# An exact solution to asymptotic Bethe equation

Yuan Miao  Institute for Theoretical Physics, University of Amsterdam, Postbus
94485, 1090 GL Amsterdam, The Netherlands

###### Abstract

We present an exact solution to the asymptotic Bethe equation of weakly
anisotropic Heisenberg spin chain, which is a set of non-linear algebraic
equations. The solution describes the low-energy excitations above
ferromagnetic ground state with fixed magnetisation, and it has a close
relation to generalised Jacobi polynomial. It is equivalent to a generalised
Stieltjes problem and in the continuous limit, it becomes a Riemann-Hilbert
problem closely related to the finite-gap solutions of classical
Landau–Lifshitz field theory.

##  I Introduction

The study of the emergence of classical dynamics of quantum interacting
systems has led to the discoveries of new phenomena recently, especially in
the regime of quantum integrable systems. One of the most notable examples is
the classical-quantum correspondence of spin transport in the domain-wall
quench between quantum anisotropic Heisenberg model and the corresponding
classical Landau-Lifshitz field theory Gamayun  _et al._ (2019); Misguich  _et
al._ (2019), closely related to semi-classical eigenstates of the spectrum
Miao  _et al._ (2020). Moreover, the superdiffusive behaviour of spin
transport in quantum isotropic Heisenberg model Žnidarič (2011); Ljubotina
_et al._ (2017) is attributed to the existence of thermally dressed
interacting ‘giant magnons’ Ilievski  _et al._ (2018); De Nardis  _et al._
(2019); Gopalakrishnan and Vasseur (2019); De Nardis  _et al._ (2020a, b), a
manifestation of semi-classical eigenstates as well.

One way to understand the problems above is to use asymptotic Bethe
ansatzSutherland (1978, 1995); Dhar and Sriram Shastry (2000), i.e. a semi-
classical limit of the Bethe ansatz 111Asymptotic Bethe ansatz here refers to
the leading order of Bethe ansatz equations with respect to the effective
Planck constant. It is not related to the Bethe–Yang equation in integrable
field theories which is usually denoted as thermodynamic Bethe ansatz.. There
are systematic explorations for the solutions to asymptotic Bethe equation of
quantum isotropic Heisenberg model Kazakov  _et al._ (2004); Bargheer  _et
al._ (2008), motivated as a limit of $\mathcal{N}=4$ supersymmetric Yang-Mills
theory Minahan and Zarembo (2003); Kruczenski (2004). In addition, the results
are helpful to obtain the three-point correlation functions in the
supersymmetric gauge theories Escobedo  _et al._ (2011); Gromov  _et al._
(2012); Kostov (2012a, b), using a semi-classical limit of the overlaps
between the Bethe states.

In this article we derive an exact solution to the asymptotic Bethe equation
of quantum weakly anisotropic XXZ model, following the example of Ref. Shastry
and Dhar (2001) of the isotropic case.

##  II Asymptotic Bethe ansatz

We focus on the ferromagnetic anisotropic Heisenberg model, also known as XXZ
model, and its Hamiltonian can be written in terms of Pauli matrices,

$H=-\frac{1}{4}\sum_{j=1}^{L}\left(\sigma^{\rm x}_{j}\sigma^{\rm
x}_{j+1}+\sigma^{\rm y}_{j}\sigma^{\rm y}_{j+1}+\Delta\sigma^{\rm
z}_{j}\sigma^{\rm z}_{j+1}\right),$ (1)

where we use periodic boundary condition
($\sigma^{\alpha}_{L+1}=\sigma^{\alpha}_{1}$), and
$\sigma^{\alpha}_{j}=\mathbb{I}^{\otimes(j-1)}\otimes\sigma^{\alpha}\mathbb{I}^{\otimes(L-j)}$.
In this article, we concentrate on $\Delta=\cosh\eta>1$ regime, also known as
gapped regime, due to the property of its antiferromagnetic ground state.

The eigenstates of the Hamiltonian can be characterised by a set of
rapidities, i.e. Bethe roots. For each eigenstates with $M$ down spins ($M$
magnons), it can be denoted by $|\\{\vartheta_{j}\\}_{j=1}^{M}\rangle$, and
the Bethe roots $\vartheta_{j}$ satisfy

$\left[\frac{\sin(\vartheta_{j}+i\frac{\eta}{2})}{\sin(\vartheta_{j}-i\frac{\eta}{2})}\right]^{L}\prod_{k\neq
j}^{M}\frac{\sin(\vartheta_{j}-\vartheta_{k}-i\eta)}{\sin(\vartheta_{j}-\vartheta_{k}+i\eta)}=1,$
(2)

i.e. Bethe equations. They are closely related to the conserved charges
associated to the eigenstates. For each magnons, the pseudo-momentum and
pseudo-energy in terms of its Bethe root are

$p(\vartheta)=-\mathrm{i}\log\frac{\sin(\vartheta+i\frac{\eta}{2})}{\sin(\vartheta-i\frac{\eta}{2})},\quad\varepsilon(\vartheta)=\frac{\sin^{2}\mathrm{i}\eta}{\cos
2\vartheta-\cos\mathrm{i}\eta},$ (3)

and the physical momentum and energy of eigenstate
$|\\{\vartheta_{j}\\}_{j=1}^{M}\rangle$ become

$P=\sum_{j=1}^{M}p(\vartheta_{j})=-\mathrm{i}\sum_{j=1}^{M}\log\frac{\sin(\vartheta_{j}+i\frac{\eta}{2})}{\sin(\vartheta_{j}-i\frac{\eta}{2})},\quad
E=\sum_{j=1}^{M}\varepsilon(\vartheta_{j})=\sum_{j=1}^{M}\frac{\sin^{2}\mathrm{i}\eta}{\cos
2\vartheta_{j}-\cos\mathrm{i}\eta}.$ (4)

We define the Baxter Q-functions for $|\\{\vartheta_{j}\\}_{j=1}^{M}\rangle$
as

$Q(\vartheta)=\prod_{j=1}^{M}\sin(\vartheta-\vartheta_{j}),$ (5)

which is a Laurent polynomial of degree $M$ with variable
$t=e^{\mathrm{i}\vartheta}$. And we could rewrite the Bethe equations (2) and
take the logarithmic form using shorthand notation
$f^{[n]}(\vartheta)=f(\vartheta+\mathrm{i}n\eta/2)$,

$\log Q_{0}^{[+1]}(\vartheta_{j})-\log
Q_{0}^{[-1]}(\vartheta_{j})=2\pi\mathrm{i}n_{j}+\log
Q_{j}^{[+2]}(\vartheta_{j})-\log
Q_{j}^{[-2]}(\vartheta_{j}),\,n_{j}\in\mathbb{Z},$ (6)

where we use the following notations

$Q_{0}(\vartheta)=\sin^{L}\vartheta,\quad Q_{j}(\vartheta)=\prod_{k\neq
j}^{M}\sin(\vartheta-\vartheta_{k}).$ (7)

We use the formalism of asymptotic Bethe ansatz Sutherland (1978, 1995), i.e.
taking the “semi-classical limit” of Bethe equations (6) Miao  _et al._
(2020), which is equivalent to the weakly anisotropic limit,

$\eta=\frac{\epsilon\ell}{L}+\mathcal{O}\left(\frac{1}{L^{2}}\right),\quad\epsilon=\sqrt{\delta},\quad\delta=\lim_{a\to
0}\frac{2(\Delta-1)}{a^{2}}\in\mathcal{O}(1),$ (8)

where classical period $\ell=La\in\mathcal{O}(1)$ as system size $L\to\infty$
and lattice spacing $a=\frac{\ell}{L}\to 0$.

In this limit, we could easily obtain that

$\log Q_{0}^{[\pm 1]}(\vartheta_{j})=\log
Q_{0}(\vartheta_{j})\pm\frac{\mathrm{i}\eta}{2}\left.\frac{\mathrm{d}}{\mathrm{d}\vartheta}\log
Q_{0}(\vartheta)\right|_{\vartheta=\vartheta_{j}}+\mathcal{O}\left(\frac{1}{L^{2}}\right),$
(9)

and

$\log Q_{j}^{[\pm 2]}(\vartheta_{j})=\log
Q_{j}(\vartheta_{j})\pm\mathrm{i}\eta\left.\frac{\mathrm{d}}{\mathrm{d}\vartheta}\log
Q_{j}(\vartheta)\right|_{\vartheta=\vartheta_{j}}+\mathcal{O}\left(\frac{1}{L^{2}}\right).$
(10)

We define variable $\mu$ as

$\mu=\frac{\eta}{\ell}\frac{\mathrm{d}}{\mathrm{d}\vartheta}\log
Q_{0}(\vartheta)=\frac{\epsilon}{\tan\vartheta},$ (11)

and the Bethe equation (6) can be recast as

$\mu_{j}=\frac{2\pi}{\ell}n_{j}-\frac{2}{L}\sum_{k\neq
j}^{M}\frac{\mu_{j}\mu_{k}+\delta}{\mu_{j}-\mu_{k}}+\mathcal{O}\left(\frac{1}{L}\right).$
(12)

Neglecting the finite-size correction, we conveniently denote Eq. (12) as
asymptotic Bethe equation. Eq. (12) can be considered as a generalised version
of equations of Stieltjes in random matrix theory Shastry and Dhar (2001). The
remaining part of the article aims at giving an exact solution to the
asymptotic Bethe equation (12) in the simplest case, when all the Bethe roots
form a “huge bound state” sharing the same mode number $n_{1}$ and filling
fraction $\nu=\frac{M}{L}\in\mathcal{O}(1)$.

We assume that the Bethe roots $\mu_{j}$ condense around certain finite
disjoint segments $\mathcal{C}_{j}$, and take the continuous limit of Eq.
(12),

$\frac{\ell\mu}{2}=\pi
n_{j}-\ell\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\mathcal{C}}\mathrm{d}\lambda\,\frac{\mu\lambda+\delta}{\mu-\lambda}\rho(\lambda)+\mathcal{O}\left(\frac{1}{L}\right),\quad\mu\in\mathcal{C}_{j},$
(13)

where $\mathcal{C}=\cup_{j}\mathcal{C}_{j}$, and density

$\rho(\mu)=\lim_{L\to\infty}\frac{1}{L}\sum_{j=1}^{M}\bm{\delta}(\mu-\mu_{j}).$
(14)

Dirac delta function is denoted as $\bm{\delta}(\mu)$. The finite-size
(quantum) correction (the $\mathcal{O}(1/L)$ correction in Eq. (13)) has been
derived in Ref. Miao  _et al._ (2020); Beisert  _et al._ (2005), namely,

$\frac{1}{L}\pi\rho^{\prime}(\mu)\ell^{2}(\mu^{2}+\delta)^{2}\coth\left[\pi\ell(\mu^{2}+\delta)\rho(\mu)\right],$
(15)

which we neglect in this article. As it turns out, when the finite-size
correction diverges due to the property of $\coth$ function, the result of
this article would require modifications, which is explained in Ref. Miao  _et
al._ (2020).

Eq. (13) is a Riemann-Hilbert problem, and it coincides with the finite-gap
solutions of classical Landau–Lifshitz field theory, which is precisely the
semi-classical limit of quantum XXZ model with weak anisotropy. The extensive
discussions of the semi-classical quantisation of quantum anisotropic
Heisenberg model have been given in Ref. Miao  _et al._ (2020), and a previous
study on the semi-classical quantisation of isotropic case has been performed
in Ref. Bargheer  _et al._ (2008).

In the scope of this article, we discuss the simplest solutions with only one
mode number $n_{1}$ and the semi-classical limit of them is the 1-cut rational
solutions, describing the precessional motion of the spin field Miao  _et al._
(2020). We will refer corresponding solutions to asymptotic Bethe equation
(12) as 1-cut solutions.

##  III Exact solution to asymptotic Bethe equation

The asymptotic Bethe equation (12) can be formulated as an equation for zeros
of orthogonal polynomials. This insight has been pointed out in Shastry and
Dhar (2001) for the 1-cut solution of the isotropic case ($\delta=0$), in
which case the solutions can be expressed as Laguerre polynomials. This
approach gives the possibility of obtaining an analytic expression on the
distribution of the Bethe roots, e.g. in Martínez-Finkelshtein  _et al._
(2001). It gives an opportunity to obtain the contour analytically where the
density is supported taking into account the “reality condition”, i.e.
$\rho(\mu)\mathrm{d}\mu\in\mathbb{R}$ in Eq. (13). More discussions can be
found in Ref. Miao  _et al._ (2020).

When $\delta>0$, the 1-cut solutions to the asymptotic Bethe equation (12) are
the zeros of generalised Jacobi polynomial in the absence of condensate Miao
_et al._ (2020), which is a manifestation of the divergence of finite-size
(quantum) correction (15). We show the derivation as follows. A similar
phenomenon also appears when taking the semi-classical limit of attractive
Lieb-Liniger model, cf. Ref. Flassig  _et al._ (2016); Piroli and Calabrese
(2016).

To begin with, we can rewrite the asymptotic Bethe equation (12) as

$L\left(\mu_{j}-\frac{2\pi n_{1}}{\ell}\right)=2\sum_{k\neq
j}^{M}\frac{1}{\frac{1}{\mu_{j}}-\frac{1}{\mu_{k}}}-2\delta\sum_{k\neq
j}^{M}\frac{1}{\mu_{j}-\mu_{k}},$ (16)

with the mode number of the branch cut $n_{1}$.

We can define the following polynomial $\mathcal{P}(\mu)$ with zeros being the
solution to asymptotic Bethe equation (12), similar to the Baxter’s Q-function
(5), i.e.

$\mathcal{P}(\mu)=\prod_{j=1}^{M}\left(\mu-\mu_{j}\right),$ (17)

and accordingly,

$\tilde{\mathcal{P}}(\mu)=\prod_{j=1}^{M}\left(\frac{1}{\mu}-\frac{1}{\mu_{j}}\right)=\frac{\mathcal{P}(\mu)}{(-1)^{M}\mu^{M}\prod_{j=1}^{M}\mu_{j}}.$
(18)

One can show that the polynomial above satisfies the following identities,

$2\sum_{j\neq
k}\frac{1}{\mu_{j}-\mu_{k}}=\left.\frac{d^{2}\mathcal{P}}{d\mu^{2}}\left/\frac{d\mathcal{P}}{d\mu}\right.\right|_{\mu=\mu_{j}},$
(19)

and

$2\sum_{j\neq
k}\frac{1}{\frac{1}{\mu_{j}}-\frac{1}{\mu_{k}}}=\left.\frac{d^{2}\tilde{\mathcal{P}}}{d(1/\mu)^{2}}\left/\frac{d\tilde{\mathcal{P}}}{d(1/\mu)}\right.\right|_{\mu=\mu_{j}}.$
(20)

Let us denote $\mathcal{P}^{\prime}(\mu)=\frac{d}{d\mu}\mathcal{P}(\mu)$, and
we have

$\frac{d\tilde{P}}{d(1/\mu)}=\frac{d\mu}{d(1/\mu)}\frac{d}{d\mu}\left(\frac{\mathcal{P}(\mu)}{(-1)^{M}\mu^{M}\prod_{j=1}^{M}\mu_{j}}\right)=\frac{M\mu\mathcal{P}(\mu)-\mu^{2}\mathcal{P}^{\prime}(\mu)}{(-1)^{M}\mu^{M}\prod_{j=1}^{M}\mu_{j}},$
(21)

and

$\frac{d^{2}\tilde{\mathcal{P}}}{d(1/\mu)^{2}}=\frac{M(M-1)\mu^{2}\mathcal{P}(\mu)-2(M-1)\mu^{3}\mathcal{P}^{\prime}(\mu)+\mu^{4}\mathcal{P}^{\prime\prime}(\mu)}{(-1)^{M}\mu^{M}\prod_{j=1}^{M}\mu_{j}}.$
(22)

Using Eqs. (21) and (22), we can rewrite the asymptotic Bethe equation (12) in
terms of polynomial $\mathcal{P}(\mu)$, i.e.

$L\left(\mu_{j}-\frac{2\pi
n_{1}}{\ell}\right)\mathcal{P}^{\prime}(\mu_{j})=2(M-1)\mu_{j}\mathcal{P}^{\prime}(\mu_{j})-\mu_{j}^{2}\mathcal{P}^{\prime\prime}(\mu_{j})-\delta\mathcal{P}^{\prime\prime}(\mu_{j}),$
(23)

or equivalently

$(\mu_{j}^{2}+\delta)\mathcal{P}^{\prime\prime}(\mu_{j})+\left[(L-2M+2)\mu_{j}-\frac{2\pi
n_{1}}{\ell}L\right]\mathcal{P}^{\prime}(\mu_{j})=0\propto\mathcal{P}(\mu_{j}),\quad\forall
j=1,\cdots M.$ (24)

By matching the coefficient of the highest order, we obtain the following
ordinary differential equation that the polynomial $\mathcal{P}(\mu)$
satisfies,

$(\mu^{2}+\delta)\mathcal{P}^{\prime\prime}(\mu)+\left[(L-2M+2)\mu-\frac{2\pi
n_{1}}{\ell}L\right]\mathcal{P}^{\prime}(\mu)-M(L-M+1)\mathcal{P}(\mu)=0.$
(25)

The solutions to this ordinary differential equation can be expressed in terms
of the renowned generalised Jacobi polynomial $P_{K}^{(a,b)}(x)$. 222The
reason why it is called “generalised” is that parameters $a$ and $b$ are
complex in our case rather than $a,b>-1$ in the conventional case.

Generalised Jacobi polynomial satisfies the following ordinary differential
equation Szegő (1959),

$\begin{split}&(1-y^{2})P^{\prime\prime}(y)+[b-a-(a+b+2)y]P^{\prime}(y)+K(K+a+b+1)P(y)=0,\\\
&P(y)=P^{(a,b)}_{K}(y).\end{split}$ (26)

Through a simple change of variable $z=\mu/(i\sqrt{\delta})$, we have

$(z^{2}-1)\mathsf{P}^{\prime\prime}(z)+\left[(L-2M+2)z-\frac{2\pi
n_{1}}{i\sqrt{\delta}\ell}L\right]\mathsf{P}^{\prime}(z)-M(L-M+1)\mathsf{P}(z)=0,$
(27)

with $\mathsf{P}(z)=\prod_{j=1}^{M}(z-z_{j})$. The following identification
can be found by matching the coefficients,

$a-b=-\frac{2\pi n_{1}}{i\sqrt{\delta}\ell}L,\quad a+b=L-2M,\quad K=M.$ (28)

Therefore, the (polynomial) solution to the original ordinary differential
equation is

$\mathcal{P}(\mu)\propto P_{M}^{a,b}\left(\frac{\mu}{i\sqrt{\delta}}\right).$
(29)

Therefore, the 1-cut solution to the asymptotic Bethe equation (12) coincide
with the zeros of the corresponding Jacobi polynomial divided by
$i\sqrt{\delta}$. One might notice that it is singular to take the isotropic
limit $\delta\to 0$. Indeed, it is not straightforward to take the limit
$a-b\to 0$ of the generalised Jacobi polynomial. The correct approach is to
start with asymptotic Bethe equation in terms of spectral parameter
$\zeta=1/\mu$ and take the isotropic limit directly so that the solution to it
can be expressed as zeros of Laguerre polynomial as shown in Shastry and Dhar
(2001). We demonstrate the procedure in Appendix A.

Figure 1: The comparison between solution to Eq. (29), the numerical solution
to a finite-size Bethe equation and the branch cut for $L=300$, $M=30$,
$n_{1}=1$. The figure is plotted in $\zeta=1/\mu$ variable. The blue dots
denote the numerical solution to Bethe equation (2). The black crosses are the
solution to Eq. (29) and the red dashed line is the location of the branch cut
in the limit $L\to\infty$.

### Remarks.

When the finite-size (quantum) correction (15) diverges, we need to take into
account the Bethe equations (2) non-perturbatively. This gives rise to the
phenomenon of condensate Miao  _et al._ (2020); Bargheer  _et al._ (2008),
i.e. Bethe roots forming a segment of logarithmic branch cut with constant
density. So long as the filling fraction $\nu=M/L$ is not too high, cf. Ref.
Miao  _et al._ (2020), we could safely omit the finite-size correction and our
result fits well with the numerical results.

### Continuous limit.

In the continuous limit, the semi-classical eigenstates that we describe using
the zeros of generalised Jacobi polynomial are the low-energy excitations of
ferromagnetic ground state with fixed magnetisation, as described in Ref. Miao
_et al._ (2020) in details. We could determine the location of the branch cut
numerically using method in Ref. Miao  _et al._ (2020), and the comparison
between Eq. (29), the numerical solution to a finite-size Bethe equation and
the branch cut is shown in Fig. 1. As we could observe in Fig. 1, the solution
to Eq. (29) can be considered to be a good approximation to the solution to
Bethe equation (2) as system size $L$ increases, and in the limit
$L\to\infty$, both of them converge to the contour of the branch cut.

##  IV Conclusion

We present an exact solution to the asymptotic Bethe equation to the weakly
anisotropic Heisenberg spin chain, i.e. 1-cut solution. In the continuous
limit 1-cut solutions become the quantised non-linear spin waves, as shown in
Ref. Miao  _et al._ (2020). The exact solution to asymptotic Bethe equation
(12) are the zeros of generalised Jacobi polynomial.

The result makes it possible to obtain the analytic expression of the contour
of spectral curve (i.e. the branch cut where the Bethe roots condense) using
the information of the distribution of zeros of generalised Jacobi polynomial
when taking the $L\to\infty$ limit, using similar method in Martínez-
Finkelshtein  _et al._ (2001). In comparison, at the moment the contour of the
branch cut is determined numerically. In addition, it would be interesting to
consider more generic scenarios, such as more than 1 branch cuts are present,
and the cases with condensate. However, to obtain exact solutions becomes more
difficult due to the interactions between cuts, and we leave it for future
investigations.

## Acknowledgements

The author thanks Oleksandr Gamayun and Enej Ilievski for helpful discussions
and collaborations. The author acknowledges the support from the European
Research Council under ERC Advanced grant 743032 dynamint.

##  Appendix A Isotropic limit

At isotropic limit ($\Delta=1$), the Bethe equation becomes

$\left(\frac{\lambda_{j}+\mathrm{i}/2}{\lambda_{j}-\mathrm{i}/2}\right)^{L}\prod_{k\neq
j}^{M}\frac{\lambda_{j}-\lambda_{k}-\mathrm{i}}{\lambda_{j}-\lambda_{k}+\mathrm{i}}=1.$
(30)

In order to obtain the asymptotic Bethe equation, we need to consider the
following scaling limit $x=\frac{\lambda\ell}{L}$, and after taking the
logarithmic form and taking the limit $L\to\infty$, we have

$\frac{\ell}{x_{j}}=2\pi n_{j}+\frac{2\ell}{L}\sum_{k\neq
j}\frac{1}{x_{j}-x_{k}},$ (31)

i.e. asymptotic Bethe equation in the isotropic limit.

First of all, we assume that we only cope with 1-cut solution, i.e. there is
only 1 mode number $n_{1}$. We introduce a new variable $X$,

$X=-\frac{2\pi n_{1}}{\ell}Lx,$ (32)

and define the following polynomial with zeros being the solution of
asymptotic Bethe equation 31,

$\mathsf{P}_{\rm iso}(X)=\prod_{j=1}^{M}\left(X-X_{j}\right).$ (33)

Using the method outlined in Section III, we obtain a differential equation
that $\mathsf{P}_{\rm iso}(X)$ satisfies,

$X\mathsf{P}_{\rm iso}^{\prime\prime}(X)+\left(-L-X\right)\mathsf{P}_{\rm
iso}^{\prime}(X)+M\mathsf{P}_{\rm iso}(X)=0.$ (34)

The solution can be expressed in terms of Laguerre polynomial
$\mathcal{L}^{b}_{a}(u)$ Szegő (1959); Dhar and Sriram Shastry (2000), i.e.

$\mathsf{P}_{\rm iso}(X)\propto\mathcal{L}_{M}^{-L-1}\left(X\right).$ (35)

In terms of $x$, we have

$\mathcal{P}_{\mathrm{iso}}(x)\propto\mathcal{L}_{M}^{-L-1}\left(-\frac{2\pi
n_{1}}{\ell}Lx\right),\quad\mathcal{P}_{\mathrm{iso}}(x)=\prod_{j=1}^{M}\left(x-x_{j}\right).$
(36)

When the filling fraction $\nu$ becomes larger, there is still the problem
with the emergence of condensate, cf. Ref. Bargheer  _et al._ (2008).

Since
$\sin\left[\eta\left(\frac{\vartheta}{\eta}+\frac{\mathrm{i}}{2}\right)\right]\simeq\eta\left(\lambda+\frac{\mathrm{i}}{2}\right)$
when $\eta$ is small, the spectral parameter $x$ can be considered as the
isotropic limit of $\zeta=1/\mu$, as explained in the Section III. We can
rewrite Eq. (12) in terms of $\zeta$, i.e.

$\frac{\ell}{\zeta_{j}}=2\pi n_{j}+\frac{2\ell}{L}\sum_{k\neq
j}^{M}\frac{1+\delta\zeta_{j}\zeta_{k}}{\zeta_{j}-\zeta_{k}},$ (37)

and taking the limit $\delta\to 0$, we recover the asymptotic Bethe equation
at isotropic limit (31).

## References

  * Gamayun  _et al._ (2019) O. Gamayun, Y. Miao, and E. Ilievski, Phys. Rev. B 99, 140301 (2019).
  * Misguich  _et al._ (2019) G. Misguich, N. Pavloff, and V. Pasquier, SciPost Phys. 7, 25 (2019).
  * Miao  _et al._ (2020) Y. Miao, E. Ilievski, and O. Gamayun, arXiv preprint arXiv:2010.07232 (2020).
  * Žnidarič (2011) M. Žnidarič, Phys. Rev. Lett. 106, 220601 (2011).
  * Ljubotina  _et al._ (2017) M. Ljubotina, M. Žnidarič, and T. Prosen, Nat. Comm. 8 (2017), 10.1038/ncomms16117.
  * Ilievski  _et al._ (2018) E. Ilievski, J. De Nardis, M. Medenjak, and T. Prosen, Phys. Rev. Lett. 121, 230602 (2018).
  * De Nardis  _et al._ (2019) J. De Nardis, M. Medenjak, C. Karrasch, and E. Ilievski, Phys. Rev. Lett. 123, 186601 (2019).
  * Gopalakrishnan and Vasseur (2019) S. Gopalakrishnan and R. Vasseur, Phys. Rev. Lett. 122, 127202 (2019).
  * De Nardis  _et al._ (2020a) J. De Nardis, M. Medenjak, C. Karrasch, and E. Ilievski, Phys. Rev. Lett. 124, 210605 (2020a).
  * De Nardis  _et al._ (2020b) J. De Nardis, S. Gopalakrishnan, E. Ilievski, and R. Vasseur, Phys. Rev. Lett. 125, 070601 (2020b).
  * Sutherland (1978) B. Sutherland, Rocky Mt. J. Math. 8, 413 (1978).
  * Sutherland (1995) B. Sutherland, Phys. Rev. Lett. 74, 816 (1995).
  * Dhar and Sriram Shastry (2000) A. Dhar and B. Sriram Shastry, Phys. Rev. Lett. 85, 2813 (2000).
  * Kazakov  _et al._ (2004) V. A. Kazakov, A. Marshakov, J. A. Minahan, and K. Zarembo, J. High Energy Phys. 2004, 024 (2004).
  * Bargheer  _et al._ (2008) T. Bargheer, N. Beisert, and N. Gromov, New J. Phys. 10, 103023 (2008).
  * Minahan and Zarembo (2003) J. A. Minahan and K. Zarembo, J. High Energy Phys. 2003, 013 (2003).
  * Kruczenski (2004) M. Kruczenski, Phys. Rev. Lett. 93, 161602 (2004).
  * Escobedo  _et al._ (2011) J. Escobedo, N. Gromov, A. Sever, and P. Vieira, J. High Energy Phys. 2011 (2011), 10.1007/jhep09(2011)028.
  * Gromov  _et al._ (2012) N. Gromov, A. Sever, and P. Vieira, J. High Energy Phys. 2012 (2012), 10.1007/jhep07(2012)044.
  * Kostov (2012a) I. Kostov, Phys. Rev. Lett. 108, 261604 (2012a).
  * Kostov (2012b) I. Kostov, J. Phys. A 45, 494018 (2012b).
  * Shastry and Dhar (2001) B. S. Shastry and A. Dhar, J. Phys. A 34, 6197 (2001).
  * Beisert  _et al._ (2005) N. Beisert, A. A. Tseytlin, and K. Zarembo, Nucl. Phys. B. 715, 190 (2005).
  * Martínez-Finkelshtein  _et al._ (2001) A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, J. Comput. Appl. Math. 133, 477 (2001).
  * Flassig  _et al._ (2016) D. Flassig, A. Franca, and A. Pritzel, Phys. Rev. A 93, 013627 (2016).
  * Piroli and Calabrese (2016) L. Piroli and P. Calabrese, Phys. Rev. A 94, 053620 (2016).
  * Szegő (1959) G. Szegő,  _Orthogonal Polynomials_ (American Mathematical Society, 1959).


analytic singular support of $(x,\omega)\mapsto Tu(x,-\omega)$. One
can also simply swap the variables $x$ and $\omega$, but this is at odds with the interpretation of $\R^n\times
S^{n-1}$ as the cosphere bundle over $\R^n$ with a natural CR
structure obtained by identifying $T^*\R^n$ and $\widetilde{\R^n}=\C^n$;
this “Grauert tube” vision of the FBI transform extends naturally to
manifolds with a real-analytic Riemannian structure
[16, 17, 28],
see also the semiclassical approach [34].

The Grauert tube approach is
essentially contained in our toolbox; the FBI transform as introduced
in [28], for instance, can be written as an
analytic Fourier Integral Operator whose
Lagrangian has similar properties to the flat case, and the definition
of the wave front set is equivalent to the one above, via analytic
charts. We will say more on this in Section <ref>.

§.§ Advanced properties of Fourier Integral Operators

With help of the FBI transform and the structure of Toeplitz
operators, we can prove that general Fourier Integral Operators push
the analytic wave front set of a distribution as expected, and also an
microlocal ellipticity result for pseudodifferential operators.

Let $a$ be a formal analytic symbol on $\R^n$. Then

\[
WF_a(u)\subset \{a_0=0\}\cup WF_a(Op(a)u).
\]

Let $(x,\omega)\in \R^n\times S^{n-1}$ and suppose that $TOp(a)u$ is
real-analytic near $(x,\omega)$ and $a_0$ is bounded away from $0$
near $(x,\omega)$. Modulo an analytic function, one has

\[
\]

where $b$ is a formal analytic amplitude such that $b_0$ is bounded
away from $0$ near $(x,\omega)$. By Proposition
<ref>, on a neighbourhood $V$ of $(x,\omega)$,
$b$ admits a formal inverse for the $I_{\psi}$ calculus. Letting $d$
denote the formal analytic amplitude of its inverse, given $U\Subset V$, one has, modulo an
analytic function,

\[
I_{\psi,U\times V,U}(d)(TOp(a)u) = I_{\psi,U}(d)I_{\psi}(b)Tu=u;
\]

now, by Proposition <ref>, since $TOp(a)u$ is
real-analytic on $V$, the left-hand side is real-analytic on $U$. This concludes the proof.

Let $(\phi,\Omega,U)$ satisfy (<ref>) and let
$a$ be a formal analytic amplitude on $\Omega$. Then

\[
WF_a(I_{\phi,\Omega,U}(a)u)\subset \{(x,\xi)\in T^*\R^{n_x}\setminus \{0\},
\exists (y,\xi)\in WF_a(u), (x,\xi,y,\eta)\in (\Lambda_{\phi})_{\R}\}.
\]

The operator $T\circ I_{\phi,\Omega,U}(a)\circ T^*$ is a Fourier
Integral Operator whose real locus of the Lagrangian projects on the
base onto

\[
\{(x,\omega,y,v)\in (\R^{n_x}\times S^{n_x-1}\times \R^{n_y}\times
S^{n_y-1})\cap (\Lambda_{\phi})_{\R}\};
\]

thus, we are left with the claim

\[
SS_a(I_{\phi,\Omega,U}(a)u)\subset \{x\in \R^{n_x}, \exists
\xi,\eta\neq 0, \exists y\in SS_a(u), (x,\xi,y,\eta)\in
\]

which was proved in Theorem <ref>.

Another important application of the FBI transform is the construction
of unitary Fourier Integral operators associated with arbitrary real
symplectic maps, also called “quantized contact transformations”.

Let $\kappa$ be a one-homogeneous symplectic transformation
between respective neighbourhoods of two open cones $U,V$ of $T^*\R^n$. Let $[U]=U\cap
(\R^n\times S^{n-1})$ and $[V]=V\cap (\R^n\times S^{n-1})$. Then there
exists $\hat{\kappa}:\mathcal{H}([V])\to \mathcal{H}([U])$ and
$\widehat{\kappa^{-1}}:\mathcal{H}([U])\to \mathcal{H}([V])$ such that
the following is true.

* for every formal analytic amplitude $a\in
S([V])$, there exists $b\in S([U])$, such
that, for every $U_0\Subset U$, there exists $V_0\Subset V$, a neighbourhood $\Omega$ of
$\diag([U_0])$ in $(\R^n\times S^{n-1})\times [U]\times (0,+\infty)$, a neighbourhood
$\Omega'$ of $\diag([V_0])$ in $(\R^n\times S^{n-1})\times
[V]\times (0,+\infty)$ with

\[
\hat{\kappa}I_{\psi,\Omega',[V_0]}(b)\widehat{\kappa^{-1}}=I_{\psi,\Omega,[U_0]}(a):\mathcal{H}([U])\to \mathcal{H}([U_0]),
\]

where $b_0=a_0\circ \kappa$; if $a$ is the amplitude of the
Szegő projector then so is $b$.

* for every formal analytic amplitude $b\in
S([U])$, there exists $a\in S([V])$ such
that, for every $V_0\Subset V$, there exists $U_0\Subset U$, a neighbourhood $\Omega$ of
$\diag([U_0])$ in $(\R^n\times S^{n-1})\times [U]\times (0,+\infty)$, a neighbourhood
$\Omega'$ of $\diag([V_0])$ in $(\R^n\times S^{n-1})\times [V]\times (0,+\infty)$with

\[
\widehat{\kappa^{-1}}I_{\psi,\Omega,[U_0]}(a)\hat{\kappa}=I_{\psi,\Omega',[V_0]}(b):\mathcal{H}([V])\to \mathcal{H}([V_0]),
\]

where $a_0=b_0\circ \kappa^{-1}$; if $b$ is the amplitude of the
Szegő projector then so is $a$.

* For every $U_0\Subset U$ and $V_0\Subset V$, the distribution
\[(x,y)\mapsto
\widehat{\kappa^{-1}}(x,y)-\overline{\hat{\kappa}(y,x)}\]
real-analytic on $V_0\times U_0$.

For convenience we drop the bracket notations and identify open sets of
$\R^n\times S^{n-1}$ with open cones in $T^*\R^n$. Thus $\kappa$ is
a contact transformation on $\R^n\times S^{n-1}$. We
let $\phi_{\kappa}$ be any function from a
neighbourhood $W$ of the graph of $\kappa$ in $V_1\times U_1$,
where $U_1\Subset U$ and $V_1\Subset V$, which is
holomorphic in $x_1-i\omega_1$ and in $x_2+i\omega_2$, and such that
\dist((x_1,\omega_1),\kappa(x_2,\omega_2))$. Such a
function exists thanks to Proposition
<ref>: it is a phase function for the
only Lagrangian containing the graph of $\kappa$.

We now set

\[
\widehat{\kappa_0}=I_{\psi_{\kappa},W,V_1}\1_{(x_1,\omega_1,x_2,\omega_2)\in
\]

and we similarly define $\widehat{\kappa_0^{-1}}$ based on

Given $a$ as in the claim, the operator
$\widehat{\kappa_0^{-1}}I_{\psi,\Omega,U_0}(a)\widehat{\kappa_0}$ is, by
Theorem <ref>, a Fourier integral operator with the
same Lagrangian as the Szegő projector (indeed, the real locus
of the Lagrangian is the diagonal, and this Lagrangian is
holomorphic for the skew CR structure on $(\R^n\times
S^{n-1})^2$). Therefore it is of the form $I_{\psi,\Omega',[V_0]}(b)$
for another analytic formal amplitude $b$ obtained from $a$ by stationary
phase. In particular, letting $S$ denote the Szegő projector,

\[
\widehat{\kappa_0^{-1}}\widehat{\kappa_0}=\widehat{\kappa_0^{-1}}S\widehat{\kappa_0}=I_{\psi,\Omega',V_1}(f),
\]

where $f_0\neq 0$ and $V_1$ is any relatively compact open set of
$V$. By Proposition <ref>, $f$ admits an
inverse for the product of analytic Fourier Integral Operators with
phase $\psi$, and therefore there exists a formal analytic amplitude $g$
on a neighbourhood of $V$, such that

\[
\mathcal{H}(V_1).
\]

We now set $\hat{\kappa}=\widehat{\kappa_0}I_{\psi,
\Omega',V_1}(\sqrt{g})$ and
$\widehat{\kappa^{-1}}=I_{\psi,\Omega',V_1}(\sqrt{g}^*)\widehat{\kappa_0^{-1}}$. In
this algebra,
the square root and adjoint of an amplitude are well-defined because, as
in the proof of
Proposition <ref>, the calculus of these
amplitudes is conjugated via $T$ to the symbol calclulus of
pseudo-differential operators. With this correction, we obtain the
last claim.

We also obtain that the second part of the claim holds with the quantifiers on $a$ and
$b$ exchanged (for every $a$, there exists $b$, …).

Given $b$, the operator

\[
\hat{\kappa}I_{\psi,\Omega',V_0}(b)\widehat{\kappa^{-1}}
\]

is also an analytic Fourier Integral Operator whose Lagrangian is
that of $\psi$, so that it is of the form
$I_{\psi,\Omega,U_0}(a)$, and we obtain the first part of the claim
with the quantifiers on $a$ and $b$ exchanged (for every $b$, there
exists $a$, …)

The particular case
$\hat{\kappa}S\widehat{\kappa^{-1}}$ is a projector, because we
already made sure that $\widehat{\kappa^{-1}}\hat{\kappa}$ acts as
identity; however it is also an invertible operator (because it is a
Fourier Integral Operator with phase $\psi$ and nonvanishing
principal amplitude). Therefore it is the identity.

To conclude, given $a\in S(V)$, one recovers $b$ so that the first
point of the proposition is true by letting $b$ be the amplitude of
$\widehat{\kappa^{-1}}I_{\psi,\Omega,U_0}(a)\hat{\kappa}$, and

To conclude this section,
we can go back to Remark
<ref>. Now that we have characterised the analytic wave front
set of the singular kernel of an analytic Fourier Integral Operator
(Proposition <ref>), we observe that these operators are
well-defined when acting on singularity hyperfunctions, that is, elements of
$\mathcal{F}_{\omega}(V):=\mathcal{O}'(\overline{V})/(\mathcal{O}'(\partial V)+ \mathcal{O}(\overline{V}))$.
Indeed, as in the smooth case the wave front set conditions (see
[23], Theorems 8.5.1 and 8.5.2) are
satisfied. Thus, in all of Section <ref>, we can now replace $\mathcal{F}$ with
$\mathcal{F}_\omega$, and $\mathcal{H}$ with
\ker_{\mathcal{O}'(\overline{U})}(\overline{\partial}_b)/(\mathcal{O}(\overline{U})+\mathcal{O}'(\partial
U))$. This replacement is crucial in the proof of Theorem <ref>.

§ THE SZEGŐ KERNEL AND TOEPLITZ OPERATORS FOR GENERAL
PSEUDOCONVEX OPEN SETS

§.§ Normal form for the $\overline{\partial}_b$ operator

The features of $\overline{\partial}_b$ on the boundary of a strongly
pseudoconvex manifold are the following: locally, it is a
vector-valued, degree $1$ analytic pseudodifferential operator
$(D_1,\ldots,D_d)$ on a $2d-1$-dimensional manifold, such that
$[D_j,D_k]=0$ for every $j\neq k$ and the matrix of principal symbols $\sigma(i[D_j,D_k^*])$
is either positive definite or negative definite on
$\Sigma=\{\sigma(D_j)=0\forall 1\leq j\leq d\}$.

All of this is also true of the following model operator $D_0$, acting
on $\R^{d-1}_x\times \R^{d}_y$:

\[
(D_0)_j=-i\frac{\partial}{\partial x_j}-x_j\frac{\partial}{\partial
\]

Our goal is to prove the following result attributed to Sato:
microlocally near any point of $\Sigma$, $\overline{\partial}_b$ is
conjugated with $D_0$ by an analytic Fourier Integral Operator (as
always, modulo
an operator continuously mapping $\mathcal{E}'$ into
real-analytic-functions). Here, as before, “microlocally” will mean
“locally after conjugation with the FBI transform”.

The first step (Proposition <ref>) is to solve the classical problem, and find a
symplectic transformation which maps the symbol of
$\overline{\partial}_b$ to the symbol of $D_0$, multiplied by an
invertible matrix; this was described in Section
<ref>. W move on to an exact conjugation in the
algebra of analytic formal symbols of pseudodifferential operators
(Proposition <ref>), and we conclude by realising
all requested transformations on the FBI side (Proposition <ref>).

Given a unitary Fourier integral operator $\hat{\kappa}$ which quantizes $\kappa$, the
principal symbol of $\widehat{\kappa^{-1}}\overline{\partial}_b\hat{\kappa}$ is precisely that of
$Op(C)D_0$. It remains to correct the subprincipal (degree $0$) remainder, which we can do by
conjugating with an elliptic, diagonal matrix-valued pseudodifferential operator

Let $r=(r_1,\ldots,r_d)$ be a formal analytic degree 0 vector-valued symbol in the neighbourhood of a point
where $\sigma(D_0)=0$. Then there exists a smaller neighbourhood of
this point and a vector of degree 0 elliptic symbols $a=(a_1,\ldots,a_d)$ such
that, on this neighbourhood,

\[
(\sigma(D_0)_j+r_j)\sharp a_j=a_j\sharp \sigma(D_0)_j \qquad
\forall 1\leq j\leq d.
\]

We first remove the order 0 term in $r$, then all other terms.

At order 0, the following equation links the principal symbol of $a_j$
to the principal symbol of $r_j$

\[
\partial_{x_j}a_j^0+ix_j\partial_{y_1}a_j^0-i\eta_1\partial_{\xi_j}a_j^0=ir_j^0a_j^0.
\]

This transport equation admits a solution, together with the Cauchy data $a_j^0|_{x_j=0}=1$, written as

\[
\]

Since $r_j^0$ is a real-analytic function, this solution is
well-defined for $x_j$ sufficiently close to $0$, and is again
real-analytic. Hence, we can solve for $a$ at principal
order. Thus

\[
(a^0_j)^{\sharp -1}\sharp (\sigma(D_0)_j+r_j)\sharp
\]
where $r_j'$ is a degree $-1$ analytic symbol.

Consider now the family

\[
[0,1]\ni t\mapsto \sigma(D_0)_j+tr_j'.
\]

We wish to find a degree $-1$ elliptic symbol $a_j'(t)$ such that

\begin{equation}\label{eq:rprimeaprime}
(\sigma(D_0)_j+tr_j')\sharp(1+a_j'(t))=(1+a_j'(t))\sharp (D_0)_j.
\end{equation}

We will find $a_j'(t)$ as a solution of the differential equation

\[\partial_ta_j'(t)=ib_j(t)\sharp (1+a_j'(t)).\]

First, differentiating (<ref>) with respect to $t$ yields

\[
\]

and we want to solve this equation for $b_j(t)$.

One has

\begin{equation}\label{eq:exact_conj_D0}
\end{equation}

From now on we drop the index $j$ and the parameter $t$ on $r'$ and $b$. We
write their full symbols as

\[
r'=\sum_{k\geq 1} r_k'\qquad \qquad b=\sum_{k\geq 1}b_k,
\]

where $b_k$ and $r_k'$ are homogeneous of degree $-k$.

The equation above yields the following family of equations on $b_k$:

\[
b-b\sharp r')_k.
\]

Note that the right-hand-side only involves $b_l$ for
$1\leq l\leq k-1$. Thus, we can find the $b_k$ by induction,
recursively solving a transport equation of the form

\[
\partial_{y_{1}}u=g,
\]

using again the method of characteristics: the solution (with Cauchy data
$u=0$ on $x_j=0$) is

\[
\int_0^{x_j}g(x-(1-t)e_j,y+i\tfrac{t^2}{2}e_1,\xi+i(1-t)\eta_1e_j,\eta)\dd t.
\]

We have to prove that the successive solutions $(b_k)_{k\in \N}$ form an
analytic symbol. It turns out that a convenient analytic class is
given by the infinite jet of the $S^{\rho,R}_m$ class at $x_j=0$; namely
we let

\[
\|a\|_{JS^{\rho,R}_m}=\sup_{\alpha,k}\frac{\sup_\Omega(|\nabla^{\alpha}a_k|)(1+|\alpha|+k)^m}{(|\alpha|+k)!\rho^{|\alpha|}R^k},
\]

\[
\Omega=\{|\eta_{1}|\in [\tfrac 14, \tfrac
\]

Since the product of symbols is an infinitesimal operation, we have
(see Proposition <ref>)

\[
\|(b\sharp r'-r'\sharp b)\|_{JS^{\rho,R}_m}\leq
24\|r'\|_{JS^{\frac{\rho}2, \frac R4}_m}\|b\|_{JS^{\rho,R}_m}.
\]

Suppose by induction that, for all $l\leq k-1$, one has

\[
\sup_{\Omega}\|\nabla^{\alpha}b_l\|\leq
\]

Then one has readily, for all $t\in [0,1]$, if $m\geq C(d)$,

\[
\sup_{\Omega}|\nabla^\alpha(r'_k+t(r'\sharp b - b \sharp r')_k)|\leq
\]

Denote $g_k=ir'_k+t(r'\sharp b - b \sharp r')_k$, and let

\[
\]

\[
\]

We just proved that

\[C(g_k,k)\leq \|r'\|_{JS^{\frac{\rho}2, \frac R4}_m}(1+24\sup_{ \ell
\leq k-1}C(b_\ell,\ell)).
\]

Let us now prove that

\[
C(b_k,k)\leq C(g_k,k).
\]

To this end, we write the transport equation

\[
\partial_{y_{1}}b_k=g_k
\]

in terms of the power series at $x_j=0$

\[
b_k=\sum_{n=1}^{+\infty} \frac{b_k^{(n)}}{n!}x_j^n \qquad \qquad g_k=\sum_{n=1}^{+\infty}
\frac{g_k^{(n)}}{n!}x_j^n.\]
One has, for all $n\geq 1$,

\begin{equation}\label{eq:transport}
\end{equation}

and for $n=0$,

\[
\]

Let us prove by induction on $n$ that

\[
\]

This is true at $n=0$. If it is true at $n-1$ and $n$, then there are
three terms in

\begin{equation}\label{eq:ctrl_b_g}
\end{equation}

First, one has directly

\begin{align*}
\frac{\left(1+\frac{1}{n+|\alpha|+k+1}\right)^m}{\rho(n+1+|\alpha|+k)}.
\end{align*}

If $\rho\geq 6\left(\frac{3}{2}\right)^m$, then we obtain

\[
\frac{1}{6}C(g_k,k)\frac{r^{(n+1+|\alpha|)}R^k(n+1+|\alpha|+k)!}{(n+|\alpha|+k+2)^m}.
\]

We can also directly control the last term in (<ref>) in the same way, using the
induction hypothesis:

\begin{align*}
\frac{\left(1+\frac{1}{n+|\alpha|+k+1}\right)^m}{\rho}\frac{n}{n+1+|\alpha|+k}.
\end{align*}

As previously, if $\rho\geq 6\left(\frac 32\right)^m$, we obtain

\[
\frac{1}{6}C(g_k,k)\frac{\rho^{(n+1+|\alpha|)}R^k(n+1+|\alpha|+k)!}{(n+|\alpha|+k+2)^m}.\]

In the second term of (<ref>), we have to distinguish between two cases,
depending on whether or not one of the differentials has hit
$\eta_1$. Thus

\[
\]

where $\gamma$ is the length $1$ polyindex corresponding to the
direction of $\eta_{1}$.

As before,

\begin{align*}
\frac{\left(1+\frac{1}{n+|\alpha|+k+1}\right)^m}{\rho}\frac{|\alpha|}{n+1+|\alpha|+k},
\end{align*}

and if $\rho\geq 6\left(\frac 32\right)^m$, we obtain

\[
\frac{1}{6}C(g_k,k)\frac{\rho^{(n+1+|\alpha|)}R^k(n+1+|\alpha|+k)!}{(n+|\alpha|+k+2)^m}.\]

The last term is

\[
\]

Recall that we are controlling derivatives on the set $\Omega$ where
$|\eta_1|\leq \frac 12$. We obtain

\[
\frac 12 C(g_k,k)\frac{\rho^{(n+1+|\alpha|)}R^k(n+1+|\alpha|+k)!}{(n+|\alpha|+k+2)^m}.\]

Altogether, the induction is complete, and we obtain that

\[
C(b_k,k)\leq C(g_k,k)\leq (24\max_{\ell\leq
k-1}C(b_{\ell},\ell)+1)\|r'\|_{JS^{\frac{\rho}2, \frac R4}_{m}},
\]

\[
m\geq C(d)\qquad \qquad \rho\geq 6\left(\frac 34\right)^m\qquad
\qquad R\geq 2^{d+1}\rho.
\]

It remains to find $\rho,R,m$ as above such that
$\|r'\|_{S^{\rho,R}_m}<\frac{1}{24}$. Since $r'$ is of degree $-1$, this can
be done by fixing $m$ and $\rho$, then choosing $R$ large enough.

Having found $b$, it remains to build $a'$ such that

\[
(1+a')(0)=0 \qquad \qquad \partial_t (1+a')(t) = b(t)\sharp (1+a')(t).
\]

We know that $b(t)\in JS^{\rho,R}_m$ for all $t\in [0,1]$, with uniformly
bounded norm. Thus, by Proposition <ref>, this Cauchy problem satisfies
the hypotheses of the Picard-Lindelöf theorem on the Banach space
$JS^{2\rho,4R}_m$. In particular, we can find a solution in this space,
and $a'(1)$ is the requested operator.

Grouping together Proposition <ref>,
Proposition <ref>, Proposition
<ref>, Theorem
<ref>, and Remark <ref>, we obtain the following quantum normal

Let $(z,\zeta)\in \Sigma^{\pm}$. Identify a neighbourhood of $z$ in
$X$ with a neighbourhood of $0$ in $\R^{2d-1}$. There exists an open
neighbourhood $U$ of $(z,\zeta)$ in $T^*\R^{2d-1}$ such that, for
every pair of open sets $z\in U_2\Subset U_1 \Subset U$, there

* open neighbourhoods $V_2\Subset V_1$ of $\{x=y=\xi=0,\eta=(\pm 1, 0,\ldots, 0)\}$ in
$T^*(\R^d_x\times \R^{d-1}_y)$

* an analytic Fourier Integral
Operator $\hat{\kappa}:\mathcal{H}_{\omega}(U_1)\to\mathcal{H}_{\omega}(V_1)$

* an analytic Fourier Integral Operator
$\widehat{\kappa^{-1}}:\mathcal{H}_{\omega}(V_2)\to \mathcal{H}_{\omega}(U_2)$

* an elliptic matrix-valued formal symbol
$C\in S(V_1)$

* a neighbourhood $\Omega$ of $\diag(V_1)\times
(0,+\infty)$ in $(\R^{2d-1}\times S^{2d-2})^2\times (0,+\infty)$

such that

\[
T\overline{\partial}_bT^*=\widehat{\kappa^{-1}}I_{\psi,\Omega,V_2}(C)TD_0T^*\widehat{\kappa} \qquad
\qquad S=\widehat{\kappa^{-1}}\hat{\kappa}
\]

as operators mapping $\mathcal{F}_{\omega}(U_1)$ into $\mathcal{F}_{\omega}(U_2)$.

Reciprocally, there exists an open neighbourhood $V$ of $\{x=y=\xi=0,\eta=(\pm 1, 0,\ldots, 0)\}$ in
$T^*(\R^d_x\times \R^{d-1}_y)$ such that, for every pair of open
sets $V_2\Subset V_1\Subset V$ containing this point, there exists

* open neighbourhoods $U_2\Subset U_1$ of $(z,\zeta)$ in

* an analytic Fourier Integral
Operator $\hat{\kappa}:\mathcal{H}_{\omega}(U_2)\to\mathcal{H}_{\omega}(V_2)$

* an analytic Fourier Integral Operator
$\widehat{\kappa^{-1}}:\mathcal{H}_{\omega}(V_1)\to \mathcal{H}_{\omega}(U_1)$

* an elliptic matrix-valued formal symbol
$C\in S(V_1)$

* a neighbourhood $\Omega$ of $\diag(V_1)\times
(0,+\infty)$ in $(\R^{2d-1}\times S^{2d-2})^2\times (0,+\infty)$

such that

\[
\widehat{\kappa}T\overline{\partial}_bT^*\widehat{\kappa^{-1}}=I_{\psi,\Omega,V_2}(C)TD_0T^* \qquad
\qquad S=\hat{\kappa}\widehat{\kappa^{-1}}
\]

as operators mapping $\mathcal{F}_{\omega}(V_1)$ into $\mathcal{F}_{\omega}(V_2)$.

§.§ The Szegő kernel parametrix and the algebra of Toeplitz operators

In this section, we broadly follow the method proposed in
[7] to obtain a parametrix for the
Szegő kernel starting with the normal form of Proposition

In this section, the index $_0$ refers to the “model” case
$\R^d\times S^{2d-1}$. In particular $S_0$ is the Szegő projector
of Proposition <ref>, and $\Sigma^{\pm}_0$ is the
positive, resp. negative, characteristic set of
$(\overline{\partial}_b)_0$, the boundary Cauchy-Riemann operator on
$\R^d\times S^{2d-1}$. By contrast, the notations $\overline{\partial}_b, S,
\Sigma^{\pm},$ refer to the boundary of a general, relatively compact, strongly
pseudo-convex open set whose boundary is denoted by $X$.

The Szegő projector
admits a (singular) integral kernel which is real-analytic away from
the diagonal. Given a defining function $\rho$
and its polarisation $\psi$, well-defined on a small neighbourhood
$\Omega$ of the diagonal in $X\times X$, there exists a realisation
of a formal analytic
amplitude $a$ on $X\times X\times (0,+\infty)$ (see Definitions
<ref>, <ref> and Proposition
<ref> for details about the nature of $a$) such that

\[
(x,y)\mapsto S(x,y)-\int_0^{+\infty}e^{it\psi(x,y)}a(x,y;t)\dd t
\]

is real-analytic.

The proof consists first in exhibiting a parametrix for
$S$ in the form of a Fourier Integral Operator as above, then
proving that this parametrix indeed approximates $S$ in the sense
that the difference is a continuous operator from $\mathcal{O}'(X)$
to $\mathcal{O}(X)$. This requires several steps, as the mere fact
that $S$ acts nicely on $\mathcal{O}'(X)$ requires a proof.

* Given $(z,\zeta)\in \Sigma^{\pm}$ and
$(z_0,\zeta_0)\in \Sigma^{\pm}_0$, applying Proposition
<ref> twice, we obtain a pair of Fourier integral
operators $\hat{\kappa}$ and $\widehat{\kappa^{-1}}$ between
neighbourhoods of $(z,\zeta)$ and $(z_0,\zeta_0)$ (such that
$\hat{\kappa}\widehat{\kappa^{-1}}$ and
$\widehat{\kappa^{-1}}\hat{\kappa}$ slightly decrease the definition
set and are equal to the restriction map) such that, for every
pair of small neighbourhoods $V_1\Subset V$ of $(z,\zeta)$ and every
pair of small neighbourhoods $U_1\Subset U$ of $(z_0,\zeta_0)$,

\begin{align*}
\hat{\kappa}T\overline{\partial}_bT^*\widehat{\kappa^{-1}}&=I_{\psi}(C)T(\overline{\partial}_b)_0T^*:\mathcal{F}_{\omega}(U)\to \mathcal{H}_{\omega}(U_1)\\
T\overline{\partial}_bT^*&=\widehat{\kappa^{-1}}I_{\psi}(C)T(\overline{\partial}_b)_0T^*\hat{\kappa}:\mathcal{F}(V)\to \mathcal{H}(V_1).
\end{align*}

We now present the following local candidate for $TST^*$:

\[
\mathcal{S}=\widehat{\kappa^{-1}}TS_0T^*\hat{\kappa}.
\]

One readily checks that

\[
\mathcal{H}_{\omega}(V_1).
\]

Moreover $\mathcal{S}$ is self-adjoint, because of the constraints on
$\widehat{\kappa}$ and $\widehat{\kappa^{-1}}$ in Proposition
<ref> (modified by Remark <ref>).

* From the definition of $\mathcal{S}$, we see that $T^*\mathcal{S}T$ is an elliptic Fourier Integral Operator
whose Lagrangian $\Lambda$ is a half-line bundle over its projection onto the
it is also a projector. When writing $T^*\mathcal{S}T$ in a
form given by Proposition <ref>, once the phase
is chosen, the fact that it is a self-adjoint projector determines
every term of the formal amplitude. Therefore different
pieces of $T^*\mathcal{S}T$ can be glued together into a global
Fourier Integral operator $\widetilde{S}$ on $X$; in fact, this Fourier Integral
Operator takes the form proposed in the statement,
with a formal analytic amplitude.

* Let us prove that $(\widetilde{S}-1)S$ is continuous from
$L^2(X)$ to $\mathcal{O}(X)$. Given $u\in L^2(X)$, the wave front of
$Su$ is included in $\Sigma$ by Proposition <ref> because
$\overline{\partial}Su=0$. Now,
given $(z,\zeta)\in \Sigma$ and small open neighbourhoods
$V_1\Subset V$, letting $v$ be the restriction of $TSu$ to $V$, we can apply the previous technique and
obtain that
as an element of $\mathcal{F}_{\omega}(V_1).$ By Proposition
<ref>, we deduce that
$\widehat{\kappa^{-1}}T(S_0-1)T^*\hat{\kappa}v=0$ as an element of
$\mathcal{F}_{\omega}(V_1)$; but this is equal to the restriction of
$(\widetilde{S}-1)S u$ to $V_1$.

* Let us similarly prove that $\widetilde{S}(1-S)$ is continuous
from $L^2(X)$ to $\mathcal{O}(X)$. By the results of Kohn [27] and Boutet de
Monvel [5], there exists
$F:L^2(X)\to L^2(X)$, such that
$I_{L^2}=S+F\overline{\partial}_b$. Therefore, as operators on
$L^2(X)$, one has

\[
\widetilde{S}=\widetilde{S}S+(F\overline{\partial}_b\widetilde{S})^*=\widetilde{S}S+\widetilde{S}(\overline{\partial}_b)^*F^*.
\]

As functions of the second variable, the phase and all the terms
of the formal amplitude of $\widetilde{S}$ belong to the kernel
of $\partial_b$, and therefore, integrating by parts, for every
$u\in L^2(X)$, $\widetilde{S}(\overline{\partial}_b)^*u\in

* Since both $\widetilde{S}S-S$ and
$\widetilde{S}-\widetilde{S}S$ are continuous from $L^2(X)$ to
$\mathcal{O}(X)$, we deduce that $\widetilde{S}-S$ is continuous from
$L^2(X)$ to $\mathcal{O}(X)$. In particular,
$S=\widetilde{S}+(S-\widetilde{S})$ is continuous from
$\mathcal{O}(X)$ to $\mathcal{O}(X)$, and therefore by duality $S$
acts continuously on $\mathcal{O}'(X)$.

* Using item 5, we can repeat the proof of item 3 starting with $u\in
\mathcal{O}'(X)$, and we now obtain that $(\widetilde{S}-1)S$ is
continuous from $\mathcal{O}'(X)$ to $\mathcal{O}(X)$.

* Using item 5 again, we obtain that, by duality, $S-\widetilde{S}$ is
continuous from $\mathcal{O}'(X)$ to $L^2(X)$, and therefore
is continuous from $\mathcal{O}'(X)$ to $\mathcal{O}(X)$. Then, by
item 6, both $\widetilde{S}S-S$ and its dual $S\widetilde{S}-S$
are continuous from $\mathcal{O}'(X)$ to
$\mathcal{O}(X)$. Therefore $S-\widetilde{S}$ is continuous from
$\mathcal{O}'(X)$ to $\mathcal{O}(X)$, and the proof is complete.

An interesting consequence of Theorem <ref> is the
analytic hypoellipticity of $\overline{\partial}_b$: if $u\in
\mathcal{O}'(X)$ is such that $\overline{\partial}_bu$ is
real-analytic on an open set $V$, then $(1-S)u$ is real-analytic on
$V$; in fact, we obtain the much stronger property

\[
WF_a((1-S)u)= WF_a(\overline{\partial}_bu).
\]

This seems to be a new result, interesting in its own right. Away from
the characteristic $\Sigma$ this property is of course the usual
ellipticity result (Proposition <ref>), and there are now
several proofs of microlocal analytic hypoellipticity on $\Sigma_-$ [37, 36, 33].

We are now able to generalise the construction of Section
<ref> to define the algebra of Toeplitz operators on

Let $S(X)$ denote the space of formal analytic
amplitudes on $X\times (0,+\infty)$. After a holomorphic extension,
these amplitudes lead to Fourier Integral Operators acting on
$\mathcal{F}_{\omega}(X)$, with singular kernel of the form

\[
\int_0^{+\infty}e^{-t\psi(x,y)}t^{n-1}\widetilde{a}(x,y;t)\dd
t\qquad \qquad a\in S(X).
\]

* These operators form an algebra under composition; the product law
takes the form

\[
(a\sharp b)_{k}=\sum_{n+\ell=0}^kB_n(a_{\ell},b_{k-n-\ell}),
\]

where $(B_n)_{n\in \N}$ is a sequence of bilinear differential
operator such that, for all $n\in \N$, $B_n$ is of total degree

* The principal symbol of $[I_{\psi}(a),I_{\psi}(b)]$ is $\{a,b\}$
(using the natural symplectic structure on $X\times (0,+\infty)$).

* If the degree of $a$ is $d$, then $I_{\psi}(a)$ continuously sends
$H^s(X)$ into $H^{s-d}(X)$ for every $s\in \R$.

* The Szegő
projector $S=I_{\psi}(s)$ is the unit for this algebra.

* Invertible
elements are exactly those for which the principal symbol $a_0$ is bounded
away from $0$.

* The space $S(X)$ is the union of Banach spaces on
which $a,b\mapsto a\sharp b$ is continuous.

Except for the last two items, all claims follow from Theorem
<ref>, Theorem <ref>, the usual
properties of Fourier Integral operators in smooth regularity, and
the results of [1]. In particular,
if $P$ is a (smooth or analytic) pseudodifferential operator on $X$,
then $SPS$ is a (smooth or analytic) Fourier Integral Operator with
phase $\psi$ and its principal symbol is the restriction of the
principal symbol of $P$ to $\Sigma_+$ (identified with
$X\times (0,+\infty)$ by the section $x\mapsto \partial \psi(x,x)$),
so we can apply [1], Chapter 2,
Corollary 2 to compute commutators at main order.

To prove invertibility of elliptic amplitudes and continuity of the
product in suitable spaces of formal analytic amplitudes, we use again Proposition
<ref>: the wave front set of these operators is
$\{(z,\zeta,z,\overline{\zeta}),(z,\zeta)\in \Sigma^+\}$ and
microlocally near any point of $\Sigma^+$, a Fourier Integral
Operators conjugates $I_{\psi}(a)$, where $a$ is elliptic, with an operator of the form of
Proposition <ref> with elliptic symbol. The
latter can be inverted, and conjugating back we obtain, locally, an
amplitude for the inverse. Since the composition rule is local,

One can quantize homogeneous contact transformations as in the flat
case; an important subclass of these transformations are fundamental
solutions to the Schrödinger equation for degree $1$ Toeplitz operators.

Given a one-homogeneous symplectic transformation $\kappa:X\times
(0,+\infty)\to X\times (0,+\infty)$, there
exists a neighbourhood $\Omega$ of the graph of $[\kappa]:X\to X$, a function
$\psi:\Omega\to \C$, CR-holomorphic in the first variable,
anti-CR-holomorphic in the second variable, such that
$\im(\psi)\equiv \dist(\cdot,{\rm Graph}([\kappa]))^2$. In particular,
the function $\Omega\times (0,+\infty)\ni (x,y,t)\mapsto t\psi(x,y)$
is a positive phase function, whose Lagrangian $\Lambda$ is CR-holomorphic in
the first variable, anti-CR-holomorphic in the second variable, and
such that $\pi(\Lambda_{\R})={\rm Graph}([\kappa])$.

We recover $\psi$ from its Lagrangian, which is unique by
Proposition <ref>. By design
$\Lambda_{\R}$ is a half-line bundle over its projection, and this
condition is open, so that $\Lambda$ satisfies the hypotheses of
Proposition <ref>. This concludes the proof.

§ A FEW APPLICATIONS

§.§ Grauert tubes, FBI transforms, and quantum propagators

Let $(M,g)$ be a
compact, real-analytic Riemannian manifold. The holomorphic extension
of the exponential map gives a natural isomorphism between small
neighbourhoods of $M$ in $\widetilde{M}$ and small neighbourhoods of
the zero section in $T^*M$.

There exists [16] a real-analytic
Kähler structure on a neighbourhood of the zero section in $T^*M$,
compatible with the complex structure (via the aforementioned
identification) and whose restriction to $M$ is $g$. The Grauert

\[
B_r=\{(x,\xi)\in T^*M, \|\xi\|_{g(x)}<r\}
\]

is then a strongly pseudoconvex open set for $r$ small, and its
boundary $X_r$ is a real-analytic submanifold of $T^*M$.

The Toeplitz operators on $X_r$ acquire a particular importance
because of the FBI transform (from $\mathcal{O}'(M)$ to $\mathcal{H}_{\omega}(X_r)$), defined as
follows: let $\Omega=\{(x,\xi,y)\in \overline{B_r}\times M, \dist(x,y)<r\}$; let
$\psi:\Omega\to \C$ be holomorphic in its
first factor and such that $\psi(x,\xi,x)=\tfrac{i}{2}(r-\|\xi\|_{g(x)}^2)$. This
function, when restricted to $(X_r\times M)\cap \Omega$, satisfies the following properties:

* $-C\dist(x,y)^2\leq \im(\psi)(x,\xi,y)\leq -c\dist(x,y)^2$ for
some $0<c<C$.

* $(\nabla_x,\nabla_\xi,\nabla_y)\psi(x,\xi,x)=(-\xi,-i\xi,\xi)$.

In particular, $X_r\times M\times (0,+\infty)\ni (x,\xi,y)\mapsto t\psi(x,\xi,y)$ is a positive phase function whose Lagrangian
$\Lambda$ is the only Lagrangian which is CR-holomorphic in the first
component and such that

\[
\Lambda_\R=\{(x,-t\omega,\omega,0,y,-t\omega),t\in \R,(x,\omega)\in
\]

We can introduce our first guess for the FBI transform:

\[
T_0(x,\xi,y)=\int e^{t\psi(x,\xi,y)}t^{\frac{n+1}{4}}\dd t.
\]

As in the flat case, the power of $t$ is chosen such that $T_0$ is
a bounded operator, with bounded inverse, between $L^2(M)$ and
$\ker_{L^2(X_r)}(\overline{\partial}_b)$; in other terms, $T_0^*T_0$ is an elliptic
pseudodifferential operator of degree $0$.

$T_0$ may not exactly be unitary, but we can remedy to this situation
in the following way: $T_0T_0^*$ has the same Lagrangian as $S$, and
therefore it is of the form $I_{\psi}(t_0)$ for some self-adjoint amplitude $t_0$;
there exists an inverse square root $r$ for $t_0$ (because the calculus of
these Toeplitz operators is, locally, conjugated to that of
pseudodifferential operators on flat space), and therefore, letting

\[
\]

then $T$ is a unitary transformation between $L^2(M)$ and
$\ker_{L^2(X_r)}(\overline{\partial}_b)$. Useful properties of $T$ are
as follows:

* If $A$ is an analytic pseudodifferential operator on $M$
, then
$TAT^*$ is a Toeplitz operator with principal symbol $(x,\omega,t)\mapsto
\sigma(A)(x,-t\omega)$.

* More generally, if $A$ is an analytic Fourier Integral operator
on $M$, with Lagrangian $\Lambda_A$, then $TAT^*$ is a Fourier
integral operator whose Lagrangian $\Lambda_{TAT^*}$ is a
complex Lagrangian, CR-holomorphic in the first factor,
anti-CR-holomorphic in the second factor, and such that

\begin{align*}
(\Lambda_A)_\R, \\ &t_1>0, t_2>0, \\ &(x_1,\omega_1)\in S^*M, \\ &(x_2,\tfrac{t_2}{t_1}\omega_2)\in S^*M\}.
\end{align*}

Existence and uniqueness of this Lagrangian are guaranteed by
Proposition <ref>. In particular, in
this case one has $\Sigma_+\simeq T^*M\setminus \{0\}$.

We can use this transformation to describe quantum
propagators as analytic Fourier Integral Operators.

Let $P$ be an analytic self-adjoint degree 1 pseudodifferential operator on
$M$. For every $s\in \R$, the fundamental solution $e^{-isP}$ of the Schrödinger equation
for $P$ is an analytic Fourier Integral Operator whose Lagrangian is
the graph of the bicharacteristic flow $\phi$ of $P$ at time $s$. In particular,
$Te^{-isP}T^*$ is a Fourier Integral operator with a singular kernel of the

\[
(x,y)\mapsto \int_0^{+\infty}e^{it\psi(s,x,y)}a(s,x,y;t)\dd t.
\]

Here, $\psi(s,x,y)$ is CR-holomorphic with respect to $x$,
anti-CR-holomorphic with respect to $y$, and satisfies $\im(\psi(x,y))\asymp
-\dist(y,\phi_s(x))$; moreover $a$ is the realisation of a formal analytic amplitude.

We will prove the claimed description of $Te^{-isP}T^*$, from which
the more abstract fact that $e^{-isP}=T^*(Te^{-isP}T^*)T$ is a
Fourier Integral Operator will follow.

By Proposition <ref>, a pseudodifferential
operator $Q$ on $X$ is such that $[Q,S]$ and
$SQS-TPT^*$ are continuous from $L^2$ to $L^2$, if and only if the
principal symbol of $Q$, near $\Sigma_+$ is the CR-holomorphic extension of the
principal symbol of $P$ (seen as a real-analytic function on the
totally real manifold $\Sigma_+$).

We know from the $C^{\infty}$ theory that $e^{-isQ}$ is, up to a
smooth remainder, a Fourier
Integral Operator whose Lagrangian is the graph of the
bicharacteristic flow of the principal symbol of $Q$. Since the
principal symbol Poisson-commutes with that of
$(\overline{\partial}_b,\partial_b)$, the (smooth) Fourier Integral
Operator $\Pi e^{-isQ}\Pi$ has exactly the requested
Lagrangian. Letting $U_0(s)$ be any elliptic degree 0 analytic Fourier Integral Operator
with the same Lagrangian (obtained by Proposition
<ref>, for instance), the principal (order 1) symbols of
$\frac{\dd}{\dd s}U_0(s)$ and $iU_0(s)TPT^*$ coincide
(alternatively, one can compute principal symbols to show
this). The phase of this Fourier Integral Operator is the same as
that of $U_0(s)$.

The operator $R(s)=U_0(s)e^{-isTPT^*}$ then solves the equation

\[
\frac{\dd}{\dd s}R(s)=\left[\tfrac{\dd}{\dd
\]

Introducing $V_0(s)$ an inverse for $U_0(s)$ (we already have an
inverse if we applied Proposition <ref>),
modulo a real-analytic remainder,

\begin{equation}\label{eq:Rs}
\frac{\dd}{\dd s}R(s)=\left[\tfrac{\dd}{\dd
\end{equation}

The right-hand side is a degree $0$ analytic Fourier Integral Operator with
the same phase as $\Pi$; it is a degree $0$ Toeplitz operator in the
sense of Proposition <ref>. Choosing a Banach
norm of formal analytic amplitudes which contains the total symbol of $\left[\tfrac{\dd}{\dd
s}U_0(s)-iU_0(s)TPT^*\right]V_0(s)$ and on which the formal
product is continuous, one can give an approximate solution for
(<ref>) by the Picard-Lindelöf
theorem; thus there exists  an analytic
Toeplitz operator $\widetilde{R(s)}$ such that, modulo an analytic remainder,

\[
\frac{\dd}{\dd s}\widetilde{R(s)}=\left[\tfrac{\dd}{\dd
\]

To conclude, we can apply the Duhamel formula (the true propagator
$e^{isTPT^*}$ preserves real-analytic functions) and we obtain that,
modulo an analytic remainder,

\[
\]

This proves the second part of the claim, and conjugating back, we
obtain that
$T^*\widetilde{R(s)}^{-1}V_0(s)T$ is a Fourier Integral Operator
which coincides with $e^{isP}$ up to an analytic remainder.

§.§ From microlocal to semiclassical analysis

In this section we broadly describe how one can obtain semiclassical results from
the description above.

In what follows, we let $X$ be the boundary of a compact, strongly pseudoconvex open set and suppose that
$X$ admits an $S^1$ action $(r_{\theta})_{\theta\in S^1}$ which preserves all the structure. Two
important examples are:

* The case where $X$ is the Grauert tube around a compact
Riemannian manifold of the form $S^1\times M$. The action of $S^1$
on $S^1\times M$ (by rotation of the first factor) is an isometry
and lifts to a symplectic transformation on
$T^*(S^1\times M)$, which is also an isometry for the Kähler
structure near the zero section.

* The case where $X$ is the circle bundle of the dual of a
positive line bundle over a Kähler manifold.

Among Toeplitz operators, those whose formal amplitude is itself invariant
under the $S^1$ action commute with the $S^1$ action, and thus one can
decompose their action over the eigenmodes $\mathcal{H}_k,k\in \Z$ of
the $S^1$ action. Changing variables, the singular kernel of a
Toeplitz operator $I_{\psi}(a)$ restricted to $\mathcal{H}_k$ is then

\[
\theta \dd s.
\]

To perform a stationary phase in the variables $(s,\theta)$, we need
to satisfy the condition

\[
\alpha(V)\neq 0
\]

where $\alpha$ is the contact 1-form on $X$ of Proposition
<ref> and $V$ is the tangent vector field
of the $S^1$ action. In case 2 above, this is always true because the
Reeb flow coincides with the $S^1$ action (in fact one can choose
$\psi$ with simple and explicit dependence on $\theta$, see for
instance the unlabelled equation following (24) in [41]). In case 1, the Reeb flow is
the geodesic flow on $S^1\times M$, so we restrict our attention to
the open set

\[
X_+=\{(y,\eta,x,\xi)\in S^*(S^1\times M), \eta>0\}.
\]

Thus, in both cases, we are in position to apply the stationary phase lemma in the
variables $(s,\theta)$ (see for instance [40]),
and obtain a description modulo errors of size $e^{-ck}$ for some
$c>0$ (indeed, elements of analytic function spaces have exponentially
fast decaying Fourier modes). In case 2 above, we obtain semiclassical
Berezin–Toeplitz operators; in case 1, after a stereographic change
of variables $X_+\to S^1\times T^*M$, we obtain semiclassical
pseudodifferential operators.

With this point of view, one can for instance recover the expressions
of the semiclassical Szegő kernel, and many properties of
semiclassical analytic pseudodifferential operators after conjugating
back by a semiclassical Bargmann transform, since we know that the
Toeplitz and pseudodifferential algebras are equivalent in the
analytic semiclassical case [30].

This procedure can be generalised to other “small parameter”
techniques, notably to study the action of (microlocal) analytic
Fourier Integral Operators on WKB states. We
hope to describe this in future work.

§ THE ALGEBRA OF SYMBOLS OF PSEUDODIFFERENTIAL OPERATORS

Define the following formal analytic symbol norm on $\R^{d}_x\times \R^d_{\xi}$, for $m\in \R, \rho>0,R>0$:

\[
\|a\|_{S^{\rho,R}_m}=\sup_{k\in \N, \alpha\in \N^{2d}, (x,\xi)\in
\R^{2d}}\frac{|\nabla^{\alpha}a_k(x,\xi)|(1+k+|\alpha|)^m}{\rho^{|\alpha|}R^k(|\alpha|+k)!}.
\]

For the star-product of left-quantization on $T^*\R^d$, there exists
$m_0(d)$ such that

\[
\|a\sharp b\|_{S^{\rho,R}_m}\leq
12\|a\|_{S^{\rho,R}_m}\|b\|_{S^{\frac{\rho}2,\frac R4}_m}
\]

\[
R\geq 2^{d+2}\rho^2\qquad \qquad m\geq m_0(d).
\]

We start with the formula

\[
\]

\[
\sum_{|\beta|=n}\frac{n!}{\beta!}=n^d\leq 2^{nd}
\]

we obtain that, for every $\alpha$ with $|\alpha|=j$,

\begin{multline*}
|\nabla^{\alpha}(a\sharp b)_k|\leq
\|a\|_{S^{\rho,R}_m}\|b\|_{S^{\frac{\rho}2,\frac
R4}_m}\frac{\rho^jR^k(j+k)!}{(j+k+1)^m}\times \\
\sum_{n=0}^k\left(\frac{2^{d+1}\rho}{R}\right)^n\sum_{l=0}^{n-k}\sum_{j_1=0}^j\sum_{\substack{|\gamma|=j_1\\
\gamma\leq
\alpha}}\frac{(n+j_1+l)!(k-l+j-j_1)!j!}{4^{k-l}2^{j-j_1-n}n!(k+j)!j_1!(j-j_1)!}\left(\frac{1+j+k}{(1+n+j_1+l)(1+k-l+j-j_1)}\right)^m.
\end{multline*}

Let us prove that, if $0\leq j_1\leq j$ and $0\leq l+n\leq k$, then

\[
\frac{(n+j_1+l)!(k-l+j-j_1)!j!}{2^{k-l+j-j_1}n!(k+j)!j_1!(j-j_1)!}\leq
\]

If the other parameters are fixed, then

\[
\frac{(n+j_1+l)!(k-l+j-j_1)!j!}{2^{k-l+j-j_1}n!(k+j)!j_1!(j-j_1)!}\]

is log-convex; there are two extremal points.

At $l=0$, we obtain

\[
\frac{(n+j_1)!(k+j-j_1)!j!}{2^{k+j-j_1}n!(k+j)!j_1!(j-j_1)!}\leq
\frac{(n+j_1)!j!k!}{n!(k+j)!j_1!}.
\]

This increasing function of $j_1$ reaches a maximum at $j_1=j$,
where we obtain

\[
\frac{(n+j)!k!}{n!(k+j)!}\leq 1
\]

since $n\leq k$.

In the other case $l=k-n$, we obtain

\[
\frac{(k+j_1)!(n+j-j_1)!j!}{2^{n+j-j_1}n!(k+j)!j_1!(j-j_1)!}\leq
\frac{(k+j_1)!j!}{(k+j)!j_1!}\leq 1
\]

since $j_1\leq j$.  It remains to bound, for fixed $l,n$ such that
$l+n\leq k$,

\[
\sum_{j_1=0}^j\sum_{\substack{|\gamma|=j_1\\ \gamma\leq
\alpha}}\left(\frac{1+j+k}{(1+n+j_1+l)(1+k-l+j-j_1)}\right)^m
\]

To this end we use the fact that

\[
\#(|\gamma|=j_1,\gamma\leq \alpha)\leq \min(j_1,(j-j_1))^{d-1};
\]

thus, this sum is bounded by

\[
\sum_{j_1=0}^j\frac{(1+j+n+k)^m\min(j_1,j-j_1)^{d-1}}{(1+n+j_1+l)^m(1+k-l+j-j_1)^m}.
\]

Now we use [11], Lemma 2.13; for $m$ large
enough (depending only on $d$), this sum is smaller than $3$.  To

\[
\sum_{l=0}^{n-k}\frac{1}{2^{k-n-l}}\leq 2
\]

and, if $R\geq 2^{d+2}\rho^2$,

\[
\sum_{n=0}^k\left(\frac{2^{d+1}\rho^2}{R}\right)^n\leq 2.
\]

§ FUNCTIONAL SPACES IN ANALYTIC REGULARITY

This section collects the basic properties of the spaces of
analytic functions and their duals.
Most technical facts claimed here are proved in
[20]; see also Chapter 6 of [38].

Let $\Omega\subset \C^n$ (or any complex paracompact manifold) be an
open set. We define $\mathcal{O}(\Omega)$ as the space of
holomorphic functions from $\Omega$ to $\C$, endowed with the
topology of local uniform convergence.

The topology on $\mathcal{O}'(\Omega)$ coincides with that of compactly
supported Radon measures, of which it is a closed subspace. Elements
of $\mathcal{O}'(\Omega)$ will be called analytic functionals.

One can generalise this definition into “germs” of holomorphic
functions and their duals.

Let $E\subset \C^n$ (or any complex paracompact manifold) be an open
set. Define $\mathcal{O}(E)$ as the colimit of the
spaces $\mathcal{O}(\Omega)$ for all $\Omega$ open and containing
$E$ (that is to say, an element of $\mathcal{O}(E)$ is
a holomorphic function on some open set containing $E$).

If $E$ is an open subset or $\R^n$, or more generally an open subset
of a real-analytic submanifold of $\R^n$, then $\mathcal{O}(E)$ is the
space of real-analytic functions on $E$. If $E$ is a compact subspace
of $\R^n$ with non-empty interior, then $\mathcal{O}(E)$ is the space
of functions on $E$ which are real-analytic up to the boundary (and, therefore, which
extend into real-analytic functions on some open neighbourhood of

Given two sets $E\subset F\subset \C^n$ one can naturally define a
restriction map from
$\mathcal{O}(F)$ to $\mathcal{O}(E)$. By duality, this defines a
natural map $\mathcal{O}'(E)\to \mathcal{O}'(F)$. This map can only be
injective when the restriction $\mathcal{O}(F)\to \mathcal{O}(E)$ has
dense image; this is called the Runge property. We will only be
interested in the case where $E$ is a compact subset of $\R^n$, in
which case the Runge property is always satisfied, by elementary
approximation theory.

Theorem 6.2.14) Let $K$ be a compact subset of $\R^n$. Then entire functions
are dense in $\mathcal{O}(K)$; that is to say, $K$ is Runge.

The proof roughly consists in showing that, with
$\rho_N:x\mapsto=c_nN^{-n}\exp(-N|x|^2)$, given $f\in \mathcal{O}(K)$
and $g$ any smooth extension of $f$ to $\R^n$, the sequence of entire
functions $(\rho_N*g)_{N\in \N}$ converges towards $f$ in the topology
of $\mathcal{O}(K)$.

One of the main useful properties of Runge sets is the fact that one
can solve the $\overline{\partial}$ problem on them.
([20], Theorem 2.7.8) Let
$K\Subset \C^n$ be compact and Runge. Then there exists a basis of
neighbourhoods $(\Omega_j)_{j\in \N}$ of $K$ in $\C^n$ such that the
following is true. For every $j\in \N$ and every
$f=(f_1,\ldots,f_n)\in C^1(\Omega_j,\C^n)$ which is
$\overline{\partial}$-closed, in the sense that

\[
\overline{\partial}_{k}f_\ell=\overline{\partial}_{\ell}f_k\qquad
\forall 1\leq k,\ell\leq n,
\]

there exists $u\in C^1(\Omega_j,\C)$ such that $\overline{\partial}u=f$.

The ability to prove the $\overline{\partial}$-problem on some
neighbourhoods of real compact sets allow us to describe how the
spaces $\mathcal{O}$ and $\mathcal{O}'$ behave under natural
opeartions on their domains.

Let $K_1,K_2$ be compact subsets of $\R^n$.

* $\mathcal{O}(K_1)\cap \mathcal{O}(K_2)=\mathcal{O}(K_1\cup

* $\mathcal{O}(K_1)+ \mathcal{O}(K_2)=\mathcal{O}(K_1\cap

* $\mathcal{O}(K_1)'\cap \mathcal{O}'(K_2)=\mathcal{O}'(K_1\cap

* $\mathcal{O}(K_1)'+ \mathcal{O}'(K_2)=\mathcal{O}'(K_1\cup

* Let $U_1,U_2$ be respective neighbourhoods of $K_1$ and $K_2$
in $\C^n$. Then holomorphic functions on $U_1\cup U_2$ are exactly
functions that are both holomorphic on $U_1$ and on $U_2$, because
holomorphy is a local property.

* The
inclusion from left to right is obvious and it remains to prove,
given $f\in \mathcal{O}(K_1\cap K_2)$, that it is of the form
$f_1+f_2$, where $f_1\in \mathcal{O}(K_1)$ and $f_2\in
\mathcal{O}(K_2)$.

Let $\Omega_1$ and $\Omega_2$ be small neighbourhoods in $\C^n$ of
respectively $K_1$ and $K_2$, so that $f$ extends to
$\widetilde{f}\in \mathcal{O}(\Omega_1\cap \Omega_2)$ (the function
$\widetilde{f}$ may or may not correspond to the usual notion of
holomorphic extension, depending on whether $K$ is the closure of
its interior in $\R^n$).

The compact sets $K_1\setminus \Omega_2$ and $K_2\setminus \Omega_1$
do not intersect and therefore lie at positive distance from each
other. We let $\chi\in C^{\infty}(\Omega_1\cup \Omega_2,\R)$ be such
that $\chi=1$ on a neighbourhood $U_1$ of $K_1\setminus
\Omega_2$ and $\chi=0$ on a neighbourhood $U_2$ of $K_2\setminus
\Omega_1$. The one-form
$\alpha=\overline{\partial}(\widetilde{f}\chi)$, well-defined on
$\Omega_1\cap \Omega_2$, is equal to $0$ on $(U_1\cup U_2)\cap
(\Omega_1\cap \Omega_2)$, and therefore can be smoothly extended by
$0$ on $\Omega=U_1\cup U_2\cup (\Omega_1\cap \Omega_2)$.

$\Omega$ is a neighbourhood of $K_1\cup K_2$, which is a Runge set
by Proposition <ref>;
by Proposition <ref>, this means that there
exists a smaller neighbourhood $\Omega'\subset \Omega$ of $K_1\cup
K_2$ and $u\in
C^{\infty}(\Omega')$ such that $\overline{\partial}u=\alpha$.

To conclude, the function $h_1=(1-\chi)\widetilde{f}+u$ is
well-defined on $(U_1\cup (\Omega_1\cap \Omega_2))\cap \Omega'$,
which is an open neighbourhood of $K_1$; it satisfies
$\overline{\partial}h_1=\alpha-\overline{\partial}u=0$; therefore
$h_1\in \mathcal{O}(K_1)$. In the same way,
$h_2=\chi\widetilde{f}-u$ is well-defined and holomorphic on
$(U_1\cup (\Omega_1\cap \Omega_2))\cap \Omega'$ which is an open
neighbourhood of $K_2$; moreover, on $K_1\cap K_2$, one has
$h_1+h_2=f$; this concludes this part of the proof.

* This statement is the dual of item 2.

* This statement is the dual of item 1.

Given two compacts $K_1\subset K_2$ of $\R^n$, there is no natural
restriction map from $\mathcal{O}'(K_2)$ to $\mathcal{O}'(K_1)$ (in
fact, we have a natural injective extension map from $\mathcal{O}'(K_1)$
to $\mathcal{O}'(K_2)$). Nevertheless, there is a well-defined notion
of support of an analytic functional, thanks to the injectivity
of this extension map and item 3 of
Proposition <ref>.

Let us remark that if we define $C^{\infty}(K)=\mathcal{E}(K)$, for $K$ any compact
of $\R^n$, following Whitney
[39], then the equivalents of all items of
Proposition <ref> are also true, with
the small caveat that the dense maps $\mathcal{E}(\R^n)\to
\mathcal{E}(K)$ fail to be injective. We will use the following
restriction theorem on locally compact distributions: they can be
restricted to a smaller subset modulo a boundary indeterminacy.

Let $K_1\subset K$ be two compact sets. The inclusion map
$\mathcal{E}'(K_1)\to \mathcal{E}'(K)$ has a left inverse modulo
$\mathcal{E}'(\partial K_1)$. This right inverse does not depend on
adding an element of $\mathcal{E}'(\partial K)$, and therefore
defines a well-defined map

\[
\rho_{K,K_1}:\mathcal{E}'(K)/\mathcal{E}'(\partial K)\to
\mathcal{E}'(K_1)/\mathcal{E}'(\partial K_1).\]

Let $K_2=\overline{K\setminus K_1}$. Recall that
$\mathcal{E}'(K)=\mathcal{E}'(K_1)+\mathcal{E}'(K_2)$, and given
$u\in \mathcal{E}'(K)$, choose $u_1\in \mathcal{E}'(K_1)$ such that
$u-u_1\in \mathcal{E}'(K_2)$. The distribution $u_1$ is not unique,
but given any other choice $v_1$, the difference $u_1-v_1\in
\mathcal{E}'(K_1)$ is such that $(u-u_1)-(u-v_1)=v_1-u_1\in
\mathcal{E}'(K_2)$. Therefore $u_1-v_1\in \mathcal{E}'(K_2\cap
K_1)=\mathcal{E}'(\partial K_1)$, and the class of $u_1$ modulo
$\mathcal{E}'(\partial K_1)$ is uniquely defined. If one had $u
in \mathcal{E}'(K_1)$ to begin with, then one can choose $u_1=u$, so
that this map is indeed a left inverse to the extension map.

If $u\in \mathcal{E}'(\partial K)$, then, since $\partial K\subset
(\partial K_1)\cup K_2$, one can choose $u_1\in
\mathcal{E}'(\partial K_1)$
in the lines above, and therefore $u$ is mapped to $0$; this
concludes the proof.

The same property holds when $\mathcal{E}'$ is replaced with
The two families of spaces $\mathcal{E}'(K)/\mathcal{E}'(\partial K)$ and
$\mathcal{O}'(K)/\mathcal{O}'(\partial K)$ are quite
practical because of this well-defined restriction map, which mimics
the ability to restrict analytic or smooth functions to a smaller
set. Elements of the spaces $\mathcal{O}'(K)/\mathcal{O}'(\partial K)$ are
called hyperfunctions. Proposition <ref> and its
equivalent for $\mathcal{O}'$ means that these families of spaces (indexed by $\mathring{K}$) form pre-sheaves.

Let $K\Subset \R^n$ and let $f\in \mathcal{O}'(K)$. The support of
$f$ is the smallest compact $K_1\subset K$ such that $f\in \mathcal{O}'(K_1)$.

This notion is well-defined: first of all “$f\in \mathcal{O}'(K_1)$”
makes sense because the extension $\mathcal{O}'(K_1)\to
\mathcal{O}'(K)$ is injective; second, if two compacts $K_1$ and $K_2$
are such that $f\in \mathcal{O}'(K_1)\cap \mathcal{O}'(K_2)$, then
$f\in \mathcal{O}'(K_1\cap K_2)$ by item 3 of
Proposition <ref>.

If $U$ is a relatively compact open set in $\R^n$ then
$\mathcal{O}(\overline{U})\subset C^{\infty}(\overline{U})$ and
therefore $\mathcal{E}'(\overline{U})\subset
\mathcal{O}'(\overline{U})$. In this case, the support of a
distribution $T\in \mathcal{E}'(\overline{U})$ coincides with its
support as an element of $\mathcal{O}'(\overline{U})$.

We now reach the main result of this appendix: one can patch together
analytic functionals defined on different compact sets which agree on
the intersection (meaning that the support of their difference lies
away from the intersection), and if they only agree on the
intersection modulo a
real-analytic function then we can patch them together modulo a
real-analytic function.

Let $K_1,K_2$ be compact subsets of $\R^n$. Let $u_1\in
\mathcal{O}'(K_1)$ and $u_2\in \mathcal{O}'(K_2)$ be such that ${\rm
supp}(u_1-u_2)\cap K_1\cap K_2=\emptyset$. Then there exists $u\in
\mathcal{O}'(K_1\cup K_2)$ such that ${\rm supp}(u-u_1)\cap K_1=\emptyset$
and ${\rm supp}(u-u_2)\cap K_2=\emptyset$. If $u_1,u_2$ belong to
$\mathcal{E}'$, then so does $u$.

More generally, if there exists $f\in \mathcal{O}(K_1\cap K_2)$ such
that ${\rm
supp}(u_1-u_2-f)\cap K_1\cap K_2=\emptyset$, then there exists
$u\in \mathcal{O}'(K_1\cup K_2)$, $g_1\in \mathcal{O}(K_1)$ and
$g_2\in \mathcal{O}(K_2)$ such that ${\rm supp}(u-u_1-g_1)\cap
K_1=\emptyset$ and ${\rm supp}(u-u_2-g_2)\cap
K_2=\emptyset$. Again if $u_1,u_2$ belong to $\mathcal{E}'$, then so
does $u$.

By hypothesis, there exists two compacts  $L_1\in K_1\setminus K_2$ and $L_2\in
K_2\setminus K_1$ such that $u_1-u_2\in \mathcal{O}'(L_1\cup
L_2)$. Using Proposition <ref>, there
exists $f_1\in \mathcal{O}'(L_1)$ and $f_2\in \mathcal{O}'(L_2)$
such that $u_1-u_2=f_1+f_2$.

Then $u=u_1-f_2=u_2+f_1$ is an element of $\mathcal{O}'(K_1\cup K_2)$
such that $u-u_1=f_2\in \mathcal{O}'(L_2)$ and $u-u_2=f_1\in
\mathcal{O}'(L_1)$. This concludes this part of the proof, and we can
seamlessly replace $\mathcal{O}'$ with $\mathcal{E}'$ in the lines above.

If now $u_1-u_2\in \mathcal{O}'(L_1\cup
L_2)+\mathcal{O}(K_1\cup K_2)$, then by writing $\mathcal{O}(K_1\cup
K_2)=\mathcal{O}(K_1)+\mathcal{O}(K_2)$ we can correct $u_1$ and
$u_2$ by respective elements of $\mathcal{O}(K_1)$ to reduce
ourselves to the previous case.

To conclude, we mention the generalisation of these results to
general paracompact real-analytic manifolds. Any paracompact
real-analytic manifold is an analytic submanifold of $\R^n$ for $n$
large enough [15], so that we only need to
restrict our attention to compact sets which belong to this
submanifold; thus there is no additional difficulty. Of course, when
considering the restriction map as in Proposition <ref>,
the boundary is then taken with respect to the topology of the
submanifold, not with respect to the ambient $\R^n$ topology.

alpha Bou74b

[1]
Louis Boutet de Monvel and Victor Guillemin.
The Spectral Theory of Toeplitz Operators.
Number 99 in Annals of Mathematics Studies. Princeton University
Press, 1981.

[2]
Louis Boutet de Monvel and Paul Krée.
Pseudo-differential operators and Gevrey classes.
Annales de l'Institut Fourier, 17(1):295–323, 1967.

[3]
Jean-Michel Bony.
Equivalence des diverses notions de spectre singulier analytique.
Séminaire Équations aux dérivées partielles
(Polytechnique) dit aussi" Séminaire Goulaouic-Schwartz", pages 1–12,

[4]
Louis Boutet de Monvel.
Intégration des équations de Cauchy-Riemann induites formelles.
Séminaire Équations aux dérivées partielles (dit
"Goulaouic-Schwartz"), pages 1–13, 1974.

[5]
Louis Boutet Boutet de Monvel.
Hypoelliptic operators with double characteristics and related
pseudo-differential operators.
Communications on Pure and Applied Mathematics, 27(5):585–639,

[6]
L. Boutet de Monvel.
Complément sur le noyau de Bergman.
Séminaire Équations aux dérivées partielles
(Polytechnique) dit aussi" Séminaire Goulaouic-Schwartz", pages 1–13,

[7]
Louis Boutet de Monvel and Johannes Sjöstrand.
Sur la singularité des noyaux de Bergman et de Szegö.
Journées équations aux dérivées partielles,
34–35:123–164, 1975.

[8]
Laurent Charles.
Berezin-Toeplitz Operators, a Semi-Classical Approach.
Communications in Mathematical Physics, 239(1-2):1–28, August

[9]
Laurent Charles.
Analytic Berezin–Toeplitz operators.
Mathematische Zeitschrift, 299(1):1015–1035, 2021.

[10]
Gregorio Chinni.
A proof of hypoellipticity for Kohn's operator via FBI.
Revista Matemática Iberoamericana, 27(2):585–604, 2011.

[11]
Alix Deleporte.
Toeplitz operators with analytic symbols.
Journal of Geometric Analysis, 31:3915–3967, 2021.

[12]
Alix Deleporte, Michael Hitrik, and Johannes Sjöstrand.
A direct approach to the analytic Bergman projection.
Annales de la Faculté des Sciences de Toulouse :
Mathématiques, In press, 2022.

[13]
Charles L. Epstein and R. B. Melrose.
Shrinking tubes and the$\partial$-Neumann problem.
preprint, 1990.

[14]
Charles Fefferman.
On the Bergman kernel and biholomorphic mappings of pseudoconvex
Annals of Mathematics, 103(3):395–416, 1976.

[15]
Hans Grauert.
On Levi's problem and the imbedding of real-analytic manifolds.
Annals of Mathematics, pages 460–472, 1958.

[16]
Victor Guillemin and Matthew Stenzel.
Grauert tubes and the homogeneous Monge-Ampere equation.
Journal of Differential Geometry, 34(2):561–570, 1991.

[17]
Victor Guillemin and Matthew Stenzel.
Grauert tubes and the homogeneous Monge-Ampere equation. II.
Journal of Differential Geometry, 35(3):627–641, 1992.

[18]
Hamid Hezari, Zhiqin Lu, and Hang Xu.
Off-diagonal asymptotic properties of Bergman kernels associated
to analytic Kähler potentials.
International Mathematics Research Notices, rny081, 2018.

[19]
Lars Hörmander.
L2 estimates and existence theorems for the
$\overline \partial$ operator.
Acta Mathematica, 113(1):89–152, July 1965.

[20]
Lars Hörmander.
An Introduction to Complex Analysis in Several Variables.
Elsevier, 1973.

[21]
Lars Hörmander.
On the existence and the regularity of solutions of linear
pseudodifferential equations.
Uspekhi Matematicheskikh Nauk, 28(6):109–164, 1973.

[22]
Lars Hörmander.
The Analysis of Linear Partial Differential Operators
Classics in Mathematics. Springer, 1985.

[23]
Lars Hörmander.
The Analysis of Linear Partial Differential Operators. I.
Springer, Berlin, 2003.

[24]
Hamid Hezari and Hang Xu.
On a property of Bergman kernels when the Kähler potential
is analytic.
Pacific Journal of Mathematics, 313(2):413–432, 2021.

[25]
Akira Kaneko.
Introduction to Kashiwara's Microlocal Analysis for
Bergman Kernel.
Korea Advanced Institute of Science and Technology, Department of
Applied Mathematics, 1989.

[26]
Masaki Kashiwara.
Analyse micro-locale du noyau de Bergman.
Séminaire Équations aux dérivées partielles
(Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz", pages 1–10,

[27]
Joseph J. Kohn.
Harmonic integrals on strongly pseudo-convex manifolds: I.
Annals of mathematics, pages 112–148, 1963.

[28]
Eric Leichtnam, François Golse, and Matthew Stenzel.
Intrinsic microlocal analysis and inversion formulae for the heat
equation on compact real-analytic Riemannian manifolds.
In Annales Scientifiques de l'Ecole Normale Supérieure,
volume 29, pages 669–736, 1996.

[29]
Anders Melin and Johannes Sjöstrand.
Fourier integral operators with complex-valued phase functions.
In Fourier Integral Operators and Partial Differential
Equations, pages 120–223. Springer, 1975.

[30]
Ophélie Rouby, Johannes Sjöstrand, and San Vũ Ngọc.
Analytic Bergman operators in the semiclassical limit.
Duke Mathematical Journal, 169(16):3033–3097, 2020.

[31]
Johannes Sjöstrand.
Parametrices for pseudodifferential operators with multiple
Arkiv för Matematik, 12(1):85–130, 1974.

[32]
Johannes Sjöstrand.
Singularites Analytiques Microlocales, volume 95 of Astérisque.
Soc. Math. de France, 1982.

[33]
Johannes Sjöstrand.
Analytic wavefront sets and operators with multiple characteristics.
Hokkaido Math. Journal, 12(3):392–433, 1983.

[34]
Johannes Sjöstrand.
Density of resonances for strictly convex analytic obstacles.
Canadian Journal of Mathematics, 48(2):397, 1996.

[35]
Mikio Sato, Takahiro Kawai, and Masaki Kashiwara.
Microfunctions and Pseudo-differential Equations.
In Hyperfunctions and Pseudo-Differential Equations, volume
287 of Lecture Notes in Mathematics, pages 265–529.
Springer, Kataka, 1972.

[36]
David Tartakoff.
Elementary proofs of analytic hypoellipticity for
$\Square$b and $\overline \partial$-Neumann
Asterisque, 89:85–116, 1981.

[37]
François Trèves.
Analytic hypoellipticity of a class of pseudodifferential operators
with double characteristics and applications to the-Neumann problem.
Communications in Partial Differential Equations,
3(6-7):475–642, 1978.

[38]
François Trèves.
Analytic Partial Differential Equations, volume 359 of Grundlehren Der Mathematischen Wissenschaften.
Springer, 2022.

[39]
Hassler Whitney.
Analytic extensions of differentiable functions defined in closed
Transactions of the American Mathematical Society,
36(1):63–89, 1934.

[40]
Steve Zelditch.
Szegö kernels and a theorem of Tian.
Int. Math. Research Notices, 6, 2000.

[41]
Steve Zelditch and Peng Zhou.
Pointwise Weyl Law for Partial Bergman Kernels.
In Michael Hitrik, Dmitry Tamarkin, Boris Tsygan, and Steve Zelditch,
editors, Algebraic and Analytic Microlocal Analysis, Springer
Proceedings in Mathematics & Statistics, pages 589–634, Cham,
2018. Springer International Publishing.