_{-\pi}^{\pi}\mathcal{C}_{+}\Bigl{(}g(n+\tfrac{1}{2},p),\partial
g(n+\tfrac{1}{2},p),\dots\Bigr{)}dp\\\
D[g]=\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\mathcal{D}\Bigl{(}g(x,p),\partial
g(x,p),\dots\Bigr{)}dpdx\,.\end{gathered}$ (2.20)

> ######  Remark 2.17.
>
> In the definition of $\mathcal{D}$ and $\mathcal{C}_{\pm}$ there are
> redundant (gauge) degrees of freedom. On the one hand, they are defined up
> to derivatives with respect to $p$; on the other hand, they are invariant
> under the following gauge transformation
>
> $\displaystyle\mathcal{D}(g,\partial g,\dots)\rightarrow$
> $\displaystyle\mathcal{D}(g,\partial
> g,\dots)+\partial_{x}\mathcal{G}(g,\partial g,\dots)$
> $\displaystyle\mathcal{C}_{\pm}(g,\partial g,\dots)\rightarrow$
> $\displaystyle\mathcal{C}_{\pm}(g,\partial g,\dots)\pm\mathcal{G}(g,\partial
> g,\dots)$
>
> for any function $\mathcal{G}$ of $g$ and derivatives. In the following we
> fix the gauge by expressing $\mathcal{D}[g]$ as follows:
>
> $D[g]=\int_{-\infty}^{0}\delta_{g}D[e^{s}g](g)ds\,,$ (2.21)
>
> where
> $\delta_{g}D[e^{s}g]:\phi\mapsto\lim_{m\rightarrow\infty}\lim_{\epsilon\rightarrow
> 0}\frac{D[e^{s}(g+\epsilon\phi_{m})]-D[e^{s}g]}{\epsilon}$ for any sequence
> $\\{\phi_{m}\\}_{\mathbb{N}}$ of functions
> $\phi_{m}:\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))$ of class
> $C^{\infty}$ approaching $\phi$ almost everywhere in
> $(\frac{1}{2},n+\frac{1}{2})$ but vanishing at both $\frac{1}{2}$ and
> $n+\frac{1}{2}$ together with all their derivatives—see Lemma 4.5.

a

We approach this programme within the framework of a perturbation theory
developed under the assumption that the matrix elements vary over long scales,
$\nu^{-1}$, along the diagonals. For the sake of simplicity, we focus on
representations of the symbol that are infinitely differentiable in
$(\frac{1}{2},n+\frac{1}{2})\times(\mathbb{R}/(2\pi\mathbb{Z}))$ and express
the result in the form of asymptotic series

$\mathcal{D}\sim\sum_{j=0}^{\infty}\mathcal{D}^{(j)}\qquad\mathcal{C}_{\pm}\sim\sum_{j=0}^{\infty}\mathcal{C}_{\pm}^{(j)}\,$
(2.22)

as $\nu\rightarrow 0$, where $\mathcal{D}^{(j)}$ and $\mathcal{C}^{(j)}_{\pm}$
represent $O(\nu^{j})$ contributions. Importantly, in this expansion $n$ plays
the role of an independent parameter, so the asymptotics of the determinant
still depends on the behaviour of the function representing the symbol in
$(\frac{1}{2},n+\frac{1}{2})\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

a

We refer the reader to Section 4 for some conditions that are sufficient for
approximating such series with finite sums even when $g$ is only finitely
differentiable. Here we report some general properties of the expansion as
well as the formulas for the first terms.

######  Theorem 2.3.

Let $a=[f_{\nu}]$ for $\nu>0$, where $f_{\nu}$ is defined by
$f_{\nu}(x,p)=e^{g(x\nu,p)}$ and $g$ is infinitely differentiable in
$\mathbb{R}_{>0}\times\mathbb{R}/(2\pi\mathbb{Z})$. Then the following
properties can be satisfied

  1. 1.

$\mathcal{D}^{(2j-1)}(g,\partial g,\dots)=0\qquad\forall j\in\mathbb{N}$

  2. 2.

$\mathcal{D}^{(2j)}(g,\partial g,\dots)$ is a multivariate polynomial of total
degree $3j$ with respect to
$\\{\partial_{1}^{m_{1}}\partial_{2}^{m_{2}}g\\}_{m_{1},m_{2}\in\\{0,1,\dots,2j\\}}$
and consists of monomials with factors including always a single $g$ (i.e., a
single factor with $m_{1}=m_{2}=0$), in which the derivative with respect to
either of the arguments appears exactly $2j$ times.

  3. 3.

$\mathcal{C}_{\pm}^{(j)}(g,\partial g,\dots)$ are multivariate polynomials
with respect to
$\\{\partial_{1}^{m_{1}}\partial_{2}^{m_{2}}g,\partial_{1}^{m_{1}}\partial_{2}^{m_{2}}\tilde{g}\\}_{m_{1}\in\\{0,1,\dots,j\\}\atop
m_{2}\in\\{0,1,\dots,j+1\\}}$ (where $\tilde{g}=g^{+}-g^{-}$) consisting of
monomials in which the derivative with respect to the first argument appears
exactly $j$ times.

In the gauge (2.21), the first bulk contributions read

$\displaystyle\mathcal{D}^{(0)}(g,\partial g,\dots)=$ $\displaystyle g$ (2.23)
$\displaystyle\mathcal{D}^{(2)}(g,\partial g,\dots)=$
$\displaystyle-\frac{1}{12}g\det{\bf H}_{g}$

where ${\bf H}_{g}$ is the Hessian $({\bf H}_{g})_{ij}=\partial_{i,j}g$, with
$i,j\in\\{1,2\\}$. The first corner contributions in the same gauge can be
defined as

$\displaystyle\mathcal{C}^{(0)}_{\pm}(g,\partial g,\dots)=$
$\displaystyle-\frac{1}{4}gi\partial_{2}\tilde{g}$ (2.24)
$\displaystyle\mathcal{C}^{(1)}_{\pm}(g,\partial g,\dots)=$
$\displaystyle\pm\frac{1}{8}\left(\partial_{1}\tilde{g}\partial_{2}g\partial_{2}\tilde{g}+\partial_{1}g\left(-\frac{1}{3}+\partial_{2,2}g+\frac{2}{3}g\partial_{2,2}g+\frac{(\partial_{2}g)^{2}-(\partial_{2}\tilde{g})^{2}}{2}\right)\right)$

> ######  Remark 2.18.
>
> If the first $2k$ derivatives of $g$ are bounded in
> $\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))$, then the error in $\log\det
> T_{n}(a)$ after truncating the asymptotic series at order $2k-2$ is
> $O(n\nu^{2k}+\nu^{2k-1})$ as $\nu\rightarrow 0$.

a

> ######  Example 2.1.
>
> Let the symbol be $a=[f_{\nu}]$, with $f_{\nu}:(x,p)\mapsto
> e^{h(x\nu)-J(x\nu)\cos p}$, for some bounded functions
> $h,J:\mathbb{R}\to\mathbb{C}$ of class $C^{4}$. We then have
>
> $g(x,p)=h(x\nu)-J(x\nu)\cos
> p\,,\qquad\qquad\tilde{g}(x,p)=h(x\nu)-iJ(x\nu)\sin p\,,$
>
> and hence
>
> $\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{D}(g(x,p),\partial
> g(x,p),\dots)dp=$ $\displaystyle
> h+\frac{\nu^{2}}{24}\left(h[JJ^{\prime\prime}+(J^{\prime})^{2}]+J^{2}h^{\prime\prime}\right)\
> \Big{|}_{x\nu}+O(\nu^{4})$
> $\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathcal{C}_{\pm}(g(x,p),\partial
> g(x,p),\dots)dp=$
> $\displaystyle\frac{J^{2}}{8}\pm\frac{\nu}{48}\left(h^{\prime}(J^{2}-2)-(3+2h)JJ^{\prime}\right)\
> \Big{|}_{x\nu}+O(\nu^{2})\,.$

a

In Ref. [10], Tilli defined locally Toeplitz sequences with respect to a pair
of functions $(\bar{a},\bar{f})$, where $\bar{a}$ has the intuitive meaning of
a weight function whereas $\bar{f}$ is a generating function (i.e., a symbol)
in the usual Toeplitz sense. In our language, that corresponds to considering
a symbol $a^{(n)}=[f_{n}]$ that explicitly depends on the size $n$ of the
matrix with the Ansatz $f_{n}(x,p)=\bar{a}(\tfrac{x}{n})\bar{f}(p)$. Tilli
also considered finite sums of locally Toeplitz sequences, which required
little modifications in his results and have subsequently become full part of
the theory under the name of “generalised locally Toeplitz sequences”[11].
Within the framework of our analysis, it is natural to borrow Tilli’s
terminology for sequences of star-Toeplitz matrices with symbols
$a^{(n)}=[f_{n}]$ that can be represented by functions
$f_{n}(x,p)=f(\tfrac{x-\frac{1}{2}}{n},p)$.

a

######  Proposition 2.4 (Locally Toeplitz sequences).

Let us represent the symbol $a^{(n)}=[f_{n}]$ by a function $f_{n}$ such that
$f_{n}(x,p)=\exp(g(\tfrac{x-\frac{1}{2}}{n},p))$ for some
$g:\mathbb{R}\times(\mathbb{C}/(2\pi\mathbb{Z}))$ analytic in a strip
surrounding the real line in the second argument and of class $C^{2}(I)$ with
respect to the first argument, where $I$ is an open interval strictly
surrounding $(0,1)$. Then, the determinant of the star-Toeplitz matrix
$T_{n}(a^{(n)})$ satisfies

$\det
T_{n}(a^{(n)})=\exp\left(\frac{E(c_{0})+E(c_{1})}{2}\right)\exp\left[\frac{n}{2\pi}\int_{0}^{1}\int_{-\pi}^{\pi}g(t,p)dpdt\right](1+O(n^{-1}))\quad\text{as
}n\rightarrow\infty\,,$ (2.25)

where $c_{0}(e^{ip})=\lim_{t\rightarrow 0^{+}}e^{g(t,p)}$,
$c_{1}(e^{ip})=\lim_{t\rightarrow 1^{-}}e^{g(t,p)}$, and $E(a)$ is defined in
(1.2).

a

This scaling limit is only sensitive to $\mathcal{D}^{(0)}$ and
$\mathcal{C}^{(0)}_{\pm}$, but alternative sequences characterised by a
different scaling with respect to the matrix size can depend also on
$\mathcal{D}^{(j)}$, with $j>0$. We distinguish the following sequences

a

######  Definition 2.16.

We call “locally-$p$ Toeplitz sequence” with respect to a function $f$, the
sequence $\\{T_{n}(a^{(n)})\\}_{n\in\mathbb{N}}$ of square matrices
characterised by the symbol $a^{(n)}=[f_{n}]$, with
$f_{n}(x,p)=f(\frac{x-\frac{1}{2}}{n^{p}},p)$.

a

######  Proposition 2.5 (Locally-$\frac{1}{2}$ Toeplitz sequences).

Let us represent the symbol $a^{(n)}=[f_{n}]$ by a function $f_{n}$ such that
$f_{n}(x,p)=\exp(g(\frac{x-\frac{1}{2}}{\sqrt{n}},p))$ for some $g$ analytic
in a strip surrounding $\mathbb{R}/(2\pi\mathbb{Z})$ with respect to the
second argument and of class $C^{4}$ with uniformly bounded derivatives in
$(0,+\infty)$ with respect to the first argument. The sequence of determinants
of $T_{n}(a^{(n)})$ has the following limit behaviour

$\det
T_{n}(a^{(n)})=\exp\left(\frac{E(c_{0})+E(c_{\sqrt{n}})}{2}\right)\exp\left[-\frac{\int_{0}^{\sqrt{n}}\int_{-\pi}^{\pi}g(t,p)\det\mathbf{H}_{g}(t,p)dpdt}{24\pi\sqrt{n}}\right]\\\
\times\exp\left[\frac{\sqrt{n}}{2\pi}\int_{0}^{\sqrt{n}}\int_{-\pi}^{\pi}g(t,p)dpdt\right](1+O(n^{-\frac{1}{2}}))\quad\text{as
}n\rightarrow\infty\,,$ (2.26)

where $c_{0}(e^{ip})=\lim_{t\rightarrow 0^{+}}e^{g(t,p)}$,
$c_{\sqrt{n}}(e^{ip})=e^{g(\sqrt{n},p)}$ and $E(a)$ is defined in (1.2).

> ######  Example 2.2.
>
> Let us consider a locally-$\frac{1}{2}$ Toeplitz sequence characterised by a
> class of symbols of the form discussed in example 2.1. Specifically,
> $a^{(n)}=[f_{n}]$, with
> $f_{n}(x,p)=\exp(g(\frac{x-\frac{1}{2}}{\sqrt{n}},p))$ and we choose
>
> $g(t,p)=1+\cos(2\omega t)-\cos(\omega t)\cos p\,.$
>
> The asymptotic behaviour can be readily computed from (2.26)
>
> $\log\det
> T_{n}(a^{(n)})=n+\frac{1}{2\omega}\sqrt{n}\sin(2\omega\sqrt{n})+\frac{3+\cos(2\omega\sqrt{n})-\omega^{2}}{16}+O(n^{-\frac{1}{2}})\,.$

###  2.3 On block matrices

In Section 2.2 we have reported some results aiming at simplifying the formal
expressions obtained for the determinant of star-Toeplitz matrices when the
matrix elements vary slowly along the diagonals. In some physical applications
such a condition is not satisfied, but a weaker one is. Specifically, the
matrix could be locally similar to a block Toeplitz matrix rather than to a
Toeplitz one. In many respects, this generalisation requires little
modification, in that the Moyal star product is already not commutative, and
we have already dealt with most of the issues related to the lack of
commutation. If the matrix has emergent $\kappa\times\kappa$ blocks, the
symbol $a$ can be defined as a $\kappa\times\kappa$ matrix in such a way that
the star-Laurent operator is represented by the doubly-infinite matrix

$L_{\kappa\ell-1+j,\kappa
n-1+k}(a)=\frac{1}{2\pi}\int_{-\pi}^{\pi}[a_{\frac{\ell+n}{2}}(e^{ip})]_{jk}e^{-ip(\ell-n)}dp\;,\quad\ell,n\in\mathbb{Z}\,,\quad
j,k=1,\dots,\kappa\,.$ (2.27)

Such a redefinition provides an alternative representation of the star
product, in particular, $L(a\star b)=L(a)L(b)$. We can then retrace the proofs
reported in Section 3 bearing in mind that the symbols are matrices. For
example, corollary 2.1 is almost unchanged, as we have

$\lim_{n\rightarrow\infty}\frac{\det T_{n}(a)}{\det
T_{n-1}(a)\det[(a_{+,n}^{L})_{0}(a_{-,n}^{L})_{0}]}=1\,.$ (2.28)

Even the stronger theorem 2.2 can be readily adapted to the block case by
adding an overall trace to the right hand side of (2.16), which is now a
$\kappa$-by-$\kappa$ matrix. Incidentally, this also implies

######  Proposition 2.6 (Determinant of block-Toeplitz matrices).

Let $T_{n}(a)$ be the Toeplitz matrix induced by a smooth symbol, $a$, with
zero winding number, then

$\log\det T_{n}(a)=n\mathrm{tr}[(\log
a)_{0}]-\frac{i}{4}\mathrm{tr}\left[\frac{1}{2\pi}\int_{-\pi}^{\pi}\beta_{R}(p)\gamma_{R}^{\prime}(p)+\beta^{\prime}_{L}(p)\gamma_{L}(p)dp\right]+O(\rho^{n})$
(2.29)

for some $0<\rho<1$, where

$\displaystyle\beta_{L}$ $\displaystyle:p\mapsto(\log a_{+}^{L}+\log
a_{-}^{L})(e^{ip})$ (2.30) $\displaystyle\beta_{R}$
$\displaystyle:p\mapsto(\log a_{+}^{R}+\log a_{-}^{R})(e^{ip})$
$\displaystyle\gamma_{L}$ $\displaystyle:p\mapsto[(\Phi_{BCH}(\log
a_{+}^{L},\log a_{-}^{L}))^{-}+(\Phi_{BCH}(-\log a_{-}^{L},-\log
a_{+}^{L}))^{+}](e^{ip})$ $\displaystyle\gamma_{R}$
$\displaystyle:p\mapsto[(\Phi_{BCH}(\log a_{-}^{R},\log
a_{+}^{R}))^{+}+(\Phi_{BCH}(-\log a_{+}^{R},-\log a_{-}^{R}))^{-}](e^{ip})\,,$

and $\Phi_{BCH}$ is defined in (2.14) with $\mathcal{M}$ the adjoint mapping
$\mathcal{M}(c)d=-i[c,d]$.

> ######  Example 2.3.
>
> Let us consider a block Toeplitz matrix $T_{n}(a)$ with a smooth $2$-by-$2$
> symbol, $a$, close to the identity. Let us parametrize the left Wiener-Hopf
> factorisation as
> $\log(a^{L}_{\pm}):e^{ip}\mapsto\lambda^{\pm}(p)+\mathbf{n}^{\pm}(p)\cdot\boldsymbol{\sigma}$,
> where $\boldsymbol{\sigma}$ denotes the vector of Pauli matrices. Prediction
> (2.29) then gives
>
> $\log\det
> T_{n}(a)=\frac{n}{\pi}\int_{-\pi}^{\pi}\lambda(p)dp-\frac{i}{2\pi}\int_{-\pi}^{\pi}\left(\lambda\tilde{\lambda}^{\prime}+\mathbf{n}\cdot\tilde{\mathbf{n}}^{\prime}+i\mathbf{n}\cdot(\tilde{\mathbf{n}}^{\prime}\times\tilde{\mathbf{n}})\right)(p)\,dp+O(\|a-\mathrm{I}\|^{4})$
>
> as $\|a-\mathrm{I}\|\rightarrow 0$, uniformly in $n$. As an explicit example
> we set $\lambda=0$ and $\mathbf{n}=\epsilon\begin{pmatrix}\cos p&\gamma\sin
> p&h\end{pmatrix}$; we then find
>
> $\log\det
> T_{n}(a)\sim\frac{\epsilon^{2}}{2}(1+\gamma^{2}+2h\gamma\epsilon)+O(\epsilon^{4})$
>
> as $\epsilon\rightarrow 0$, uniformly in $n$.

a

The complications underlying the block structure are more evident when trying
to generalise the results reported in Section 2.2. Although the truncation of
the Moyal product and of the Wiener-Hopf star factorizations could be
justified in similar ways, the lack of a systematic procedure for the
factorisation of a block Toeplitz matrix undermines the usefulness of the
expansion. The final result would therefore rely on the knowledge of the
Wiener-Hopf factorisations for the block-Toeplitz symbols associated with the
local structure of the star block Toeplitz matrix (even in proposition 2.6 the
constant is expressed in terms of the factors of both left and right Wiener-
Hopf factorisations).

##  3 Limit behavior of determinants

In this section we prove the results reported in Section 2.1. We start by
deriving a Borodin-Okounkov type formula for star-Toeplitz matrices. We
highlight the analogy with the theory of Toeplitz operators by following the
same steps of standard derivations [15, 48, 24].

###  3.1 A Borodin-Okounkov-Case-Geronimo type formula—proof of proposition
2.1

######  Definition 3.1 (Projections).

We denote by $P$ the orthogonal projection of $l^{2}(\mathbb{Z})$ onto
$l^{2}(\mathbb{N})$, i.e., if $f$ has components $f_{j},\ j\in\mathbb{Z}$, $P$
acts as follows

$(Pf)_{j}=\left\\{\begin{array}[]{cc}f_{j}&j\geq 1\\\
0&j<1\,.\end{array}\right.$ (3.1)

Analogously, we define $Q$ and $J$ in $l^{2}(\mathbb{Z})$ as follows:

$(Qf)_{j}=\left\\{\begin{array}[]{cc}0&j\geq 1\\\
f_{j}&j<1\end{array}\right.\qquad\quad(Jf)_{j}=f_{-j+1}\,.$ (3.2)

We also denote by $P_{n}$ and $Q_{n}$ the projection on $\ell^{2}(\mathbb{N})$
acting by the rules

$(P_{n}f)_{j}=\left\\{\begin{array}[]{cc}f_{j}&1\leq j\leq n\\\
0&\textrm{otherwise}\end{array}\right.\qquad(Q_{n}f)_{j}=\left\\{\begin{array}[]{cc}0&1\leq
j\leq n\\\ f_{j}&\textrm{otherwise.}\end{array}\right.$ (3.3)

> ######  Remark 3.1.
>
> By definition, $P+Q=I$ and $J^{2}=I,P^{2}=P,Q^{2}=Q$. In addition, the star-
> Toeplitz and star-Hankel operators are expressed in terms of the
> corresponding star-Laurent operators as follows
>
> $\displaystyle T(a)=PL(a)P$ $\displaystyle H(a)=PL(a)QJ\qquad
> H(a^{\sim})=JQL(a)P\,.$ (3.4)
>
> The star-Toeplitz matrix $T_{n}$ is instead given by $P_{n}T(a)P_{n}$.

######  Lemma 3.1.

For $a,b\in V$ the following decomposition holds:

$T(a\star b)=T(a)T(b)+H(a)H(b^{\sim})$ (3.5)

Proof .

$T(a\star b)=PL(a\star
b)P=PL(a)L(b)P=PL(a)(P+Q)L(b)P=PL(a)PL(b)P+PL(a)QL(b)P\\\
=PL(a)PPL(b)P+PL(a)QJJQL(b)P=T(a)T(b)+H(a)H(b^{\sim})$ (3.6)

a

######  Lemma 3.2.

If $a,b\in V$ are $\pm$ symbols then $a\star b$ is also a $\pm$ symbol.

###### Proof.

A symbol $a$ is of $+$ ($-$) type if and only if the matrix $L(a)$ is lower
(upper) triangular. The claim follows from the fact that the product of lower
(upper) triangular matrices is lower (upper) triangular. ∎

######  Lemma 3.3.

If $a_{-},c_{+}\in V$ are $-$ and $+$ symbols, respectively, then for any
$b\in V$ we have

$T(a_{-}\star b\star c_{+})=T(a_{-})T(b)T(c_{+})\,,$ (3.7)

and hence

$T(a_{-}\star b)=T(a)T(b),\qquad T(b\star c_{+})=T(b)T(c_{+}).$ (3.8)

Proof .

The assumptions imply $H(a_{-})=0$ and $H((c_{+})^{\sim})=0$. The result
follows from applying formula (3.5) twice in succession.

a

######  Lemma 3.4.

If $a\in V$ has zero star winding number then

  1. 1.

$T(a)=T(a_{-}^{R})T(a_{+}^{R})$

  2. 2.

$T(a^{-1_{\star}})=T((a_{-}^{L})^{-1_{\star}})T((a_{+}^{L})^{-1_{\star}})$

  3. 3.

$T(a)$ is invertible and
$T^{-1}(a)=T((a_{+}^{R})^{-1_{\star}})T((a_{-}^{R})^{-1_{\star}})$

Proof .

The first claim is a trivial consequence of lemma 3.3. Since
$a^{-1_{\star}}=(a_{-}^{L})^{-1_{\star}}\star(a_{+}^{L})^{-1_{\star}}$, the
second claim is a consequence of the first one. Since $a=a_{-}^{R}\star
a_{+}^{R}$ we have
$a^{-1_{\star}}=(a_{+}^{R})^{-1_{\star}}\star(a_{-}^{R})^{-1_{\star}}$. Lemma
3.3 then gives $\displaystyle
T((a_{-}^{R})^{-1_{\star}})T(a_{-}^{R})=T((a_{-}^{R})^{-1_{\star}}\star
a_{-}^{R})=I=T(a_{-}^{R}\star(a_{-}^{R})^{-1_{\star}})=T(a_{-}^{R})T((a_{-}^{R})^{-1_{\star}})$
(3.9) $\displaystyle
T((a_{+}^{R})^{-1_{\star}})T(a_{+}^{R})=T((a_{+}^{R})^{-1_{\star}}\star
a_{+}^{R})=I=T(a_{+}^{R}\star(a_{+}^{R})^{-1_{\star}})=T(a_{+}^{R})T((a_{+}^{R})^{-1_{\star}})\;.$
(3.10) The third claim readily follows.

a

######  Lemma 3.5.

Let $b$ be in $V_{\rho}$. The following asymptotic behaviour holds

$\det\left(\mathrm{I}-H(z^{-n}\star b)H((b^{-1_{\star}})^{\sim}\star
z^{-n})\right)=1+O(\rho_{1}^{2n})\;,$ (3.11)

for any $\rho_{1}\in(\rho,1)$.

Proof .

The analyticity and uniform boundedness of $b$ in the annulus implies that,
for any $\rho_{1}\in(\rho,1)$, there is $M>0$ such that $|H_{j,k}(z^{-n}\star
b)|=|(b_{\frac{j+n-k+1}{2}})_{j+n+k-1}|\leq M\rho_{1}^{j+k-1}\rho_{1}^{n}$ for
every $j,k\in\mathbb{N}$ (see lemma A.1). Consequently, the Hilbert-Schmidt
norm of $H(z^{-n}\star b)$ is bounded as $\|H(z^{-n}\star
b)\|_{2}=O(\rho_{1}^{n})$, and similarly $\|H((b^{-1_{\star}})^{\sim}\star
z^{-n})\|_{2}=O(\rho_{1}^{n})$. This implies the following bound for the trace
norm of the product of star Hankel operators $\displaystyle\|H(z^{-n}\star
b)H((b^{-1_{\star}})^{\sim}\star z^{-n})\|_{1}\leq\|H(z^{-n}\star
b)\|_{2}\|H((b^{-1_{\star}})^{\sim}\star z^{-n})\|_{2}=O(\rho_{1}^{2n})$
(3.12) and hence the claim.

a

###### Proof of Proposition 2.1.

A proof of the Boroding-Okounkov type formula for star-Toeplitz matrices as
stated in proposition 2.1 follows the same steps as in the (block-)Toeplitz
case. Consider a function $b\in V_{\rho}$ with zero star winding number.
Applying lemma 3.1 to the trivial symbol $b\star b^{-1_{\star}}$ gives
($b=b_{-}^{R}\star b_{+}^{R}=b_{+}^{L}\star b_{-}^{L}$)

$\displaystyle\mathrm{I}-H(b)H((b^{-1_{\star}})^{\sim})$ $\displaystyle=$
$\displaystyle T(b)T(b^{-1_{\star}})$ (3.13) $\displaystyle=$ $\displaystyle
T(b_{-}^{R})T(b_{+}^{R})T((b_{-}^{L})^{-1_{\star}})T((b_{+}^{L})^{-1_{\star}})$
$\displaystyle=$ $\displaystyle
T(b_{-}^{R})T^{-1}(b_{-}^{L}\star(b_{+}^{R})^{-1_{\star}})T((b_{+}^{L})^{-1_{\star}})$

and hence

$(\mathrm{I}-H(b)H((b^{-1_{\star}})^{\sim}))^{-1}=T(b_{+}^{L})T(b_{-}^{L}\star(b_{+}^{R})^{-1_{\star}})T((b_{-}^{R})^{-1_{\star}})$
(3.14)

Let us then enforce the identity [24]

$\det\left(P_{n}(\mathrm{I}-K)^{-1}P_{n}\right)=\frac{\det\left(\mathrm{I}-Q_{n}KQ_{n}\right)}{\det\left(\mathrm{I}-K\right)}\,,$
(3.15)

valid whenever $K$ is trace-class and $I-K$ is invertible. Choosing
$K=H(b)H((b^{-1_{\star}})^{\sim})$ gives

$\det\left(P_{n}T(b_{+}^{L})T(b_{-}^{L}\star(b_{+}^{R})^{-1_{\star}})T((b_{-}^{R})^{-1_{\star}})P_{n}\right)=\frac{\det\left(\mathrm{I}-H(z^{-n}\star
b)H((b^{-1_{\star}})^{\sim}\star
z^{-n})\right)}{\det\left(T(b)T(b^{-1_{\star}})\right)}\,.$ (3.16)

Since $T(b_{+}^{L})$ and $T((b_{-}^{R})^{-1_{\star}})$ are lower and upper
triangular operators, respectively, we have

$P_{n}T(b_{+}^{L})=P_{n}T(b_{+}^{L})P_{n},\quad
T((b_{-}^{R})^{-1_{\star}})P_{n}=P_{n}T((b_{-}^{R})^{-1_{\star}})P_{n}$ (3.17)

so the determinant on the left side of eq. (3.16) can be written in terms of
finite sections of star Toeplitz operators. Moreover, for the same reason it
holds

$T_{n}(b_{+}^{L})=T_{n}^{-1}((b_{+}^{L})^{-1_{\star}}),\quad
T_{n}((b_{-}^{R})^{-1_{\star}})=T_{n}^{-1}(b_{-}^{R})\;.$ (3.18)

This allows us to rewrite (3.16) as follows

$\det
T_{n}(b_{-}^{L}\star(b_{+}^{R})^{-1_{\star}})=\frac{\det\left(\mathrm{I}-H(z^{-n}\star
b)H((b^{-1_{\star}})^{\sim}\star
z^{-n})\right)}{\det\left(T(b)T(b^{-1_{\star}})\right)}\det
T_{n}(b_{-}^{R})\det T_{n}\left((b_{+}^{L})^{-1_{\star}}\right)\,.$ (3.19)

The determinant of the star Toeplitz matrices on the right hand side is the
product of their diagonal elements, thus we have

$\det T_{n}(b_{-}^{L}\star(b_{+}^{R})^{-1_{\star}})\\\
=\frac{\det\left(\mathrm{I}-H(z^{-n}\star b)H((b^{-1_{\star}})^{\sim}\star
z^{-n})\right)}{\det\left(T(b)T(b^{-1_{\star}})\right)}\exp\left(\sum_{j=1}^{n}\log\left[\left(\left(b_{-}^{R}\right)_{j}\right)_{0}\left(\left(b_{+}^{L}\right)^{-1_{\star}}_{j}\right)_{0}\right]\right)\,.$
(3.20)

The final statement follows from the identification $a_{-}^{R}\equiv
b_{-}^{L}$, $a_{+}^{R}\equiv(b_{+}^{R})^{-1_{\star}}$, and hence
$a_{+}^{L}=(b_{+}^{L})^{-1_{\star}}$, $a_{-}^{L}=b_{-}^{R}$. Note that the
numerator of the fraction on the right hand side of (3.20) can be bounded
using lemma 3.5. ∎

###  3.2 Szegö type limit theorems—proof of theorem 2.2

######  Lemma 3.6.

Let $a_{\pm}$ and $b_{\pm}$ be $\pm$ symbols in $V$ such that
$T(a_{\pm}),T(b_{\pm})$ are trace class. Then the following identities hold:

$\displaystyle\mathrm{tr}\,T(\\{a_{-},b_{-}\\}_{M})=\mathrm{tr}\,T(\\{a_{+},b_{+}\\}_{M})=0$
(3.21) $\displaystyle\mathrm{tr}\
T(\\{a_{+},b_{-}\\}_{M})=-i\sum_{m=1}^{\infty}\sum_{j=1}^{m}(a_{+,j-\frac{m}{2}})_{m}(b_{-,j-\frac{m}{2}})_{-m}\,,$
(3.22)

where $\\{c_{1},c_{2}\\}_{M}$ is a shorthand for $-i(c_{1}\star
c_{2}-c_{2}\star c_{1})$ and is known as Moyal bracket.

Proof .

The identities follow from lemma 3.3 and from the fact that the trace of the
commutator of two trace class operators vanishes. Specifically, we have
$T(\\{a_{-},b_{-}\\}_{M})=-i[T(a_{-}),T(b_{-})]$ (3.23) which has zero trace.
Analogously we have
$T(\\{a_{+},b_{-}\\}_{M})=-i[T(a_{+}),T(b_{-})]-iH(a_{+})H(\tilde{b}_{-}),$
(3.24) where the first term has zero trace and the trace of the second one can
be evaluated in the canonical basis, giving Eq. (3.22).

a

To simplify the expressions appearing in proposition 2.1 it would be
convenient to work with trace-class operators, for which results such as lemma
3.6 could be applied. By contrast, for a given symbol $a\in V$ with zero star
winding number and factorized as
$a=\exp_{\star}(b^{R}_{-})\star\exp_{\star}(b^{R}_{+})=\exp_{\star}(b^{L}_{-})\star\exp_{\star}(b^{R}_{+})$,
the operators $T(b^{R/L}_{\pm})$ are not necessarily trace class. Enforcing
them to be trace class might seem a strong assumption, but, in fact, it can be
done without loss of generality. This is possible because the determinants of
finite sections do not depend on the behavior of the operator at infinity,
which instead determines the trace-class property. We are then free to deform
the operators so as to make them trace class without affecting the determinant
of the finite sections. We formalize this intuition by introducing a
regularisation of the symbol, i.e., a sequence of symbols converging to the
one of interest with respect to a suitable metric, introduced in the following
definition, that is not sensitive to irrelevant details at infinity.

a

######  Definition 3.2.

For $a\in V$, we define the norm $\parallel\cdot\parallel_{loc}$ as

$\parallel a\parallel_{loc}=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\parallel
T_{2n+1}(\texttt{T}^{-n}(a))\parallel\,.$ (3.25)

> ######  Remark 3.2.
>
> The norm satisfies $\parallel a\parallel_{loc}\leq\|L(a)\|$.

a

######  Lemma 3.7.

The following functions are continuous in the metric space
$(V,\|\cdot\|_{loc})$:

  1. 1.

$\det T_{n}:a\mapsto\det T_{n}(a)$ for every $n\in\mathbb{N}$

  2. 2.

$F^{l}_{b}:V\rightarrow V,a\mapsto b\star a$ and $F^{r}_{b}:V\rightarrow
V,a\mapsto a\star b$

  3. 3.

$\Phi:V^{2}\rightarrow V:(a,b)\mapsto\Phi_{BCH}(a,b)$ with $\Phi_{BCH}$
defined in (2.13).

Proof .

Let us introduce the projections $\tilde{P}_{n}$ and $\tilde{Q}_{n}$ on
$\ell^{2}(\mathbb{Z})$ acting by the rules
$(\tilde{P}_{n}f)_{j}=\left\\{\begin{array}[]{cc}f_{j}&-n\leq j\leq n\\\
0&\textrm{otherwise}\end{array}\right.\qquad(\tilde{Q}_{n}f)_{j}=\left\\{\begin{array}[]{cc}0&-n\leq
j\leq n\\\ f_{j}&\textrm{otherwise.}\end{array}\right.$ (3.26) Norm (3.25) can
be written as
$\|a\|_{loc}=\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\|\tilde{P}_{n}L(a)\tilde{P}_{n}\|$.
For any $a,b\in V$ we have
$\|T_{n}(a-b)\|\leq\|\tilde{P}_{n}L(a-b)\tilde{P}_{n}\|\leq
2^{n+1}\|a-b\|_{loc}\ ,$ (3.27) which implies $\|T_{n}(a)-T_{n}(b)\|\to 0$ as
$a\to b$. Since $\det T_{n}$ is continuous with respect to the metric induced
by $\|\cdot\|$ we also have $\det T_{n}(a)\to\det T_{n}(b)$ as $a\to b$. The
functions $F_{b}^{l},F_{b}^{r}$ are linear, so proving continuity is
equivalent to prove that they tend to zero as their argument approaches zero.
To that aim, note that, for any $n,m\in\mathbb{N}$, we have
$\begin{split}&\tilde{P}_{n}L(a\star
b)\tilde{P}_{n}=\tilde{P}_{n}L(a)L(b)\tilde{P}_{n}=\tilde{P}_{n}L(a)(\tilde{P}_{n+m}+\tilde{Q}_{n+m})L(b)\tilde{P}_{n}\\\
&=\tilde{P}_{n}\tilde{P}_{n+m}L(a)\tilde{P}_{n+m}L(b)\tilde{P}_{n+m}\tilde{P}_{n}+\tilde{P}_{n}L(a)\tilde{Q}_{n+m}L(b)\tilde{P}_{n},\end{split}$
(3.28) where in the last equality we used
$\tilde{P}_{n}\tilde{P}_{n+m}f=\tilde{P}_{n}f$ and
$f\tilde{P}_{n+m}\tilde{P}_{n}=f\tilde{P}_{n}$. From this it follows $\|a\star
b\|_{loc}\leq\|L(b)\|\left(2^{m+1}\|a\|_{loc}+\|\tilde{P}_{n}L(a)\tilde{Q}_{n+m}\|\right)$
(3.29) Now, by lemma A.3, we have
$\|\tilde{P}_{n}L(a)\tilde{Q}_{n+m}\|=O(\rho_{1}^{m})$ for some
$\rho_{1}\in(0,1)$. Thus for every $\varepsilon>0$ we can take
$m\in\mathbb{N}$ sufficiently large that
$\|\tilde{P}_{n}L(a)\tilde{Q}_{n+m}\|\|L(b)\|<\varepsilon/2$. Taking
$\delta>0$ such that $2^{m+1}\|L(b)\|\delta<\varepsilon/2$ we obtain the
implication $\|a\|_{loc}<\delta\Rightarrow\|a\star b\|_{loc}<\varepsilon$.
Thus $\lim_{a\to 0}F_{b}^{r}(a)=0$ and, by linearity, $F_{b}^{r}$ is
continuous on its domain. The proof for $F_{b}^{l}$ is analogous. Finally, the
function $\Phi_{BCH}$ defined in (2.13) is a uniformly convergent series of
Moyal products, and we have just shown that the Moyal product is a continuous
function of its arguments; thus, $\Phi_{BCH}$ is continuous.

a

######  Definition 3.3 (Regularisation).

Let $\\{\mathfrak{a}_{k}\\}_{k=1}^{\infty}$ be a sequence of symbols
$\mathfrak{a}_{k}$ with zero star winding number in $(V,\|\cdot\|_{loc})$ ,
and let $\mathfrak{b}_{k,\pm}^{L/R}$ denote
$\log_{\star}\mathfrak{a}_{k,\pm}^{L/R}$. We say that
$\\{\mathfrak{a}_{k}\\}_{k=1}^{\infty}$ regularises the symbol $a$ if

  * -

$a=\lim_{k\rightarrow\infty}\mathfrak{a}_{k}$;

  * -

$L(\mathfrak{b}_{k,\pm}^{L/R})$ is uniformly bounded in $k$;

  * -

For any $k\in\mathbb{N}$, $T(\mathfrak{b}_{k,\pm}^{L/R})$ are trace class.

> ######  Remark 3.3.
>
> For example, a symbol $a$ could be regularised by the sequence
> $\\{\mathfrak{a}_{k}\\}_{k=1}^{\infty}$ with
> $\mathfrak{a}_{k}=1+g_{k}\star(a-1)$ and $g_{k}:(x,z)\mapsto e^{-x^{2}/k}$.

a

######  Lemma 3.8.

Let $a\in(V,\|\cdot\|_{loc})$ be a symbol with zero star winding number and
define its Wiener-Hopf star factorisation
$a_{\pm}^{R/L}\equiv\exp_{\star}(b^{R/L}_{\pm})$ in such a way that all the
diagonal matrix elements of $L(a_{-}^{R/L})$ are equal to $1$. If
$\\{\mathfrak{a}_{k}\\}_{k=1}^{\infty}$ is a regularisation for $a$ and
$\mathfrak{a}_{k,\pm}^{R/L}\equiv\exp_{\star}(\mathfrak{b}^{R/L}_{\pm})$ are
the corresponding factors with the same convention, then
$\lim_{k\to\infty}\mathfrak{b}^{R/L}_{k,\pm}=b^{R/L}_{\pm}$.

Proof .

Let us focus first on the right factorization and write
$\mathfrak{a}_{k}=e^{\mathfrak{b}^{-}_{k}}_{\star}\star
e^{\mathfrak{b}^{+}_{k}}_{\star}$, $a=e^{b^{-}}_{\star}\star
e^{b^{+}}_{\star}$. This implies $e^{\mathfrak{b}^{+}_{k}}_{\star}\star
e_{\star}^{-\mathfrak{b}^{+}}-e_{\star}^{-\mathfrak{b}^{-}_{k}}e_{\star}^{b^{-}}=e^{-\mathfrak{b}^{-}_{k}}_{\star}\star(\mathfrak{a}_{k}-a)\star
e_{\star}^{b^{+}}$ (3.30) Using lemma 3.3 and remark 3.2 we have
$\|T_{2n+1}(\texttt{T}^{-n}(e^{-\mathfrak{b}^{-}_{k}}_{\star}\star(\mathfrak{a}_{k}-a)\star
e_{\star}^{-b^{+}}))\|\leq\|T_{2n+1}(\texttt{T}^{-n}(\mathfrak{a}_{k}-a))\|\|L(e^{-\mathfrak{b}^{-}_{k}}_{\star})\|\|L(e_{\star}^{-b^{+}})\|,$
(3.31) giving us the bound $\|e^{\mathfrak{b}^{+}_{k}}_{\star}\star
e_{\star}^{-b^{+}}-e_{\star}^{-\mathfrak{b}^{-}_{k}}\star
e_{\star}^{b^{-}}\|_{loc}\leq\|\mathfrak{a}_{k}-a\|_{loc}e^{\|L(\mathfrak{b}^{-}_{k})\|+\|L(b^{+})\|}$
(3.32) It follows $\lim_{k\to\infty}\|e^{\mathfrak{b}^{+}_{k}}_{\star}\star
e_{\star}^{-b^{+}}-e_{\star}^{-\mathfrak{b}^{-}_{k}}\star
e_{\star}^{b^{-}}\|_{loc}=0$ (3.33) Since the $\pm$ components are split here
we have $\lim_{k\to\infty}\mathfrak{b}_{k}^{\pm}=b^{\pm}$. Similarly, for the
left factorization we have
$\mathfrak{a}_{k}^{-1_{\star}}=e^{-\mathfrak{b}^{-}_{k}}_{\star}\star
e^{-\mathfrak{b}^{+}_{k}}_{\star}$, $a^{-1_{\star}}=e^{-b^{-}}_{\star}\star
e^{-b^{+}}_{\star}$, giving us the bound
$\|e^{-\mathfrak{b}^{+}_{k}}_{\star}\star
e_{\star}^{b^{+}}-e_{\star}^{\mathfrak{b}^{-}_{k}}\star
e_{\star}^{-b^{-}}\|_{loc}\leq\|\mathfrak{a}_{k}^{-1_{\star}}-a^{-1_{\star}}\|_{loc}e^{\|L(\mathfrak{b}^{-}_{k})\|+\|L(b^{+})\|\|}$
(3.34) Again, we get $\lim_{k\to\infty}\mathfrak{b}_{k}^{\pm}=b^{\pm}$.

a

a

###### Proof of theorem 2.2.

Let us focus on a particular symbol $\mathfrak{a}$ belonging to a sequence
regularising the symbol $a$. The denominator of the BOCG formula can be
manipulated as follows

$\begin{split}\det(T(\mathfrak{a}_{-}^{L}\star(\mathfrak{a}_{+}^{R})^{-1_{\star}})T((\mathfrak{a}_{-}^{R})^{-1_{\star}}\star\mathfrak{a}_{+}^{L}))&=\det\left(T(\mathfrak{a}_{-}^{L})T((\mathfrak{a}_{+}^{R})^{-1_{\star}})T((\mathfrak{a}_{-}^{R})^{-1_{\star}})T(\mathfrak{a}_{+}^{L})\right)\\\
&=\det\left(e^{T(\mathfrak{b}^{L}_{-})}\right)\det\left(e^{-T(\mathfrak{b}^{R}_{+})}\right)\det\left(e^{-T(\mathfrak{b}^{R}_{-})}\right)\det\left(e^{T(\mathfrak{b}^{L}_{+})}\right)\\\
&=\exp\left(\mathrm{tr}\
T(\mathfrak{b}_{+}^{L}+\mathfrak{b}_{-}^{L}-\mathfrak{b}_{-}^{R}-\mathfrak{b}_{+}^{R})\right),\end{split}$
(3.35)

where $\mathfrak{b}_{\pm}^{L/R}=\log_{\star}\mathfrak{a}_{\pm}^{L/R}$.
Combining this with the numerator of the BOCG formula we have

$\begin{split}&\frac{\exp\left(\sum_{j=1}^{n}\log\bigl{(}(\mathfrak{a}_{+,j}^{L})_{0}(\mathfrak{a}_{-,j}^{L})_{0}\bigr{)}\right)}{\det(T(\mathfrak{a}_{-}^{L}\star(\mathfrak{a}_{+}^{R})^{-1_{\star}})T((\mathfrak{a}_{-}^{R})^{-1_{\star}}\star\mathfrak{a}_{+}^{L}))}\\\
&=\exp\left[\left(\sum_{j=1}^{n}(\log_{\star}\mathfrak{a})_{j}+\sum_{j=n+1}^{\infty}\left(\log_{\star}\mathfrak{a}-\mathfrak{b}_{-}^{L}-\mathfrak{b}_{+}^{L}\right)_{j}-\sum_{j=1}^{\infty}\left(\log_{\star}\mathfrak{a}-\mathfrak{b}_{-}^{R}-\mathfrak{b}_{+}^{R}\right)_{j}\right)_{0}\right],\end{split}$
(3.36)

trivially obtained using
$((\log\mathfrak{a}_{\pm}^{L})_{j})_{0}=((\log_{\star}\mathfrak{a}_{\pm}^{L})_{j})_{0}$.
Now we can write

$\begin{split}&\log_{\star}\mathfrak{a}-\mathfrak{b}_{+}^{L}-\mathfrak{b}_{-}^{L}=\frac{1}{4}\left(i\\{\mathfrak{b}_{+}^{L},\mathfrak{d}_{(-)}^{L}\\}_{M}+i\\{\mathfrak{d}_{(+)}^{L},\mathfrak{b}_{-}^{L}\\}_{M}\right),\quad\\\
&\log_{\star}\mathfrak{a}-\mathfrak{b}_{+}^{R}-\mathfrak{b}_{-}^{R}=\frac{1}{4}\left(i\\{\mathfrak{b}_{-}^{R},\mathfrak{d}^{R}_{(+)}\\}_{M}+i\\{\mathfrak{d}^{R}_{(-)},\mathfrak{b}_{+}^{R}\\}_{M}\right)\end{split}$
(3.37)

with

$\displaystyle\mathfrak{d}^{L}_{(-)}=$
$\displaystyle\Phi(\mathfrak{b}_{+}^{L},\mathfrak{b}_{-}^{L})$
$\displaystyle\mathfrak{d}^{R}_{(+)}=$
$\displaystyle\Phi(\mathfrak{b}_{-}^{R},\mathfrak{b}_{+}^{R})$ (3.38)
$\displaystyle\mathfrak{d}^{L}_{(+)}=$
$\displaystyle-\Phi(-\mathfrak{b}_{-}^{L},-\mathfrak{b}_{+}^{L})$
$\displaystyle\mathfrak{d}^{R}_{(-)}=$
$\displaystyle-\Phi(-\mathfrak{b}_{+}^{R},-\mathfrak{b}_{-}^{R})\,.$

where $\Phi$ is defined in (2.15). This can be readily obtained from the
symmetrised BCH integral formula $\log(e^{A}e^{B})=\frac{Z(A,B)-Z(-B,-A)}{2}$,
where

$Z(A,B)=A+\left(\int_{0}^{1}\psi(e^{\mathrm{ad}_{A}}e^{t\mathrm{ad}_{B}})\,dt\right)B$
(3.39)

and $\psi(x)=\frac{x\log x}{x-1}$. The final expression can be obtained using
lemma 3.6 and taking the limit of the sequence regularising $a$, in light of
the continuity properties given in lemmas 3.7 and 3.8 ∎

##  4 Matrices with slowly varying elements along the diagonals

Here we refine the results obtained in the previous section when there is a
flow that connects the terms of the asymptotic expansion of $\log\det
T_{n}(a)$ for large $n$ with those corresponding to the symbol of a Toeplitz
operator. The aim of the connection is to work out the Wiener-Hopf star
factorisations. We restrict ourselves to the case in which the flow can be
parametrised by a scaling variable, $s\in[0,1]$, in the sense that each symbol
of the family is represented by a function $f_{s\nu}:(x,p)\rightarrow
f(xs\nu,p)$, where $\nu$ is an auxiliary parameter quantifying the contraction
of the image of $f_{\nu}$ associated with taking a derivative with respect to
its first argument. The ultimate goal would be to express the strong Szegö
limit theorem 2.2 in the form of a series in $\nu$ whose terms can be computed
systematically. A close inspection to theorem 2.2 reveals that one needs the
asymptotic expansion of

  1. 1.

sums

  2. 2.

the Moyal star product

  3. 3.

left and right Wiener-Hopf star factorisations

  4. 4.

the star logarithm

  5. 5.

the function $\Phi_{BCH}$ in (2.13).

The former ingredient is required because the distinction between corner and
bulk terms envisaged in (2.19) and (2.20) is not yet manifest in (2.16), in
which the (pretended) bulk term is expressed as a sum rather than as an
integral. To that aim, we adapt the Euler-Maclaurin formula to our case. The
asymptotic expansion of the Moyal star product in the order of the derivatives
is well known and Section 4.1 is mainly focussed on providing conditions that
are sufficient for truncating the series. We are not aware of analogous
results for the left and right Wiener-Hopf star factorisations, so Section 4.2
is also devoted to develop a systematic asymptotic expansion. The expansion of
the star logarithm is worked out in Section 4.3 with the standard method of
the resolvent. Finally, we will be in a position to use the Baker-Campbell-
Hausdorff formula for the asymptotic expansion of $\Phi_{BCH}$ as the series
is uniformly convergent for small enough $\nu$.

We collect the main asymptotic formulas in Table 1 for quick reference.

| asymptotic series |
---|---|---
$\sum_{j=1}^{n}g(j\nu)$ | $\frac{1}{\nu}\int_{\frac{\nu}{2}}^{(n+\frac{1}{2})\nu}\left[g(y)-\sum_{j=1}^{\infty}\frac{\nu^{2j}}{(2j)!}(1-2^{1-2j})B_{2j}(1)g^{(2j)}(y)\right]dy$ | Euler-Maclaurin
$(f_{\nu}\star g_{\nu})(x,p)$ | $\bigl{(}fe^{i\nu\frac{\overleftarrow{\partial}_{p}\overrightarrow{\partial}_{x}-\overleftarrow{\partial}_{x}\overrightarrow{\partial}_{p}}{2}}g\bigr{)}(x\nu,p)$ | Groenewold
$f_{\pm}^{R/L}(\nu)$ | $e^{(\log f)^{\pm}}\sum_{j=0}^{\infty}(\psi_{j}^{R/L}[\nabla\log f])^{\pm}\frac{\nu^{j}}{j!}$ | Lemma 4.3 with def. 4.3
$(\zeta-f)^{-1_{\star}}$ | $\sum_{j=0}^{\infty}\frac{R_{j}^{(\zeta)}[f]}{(2j)!}\nu^{2j}$ | Lemma 4.4 with def. 4.5
$\nu^{-1}\Phi_{BCH}(\nu c,\nu d)$ | $d+\frac{1}{3}i\nu\\{c,d\\}_{M}+\frac{1}{12}\nu^{2}\\{\\{c,d\\}_{M},d\\}_{M}+\dots$ | Baker-Campbell-Hausdorff
Table 1: Outline of the asymptotic series used or worked out in this section.
Here $f_{\nu}(x,p):(x,p)\mapsto f(x\nu,p)$ is a smooth enough function
representing the symbol and $f_{\pm}^{R/L}(\nu)$ represent the factors of the
right and left factorisations.

###  4.1 Truncated Moyal product

When two symbols characterised by the same value of $\nu$ are (star)
multiplied, $\nu$ effectively enters the star product as the reduced Planck’s
constant $\hbar$ does in phase-space quantum mechanics. This opens the door to
asymptotic expansions in the limit of small $\nu$ analogous to semiclassical
approximations based on the expectation that, close to classical settings,
$\hbar$ can be treated as a small parameter. The following lemma provide a
justification for the truncation of the star product.

######  Definition 4.1.

We denote by $\Sigma_{\rho}$ the horizontal strip in the complex plane
$\\{z\in\mathbb{C}:\ \log\rho<\mathrm{Im}z<-\log\rho\\}$ for some
$\rho\in(0,1)$. We also use $\Sigma_{\rho}^{(2\pi)}$ as a shorthand for
$\Sigma_{\rho}/(2\pi\mathbb{Z})$.

######  Lemma 4.1.

Let $f,g$ be two functions on $\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}$ that
are smooth in the second argument and of class $C^{k}$ in the first argument
for some $k\in\mathbb{N}$. Given two symbols $a=[f_{\nu}],\ b=[g_{\nu}]$ with
$f_{\nu}(x,p)=f(x\nu,p),\ g_{\nu}(x,p)=g(x\nu,p)$, we have $a\star
b=[h_{\nu}]$, where $h_{\nu}$ can be approximated by

$h(\nu):(y,p)\mapsto\sum_{j=0}^{k-1}\frac{1}{j!}\left.\Bigl{(}i\nu\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\Bigr{)}^{j}f(y_{1},p_{1})g(y_{2},p_{2})\right|_{p_{1}=p_{2}=p\atop
y_{1}=y_{2}=y}$ (4.1)

with an error that can be bounded from above as follows

$\left|h_{\nu}(x,p)-(h(\nu))(x\nu,p)\right|\leq\\\
\left(\frac{\nu}{2}\right)^{k}\sum_{j=0}^{k}\frac{K_{k-j,j}^{\Lambda}(x\nu)K_{j,k-j}^{\Lambda}(x\nu)}{j!(k-j)!}+\rho^{\frac{2\Lambda}{\nu}}\left[2K^{2}+3\sum_{j=0}^{k-1}\sum_{j^{\prime}=0}^{j}\frac{K^{0}_{j^{\prime},j-j^{\prime}}(x\nu)}{j^{\prime}!(j-j^{\prime})!}\Bigl{(}\frac{\nu}{2}\Bigr{)}^{j}H^{(x\nu)}_{j-j^{\prime},j^{\prime}}(\tfrac{2\Lambda}{\nu})\right]$
(4.2)

for any $\Lambda\in\mathbb{N}$. Here $K$ is a finite constant that depends on
the global behaviour of $f$ and $g$ and satisfies

$\sup_{y\in\mathbb{R}}\sum_{|m|\geq\Lambda}|(f_{1}(y))_{m}|\leq
K\rho^{\Lambda}\qquad\sup_{y\in\mathbb{R}}\sum_{|m|\geq\Lambda}|(g_{1}(y))_{m}|\leq
K\rho^{\Lambda}\,;$ (4.3)

$K^{\Lambda}_{j,k}(y)$ depend on the local behaviour of $f$ and $g$ in an
interval of extent $2\Lambda+1$ around $y$:

$\sum_{m\in\mathbb{Z}}|m|^{j}\left<\partial_{1}^{k}f_{1}(y)\right>_{m}^{\Lambda}\leq
K_{j,k}^{\Lambda}(y)\qquad\sum_{m\in\mathbb{Z}}|m|^{j}\left<\partial_{1}^{k}g_{1}(y)\right>_{m}^{\Lambda}\leq
K_{j,k}^{\Lambda}(y)\,,$ (4.4)

with

$\left<f_{1}(y)\right>_{k}^{\Lambda}=\sup_{z\in
y+[-\Lambda,\Lambda]}|(f_{1}(z))_{k}|\,;$ (4.5)

$H_{j,k}^{(y)}(\Lambda)$ are polynomials in $\Lambda$ that depend on the local
behaviour of $f$ and $g$ about $y$:

$\sum_{|m|\geq\Lambda}|m|^{j}|(\partial_{1}^{k}f_{1}(y))_{m}|\leq
H^{(y)}_{j,k}(\Lambda)\rho^{\Lambda}\qquad\sum_{|m|\geq\Lambda}|m|^{j}|(\partial_{1}^{k}g_{1}(y))_{m}|\leq
H^{(y)}_{j,k}(\Lambda)\rho^{\Lambda}\,.$ (4.6)

Proof .

A proof is reported in Appendix B.

a

######  Corollary 4.1.

Under the assumptions of the lemma, for any compact $\Omega\subset\mathbb{R}$
we have

$h_{\nu}:(x,p)\mapsto(h(\nu))(x\nu,p)+O(\nu^{k})$ (4.7)

as $\nu\rightarrow 0$ uniformly in
$\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$

Proof .

This is because $K_{j,j^{\prime}}^{\Lambda}(y)$ and $H^{y}_{j,j^{\prime}}(M)$
are bounded from above by $\sup_{y\in\Omega}K_{j,j^{\prime}}^{\Lambda}(y)$ and
$\sup_{y\in\Omega}H^{y}_{j,j^{\prime}}(M)$, which are both finite since
$\Omega$ is compact, and $\rho^{M}\sup_{y\in\Omega}H^{y}_{j,j^{\prime}}(M)$
approaches $0$ faster than any power of $M$ as $M\rightarrow\infty$. Thus, the
upper bound in (4.2) is bounded from above by a constant multiplied by
$\nu^{k}$,

a

######  Corollary 4.2.

If $f,g$ are smooth on $\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}$, for any
compact $\Omega\subset\mathbb{R}$, $h_{\nu}(x,p)$ is captured by the
asymptotic series

$\sum_{j=0}^{\infty}\frac{1}{j!}\left.\Bigl{(}i\nu\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\Bigr{)}^{j}f(y_{1},p_{1})g(y_{2},p_{2})\right|_{p_{1}=p_{2}=p\atop
y_{1}=y_{2}=x\nu}$ (4.8)

as $\nu\rightarrow 0$, uniformly in
$\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

Proof .

This follows from the possibility of applying corollary 4.1 for any
$k\in\mathbb{N}$.

a

> ######  Remark 4.1.
>
> If $f$ and $g$ are entire with respect to their first argument and their
> derivatives are bounded from above, then the double series is absolutely
> convergent—see lemma B.1.

###  4.2 Truncated Wiener-Hopf star factorisation

In this section we take advantage of the possibility to truncate the star
product to work out an explicit approximation for the Wiener-Hopf star
factorisations. An approximation for the product provides indeed an
approximation for the factors:

######  Lemma 4.2.

Consider a symbol $a\in V$ with zero star winding number. Suppose
$a=a_{-}^{\epsilon}\star a_{+}^{\epsilon}+O(\epsilon)$ as $\epsilon\rightarrow
0$ for given $\pm$ symbols $a^{\epsilon}_{\pm}$ with
$\log_{\star}a_{\pm}^{\epsilon}\in V$ uniformly bounded in $\epsilon$, where
$\epsilon$ is some auxiliary parameter. Then, there is a right Wiener-Hopf
star factorisation of $a$ such that

$a_{\pm}^{R}=a_{\pm}^{\epsilon}+O(\epsilon)\qquad\text{as $\epsilon\rightarrow
0$.}$ (4.9)

An analogous result applies to the factors of the left Wiener-Hopf star
factorisation if $a=a_{+}^{\epsilon}\star a_{-}^{\epsilon}+O(\epsilon)$ as
$\epsilon\rightarrow 0$.

Proof .

By assumptions the symbol satisfies $a=a_{-}^{R}\star
a_{+}^{R}=a_{-}^{\epsilon}\star a_{+}^{\epsilon}+O(\epsilon)$, and hence
$a_{+}^{R}\star(a_{+}^{\epsilon})^{-1_{\star}}=(a_{-}^{R})^{-1_{\star}}\star
a_{-}^{R}\star
a_{+}^{R}\star(a_{+}^{\epsilon})^{-1_{\star}}=(a_{-}^{R})^{-1_{\star}}\star
a_{-}^{\epsilon}+O(\epsilon)$ (4.10) where we used that
$(a_{-}^{R})^{-1_{\star}}$ and $(a_{+}^{\epsilon})^{-1_{\star}}$ are uniformly
bounded in $\epsilon$. Extracting the $+$ part of the members of the equation
gives
$a_{+}^{R}\star(a_{+}^{\epsilon})^{-1_{\star}}=((a_{-}^{R})^{-1_{\star}}\star
a_{-}^{\epsilon})_{0}+O(\epsilon)=\frac{(a_{-}^{\epsilon})_{0}}{(a_{-}^{R})_{0}}+O(\epsilon)\,.$
(4.11) The first identity follows from the fact that
$(a_{-}^{R})^{-1_{\star}}\star a_{-}^{\epsilon}$ is a $-$ symbol and hence its
$+$ part consists of solely the zeroth Fourier coefficient; the second
identity is instead a special case of $(a_{-}\star
b_{-})_{0}=(a_{-})_{0}(b_{-})_{0}$. The star product with $a_{+}^{\epsilon}$
then gives $a_{+}^{R}=\frac{(a_{-}^{\epsilon})_{0}}{(a_{-}^{R})_{0}}\star
a_{+}^{\epsilon}+O(\epsilon)\,,$ (4.12) where we used again the boundedness
properties. Analogously we have
$a_{-}^{R}=a_{-}^{\epsilon}\star\frac{(a_{-}^{R})_{0}}{(a_{-}^{\epsilon})_{0}}+O(\epsilon)\,.$
(4.13) The result of the lemma applies to the alternative factorisation
$a=\tilde{a}_{-}^{R}\star\tilde{a}_{+}^{R}$, with
$\tilde{a}_{-}^{R}=a_{-}^{R}\star\frac{(a_{-}^{\epsilon})_{0}}{(a_{-}^{R})_{0}}$
and $\tilde{a}_{+}^{R}=\frac{(a_{-}^{R})_{0}}{(a_{-}^{\epsilon})_{0}}\star
a_{+}^{R}$, indeed (4.12) and (4.13) imply
$\tilde{a}_{-}^{R}=a_{-}^{\epsilon}+O(\epsilon)$ and
$\tilde{a}_{+}^{R}=a_{+}^{\epsilon}+O(\epsilon)$. The proof for the left
factorisation is analogous.

a

The idea is then to construct an asymptotic series for the factors based on
the knowledge of the case $\nu=0$, which corresponds to a Toeplitz matrix.

######  Definition 4.2.

Let $\\{a\\}=\\{a_{s}\in V:s\in[0,1]\\}$ be a one-parameter family of symbols.
If $a_{0}$ induces a Toeplitz operator then we say that $\\{a\\}$ is a star-
Toeplitz flow of symbols in $V$. If there exists also the one-parameter family
$\\{a^{-1_{\star}}\\}=\\{a^{-1_{\star}}_{s}\in V:s\in[0,1]\\}$ of star
inverses ($a_{s}\star a_{s}^{-1_{\star}}=1$), then we say that $\\{a\\}$ is a
star-Toeplitz flow of star-invertible symbols in $V$. If there exist also the
one-parameter families $\\{a^{R/L}_{\pm}\\}=\\{a^{R/L}_{s,\pm}\in
V:s\in[0,1]\\}$ of $\pm$ symbols such that $a_{s}=a_{s,-}^{R}\star
a_{s,+}^{R}=a_{s,+}^{L}\star a_{s,-}^{L}$ and $\log_{\star}a^{R/L}_{s,\pm}\in
V$ then we say that $\\{a\\}$ is a star-Toeplitz flow of symbols in $V$ with
zero star winding number.

By imposing that the star product of the factors is the symbol, we obtain a
recurrence relation between the terms of the asymptotic series of the factors,
which is related to the following

######  Definition 4.3.

Let $v_{1}\partial_{1}+v_{2}\partial_{2}$ be a vector field on an open subset
$S$ of $\mathbb{C}^{2}$ and $v_{1}(\vec{\varphi}),v_{2}(\vec{\varphi})$, with
$\vec{\varphi}\in S$, be the coordinates of $\vec{v}(\vec{\varphi})$. Let
$\vec{v}$ be of class $C^{k}(S)$. For $m\leq k$, we define the functions
$\psi^{H}_{m}[\vec{v}]$ by the recursive equation

$\displaystyle\psi^{H}_{0}[\vec{v}]:$ $\displaystyle\vec{\varphi}\mapsto 1$
(4.14) $\displaystyle\psi^{H}_{m}[\vec{v}]:$
$\displaystyle\vec{\varphi}\mapsto-\sum_{j^{\prime},j^{\prime\prime}=0\atop{j^{\prime}+j^{\prime\prime}\leq
m}}^{m-1}\binom{m}{j^{\prime},j^{\prime\prime}}\left(i\tfrac{\sigma_{H}}{2}\det\vec{D}^{v}_{\vec{\varphi},\vec{\varphi}^{\prime}}\right)^{m-j^{\prime}-j^{\prime\prime}}(\psi_{j^{\prime}}[\vec{v}(\vec{\varphi})])^{-}(\psi_{j^{\prime}}[\vec{v}(\vec{\varphi}^{\prime})])^{+}\Bigr{|}_{\vec{\varphi}^{\prime}=\vec{\varphi}}$

where $H\in\\{L,R\\}$, $\sigma_{R}=-\sigma_{L}=1$,
$\binom{m}{j_{1},j_{2}}=\frac{m!}{j_{1}!j_{2}!(m-j_{1}-j_{2})!}$, and

$D^{v}_{\vec{\varphi},\vec{\varphi}^{\prime}}=\begin{pmatrix}\partial_{\varphi_{1}^{\prime}}+v_{1}^{+}(\vec{\varphi}^{\prime})&\partial_{\varphi_{1}}+v_{1}^{-}(\vec{\varphi})\\\
\partial_{\varphi_{2}^{\prime}}+v_{2}^{+}(\vec{\varphi}^{\prime})&\partial_{\varphi_{2}}+v_{2}^{-}(\vec{\varphi})\end{pmatrix}\,.$

######  Lemma 4.3.

Let $f$ be a function on $\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}$ that is
smooth in the second argument and of class $C^{k+1}$ in the first argument for
some $k\in\mathbb{N}_{0}$, and $f_{s\nu}:(x,p)\mapsto f(xs\nu,p)$, with
$s\in[0,1]$, parametrises a star-Toeplitz flow of symbols in $V$ with zero
star winding number connecting $a=[f_{\nu}]$ with the Toeplitz symbol
$e^{ip}\mapsto f(0,p)$. Then, for any compact $\Omega\subset\mathbb{R}$, the
factors $a^{R/L}_{\pm}$ of the right and left Wiener-Hopf star factorisations
of $a$ are represented by functions
$f^{R/L}_{\nu\pm}:(x,p)\mapsto(f_{\pm}^{R/L}(\nu))(x\nu,p)$ where
$f^{R/L}_{\pm}(\nu):\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))\to\mathbb{C}$
exhibit the following asymptotic behaviour

$f^{R/L}_{\pm}(\nu)=e^{(\log f)^{\pm}}\sum_{j=0}^{k}(\psi_{j}^{R/L}[\nabla\log
f])^{\pm}\frac{\nu^{j}}{j!}+O(\nu^{k+1})$ (4.15)

as $\nu\to 0$, uniformly in
$\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

Proof .

A proof is reported in Appendix B.

a

> ######  Remark 4.2.
>
> By corollary 4.2, if the function $f$ considered in lemma 4.3 is smooth with
> respect to its first argument then
>
> $f^{R/L}_{\pm}(\nu)\sim e^{(\log
> f)^{\pm}}\sum_{j=0}^{\infty}(\psi_{j}^{R/L}[\nabla\log
> f])^{\pm}\frac{\nu^{j}}{j!}\,,$ (4.16)
>
> asymptotically as $\nu\rightarrow 0$, uniformly in
> $\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

a

######  Definition 4.4.

Given two functions on an open subset $S$ of $\mathbb{C}^{2}$,
$f,g:S\rightarrow\mathbb{C}$, of class $C^{1}(S)$, the Poisson brackets read

$\\{f,g\\}:(y,p)\mapsto\frac{\partial f(y,p)}{\partial y}\frac{\partial
g(y,p)}{\partial p}-\frac{\partial g(y,p)}{\partial y}\frac{\partial
f(y,p)}{\partial p}\ .$ (4.17)

######  Corollary 4.3.

Let $f$ be a function as in lemma 4.3 and $g=\log f$ smooth in the second
argument and of class $C^{3}$ in the first argument. Then, for any compact
$\Omega\subset\mathbb{R}$, we have

$\log f^{H}_{+}(\nu)+\log f^{H}_{-}(\nu)=\log
f+\frac{i}{2}\sigma_{H}\nu\\{g^{-},g^{+}\\}\\\
+\frac{\nu^{2}}{8}\Bigl{(}2\\{g^{+},\\{g^{-},g^{+}\\}^{-}\\}-2\\{g^{-},\\{g^{-},g^{+}\\}^{+}\\}-\partial_{p}g^{-}\\{g^{-},\partial_{y}g^{+}\\}+\partial_{y}g^{-}\\{g^{-},\partial_{p}g^{+}\\}\\\
-\partial_{y}g^{+}\\{\partial_{p}g^{-},g^{+}\\}+\partial_{p}g^{+}\\{\partial_{y}g^{-},g^{+}\\}-\\{\partial_{p}g^{-},\partial_{y}g^{+}\\}+\\{\partial_{y}g^{-},\partial_{p}g^{+}\\}\Bigr{)}+O(\nu^{3})$
(4.18)

asymptotically as $\nu\to 0$, uniformly in
$\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

Proof .

This can be obtained by expanding the logarithm of $f_{\pm}^{H}(\nu)$, as
defined in (4.15), at the second order in $\nu$, and using (4.14) with
$m=0,1,2$, which can be explicitly written as follows:
$\displaystyle\psi^{H}_{0}[\nabla g]$ $\displaystyle=1$ (4.19)
$\displaystyle\psi^{H}_{1}[\nabla g]$
$\displaystyle=\frac{i}{2}\sigma^{H}\\{g^{-},g^{+}\\}$ (4.20)
$\displaystyle\psi^{H}_{2}[\nabla g]$
$\displaystyle=\frac{1}{2}\\{g^{-},g^{+}\\}^{-}\\{g^{-},g^{+}\\}^{+}-\frac{1}{4}\\{g^{-},g^{+}\\}^{2}+\frac{1}{2}\\{g^{+},\\{g^{-},g^{+}\\}^{-}\\}-\frac{1}{2}\\{g^{-},\\{g^{-},g^{+}\\}_{+}\\}$
$\displaystyle-\frac{1}{4}(\partial_{p}g^{-}\\{g^{-},\partial_{y}g^{+}\\}-\partial_{y}g^{-}\\{g^{-},\partial_{p}g^{+}\\}+\partial_{y}g^{+}\\{\partial_{p}g^{-},g^{+}\\}-\partial_{p}g^{+}\\{\partial_{y}g^{-},g^{+}\\})$
$\displaystyle-\frac{1}{4}(\\{\partial_{p}g^{-},\partial_{y}g^{+}\\}-\\{\partial_{y}g^{-},\partial_{p}g^{+}\\})\,.$
(4.21)

a

###  4.3 Truncated star functions

The main ingredient for the calculation of star functions is the star inverse
of $\zeta-a$ with $\zeta\in\mathbb{C}$ (definition 2.6). This problem can be
solved asymptotically just as the Wiener-Hopf star factorisations, indeed the
starting point is again an identity for a star product

$(\zeta-a)\star(\zeta-a)^{-1_{\star}}=1\,.$ (4.22)

Under the assumptions of lemma 4.1, we can truncate the Moyal product and
follow the lines of the proof of lemma 4.3. Note that this inversion is
simpler than for the Wiener-Hopf star factorisations, as one of the factors of
the star product (4.22) is known and there is no need to extract $\pm$ parts
of symbols. We start by defining the auxiliary functions that play the role of
those in definition 4.3.

######  Definition 4.5.

Let $f:S\to\mathbb{C}$ be a function of class $C^{k}(S)$ on a subset $S$ of
$\mathbb{R}^{2}$. For $\zeta\notin\mathrm{ran}f$, where $\mathrm{ran}f$
denotes the image of $f$, and $m\leq k$, we define the functions
$R_{m}^{(\zeta)}[f]$ of $f$ and derivatives by the recursive equation

$\displaystyle R^{(\zeta)}_{0}[f]:$
$\displaystyle\vec{\varphi}\mapsto\frac{1}{\zeta-f(\vec{\varphi})}$ (4.23)
$\displaystyle R^{(\zeta)}_{m}[f]:$
$\displaystyle\vec{\varphi}\mapsto\frac{1}{\zeta-f(\vec{\varphi})}\sum_{j=0}^{m-1}\binom{2m}{2j}(-1)^{m-j}\left.\Bigl{(}\frac{\partial_{\varphi_{1}}\partial_{\varphi_{2}^{\prime}}-\partial_{\varphi_{2}}\partial_{\varphi_{1}^{\prime}}}{2}\Bigr{)}^{2(m-j)}f(\vec{\varphi}^{\prime})R^{(\zeta)}_{j}[f(\vec{\varphi})]\right|_{\vec{\varphi}^{\prime}=\vec{\varphi}}$

where $\vec{\varphi}\in S$.

######  Lemma 4.4.

Let $f:\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}\to\mathbb{C}$ be of class
$C^{2(k+1)}$, for some $k\in\mathbb{N}_{0}$, with respect to the first
argument and smooth with respect to the second one. Let $\zeta$ be a complex
number $\notin\mathrm{ran}f$ and $\Omega$ be a compact subset of $\mathbb{R}$.
If $\zeta-f_{s\nu}$, with $f_{s\nu}:(x,p)\mapsto f(xs\nu,p)$ and $s\in[0,1]$,
parametrises a star-Toeplitz flow of star-invertible symbols in $V_{\rho}$
connecting $\zeta-a=[\zeta-f_{\nu}]$ with the Toeplitz symbol
$e^{ip}\mapsto\zeta-f(0,p)$, then

$(\zeta-a)^{-1_{\star}}=[R^{(\zeta,f)}_{\nu}]$ (4.24)

where $R^{(\zeta,f)}_{\nu}:(x,p)\mapsto(R^{(\zeta,f)}(\nu))(x\nu,p)$ and
$R^{(\zeta,f)}(\nu):\mathbb{R}\times(\mathbb{R}/(2\pi\mathbb{Z}))\rightarrow\mathbb{C}$
exhibits the asymptotic behaviour

$R^{(\zeta,f)}(\nu)=\sum_{j=0}^{k}\frac{R_{j}^{(\zeta)}[f]}{(2j)!}\nu^{2j}+O(\nu^{2k+2})$
(4.25)

as $\nu\rightarrow 0$, uniformly in
$\Omega\times(\mathbb{R}/(2\pi\mathbb{Z}))$. In particular, we have

$R^{(\zeta,f)}_{\nu}=\begin{cases}\frac{1}{\zeta-f_{\nu}}+O(\nu^{2})&k=0\\\
\frac{1}{\zeta-
f_{\nu}}+\frac{1}{4}\frac{(\partial_{1,2}f_{\nu})^{2}-\partial_{1}^{2}f_{\nu}\partial_{2}^{2}f_{\nu}}{(\zeta-
f_{\nu})^{3}}-\frac{1}{4}\frac{\partial_{2}^{2}f_{\nu}(\partial_{1}f_{\nu})^{2}+\partial_{1}^{2}f_{\nu}(\partial_{2}f_{\nu})^{2}-2\partial_{1}f_{\nu}\partial_{2}f_{\nu}\partial_{1,2}f_{\nu}}{(\zeta-
f_{\nu})^{4}}+O(\nu^{4})&k=1\end{cases}$ (4.26)

Proof .

The proof is analogous to that of lemma 4.3. Just note that, in this case, the
expansion of the Moyal product can be restricted to even powers (even $j$ in
(4.1)) as $\zeta-a$ star commutes with its star inverse.

a

> ######  Remark 4.3.
>
> By construction and in view of corollary 4.2, if the function $f$ considered
> in lemma 4.4 is smooth also with respect to its first argument then
>
>
> $R^{(\zeta,f)}(\nu)\sim\sum_{j=0}^{\infty}\frac{R_{j}^{(\zeta)}[f]}{(2j)!}\nu^{2j}\,,$
> (4.27)
>
> asymptotically as $\nu\rightarrow 0$.

a

######  Corollary 4.4.

Let $f$ be a function on $\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}$ that is
smooth in the second argument and of class $C^{2(k+1)}$ in the first argument
for some $k\in\mathbb{N}_{0}$. Let $f_{s\nu}:(x,p)\mapsto f(xs\nu,p)$, with
$s\in[0,1]$, parametrise a star-Toeplitz flow in $V$, and the symbol $a$ be
represented by $f_{\nu}$. If $F:\mathbb{C}\rightarrow\mathbb{C}$ is a function
analytic inside a closed simple curve $C$ of the complex plane strictly
surrounding the spectrum of $L(a^{\prime})$ for any $a^{\prime}$ in the orbit
of the flow, then for any compact $\Omega\in\mathbb{R}$ the star function
$F_{\star}(a)$ is represented by a function $F_{\star}[f_{\nu}]$ of $f_{\nu}$
and derivatives with the asymptotic behaviour

$F_{\star}[f_{\nu}]=\begin{cases}F(f_{\nu})+O(\nu^{2})&k=0\\\
F(f_{\nu})+F^{\prime\prime}(f_{\nu})\frac{(\partial_{1,2}f_{\nu})^{2}-\partial_{1}^{2}f_{\nu}\partial_{2}^{2}f_{\nu}}{8}\\\
\qquad\qquad-F^{\prime\prime\prime}(f_{\nu})\frac{\partial_{2}^{2}f_{\nu}(\partial_{1}f_{\nu})^{2}+\partial_{1}^{2}f_{\nu}(\partial_{2}f_{\nu})^{2}-2\partial_{1}f_{\nu}\partial_{2}f_{\nu}\partial_{1,2}f_{\nu}}{24}+O(\nu^{4})&k=1\,,\end{cases}$
(4.28)

as $\nu\rightarrow 0$, uniformly in
$\tfrac{\Omega}{\nu}\times(\mathbb{R}/(2\pi\mathbb{Z}))$.

Proof .

This follows from plugging (4.26) into the Cauchy integral formula (2.5).

a

######  Proposition 4.1.

With the same notations of corollary 4.4, let $f:\mathbb{R}\times
S_{\rho}^{(2\pi)}\rightarrow\mathbb{C}$ be smooth with respect to either of
the arguments, the flow be with zero star winding number, and $F$ be the
logarithm. Then, the asymptotic expansion induced by (4.27) can be written as

$\log_{\star}a\sim\left[g+\sum_{j=1}^{\infty}\frac{L_{j}[g]}{(2j)!}\right]\,,\qquad\text{with}\quad
L_{j}[g]=\nu^{2j}\oint_{C}\frac{\mathrm{d}\zeta}{2\pi
i}R_{j}^{(\zeta)}[e^{g}]\,,$ (4.29)

$g=\log f_{\nu}$, and

  1. 1.

$L_{j}[g]$ is a multivariate polynomial of the derivatives of $g$ of total
degree $3j$.

  2. 2.

in each monomial of $L_{j}[g]$, the derivative with respect to each argument
appears exactly $2j$ times.

Proof .

The second property is true by construction: $L_{j}[g]$ is proportional to
$R_{j}^{(\zeta)}[f]$, which is written in terms of $2j$ derivatives with
respect to the first argument and $2j$ derivatives with respect to the second
one. The first property is a direct consequence of the following observations
- for every $j>0$, $R_{j}^{(\zeta)}[f]$ is a linear combination of terms of
the form
$\frac{\overbrace{(\partial_{\dots}f)\cdots(\partial_{\dots}f)}^{k-1\text{
factors}}}{(\zeta-f)^{k}}$, with $k>1$; - the contour integration in $\zeta$
of each term is a multivariate polynomial in the derivatives of $g$; - the
monomial with the largest degree corresponds to the term in which all the
derivatives to $R$ in the second line of (4.23) are applied to the poles
$\frac{1}{(\zeta-f)^{k}}$. By solving the recursive equation induced by (4.23)
to the total degree of the polynomial, we readily find that $L_{j}[g]$ has
total degree $3j$.

a

> ######  Remark 4.4.
>
> If $f$ is of class $C^{4}$, we have
>
>
> $\log_{\star}a=\left[g-\frac{2\partial_{1}g\partial_{2}g\partial_{1,2}g+3(\partial_{1,2}g)^{2}-(\partial_{2}g)^{2}\partial_{1}^{2}g-(\partial_{1}g)^{2}\partial_{2}^{2}g-3\partial^{2}_{1}g\partial^{2}_{2}g}{24}+O(\nu^{4})\right]\,.$
> (4.30)

###  4.4 Asymptotic expansion—proof of theorem 2.3

In this section we put all pieces together and work out the first terms of the
asymptotic expansion of (2.20) in the limit of small $\nu$. We start by
showing that there is a gauge in which (2.21) holds.

######  Definition 4.6 (Bulk function).

We say that an infinitely differentiable function $f$ is a bulk function in
its domain if both the function and its derivatives vanish approaching the
boundaries of the domain.

######  Definition 4.7 (Boundary functional).

We say that a functional $F[g]$ of an infinitely differentiable function is a
boundary functional if, for every bulk function,

$\lim_{\epsilon\rightarrow 0}\frac{F[g+\epsilon\phi]-F[g]}{\epsilon}=0$ (4.31)

######  Lemma 4.5.

Let $g$ be infinitely differentiable in
$(\frac{1}{2},n+\frac{1}{2})\times(\mathbb{R}/(2\pi\mathbb{Z}))$ and $F[g]$ be
a functional of the form

$F[g]=\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\left(\prod_{i=1}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx$
(4.32)

with $c_{i},d_{i}\in\mathbb{N}_{0}$. Let $\\{g_{m}\\}$ be a sequence of bulk
functions converging to $g$ almost everywhere in
$(\frac{1}{2},n+\frac{1}{2})\times(\mathbb{R}/(2\pi\mathbb{Z}))$. Let $G[g]$
be the functional

$G[g]=\int_{-\infty}^{0}\bar{\delta}_{g}F[e^{s}g](\\{g_{m}\\})ds$ (4.33)

where

$\bar{\delta}_{g}F[g]:\\{\phi_{m}\\}\mapsto\lim_{m\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\frac{F[g+\epsilon\phi_{m}]-F[g]}{\epsilon}\,.$ (4.34)

Then

  1. 1.

for any sequence $\\{\phi_{m}\\}$ of bulk functions converging almost
everywhere to some infinitely differentiable function $\phi$,
$\bar{\delta}_{g}F[g](\\{\phi_{m}\\})$ depends on the sequences only through
$\phi$: $\bar{\delta}_{g}F[g](\\{\phi_{m}\\})\equiv\delta_{g}F[g](\phi)$

  2. 2.

the functional $G[g]$ is given by

$G[g]=\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}g(x,p)\frac{1}{k}\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx$ (4.35)

  3. 3.

for any bulk function $\phi$, $\delta_{g}G[g](\phi)=\delta_{g}F[g](\phi)$

Proof .

A proof is reported in Appendix B.

a

######  Corollary 4.5.

A functional $F[g]$ as in (4.32) can be decomposed in the sum of a bulk
contribution, captured by $G[g]$ and given by (4.35), and boundary
contributions written in terms of $g$ and derivatives evaluated at
$\frac{1}{2}$ and $n+\frac{1}{2}$. Specifically, we have

$F[g]-G[g]=\left.\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{k}\sum_{j=1}^{k}\sum_{\eta=0}^{c_{j}-1}\partial_{1}^{\eta}\partial_{2}^{d_{j}}g(x,p)(-\partial_{1})^{c_{j}-\eta-1}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dp\right|_{x=\frac{1}{2}}^{x=n+\frac{1}{2}}$
(4.36)

Proof .

The lemma shows that $F[g]-G[g]$ is a boundary contribution, hence the main
statement. To compute the boundary contributions, it is sufficient to
integrate (4.35) by parts $c_{j}$ times with respect to $x$ and $d_{j}$ times
with respect to $p$. The boundary terms of the integrations by parts with
respect to $p$ vanish by periodicity, whereas the other integrations by parts
give (4.36).

a

######  Lemma 4.6 (Euler-Maclaurin).

Let $g(x\nu)$ be a function of class $C^{2k}$, then we have

$\sum_{j=1}^{n}g(j\nu)=\frac{1}{\nu}\int_{\frac{\nu}{2}}^{(n+\frac{1}{2})\nu}\left[g(y)-\sum_{j=1}^{k}\frac{\nu^{2j}}{(2j)!}(1-2^{1-2j})B_{2j}(1)g^{(2j)}(y)\right]\,dy\\\
-\frac{\nu^{2k}}{(2k)!}\int_{-\frac{1}{2}}^{\frac{1}{2}}B_{2k}(t-[t])\left\\{\sum_{j=1}^{n}g^{(2k)}(\nu(j-t))\right\\}dt$
(4.37)

where $B_{2j}$ are the Bernoulli polynomials.

> ######  Remark 4.5.
>
> Let $g(x\nu)$ be an infinitely differentiable function, then the following
> asymptotic formula holds
>
>
> $\sum_{j=1}^{n}g(j\nu)\sim\frac{1}{\nu}\int_{\frac{\nu}{2}}^{(n+\frac{1}{2})\nu}\left[g(y)-\sum_{j=1}^{\infty}\frac{\nu^{2j}}{(2j)!}(1-2^{1-2j})B_{2j}(1)g^{(2j)}(y)\right]\,dy$
> (4.38)
>
> as $\nu\rightarrow 0$.

a

######  Lemma 4.7.

Let the symbol be $a=[f_{\nu}]$ with $f_{\nu}(x,p)=f(x\nu,p)$ and $f$ being a
function on $\mathbb{R}\times\Sigma_{\rho}^{(2\pi)}$ that is smooth in the
second argument and of class $C^{\infty}$ in the first argument. Let us also
use the usual notations $b_{\pm}^{R/L}=\log_{\star}a_{+}^{R/L}$. The boundary
terms $E^{R/L}(x)$ defined in (2.17) are captured by the following asymptotic
series

$E^{R/L}(x)\sim\frac{1}{4\pi}\int_{-\pi}^{\pi}\left.\frac{\sinh(\nu\frac{\partial_{y,p^{\prime}}}{2})}{\sinh(\nu\frac{i\partial_{y}}{2})}(g^{R/L}(\nu))(y,p)(\tilde{h}^{R/L}(\nu))(y,p^{\prime})\right|_{y=x\nu\atop
p^{\prime}=p}\,dp=\\\
-\frac{i}{2}(g^{R/L}(\nu)\partial_{2}\tilde{h}^{R/L}(\nu))_{0}-\frac{i\nu^{2}}{48}\partial_{1}^{2}\left(g^{R/L}(\nu)(\partial_{2}+\partial_{2}^{3})\tilde{h}^{R/L}(\nu)\right)_{0}+O(\nu^{4})$
(4.39)

as $\nu\rightarrow 0$. Here $g^{R/L}(\nu)$ is defined in such a way that

$b_{+}^{R/L}+b_{-}^{R/L}=[g^{R/L}_{\nu}]$ (4.40)

with $g^{R/L}_{\nu}(x,p)=(g^{R/L}(\nu))(x\nu,p)$, whereas
$\tilde{h}^{R/L}(\nu)$ is defined in such a way that

$\begin{gathered}(\Phi_{BCH}(b_{-}^{R},-b_{+}^{R}))^{+}+(\Phi_{BCH}(-b_{+}^{R},-b_{-}^{R}))^{-}=[\tilde{h}^{R}_{\nu}]\\\
-(\Phi_{BCH}(-b_{-}^{L},-b_{+}^{L}))^{+}-(\Phi_{BCH}(b_{+}^{L},b_{-}^{L}))^{-}=[\tilde{h}^{L}_{\nu}]\end{gathered}$
(4.41)

with $\tilde{h}^{R/L}_{\nu}(x,p)=(\tilde{h}^{R/L}(\nu))(x\nu,p)$.

Proof .

This is readily obtained by replacing $b_{\pm,x+j-\frac{m+1}{2}}^{R/L}$ and
$d_{(\pm),x+j-\frac{m+1}{2}}^{R/L}$ by their Taylor expansion about $x$ and
using that the expression depends only on the positive (negative) Fourier
coefficients of $d_{(+),x+j-\frac{m+1}{2}}^{R/L}$
($d_{(-),x+j-\frac{m+1}{2}}^{R/L}$).

a

###### Proof of theorem 2.3.

We start by proving the properties of the functions $\mathcal{D}^{(j)}$ and
$\mathcal{C}^{(j)}$. First of all we note that bulk contributions come only
from the sum of $\log_{\star}a$ in (2.16). We extract them following the
following procedure:

  * -

We apply the Euler-Maclaurin formula (4.38) to the sum and trasform it into an
integral.

  * -

Only the first term in the expansion captures bulk contributions, so we can
focus on that term and replace $\log_{\star}a$ by the asymptotic expansion in
(4.29).

  * -

We fix the gauge using the convention of corollary (4.5).

The first two properties highlighted in the theorem follow from proposition
4.1.

The boundary contributions have different origins. Some of them come the
Euler-Maclaurin formula and correspond to derivatives of
$(\log_{\star}a)_{x}(e^{ip})$ with respect to $x$, which therefore are
characterised by a lower number of derivatives with respect to $p$ rather than
$x$. Other boundary contributions come from the asymptotic series describing
the star logarithm $b_{\pm}$ of the factors of the Wiener-Hopf star
factorisations as well as the associated functions $\Phi_{BCH}(b_{-},b_{+})$
and $\Phi_{BCH}(b_{+},b_{-})$ emerging from the Baker-Campbell-Hausdorff
formula (2.14). Independently of how complicated such terms are, they are
always characterised by the same number of derivatives with the respect to the
two variables. They enter the formula, however, through $E^{R/L}(x)$, which we
replace by the asymptotic series (4.39) and is characterised by one extra
derivative with respect to $p$. This substantiates the third property of the
theorem.

The first orders of the bulk terms exhibited in (2.23) readily follow from
applying (4.35) to (4.30). The boundary terms exhibited in (2.24) are instead
obtain by

  * -

taking the first correction to the Euler-Maclaurin formula ($j=1$ in (4.38))

  * -

collecting the boundary terms coming from the asymptotic expansion of
$\log_{\star}a$, which can be obtained by applying (4.36) to (4.30);

  * -

taking the leading order in the expansion (4.39) of $E^{R/L}(x)$

  * -

truncating the expansion of $\Phi_{BCH}$ as in (2.15) to $O(\nu)$;

  * -

expanding the star logarithm to leading order (i.e., replacing it with a
conventional logarithm) for the factors of the factorisation

  * -

truncating the asymptotic expansion of the Wiener-Hopf star factorisation to
first order in $\nu$, i.e., keeping only the first line of (4.18).

∎

##  5 Discussion

We conclude by pointing out some physical applications of the formulas that we
derived. We have already highlighted in Section 2.3 that the determinant
formula provided by theorem 2.2 can be used to infer an explicit
representation of the constant contribution to the determinant of a block-
Toeplitz matrix, which appears, for example, in the calculation of correlation
functions in translationally invariant quantum spin models. Unfortunately,
however, the formula relies on the knowledge of both left and right
factorizations, which is often beyond reach. In inhomoegeneous physical
settings, instead, theorem 2.3 provides the formulas with more direct
applications. The asymptotic expansion that we worked out can indeed be
applied, for example, to the calculation of correlation functions and
entanglement entropies in systems prepared in smooth traps both in and out of
equilibrium. Arguably, some derivations could also be adapted to capture more
complicated bipartitioning protocols, generalising, for example, the results
of Ref. [49] to operators with semilocal fermionic representations.

The strong assumptions we made constitute, however, an obstacle to
applications. While it is reasonable to expect that symbols with a nonzero
star winding number could be easily accommodated in the theory, many
interesting physical problems are characterised by singularities that would
require generalisations of the Fisher-Hartwig formulas which go far beyond the
plain framework of this work. In that respect, even if the course of action is
not obvious, we believe that symbol regularisation could still be the key to
obtain explicit expressions.

Acknowledgements

We thank Luca Capizzi and Giuseppe Del Vecchio Del Vecchio for discussions.
This work was supported by the European Research Council under the Starting
Grant No. 805252 LoCoMacro.

##  Appendix A Basic results

######  Lemma A.1.

If $a\in V_{\rho}$ then for any $\rho_{1}\in(\rho,1)$ we have

$|(a_{x})_{j}|\leq M\rho_{1}^{|j|}\ ,$ (A.1)

where $M=\sup\\{a_{x}(z):\ x\in\frac{1}{2}\mathbb{Z},|z|\leq\rho_{1}\\}$.

Proof .

This is an elementary claim. Using analyticity we can deform the integration
contour from $|z|=1$ to $|z|=\rho^{\prime}$ for any
$\rho^{\prime}\in(\rho,\rho^{-1})$ giving
$|(a_{x})_{j}|=\left|\oint_{|z|=\rho^{\prime}}a_{x}(z)z^{-j}\frac{dz}{2\pi
iz}\right|\leq(\rho^{\prime})^{-j}\sup_{|z|=\rho^{\prime}}|a_{x}(z)|\;.$ (A.2)
The claim follows by choosing $\rho^{\prime}=\rho_{1}$ for $j<0$ and
$\rho^{\prime}=\rho_{1}^{-1}$ for $j>0$ (and say $\rho^{\prime}=1$ for $j=0$)
and using the uniform boundedness in $x$ in the annulus.

a

######  Lemma A.2.

Let $A=\left(A_{j,k}\right)_{j,k\in\mathbb{Z}}$ be an infinite matrix. The
matrix defines a bounded linear operator on $\ell(\mathbb{Z})$ with the norm
satisfying

$\|A\|\leq\sqrt{\sup\limits_{j\in\mathbb{Z}}\left(\sum_{k\in\mathbb{Z}}|A_{j,k}|\right)\sup\limits_{k^{\prime}\in\mathbb{Z}}\left(\sum_{j^{\prime}\in\mathbb{Z}}|A_{j^{\prime},k^{\prime}}|\right)},$
(A.3)

provided this bound is finite.

Proof .

Denote
$K_{1}=\sup\limits_{j\in\mathbb{Z}}(\sum_{k\in\mathbb{Z}}|A_{j,k}|),K_{2}=\sup\limits_{k^{\prime}\in\mathbb{Z}}(\sum_{j^{\prime}\in\mathbb{Z}}|A_{j^{\prime},k^{\prime}}|)$.
Let $v\in l^{2}(\mathbb{Z})$ be an arbitrary vector. We have
$\displaystyle\|Av\|^{2}=\sum_{j\in\mathbb{Z}}|\left(Av\right)_{j}|^{2}=\sum_{j\in\mathbb{Z}}|\sum_{k\in\mathbb{Z}}A_{j,k}v_{k}|^{2}$
(A.4) Writing
$A_{j,k}(a)v_{k}=\sqrt{A_{j,k}}\big{(}\sqrt{A_{j,k}}v_{k}\big{)}$, for any
choice of the complex square root, and applying the Cauchy-Schwarz inequality
we get
$\begin{split}\|Av\|^{2}&\leq\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}|A_{j,k}|\sum_{\ell\in\mathbb{Z}}|A_{j,\ell}||v_{\ell}|^{2}\leq
K_{1}\sum_{j\in\mathbb{Z}}\sum_{\ell\in\mathbb{Z}}|A_{j,\ell}||v_{\ell}|^{2}\\\
&=K_{1}\sum_{\ell\in\mathbb{Z}}|v_{\ell}|^{2}\sum_{j\in\mathbb{Z}}|A_{j,\ell}|\leq
K_{1}K_{2}\|v\|^{2}<\infty\>.\end{split}$ (A.5) Therefore
$\|A\|\leq\sqrt{K_{1}K_{2}}$.

a

a

######  Lemma A.3.

For $a\in V$ we have

  1. 1.

$L(a)$ is a bounded linear operator on $l^{2}(\mathbb{Z})$

  2. 2.

$T(a)$ is a bounded linear operator on $l^{2}(\mathbb{N})$

  3. 3.

$H(a)$ is a Hilbert-Schmidt operator

Proof .

We have $a\in V_{\rho}$ for some $0<\rho<1$. By lemma A.1 there is $M>0$ such
that $|L_{j,k}(a)|\leq M\rho_{1}^{|j-k|}\;.$ (A.6) for any
$\rho_{1}\in(\rho,1)$. This implies, by summing the geometric series,
$\sum_{j\in\mathbb{Z}}|L_{j,k}(a)|\leq
M\frac{1+\rho_{1}}{1-\rho_{1}},\qquad\sum_{j\in\mathbb{Z}}|L_{k,j}(a)|\leq
M\frac{1+\rho_{1}}{1-\rho_{1}}\qquad\forall k\in\mathbb{Z}.$ (A.7) By lemma
A.2 we have that $L(a)$ is bounded. The Toeplitz part of the lemma follows
from $\|T(a)\|=\|PL(a)P\|\leq\|P\|\|L(a)\|\|P\|=\|L(a)\|.$ (A.8) To prove the
Hankel part of the lemma we notice $|H_{j,k}(a)|=|L_{j,-k+1}(a)|\leq
M\rho_{1}^{j+k-1},\quad j,k\in\mathbb{N}\;.$ (A.9) Thus, summing the geometric
series, we have for the squared Hilbert-Schmidt norm
$\sum_{j,k=1}^{\infty}|H_{j,k}(a)|^{2}\leq\frac{M\rho_{1}}{(1-\rho_{1})^{2}}<\infty\>.$
(A.10)

a

###### Proof of Remark 2.3.

By assumption $a,b\in V_{\rho}$ for some $\rho\in(0,1)$. Let
$\rho<\rho_{1}<\rho_{2}<1$. The assumptions imply that there is $M>0$ such
that $|(a_{x})_{j}|\leq M\rho_{1}^{|j|}$, $|(b_{x})_{j}|\leq M\rho_{1}^{|j|}$.
Therefore on the annulus $\rho_{2}<|z|<\rho_{2}^{-1}$ we have

$\displaystyle|(a\star
b)_{x}(z)|\leq\sum\limits_{m,n\in\mathbb{Z}}|z|^{m+n}|(a_{x+\frac{m}{2}})_{n}||(b_{x-\frac{n}{2}})_{m}|\leq
M^{2}\sum\limits_{m,n\in\mathbb{Z}}\left(|z|^{m}\rho_{1}^{|m|}\right)\left(|z|^{n}\rho_{1}^{|n|}\right)$
(A.11)
$\displaystyle=M^{2}\left(\frac{1}{1-|z|\rho_{1}}+\frac{1}{1-\frac{\rho_{1}}{|z|}}-1\right)\leq
M^{2}\left(\frac{1+\frac{\rho_{1}}{\rho_{2}}}{1-\frac{\rho_{1}}{\rho_{2}}}\right)^{2}<\infty\;.$
(A.12)

Clearly, the bound we have obtained is independent of $x$. Since $\rho_{1}$
and $\rho_{2}$ are arbitrary the double series is absolutely convergent on the
whole annulus $\rho<|z|<\rho^{-1}$. Absolute convergence allows to interchange
the sum over $m$ and $n$ in the definition (2.4) of the star product. Making
also the shift of indices $m\to m-n$, we can write the star product as Laurent
series

$(a\star b)_{x}(z)=\sum_{m\in\mathbb{Z}}((a\star
b)_{x})_{m}z^{m}\;,\quad((a\star
b)_{x})_{m}=\sum_{n\in\mathbb{Z}}\big{(}a_{x+\frac{m-n}{2}}\big{)}_{n}\big{(}b_{x-\frac{n}{2}}\big{)}_{m-n}\;.$
(A.13)

The associativity of the star product can now be readily checked by comparing
$((a\star b)\star c)_{x})_{m}$ with $((a\star(b\star c))_{x})_{m}$ and the
claim $L(a\star b)=L(a)L(b)$ can be checked by comparing the matrix elements
of $L(a\star b)$ with the matrix elements of $L(a)L(b)$.

∎

##  Appendix B Additional details

###### Proof of lemma 4.1.

We start from (2.4), from which we get

$\left|h_{\nu}(x,p)-\frac{1}{(2\pi)^{2}}\sum_{m\in\frac{2B}{\nu}}\sum_{n\in\frac{2B}{\nu}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}f(\nu(x+\tfrac{m}{2}),q_{1})g(\nu(x-\tfrac{n}{2}),q_{2})e^{-i(nq_{1}+mq_{2})}d^{2}q\right|=\\\
\left|\frac{1}{(2\pi)^{2}}\sum_{m\notin\frac{2B}{\nu}}\sum_{n\in\mathbb{Z}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}f(\nu(x+\tfrac{m}{2}),q_{1})g(\nu(x-\tfrac{n}{2}),q_{2})e^{-i(nq_{1}+mq_{2})}d^{2}q\right.\\\
\left.+\frac{1}{(2\pi)^{2}}\sum_{m\in\frac{2B}{\nu}}\sum_{n\notin\frac{2B}{\nu}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}f(\nu(x+\tfrac{m}{2}),q_{1})g(\nu(x-\tfrac{n}{2}),q_{2})e^{-i(nq_{1}+mq_{2})}d^{2}q\right|\,.$
(B.1)

Let us first bound the last two lines. For the last line we have

$\sum_{m\notin\frac{2B}{\nu}}\sum_{n\in\mathbb{Z}}e^{i(m+n)p}(f_{1}(\nu(x+\tfrac{m}{2})))_{n}(g_{1}(\nu(x-\tfrac{n}{2})))_{m}\leq\sum_{m\notin\frac{2B}{\nu}}\sum_{n\in\mathbb{Z}}|(f_{1}(\nu(x+\tfrac{m}{2})))_{n}||(g_{1}(\nu(x-\tfrac{n}{2})))_{m}|\\\
\leq\sup_{y\in\mathbb{R}}\sum_{n\in\mathbb{Z}}|(f_{1}(y))_{n}|\sup_{y\in\mathbb{R}}\sum_{m\notin\frac{2B}{\nu}}|(g_{1}(y))_{m}|\leq
K^{2}\rho^{\frac{2\Lambda}{\nu}}\,,$ (B.2)

which is also an upper bound for the other line (after having interchanged $f$
and $g$). Thus the left hand side of (B.1) is upper bounded by
$2K^{2}\rho^{\frac{2\Lambda}{\nu}}$. We can now use the Taylor’s theorem and
expand $f$ up to order $k-1$ with the integral form of the reminder. This
gives

$\left|h_{\nu}(\tfrac{y}{\nu},p)-\frac{1}{(2\pi)^{2}}\sum_{m,n\in\frac{2B}{\nu}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}\sum_{j=0}^{k-1}\tfrac{\partial_{1}^{j}f(y,q_{1})(\nu\frac{m}{2})^{j}e^{-inq_{1}}}{j!}g(y-\tfrac{\nu
n}{2},q_{2})e^{-imq_{2}}d^{2}q\right|\leq\\\
2K^{2}\rho^{\frac{2\Lambda}{\nu}}+\left|\sum_{m,n\in\frac{2B}{\nu}}e^{i(m+n)p}\int_{0}^{\frac{m}{2}\nu}\frac{(\partial_{1}^{k}f_{1}(y+t))_{n}(\frac{m}{2}\nu-t)^{k-1}}{(k-1)!}dt(g_{1}(y-\tfrac{\nu
n}{2}))_{m}\right|\leq\\\
2K^{2}\rho^{\frac{2\Lambda}{\nu}}+\frac{(\frac{\nu}{2})^{k}}{k!}\sum_{n\in\mathbb{Z}}\left<\partial_{1}^{k}f_{1}(y)\right>^{B}_{n}\sum_{m\in\mathbb{Z}}|m|^{k}\left<g_{1}(y)\right>_{m}^{B}=2K^{2}\rho^{\frac{2\Lambda}{\nu}}+\frac{(\frac{\nu}{2})^{k}}{k!}K_{0,k}^{\Lambda}(y)K_{k,0}^{\Lambda}(y)\,.$
(B.3)

By Taylor expanding also $g$ up to order $k-1-j$ in the first sum we get

$\Bigl{|}h_{\nu}(\tfrac{y}{\nu},p)\\\
-\frac{1}{(2\pi)^{2}}\sum_{m,n\in\frac{2B}{\nu}}e^{i(m+n)p}\int\limits_{[-\pi,\pi]^{2}}\sum_{j=0}^{k-1}\tfrac{\partial_{1}^{j}f(y,q_{1})(\nu\frac{m}{2})^{j}e^{-inq_{1}}}{j!}\sum_{j^{\prime}=0}^{k-1-j}\tfrac{\partial_{1}^{j^{\prime}}g(y,q_{2})(-\nu\frac{n}{2})^{j^{\prime}}e^{-imq_{2}}}{j^{\prime}!}d^{2}q\Bigr{|}\leq\\\
2K^{2}\rho^{\frac{2\Lambda}{\nu}}+\left(\frac{\nu}{2}\right)^{k}\sum_{j=0}^{k}\frac{K_{k-j,j}^{\Lambda}(y)K_{j,k-j}^{\Lambda}(y)}{j!(k-j)!}$
(B.4)

The expression that approximates $h_{\nu}(\tfrac{y}{\nu},p)$ can be
manipulated by moving the sums in front of everything else and integrate by
parts $j^{\prime}$ times in $q_{1}$ and $j$ times in $q$, each time
integrating out the phases $e^{-inq_{1}}$ and $e^{-imq_{2}}$. This results in

$\Bigl{|}h_{\nu}(\tfrac{y}{\nu},p)\\\
-\sum_{m,n\in\frac{2B}{\nu}}\int\limits_{[-\pi,\pi]^{2}}\frac{e^{inq_{1}}e^{imq_{2}}}{(2\pi)^{2}}\left.\sum_{j=0}^{k-1}\tfrac{(i\nu\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2})^{j}}{j!}f(y_{1},p_{1}-q_{1})g(y_{2},p_{2}-q_{2})d^{2}q\right|_{p_{1}=p_{2}=p\atop
y_{1}=y_{2}=y}\Bigr{|}\leq\\\
2K^{2}\rho^{\frac{2\Lambda}{\nu}}+\left(\frac{\nu}{2}\right)^{k}\sum_{j=0}^{k}\frac{K_{k-j,j}^{\Lambda}(y)K_{j,k-j}^{\Lambda}(y)}{j!(k-j)!}\,.$
(B.5)

Analogously we can upper bound the discrepancy when extending the sums over
$m$ and $n$ to $\mathbb{Z}$, which finally gives (4.2). ∎

###### Proof of lemma 4.3.

By assumption all the symbols in the orbit of the flow have left and right
factorisations with bounded star logarithm. Let us then impose the Ansatz

$f^{R/L}_{\pm}(x,p)=\sum_{j=0}^{k}f^{R/L\pm}_{j}(x\nu,p)\frac{\nu^{j}}{j!}+O(\nu^{k+1})$
(B.6)

for some functions $f^{R/L\pm}_{j}(y,p)$, $j=0,1,\ldots,k$ and assume that
$f^{R/L\pm}_{j}(y,p)$ are of class $C^{k+1-j}$ in the first argument and
smooth in the second. This assumption will be shown to hold by construction.
We are in a position to apply Lemma 4.1, which gives

$f(x\nu,p)=\\\
\sum_{m=0}^{k}\frac{\nu^{m}}{m!}\sum_{j_{1}=0}^{m}\sum_{j_{2}=0}^{m-j_{1}}\binom{m}{j_{1},j_{2}}\left.\left(i\sigma_{H}\tfrac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\right)^{m-j_{1}-j_{2}}f_{j_{1}}^{H-}(y_{1},p_{1})f^{H+}_{j_{2}}(y_{2},p_{2})\right|_{y_{1}=y_{2}=x\nu\atop
p_{1}=p_{2}=p}\\\ +O(\nu^{k+1})$ (B.7)

uniformly in $x\nu\in\Omega$ and $p\in\mathbb{R}/(2\pi\mathbb{Z})$, where
$H\in\\{L,R\\}$ and $\sigma_{R}=-\sigma_{L}=1$. Treating $\nu$ and $x\nu$ as
independent variables and imposing that the coefficients of the asymptotic
polynomial in $\nu$ shown in (B.7) are zero, we obtain the recurrence equation

$\delta_{m,0}f(y,p)=\sum_{j_{1}=0}^{m}\sum_{j_{2}=0}^{m-j_{1}}\binom{m}{j_{1},j_{2}}\left.\left(i\sigma_{H}\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\right)^{m-j_{1}-j_{2}}f_{j_{1}}^{H-}(y_{1},p_{1})f_{j_{2}}^{H+}(y_{2},p_{2})\right|_{y_{1}=y_{2}=y\atop
p_{1}=p_{2}=p}$ (B.8)

For $m=0$, it reads $f=f_{0}^{H-}f_{0}^{H+}$ and is solved by

$f_{0}^{H\pm}=e^{(\log f)^{\pm}}$ (B.9)

For $m=1,2,\ldots,k$, we can rewrite Eq. (B.8) as

$0=f_{m}^{H-}f_{0}^{H+}+f_{0}^{H-}f_{m}^{H+}-f\Omega_{m}^{H}\ ,$ (B.10)

where we have defined the functions

$\Omega_{m}^{H}(y,p)=-\frac{1}{f}\sum_{j_{1}=0}^{m-1}\sum_{j_{2}=0\atop
j_{2}\neq
m}^{m-j_{1}}\binom{m}{j_{1},j_{2}}\left.\left(i\sigma_{H}\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\right)^{m-j_{1}-j_{2}}f_{j_{1}}^{H-}(y_{1},p_{1})f_{j_{2}}^{H+}(y_{2},p_{2})\right|_{y_{1}=y_{2}=y\atop
p_{1}=p_{2}=p}\,.$ (B.11)

Dividing by $f=f_{0}^{H-}f_{0}^{H+}$ we get

$e^{-(\log f)^{-}}f_{m}^{H-}+e^{-(\log f)^{+}}f_{m}^{H+}=\Omega_{m}^{H}\ ,$
(B.12)

which is solved by

$f_{m}^{H\pm}=e^{(\log f)^{\pm}}\left(\Omega_{m}^{H}\right)^{\pm},\quad
m=1,2,\ldots,k$ (B.13)

We can then rewrite $\Omega_{m}^{H}(y,p)$ as

$\Omega_{m}^{H}(y,p)=-e^{-(\log f)^{-}(y_{1},p_{1})-(\log
f)^{+}(y_{2},p_{2})}\sum_{j_{1}=0}^{m-1}\sum_{j_{2}=0\atop j_{2}\neq
m}^{m-j_{1}}\binom{m}{j_{1},j_{2}}\left(i\sigma_{H}\frac{\partial_{p_{1}}\partial_{y_{2}}-\partial_{y_{1}}\partial_{p_{2}}}{2}\right)^{m-j_{1}-j_{2}}\\\
\left.e^{(\log f)^{-}(y_{1},p_{1})}\Omega_{j_{1}}^{H-}(y_{1},p_{1})e^{(\log
f)^{+}(y_{2},p_{2})}\Omega_{j_{2}}^{H+}(y_{2},p_{2})\right|_{y_{1}=y_{2}=y\atop
p_{1}=p_{2}=p}=\\\ -\sum_{j_{1}=0}^{m-1}\sum_{j_{2}=0\atop j_{2}\neq
m}^{m-j_{1}}\binom{m}{j_{1},j_{2}}\Bigl{(}i\tfrac{\sigma_{H}}{2}\bigl{(}e^{-(\log
f)^{-}(y_{1},p_{1})}\partial_{p_{1}}e^{(\log f)^{-}(y_{1},p_{1})}e^{-(\log
f)^{+}(y_{2},p_{2})}\partial_{y_{2}}e^{(\log f)^{+}(y_{2},p_{2})}\\\
-e^{-(\log f)^{-}(y_{1},p_{1})}\partial_{y_{1}}e^{(\log
f)^{-}(y_{1},p_{1})}e^{-(\log f)^{+}(y_{2},p_{2})}\partial_{p_{2}}e^{(\log
f)^{+}(y_{2},p_{2})}\bigr{)}\Bigr{)}^{m-j_{1}-j_{2}}\\\
\left.\Omega_{j_{1}}^{H-}(y_{1},p_{1})\Omega_{j_{2}}^{H+}(y_{2},p_{2})\right|_{y_{1}=y_{2}=y\atop
p_{1}=p_{2}=p}\,,$ (B.14)

which, using $e^{-g}\partial e^{g}=\partial+(\partial g)$, reduces to
$\psi_{m}^{H}[\nabla\log f](y,p)$, as defined in (4.14), and hence (B.13)
implies (4.15). Finally, $f^{R/L\pm}_{j}(x\nu,p)$ as defined are $C^{k+1-j}$
by construction. We have thus shown that the symbols $d^{R/L}_{\pm}$
represented by the functions
$(x,p)\mapsto\sum_{j=0}^{k}f^{R/L\pm}_{j}(x\nu,p)\frac{\nu^{j}}{j!}$ satisfy
$d^{L}_{+}\star d_{-}^{L}=a+O(\nu^{k+1})$, $d^{R}_{-}\star
d_{+}^{R}=a+O(\nu^{k+1})$. Lemma 4.2 implies then
$a_{\pm}^{R/L}=d_{\pm}^{R/L}+O(\nu^{k+1})$. ∎

###### Proof of lemma 4.5.

The first property follows from the chain of identities

$\lim_{m\rightarrow\infty}\lim_{\epsilon\rightarrow
0}\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\frac{\prod_{i=1}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}(g(x,p)+\epsilon\phi_{m}(x,p))-\prod_{i=1}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}(g(x,p))}{\epsilon}dpdx=\\\
\lim_{m\rightarrow\infty}\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\sum_{j=1}^{k}\partial_{1}^{c_{j}}\partial_{2}^{d_{j}}\phi_{m}(x,p)\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)dpdx=\\\
\lim_{m\rightarrow\infty}\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\sum_{j=1}^{k}\phi_{m}(x,p)(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx=\\\
\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\sum_{j=1}^{k}\phi(x,p)(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx$ (B.15)

with the identification of $\delta_{g}F[g](\phi)$ with the last line. For the
second identity, we plug $\delta_{g}F[g](\phi)$ in (4.33) and integrate over
$s$, which gives (4.35). We can now compute the variation under a perturbation
with a bulk function $\phi$

$\delta_{g}G[g](\phi)=\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\phi(x,p)\frac{1}{k}\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx\\\
+\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}g(x,p)\frac{1}{k}\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\sum_{\ell\neq
j}^{k}\partial_{1}^{c_{l}}\partial_{2}^{d_{l}}\phi(x,p)\prod_{i\neq
j,\ell}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx$ (B.16)

If we integrate the integral in the second line by parts $c_{j}$ times with
respect to $x$ and $d_{j}$ times with respect to $p$, the boundary terms
vanish because the derivatives of the bulk function are bulk functions. Thus
we have

$\delta_{g}G[g](\phi)=\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\phi(x,p)\frac{1}{k}\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx\\\
+\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\frac{1}{k}\sum_{j=1}^{k}\partial_{1}^{c_{j}}\partial_{2}^{d_{j}}g(x,p)\sum_{\ell\neq
j}^{k}\partial_{1}^{c_{l}}\partial_{2}^{d_{l}}\phi(x,p)\prod_{i\neq
j,\ell}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)dpdx=\\\
\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\phi(x,p)\frac{1}{k}\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx\\\
+\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\phi(x,p)\frac{1}{k}\sum_{j=1}^{k}\sum_{\ell\neq
j}^{k}(-\partial_{1})^{c_{l}}(-\partial_{2})^{d_{l}}\left(\prod_{i\neq\ell}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx=\\\
\frac{1}{2\pi}\int_{\frac{1}{2}}^{n+\frac{1}{2}}\int_{-\pi}^{\pi}\phi(x,p)\sum_{j=1}^{k}(-\partial_{1})^{c_{j}}(-\partial_{2})^{d_{j}}\left(\prod_{i\neq
j}^{k}\partial_{1}^{c_{i}}\partial_{2}^{d_{i}}g(x,p)\right)dpdx\,,$ (B.17)

quod erat demonstrandum. ∎

######  Lemma B.1.

Consider two smooth functions
$f,g:\mathbb{C}\times\Sigma_{\rho}^{(2\pi)}\rightarrow\mathbb{C}$ that are
entire with respect to their first argument and satisfy

$\left|\frac{\partial^{n}}{\partial x^{n}}f(x,p)\right|\leq
M^{n},\quad\left|\frac{\partial^{n}}{\partial x^{n}}g(x,p)\right|\leq
M^{n}\qquad\forall x\in\mathbb{R},\quad\rho<|e^{ip}|<\rho^{-1},\ \forall
n\in\mathbb{N}\ ,$ (B.18)

for some $M>0$. If $M<2|\log\rho|$ then the star product is equal to

$\begin{split}(f\star
g)(x,p)&=\left.e^{i\frac{\partial_{p}\partial_{y}-\partial_{q}\partial_{x}}{2}}f(x,p)g(y,q)\right|_{q=p\atop
y=x}\\\
&=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{i^{m-n}}{2^{m+n}m!n!}\frac{\partial^{m}}{\partial
p^{m}}\frac{\partial^{m}}{\partial y^{m}}\frac{\partial^{n}}{\partial
q^{n}}\frac{\partial^{n}}{\partial x^{n}}\left[f(x,p)g(y,q)\right]_{q=p\atop
y=x}\ ,\end{split}$ (B.19)

where the double series is absolutely convergent.

Proof .

We start from the right hand side of Eq. (B.19) and show that it equals the
star product. By assumption we have
$\left|\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}e^{-ijq}\frac{\partial^{n}}{\partial
x^{n}}f(x,q)dq\right|=\left|\frac{1}{2\pi
i}\oint_{|z|=\rho_{1}^{-\mathrm{sgn}(j)}}z^{-j-1}\frac{\partial^{n}}{\partial
x^{n}}f(x,p)|_{e^{ip}=z}dz\right|\leq M^{n}\rho_{1}^{|j|}$ (B.20) for any
$\rho_{1}\in(\rho,1)$. By Fourier expanding $f$ and $g$ we obtain terms of the
form $\frac{\partial^{m}}{\partial p^{m}}\frac{\partial^{n}}{\partial
x^{n}}\sum_{j\in\mathbb{Z}}e^{ijp}\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x,q_{1})e^{-iq_{1}j}dq_{1}=\sum_{j\in\mathbb{Z}}(ij)^{m}e^{ijp}\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\partial^{n}}{\partial
x^{n}}f(x,q_{1})e^{-iq_{1}j}dq_{1}\ ,$ (B.21) where that the derivatives and
the series can be interchanged because the resulting series are uniformly
convergent (this can be seen, e.g., using the bound (B.20) to set a
Weierstrass M-test). We are thus left with
$\begin{split}&\left.e^{i\frac{\partial_{p}\partial_{y}-\partial_{q}\partial_{x}}{2}}f(x,p)g(y,q)\right|_{q=p\atop
y=x}\\\
&=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\frac{(-j)^{m}k^{n}}{m!n!2^{m+n}}e^{i(j+k)p}\frac{1}{(2\pi)^{2}}\\!\\!\\!\int\limits_{[-\pi,\pi]^{2}}\frac{\partial^{n}}{\partial
x^{n}}f(x,q_{1})\frac{\partial^{m}}{\partial
x^{m}}g(x,q_{2})e^{-i(q_{1}j+q_{2}k)}d^{2}q\end{split}$ (B.22) This series is
absolutely convergent, indeed, using (B.20), we have
$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\frac{1}{m!n!}\left|\frac{j}{2}\right|^{m}\left|\frac{k}{2}\right|^{n}\left|\
\frac{1}{(2\pi)^{2}}\int\limits_{[-\pi,\pi]^{2}}\frac{\partial^{n}}{\partial
x^{n}}f(x,q_{1})\frac{\partial^{m}}{\partial x^{m}}g(x,q_{2})d^{2}q\right|$
(B.23)
$\displaystyle\leq\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\frac{1}{m!n!}\left|\frac{j}{2}\right|^{m}\left|\frac{k}{2}\right|^{n}\rho_{1}^{|j|+|k|}M^{m+n}$
(B.24)
$\displaystyle=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}\exp\left(\frac{M}{2}|j|+\frac{M}{2}|k|\right)\rho_{1}^{|j|+|k|},$
(B.25) which converges for any $\rho_{1}$ such that $M/2+\log\rho_{1}<0$. We
can therefore interchange the order of summation in (B.22), which gives
$\left.e^{i\frac{\partial_{p}\partial_{y}-\partial_{q}\partial_{x}}{2}}f(x,p)g(y,q)\right|_{q=p\atop
y=x}=\\\
\sum_{j,k\in\mathbb{Z}}e^{i(j+k)p}\frac{1}{(2\pi)^{2}}\int\limits_{[-\pi,\pi]^{2}}e^{-i(q_{1}j+q_{2}k)}\sum_{n=0}^{\infty}\frac{(\frac{k}{2})^{n}}{n!}\frac{\partial^{n}}{\partial
x^{n}}f(x,q_{1})\sum_{m=0}^{\infty}\frac{(-\frac{j}{2})^{m}}{m!}\frac{\partial^{m}}{\partial
x^{m}}g(x,q_{2})d^{2}q=\\\
\sum_{j,k\in\mathbb{Z}}e^{i(j+k)p}\frac{1}{(2\pi)^{2}}\int\limits_{[-\pi,\pi]^{2}}e^{-i(q_{1}j+q_{2}k)}f(x+\tfrac{k}{2},q_{1})g(x-\tfrac{j}{2},q_{2})d^{2}q=(f\star
g)(x,p),$ (B.26) where in the third equality we used that the function is
entire and hence equal to its Taylor series (uniformly convergent in all
variables), and in the fourth equality we recognised the star product (2.4).

a

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