X}X}\quad(=\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}).$

We set
$\overline{\mathcal{W}}_{\psi}=\widetilde{\mathcal{W}}_{\psi}\cup\widetilde{\mathcal{W}}_{\psi
e}$, respectively
$\overline{\mathcal{W}}_{e}=\widetilde{\mathcal{W}}_{e}\cup\widetilde{\mathcal{W}}_{\psi
e}$. In particular, $\overline{\mathcal{W}}_{e}$ and
$\overline{\mathcal{W}}_{\psi}$ are the boundary hyper-surfaces of
${}^{sc}{\overline{T}^{\ast}X}$, both manifolds with boundary,
$\widetilde{\mathcal{W}}_{e}$ and $\widetilde{\mathcal{W}}_{\psi}$ are the
respective interiors, and $\widetilde{\mathcal{W}}_{\psi e}$ is the corner.

Finally, we will also find use for the following spaces:

$\displaystyle\mathcal{W}_{\psi}$
$\displaystyle=\widetilde{\mathcal{W}}_{\psi}\times\mathbb{R}^{+}\cong
T^{\ast}X\setminus\\{0\\}\quad(=\mathring{\mathbb{S}_{+}^{n}}\times\mathbb{R}^{n}_{0}),$
$\displaystyle\mathcal{W}_{e}$
$\displaystyle=\mathbb{R}^{+}\times\widetilde{\mathcal{W}}_{e}\cong
T^{\ast}(\mathbb{R}^{+}\times\partial
X)\quad(=\mathbb{R}^{n}_{0}\times\mathring{\mathbb{S}_{+}^{n}}),$
$\displaystyle\mathcal{W}_{\psi e}$
$\displaystyle=\mathbb{R}^{+}\times\widetilde{\mathcal{W}}_{\psi
e}\times\mathbb{R}^{+}\cong T^{\ast}(\mathbb{R}^{+}\times\partial
X)\setminus\\{0\\}\quad(=\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}_{0}).$

In the above formulæ, $\\{0\\}$ denotes the zero section of the involved
cotangent bundles. Also notice that $\mathbb{R}^{+}\times\partial X$ is,
topologically, the (interior of) a collared neighbourhood of $\partial X$,
seen however with the metric structure of a cone over $\partial X$.

######  Proposition 3.32.

Let
$a_{\bullet}\in\mathcal{C}^{\infty}(\overline{\mathcal{W}}_{\bullet}),\bullet\in\\{e,\psi\\}$
and $a_{\psi e}\in\mathcal{C}^{\infty}(\widetilde{\mathcal{W}}_{\psi e})$ be
smooth functions satisfying
$a_{e}|_{\widetilde{\mathcal{W}}_{\psi}e}=a_{\psi}|_{\widetilde{\mathcal{W}}_{\psi
e}}=a_{\psi e}$. There exists then a function
$a\in\mathcal{C}^{\infty}(\mathbb{B}^{n}\times\mathbb{B}^{n})$ such that
$a|_{\overline{\mathcal{W}}_{\bullet}}=a_{\bullet}$.

Under pull-back with the radial compactification map, the associated symbol
$\check{p}$ of (2.19) is of course nothing else than a particular choice of
such an extension to the interior. There is a similar extension result for
maps between the boundary faces, namely Theorem 3.34 below (as formulated in
[CDS19], Proposition 1.30). To state it we need to discuss scattering maps
between scattering manifolds.

######  Definition 3.33.

Given two scattering manifolds $X,Y$ and a smooth map $C\colon X\rightarrow
Y$, we say that $C$ is a scattering map if, for any given $m\in\mathbb{R}$ and
any $a\in\rho_{Y}^{-m}\mathcal{C}^{\infty}(Y)$ it holds true that

  1. (1)

$C^{\ast}a\in\rho_{X}^{-m}\mathcal{C}^{\infty}(X)$;

  2. (2)

if $q=C(p)$ for $p\in X$ and $\rho_{Y}^{-m}a(q)>0$, then
$\rho_{X}^{-m}C^{\ast}a(p)>0$.

Locally, this corresponds to the fact that scattering maps ($sc$-maps for
short) are exactly those maps which pull back $\rho_{Y}$ to $\rho_{X}h$ for
some positive function $h\in\mathcal{C}^{\infty}(X)$. We call maps satisfying
this condition local sc-maps and extend their definition to manifolds with
corners as follows. Given complete sets of boundary-defining functions
$(\rho_{i})$ for $X$ and $(r_{i})$ for $Y$ (that is, such that the whole
boundary of the manifold can be identified with $\prod\rho_{i}=0$ or $\prod
r_{i}=0$ respectively), $C$ is a local $sc$-map if there exist positive
functions $h_{i}\in\mathcal{C}^{\infty}(X)$ for which it holds true
$C^{\ast}r_{i}=h_{i}\rho_{i}$.

In [CDS19] it is proven that $sc$-maps are really morphisms in the category of
scattering manifolds. We will not need many facts from the theory explored
there, so we’ll only state what we’ll need in the following. In particular,
the next theorem is the aforementioned extension result near the corner of a
product $X\times Y$ of manifolds with boundary.

######  Theorem 3.34.

Consider manifolds with boundary $X_{i},Y_{i},i=1,2$, with boundary defining
functions $\rho_{X_{i}},\rho_{Y_{i}}$, and the products $B_{i}=X_{i}\times
Y_{i}$. Consider, for $\bullet\in\\{e,\psi\\}$, local $sc$-maps
$C_{\bullet}\colon\overline{\mathcal{W}}^{1}_{\bullet}\rightarrow\overline{\mathcal{W}}^{2}_{\bullet}$
defined near a point $p\in\partial X_{1}\times\partial Y_{1}$, such that
$C_{e}|_{\partial X_{1}\times\partial Y_{1}}=C_{\psi}|_{\partial
X_{1}\times\partial Y_{1}}$. There exists then a local $sc$-map $C$ on a
neighbourhood $p\dot{\ni}U\subset B_{1}$ such that $C_{e}=C|_{\partial
X_{1}\times Y_{1}}$, $C_{\psi}=C|_{X_{1}\times\partial Y_{1}}$ and

(3.29) $\frac{\partial
C^{\ast}\rho_{Y_{2}}}{\partial\rho_{X_{1}}}=\frac{\partial
C_{\ast}\rho_{X_{2}}}{\partial\rho_{Y_{1}}}=0.$

Moreover, provided that both $C_{\bullet}$ are local diffeomorphisms near $p$,
then $C$ is also a local diffeomorphism near $p$. If both $C_{\bullet}$ are
diffeomorphisms defined in a neighbourhood of the whole corner in the
respective boundary hyper-surface, then we can pick $C$ to be a diffeomorphism
of a neighbourhood of the corner in $B_{1}$ onto a neighbourhood of the corner
in $B_{2}$.

###  3.2. Symplectic and contact properties of the scattering bundle

Recall that on every cotangent bundle $T^{\ast}X$ a canonical 1-form, known as
the Liouville form $\lambda$, is defined. If $x$ are local coordinates on $X$
and $(x,\xi)$ are the induced canonical coordinates on the cotangent bundle,
$\lambda$ takes the form $\xi\operatorname{\mathrm{d}\\!}x$. The differential
$\omega=\operatorname{\mathrm{d}\\!}\lambda=\operatorname{\mathrm{d}\\!}\xi_{i}\wedge\operatorname{\mathrm{d}\\!}x^{i}$
is a symplectic form on $T^{\ast}X$, which is therefore an exact symplectic
manifold. In the classical theory of FIOs, the 1-form $\lambda$ plays a
crucial role, in that it determines the conic Lagrangian submanifolds and,
therefore, the microlocal form of the operators. In the global scattering
calculus, it turns out that $\lambda$ does not suffice to describe the
peculiar features that appear at “spatial infinity” $\partial X$. In the
existing literature, the problem has been obviated to by introducing a similar
structure on a collared neighbourhood of $\partial X$. Since this is paramount
for our future discussion, we recall hereafter, following [CS17] and [MZ96],
the important concepts, all the while introducing the notation we shall refer
to.

First, we note that the Poisson brackets $\\{\cdot,\cdot\\}$ associated with
the symplectic form $\omega$ extends to $B_{sc}X$. If we keep in mind Theorem
3.29, we understand that, in the model case $X=\mathbb{S}_{+}^{n}$, we can
reformulate Proposition 2.34 in the language of scattering geometry (cf.
[Mel94], Proposition 4).

######  Proposition 3.35.

The Poisson structure on $T^{\ast}\mathring{X}$, induced by the canonical
symplectic form $\operatorname{\mathrm{d}\\!}\lambda$, extends to
$B_{sc}X=\partial({}^{sc}{\overline{T}^{\ast}X})$ as a filtered bracket
between (weighted) smooth functions on $B_{sc}$. More precisely, the Poisson
bracket extends to a map

$\\{\cdot,\cdot\\}\colon\mathcal{C}^{\infty}(B_{sc}X;S^{l_{1},m_{1}})\times\mathcal{C}^{\infty}(B_{sc}X;S^{l_{2},m_{2}})\rightarrow\mathcal{C}^{\infty}(B_{sc}X;S^{l_{1}+l_{2}-1,m_{1}+m_{2}-1}),$

where we understand that we obtain such a map on each boundary hyper-surface
and that $\\{\cdot,\cdot\\}$ preserves the compatibility condition in the
corner. Namely, if we have tuples $(a_{\bullet}),(b_{\bullet})$, extended to
functions $a,b\in\mathcal{C}^{\infty}({}^{sc}{\overline{T}^{\ast}X})$
according to Proposition 3.32, then the Poisson brackets of the tuples
$(\\{a_{\bullet},b_{\bullet}\\})$ satisfy the same conditions and admit
therefore an extension to a smooth function
$c\in\mathcal{C}^{\infty}({}^{sc}{\overline{T}^{\ast}X})$. In particular, the
extensions $a,b$ and $c$ can be chosen so that $c=\\{a,b\\}$.

Second, denote the Liouville 1-form by $\lambda_{\psi}$ and recall that
$\lambda_{\psi}$, being homogeneous of degree $1$ in the fibres of
$T^{\ast}X$, induces a contact structure on $\mathbb{S}^{\ast}X$. In
particular, $\lambda_{\psi}$ restricts to a contact form there, so it can be
rescaled by a conformal factor without changing the actual structure. Of
course, since $\widetilde{\mathcal{W}}_{\psi}$ is diffeomorphic to
$\mathbb{S}^{\ast}\mathring{X}$, it also is a contact manifold and we give it
a specific contact structure as follows. For
$\rho_{\sigma}=R_{\ast}\braket{\xi}^{-1}$, our standard boundary-defining
function, consider the inward-pointing radial vector field
$\rho_{\sigma}\partial_{\rho_{\sigma}}$. This is just the Euler vector field
$\xi_{j}\partial_{\xi_{j}}$ expressed in adapted coordinates at the boundary.
Then, since
$\lambda_{\psi}=\rho_{\sigma}\partial_{\rho_{\sigma}}\lrcorner\omega$, and
$\rho_{\sigma}$ is positive in the interior, we can consider the rescaled
radial vector $\rho_{\sigma}^{2}\partial_{\rho_{\sigma}}$, which, inserted
into $\omega$, gives the 1-form
$\alpha_{\psi}\equiv\rho_{\sigma}^{2}\partial_{\rho_{\sigma}}\lrcorner\omega$.
This form is conformally equivalent to $\lambda_{\psi}$ in the interior, and
we consider it as the “standard” contact form on
$\widetilde{\mathcal{W}}_{\psi}$, the co-sphere bundle at infinity.

A similar process produces our “standard” contact form on
$\widetilde{\mathcal{W}}_{e}$. Choosing a collared neighbourhood and canonical
coordinates $(\rho_{N},z,\tau,\mu)$ near $\partial{}^{sc}{T^{\ast}X}$, with
$\rho_{N}$ a boundary-defining function, $\omega$ is the differential of a
1-form $\lambda_{e}$ given by

(3.30)
$\lambda_{e}=\frac{1}{\rho_{N}}(\operatorname{\mathrm{d}\\!}\tau+\mu_{k}\operatorname{\mathrm{d}\\!}z^{k}).$

Notice that
$\lambda_{e}=\rho_{N}\partial_{\rho_{N}}\operatorname{\lrcorner}\omega$ is not
a smooth 1-form on $T^{\ast}X$, since it blows up at $\partial X$. It is
however smooth as a scattering 1-form on ${}^{sc}{T^{\ast}X}$ and can be
rescaled to give a smooth 1-form
$\alpha_{e}=\rho_{N}^{2}\partial_{\rho_{N}}\operatorname{\lrcorner}\omega$
near the interior of the boundary hyper-surface. Since the collared
neighbourhood gives a conic structure near $\widetilde{\mathcal{W}}_{e}$, we
obtain a contact structure on the boundary hyper-surface as above. Moreover,
the contact distribution over $\widetilde{\mathcal{W}}_{e}$ is unambiguously
determined by the restriction of $\alpha_{e}$ to
$\widetilde{\mathcal{W}}_{e}$, and $\alpha_{e}$ is a contact form there.
Notice again that this is conformally equivalent to $\lambda_{e}$ in the
interior but not at “spatial infinity” $\partial X$.

We have then a pair of 1-forms $\alpha_{\psi},\alpha_{e}$ on $T^{\ast}X$ which
determine the symplectic structure at either spatial infinity or fibre
infinity. In particular, since they induce contact distributions of
$\widetilde{\mathcal{W}}_{\psi}$ and $\widetilde{\mathcal{W}}_{e}$, we can
speak of Legendrian submanifolds at infinity. Recall that, classically, FIOs
are operators whose singularities are contained in conic Lagrangian
submanifolds of $T^{\ast}X\setminus\\{0\\}$, which in turn can be identified
with Legendrian submanifolds of the co-sphere bundle by restriction. Parallel
to this situation, Melrose and Zworski [MZ96] introduced a class of Legendrian
distributions in $\widetilde{\mathcal{W}}_{e}$, which has been subsequently
generalised by Coriasco, Doll and Schulz in [CDS19] to include singularities
in the whole $B_{sc}X$. As a last step, before introducing our notion of
singular symplectomorphism, we recall the notion of the so-called
$SG$/$sc$-Lagrangians/Legendrians, that have been studied by Coriasco and
collaborators, and refer to the cited literature for more details.

######  Definition 3.36 ($SG$-Legendrian).

Let $\Lambda=\overline{\Lambda}_{e}\cup\overline{\Lambda}_{\psi}\subset
B_{sc}X$ be a closed submanifold, where
$\Lambda_{e}\subset\overline{\mathcal{W}}_{e}$ and
$\Lambda_{\psi}\subset\overline{\mathcal{W}}_{\psi}$ and
$\overline{\Lambda}_{\bullet}$ denotes the closure of $\Lambda_{\bullet}$.
$\Lambda$ is called an $SG$-Legendrian submanifold if it satisfies the
following conditions:

  1. (1)

$\Lambda_{\psi}$ is Legendrian in $\widetilde{\mathcal{W}}_{\psi}$;

  2. (2)

$\Lambda_{e}$ is Legendrian in $\widetilde{\mathcal{W}}_{e}$;

  3. (3)

$\overline{\Lambda}_{e}$ has boundary if and only if
$\overline{\Lambda}_{\psi}$ has boundary, in which case

$\Lambda_{\psi
e}=\partial\overline{\Lambda}_{e}=\partial\overline{\Lambda}_{\psi}=\overline{\Lambda}_{e}\cap\overline{\Lambda}_{\psi}$

with clean intersection.

Having finished our recap, we build upon this structure to introduce our
notion of symplectomorphism. Here and later, $X,Y$ are scattering manifolds
and we use the notation of Definition 3.31 to refer to subsets of $B_{sc}X$
and $B_{sc}Y$ indifferently (no confusion should arise, in particular, in view
of the fact that all statements to come can be checked in local coordinates).

######  Definition 3.37.

Let $U$, respectively $V$, be open in $B_{sc}X$, respectively $B_{sc}Y$.
Assume $\chi\colon U\rightarrow V$ is a diffeomorphism (in particular, a
$sc$-map), given as a pair of maps $\chi=(\chi_{e},\chi_{\psi})$ defined on
$U\cap\overline{\mathcal{W}}_{e}$ and $U\cap\overline{\mathcal{W}}_{\psi}$,
respectively (if
$U\cap\overline{\mathcal{W}_{\bullet}}=\emptyset,\bullet\in\\{e,\psi\\},$ we
understand that $\chi_{\bullet}$ is not present). We define the notion of
scattering symplectomorphism or scattering-canonical transformation (SCT)
depending on whether $U$ and (consequently) $V$ intersect the corner: If
$U\cap\widetilde{\mathcal{W}}_{\psi e}=\emptyset$, we say that $\chi$ is an
SCT if $\chi_{\bullet}$ is a contact diffeomorphism with respect to
$\alpha_{\bullet}$; else, $\chi$ is an SCT if both $\chi_{e}$ and
$\chi_{\psi}$ are contact diffeomorphism in the interior of the respective
boundary face and $\chi$ preserves the Poisson bracket on
$\mathcal{C}^{\infty}(U;S^{l,m})$ across the corner.

We have the following easy-to-prove properties of a scattering-
symplectomorphism, which reflect classical behaviour of regular
symplectomorphisms.

######  Lemma 3.38.

Let $\chi\colon U\rightarrow V$ be an SCT. Then $\chi$ maps $SG$-Legendrian
submanifolds in $U$ to $SG$-Legendrian submanifolds in $V$. Moreover, if
$U\cap\widetilde{\mathcal{W}}_{\psi e}=\emptyset$, then $\chi_{e}$,
respectively $\chi_{\psi}$, extends to a homogeneous symplectomorphism on a
collared neighbourhood of $\partial T^{\ast}X$, respectively on
$T^{\ast}_{0}X$, which admits a local parametrisation via a $e$-homogeneous,
respectively $\psi$-homogeneous, phase function. Finally $\chi$ induces
Poisson maps on the boundaries and corners (i.e. it preserves the Poisson
structure in Proposition 3.35).

###### Proof.

First, recall that a diffeomorphism between contact manifolds $X$ and $Y$ is
contact if and only if it preserves each Legendrian submanifold (cf. [Sas64]).
Then, since $\chi$ is a diffeomorphism by definition, we obtain at once that
$SG$-Legendrians are preserved if $U\cap\widetilde{\mathcal{W}}_{\psi
e}=\emptyset$. On the other hand $\chi$ preserves clean intersection so the
preservation of $SG$-Legendrians for $U\cap\widetilde{\mathcal{W}}_{\psi
e}\neq\emptyset$ also follows.

The second statement is obtained by applying the classical procedure of
symplectisation to the contact manifolds $\widetilde{\mathcal{W}}_{\psi}$ and
$\widetilde{\mathcal{W}}_{e}$ separately. Indeed notice that, if $U$ does not
intersect the corner, then $U$ can be “conified” to a subset of $T^{\ast}X$.
Near the boundary we just have to pick a collared neighbourhood $\partial
X\times[0,1)$ and pull-back $(U\cap\widetilde{\mathcal{W}}_{e})\times[0,1)$
with $R\times\operatorname{\mathrm{id}}$, while on the co-sphere at $\infty$
we just identify $U\cap\mathcal{W}_{\psi}$ with a subset of
$\mathbb{S}^{\ast}X$ and consider, as in the classical theory, the associated
conic neighbourhood. Let us spend a few extra words to describe, for the
$e$-component, how one obtains a symplectic form and can extend $\chi_{e}$ to
a homogeneous symplectomorphism. More details can be found in [MS17], Section
<<Symplectization of contact manifolds>>, or in [Arn89], Appendix 4. We are
given the contact form
$\alpha_{e}=\operatorname{\mathrm{d}\\!}\tau+\mu_{k}\operatorname{\mathrm{d}\\!}z^{k}$
on $\widetilde{\mathcal{W}}_{e}$. In the conified neighbourhood
$\mathbb{R}^{+}\times U_{e}$, we are introducing the new coordinate
$\rho_{N}$, effectively identifying $U_{e}$ with $\\{1\\}\times U_{e}$, and
can consider

$\omega_{e}=-\operatorname{\mathrm{d}\\!}\,(\rho_{N}\alpha_{e})=-\rho_{N}\left(\frac{\operatorname{\mathrm{d}\\!}\rho_{N}}{\rho_{N}}\wedge\alpha_{e}+\operatorname{\mathrm{d}\\!}\alpha_{e}\right).$

It is readily checked that this is now a symplectic form on the conified
neighbourhood. By definition of contact transformation,
$\chi^{\ast}\alpha_{e}=g\alpha_{e}$ for a positive smooth function $g$. Then,
the extension
$C_{e}(\rho_{N},z,\tau,\mu)\equiv(\rho_{N}/g(z,\tau,\mu),\chi(z,\tau,\mu))$ is
homogeneous symplectic. Indeed,

$C_{e}^{\ast}\omega_{e}=-\operatorname{\mathrm{d}\\!}C_{e}^{\ast}(\rho_{N}\alpha_{e})=-\operatorname{\mathrm{d}\\!}\left(\frac{\rho_{N}}{g}g\alpha_{e}\right)=\omega_{e},$

proving that $C_{e}$ is symplectic. The homogeneity is, on the other hand,
manifest.

We have now the homogeneous symplectic extensions $C_{e}$ and $C_{\psi}$. Now,
the local parametrisation of $C_{\psi}$ is the classical result of Hörmander,
Proposition 25.3.3 in [H0̈9]. On the other hand, for $C_{e}$ it suffices to
exchange the rôles of variables and covariables (also cf. [MZ96], Section 6).

Concerning the last statement, observe that, at the corner, the Poisson
structure is preserved by definition. On the other hand, the homogeneous
symplectic extensions just constructed guarantee that $\\{,\\}$ is preserved
away from the corner. ∎

In the next Theorem 3.39 we present a more thorough analysis of the structure
of a $sc$-symplectomorphism defined near the corner. To avoid overburdening
the notation, let us first clarify that the local expressions given below hold
true in coordinates $(x,\xi)$, obtained as the pull-back of standard systems
of coordinates near the boundary faces (or possibly the corner). In
particular, the $\alpha$’s are angular coordinates on $\mathbb{S}^{n-1}$ and
the boundary-defining function is the inverse of the radial coordinate in
polar coordinates. That is, $x^{i}=\left|x\right|X^{i}(\alpha)$ for smooth
functions $X^{i}$ such that $(X^{1})^{2}+\dots+(X^{n})^{2}=1$ and
$\rho_{\partial}=1/\left|x\right|$. We employ the same convention for $\xi$’s
and $\beta$’s. Also, in this notation we will consider homogeneous extensions
of functions $\mathbb{S}^{n-1}\times\mathbb{R}^{n}$ to
$\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}$. To be precise, we will look for
$\mathbb{R}^{+}$-equivariant maps
$C_{\bullet}\colon\mathcal{W}_{\bullet}\rightarrow\mathcal{W}_{\bullet}$
agreeing with $\chi_{\bullet}$ on $\widetilde{\mathcal{W}}_{\bullet}$. Any
such map is of the form (for example $\bullet=e$, w.l.o.g.)

(3.31) $C_{e}(r,\alpha,\xi)=(f_{e}(\alpha,\xi)r,\chi_{e}(\varphi,\xi))$

for some smooth $f_{e}\in\mathcal{C}^{\infty}(\widetilde{\mathcal{W}}_{e})$,
where $(r=\left|x\right|,\alpha)$ are the above polar coordinates on
$\mathbb{R}^{n}_{0}$ and $\xi$ are coordinates on $\mathbb{R}^{n}$. The
inverse of such map, again taking polar coordinates on the first factor and
global coordinates on the second, is given by

(3.32)
$C_{e}^{-1}(s,\alpha,\eta)=\left(\frac{s}{f_{e}(\chi_{e}^{-1}(\alpha,\eta))},\chi_{e}^{-1}(\alpha,\eta)\right).$

Recalling again Section <<Symplectization of contact manifolds>> in [MS17], it
must be possible to choose $f_{e}$ appropriately to ensure that $C_{e}$ so
extended is symplectic. We will find the explicit form of the section $f_{e}$
(and $f_{\psi}$ too, of course) in the next chapter, in the course of the
proof of Lemma 4.5. For the moment, we content ourselves with saying that such
a choice is possible.

######  Theorem 3.39.

Let $\chi$ be a $sc$-canonical transformation, between open sets $U,V$ as
above, with $U\cap\widetilde{\mathcal{W}}_{\psi e}\neq\emptyset$. Then $\chi$
is given as the datum of a triple of diffeomorphisms
$(\chi_{e},\chi_{\psi},\chi_{\psi e})$, for
$\chi_{\bullet}\colon\widetilde{\mathcal{W}}_{\bullet}\rightarrow\widetilde{\mathcal{W}}_{\bullet}$,
such that

  1. (1)

If $\chi_{e}(\alpha,\xi)=(T(\alpha,\xi),H(\alpha,\xi))$ for
$T\colon\mathbb{S}^{n-1}\times\mathbb{R}^{n}\rightarrow\mathbb{S}^{n-1},H\colon\mathbb{S}^{n-1}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$,
then the components of $H$ are elements of
$\mathcal{C}^{\infty}(\mathbb{S}^{n-1};\mathit{S}^{1}(\mathbb{R}^{n}))$;

  2. (2)

If $\chi_{\psi}(x,\beta)=(Y(x,\beta),G(x,\beta))$ for
$G\colon\mathbb{R}^{n}\times\mathbb{S}^{n-1}\rightarrow\mathbb{S}^{n-1},Y\colon\mathbb{R}^{n}\times\mathbb{S}^{n-1}\rightarrow\mathbb{R}^{n}$
then the components of $Y$ are elements of
$\mathcal{C}^{\infty}(\mathbb{S}^{n-1};\mathit{S}^{1}(\mathbb{R}^{n}))$;

  3. (3)

If $\chi_{\psi e}(\alpha,\beta)=(A(\alpha,\beta),B(\alpha,\beta))$ and we
write $\chi_{e},\chi_{\psi}$ as above, the principal symbol of $Y$,
respectively $H$, restricted to $\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}$
coincides with $T$, respectively $F$. More generally, it holds true that

(3.33) $\displaystyle A(\alpha,\beta)$
$\displaystyle=\lim_{\lambda\rightarrow+\infty}T(\alpha,\lambda\xi)=\lim_{\lambda\rightarrow+\infty}\frac{1}{\lambda}Y(\lambda
x,\beta),$ (3.34) $\displaystyle B(\alpha,\beta)$
$\displaystyle=\lim_{\lambda\rightarrow+\infty}\frac{1}{\lambda}H(\alpha,\lambda\xi)=\lim_{\lambda\rightarrow+\infty}G(\lambda
x,\beta);$

  4. (4)

We can pick homogeneous extensions $C_{e}$ of $\chi_{e}$ in $\alpha$,
$C_{\psi}$ of $\chi_{\psi}$ in $\beta$ and $C_{\psi e}$ of $\chi_{\psi e}$ in
$\alpha$ and $\beta$ separately, so that $C_{\bullet}$ is a symplectomorphism,
homogeneous in the respective variables;

  5. (5)

Writing these extensions as
$C_{\bullet}(x,\xi)=(Y_{\bullet}(x,\xi),H^{\bullet}(x,\xi))$ for
$Y_{\bullet}=(Y_{\bullet}^{1},\dots,Y_{\bullet}^{n})$,
$H^{\bullet}=(H^{\bullet}_{1},\dots,H^{\bullet}_{n})$, we have that each
triple $(Y_{\bullet}^{j})$, $(H^{\bullet}_{k})$ can be continued to a
classical $SG$-symbol near “infinity”. In particular there is a diffeomorphism
$C(x,\xi)=(Y^{j}(x,\xi),H_{k}(x,\xi))$ near “infinity” having
$(C_{e},C_{\psi},C_{\psi e})$ as “principal symbol”.

###### Proof.

For 1., notice that $\chi_{e}$ is given as a diffeomorphism of
$\mathbb{S}^{n-1}\times\mathbb{B}^{n}$. We pull it back to a diffeomorphism of
$\mathbb{S}^{n-1}\times\mathbb{R}^{n}$ using $\operatorname{\mathrm{id}}\times
R$. But then the $\mathbb{R}^{n}$-components of $\chi_{e}$ must be classical
symbols of order 1 in $\xi$, depending in a smooth way on
$\alpha\in\mathbb{S}^{n-1}$. This is exactly the claim.

For 2., we argue exactly as in 1., exchanging the rôles of the variables.

To prove 3. notice that the expressions involving $H$ and $Y$ are just the
standard formulæ to compute the principal symbol for the classes
$S^{1}(\mathbb{R}^{n})$, depending on a parameter on $\mathbb{S}^{n-1}$.
Recalling Theorem 3.29 we see immediately that we can compute it also by
restriction to $\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}$. Now, $\chi_{\bullet}$
is obtained as a diffeomorphism of
$\partial(\mathbb{S}^{n}_{+}\times\mathbb{S}^{n}_{+})$, so in the corner
$\chi_{e}=\chi_{\psi}$. Pulling this back with
$\operatorname{\mathrm{id}}\times R$ and $R\times\operatorname{\mathrm{id}}$
and comparing the respective components gives the claimed formulas.

4\. is clear if one exploits the close relation between canonical
transformations and contact diffeomorphisms. However, a more explicit
construction will be given in the proof of Lemma 4.5, where we will see that
the choice of order reductions uniquely determines the homogeneous extensions
to be symplectic.

5\. is now a consequence of the above facts. Indeed, the components of the
homogeneous extensions $C_{\bullet}$ satisfy symbol estimates in the non-
homogeneous variables. In particular, each pair of components
$(Y_{e}^{j},Y_{\psi}^{j})$, respectively $(H^{e}_{k},H^{\psi}_{k})$, can be
continued to a symbol
$Y^{j}\in\mathit{SG}^{\mathbbm{1}_{e}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$,
respectively
$H_{k}\in\mathit{SG}^{\mathbbm{1}_{\psi}}(\mathbb{R}^{n}\times\mathbb{R}^{n})$.
We can choose them so that the resulting map $C(x,\xi)=(Y(x,\xi),H(x,\xi))$ is
a diffeomorphism, in view of Theorem 3.34. Indeed, our maps are all globally
defined on the boundary hyper-surfaces and the corner, so they can be patched
together correctly. ∎

######  Remark 3.40.

One would certainly hope that the extension in 5. of the above theorem could
be achieved symplectic. However, the best of our efforts could not deduce this
desirable fact from the properties of $C_{e},C_{\psi}$ and $C_{\psi e}$.

######  Remark 3.41.

Notice that, for a scattering map on a manifold with corners, one preassigns
an ordering on the set of the boundary-defining functions, so that at a corner
we are specifying which boundary hyper-surface is mapped to which. For example
on $\mathbb{S}^{n}_{+}\times\mathbb{S}^{n}_{+}$, seen as
${}^{sc}{\overline{T^{\ast}\mathbb{S}^{n}_{+}}}$, we use the ordered set of
boundary-defining functions $(\rho_{N},\rho_{\sigma})$. Then, it is easily
seen that the symplectic rotation $F\colon(x,\xi)\rightarrow(\xi,-x)$,
extended to the compactification as in Theorem 3.29, is not a scattering map
in this sense, since $F^{\ast}\rho_{N}=\rho_{\sigma}$ and vice-versa. This
reflects the fact that, on a manifold with boundary $X$, the two components of
the joint scattering symbol live as smooth functions on two in principle
different compact manifolds, namely, the scattering co-sphere bundle and the
boundary of $X$ (pulled back to the compactified scattering cotangent bundle).
Of course, nothing in principle prevents us from considering $F$ as some sort
of “generalised scattering map” on the model case $\mathbb{S}_{+}^{n}$.
However we notice that pull-back along $F$ does not preserve $SG$-classes,
since it exchanges the two filtrations as in Proposition 3.28. We will
therefore assume that SCTs cannot exhibit this kind of behaviour, although we
will comment again on this point at the very end of Chapter 4.

We now come to the core of this section: the relation between scattering-
symplectic maps and the classical $SG$-phase functions.

######  Theorem 3.42 (Parametrising $sc$-symplectomorphisms).

Let
$\chi\colon\partial(\mathbb{S}_{+}^{n}\times\mathbb{S}_{+}^{n})\rightarrow\partial(\mathbb{S}_{+}^{n}\times\mathbb{S}_{+}^{n})$
be a (possibly only locally defined) scattering canonical transformation.
Then, at each point $(p,q)$ on the graph of $\chi$, we can find a
neighbourhood $\tilde{U}$ of $p$, a neighbourhood $\tilde{V}$ of $q$ and an
$SG$-phase function
$\varphi(x,y,\xi)\in\mathit{SG}^{\mathbbm{1}}_{(x,y),\xi}$, parametrising a
neighbourhood of $(p,q)$. More explicitly, if $p$ does not lie on the corner,
then we can parametrise the homogeneous symplectic extension $C$ of $\chi$
near $p$ via a homogeneous phase function in the classical sense. On the other
hand, if $p$ is in the corner then there is a conic neighbourhood $U_{e}$,
respectively $U_{\psi}$, associated with a neighbourhood $\tilde{U}_{e}$,
respectively $\tilde{U}_{\psi}$, of $p$ in
$\mathbb{S}^{n-1}\times\mathbb{S}_{+}^{n}$, respectively
$\mathbb{S}_{+}^{n}\times\mathbb{S}^{n-1}$, and we find a phase $\varphi$ as
above such that $\varphi_{e}=\sigma_{e}(\varphi)$, respectively
$\varphi_{\psi}=\sigma_{\psi}(\varphi)$, parametrises the graph of $C_{e}$,
respectively $C_{\psi}$, in the usual sense for conic Lagrangians, and
$\varphi_{\psi e}$ parametrises the bi-homogeneous extension $C_{\psi e}$ of
$\chi_{\psi e}$.

###### Proof.

The case
$(p,q)\in\widetilde{\mathcal{W}}_{\psi}\times\widetilde{\mathcal{W}}_{\psi}$
is just an instance of the classical parametrization result for homogeneous
symplectomorphisms of Hörmander, namely Proposition 25.3.6 in [H0̈9]. To see
this, consider neighbourhoods $U$ of $p$ and $V$ of $q$ which are away from
the corner $\tilde{\mathcal{W}}_{\psi e}$. Then, we can exploit the triviality
of the bundle $\pi\colon U\times\mathbb{R}^{n}_{0}\rightarrow
U\times\mathbb{S}^{n}$ to define a homogeneous extension $C$ of $\chi$. By
picking the correct section of $\pi$, we can ensure that $C$ is actually a
homogeneous canonical transformation (namely it preserves the canonical 1-form
$\lambda_{\psi}$) and apply Hörmander’s result. Similarly, if
$(p,q)\in\widetilde{\mathcal{W}}_{e}\times\widetilde{\mathcal{W}}_{e}$, we can
consider the trivial bundle
$\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}\rightarrow\mathbb{S}^{n-1}\times\mathbb{R}^{n}$.
Here we can pick homogeneous extensions in $x$ and reproduce the proof of
Hörmander (notice also [MZ96], Section 6) exchanging the rôles of $x$ and
$\xi$ (in particular, using the fact that $C^{\ast}\lambda_{e}=\lambda_{e}$
for $\lambda_{e}=x^{j}\operatorname{\mathrm{d}\\!}\xi_{j}$). The case
$(p,q)\in\widetilde{\mathcal{W}}_{\psi e}\times\widetilde{\mathcal{W}}_{\psi
e}$ is a bit more involved and we adapt the careful analysis of [CS17].

Mimicking the construction there, we work in a chart
$\widetilde{U}\subset\widetilde{\mathcal{W}}$ around $p$ where on
$\widetilde{U}\cap\widetilde{\mathcal{W}}_{\psi e}$ we have coordinates in the
form

(3.35)
$(\alpha^{1},\dots,\alpha^{n-1},\sqrt{1-(\alpha^{1})^{2}-\dots-(\alpha^{n-1})^{2}},\sqrt{1-(\beta_{2})^{2}-\dots-(\beta_{n})^{2}},\beta_{2},\dots,\beta_{n}).$

On $\widetilde{\mathcal{W}}_{e}$ and $\widetilde{\mathcal{W}}_{\psi}$ we use
adapted coordinates

(3.36)
$\displaystyle(\alpha^{1},\dots,\alpha^{n-1},\sqrt{1-(\alpha^{1})^{2}-\dots-(\alpha^{n-1})^{2}},\rho_{2},\beta_{2},\dots,\beta_{n})$
$\displaystyle\in\widetilde{\mathcal{W}}_{\psi},$
$\displaystyle(\alpha^{1},\dots,\alpha^{n-1},\rho_{1},\sqrt{1-(\beta_{2})^{2}-\dots-(\beta_{n})^{2}},\beta_{2},\dots,\beta_{n})$
$\displaystyle\in\widetilde{\mathcal{W}}_{e},$

where $\rho_{1}=\sqrt{1-(\alpha^{1})^{2}-\dots-(\alpha^{n-1})^{2}}$ and
$\rho_{2}=\sqrt{1-(\beta_{2})^{2}-\dots-(\beta_{n})^{2}}$ are defining
equations for the common boundary
$\widetilde{U}\cap\widetilde{\mathcal{W}}_{\psi e}$. We can similarly choose
coordinates $(\theta,r_{1},r_{2},\gamma)$ satisfying the same relations in a
chart $\tilde{V}$ around $q$. In these coordinates the map
$\tilde{C}=(\tilde{C}_{e},\tilde{C}_{\psi})$ can be expressed as

(3.37)
$\tilde{C}_{\bullet}(\alpha,\rho_{1},\rho_{2},\beta)=(T_{\bullet}(\alpha,\rho_{1},\rho_{2},\beta),r_{1}^{\bullet}(\alpha,\rho_{1},\beta),r_{2}^{\bullet}(\alpha,\rho_{2},\beta),G^{\bullet}(\alpha,\rho_{1},\rho_{2},\beta)),$

namely $\theta=T_{\bullet}$ and $\gamma=G^{\bullet}$ are equations defining
the graph of $\tilde{C}_{\bullet}$ in $\tilde{U}\times\tilde{V}$. Let
$\widetilde{U_{\bullet}}=\widetilde{U}\cap\widetilde{\mathcal{W}}_{\bullet}$
and $U_{\bullet}$ be the conic set associated with $\widetilde{U_{\bullet}}$
under inverse radial compactification. Then, on $U_{\bullet}$ we can introduce
“polar coordinates” and extend $C_{\bullet}$ homogeneously. For example, on
$U_{e}$ we choose a section
$f_{e}(\alpha,\rho_{2},\beta)\colon\mathbb{S}^{n-1}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}$,
pull back the covariables using $R$, and set, for $\mu>0$ and $\rho_{2}>0$,

(3.38)
$\displaystyle(x,\xi)\equiv(\mu\alpha^{1},\dots,\mu\alpha^{n-1},\mu\sqrt{1-\left|\alpha\right|^{2}},R^{-1}(\rho_{2}\beta_{1},\dots,\rho_{2}\beta_{n})),$
$\displaystyle
C_{e}(x,\xi)=\left(T_{e}\left(\frac{x}{\mu},\mu,R(\xi)\right),r_{1}^{e}\left(\frac{x}{\mu},R(\xi)\right),r_{2}^{e}\left(\frac{x}{\mu},R(\xi)\right),G^{e}\left(\frac{x}{\mu},\mu,R(\xi)\right)\right).$

Again as in [MS17], the section $f_{e}$ can be appropriately chosen to ensure
that $C_{e}$ is symplectic and homogeneous in the $x$ variables. Therefore, it
preserves the 1-form $\lambda_{e}$. Similarly, we have an extension $C_{\psi}$
which preserves the Liouville 1-form $\lambda_{\psi}$ (using a section
$f_{\psi}\colon\mathbb{S}^{n-1}\times
U_{\psi}\rightarrow\mathbb{R}^{n}_{0}\times U_{\psi}$), and we can also define
a map $C_{\psi e}$ by extending $\chi_{\psi e}$ using both sections
$f_{e},f_{\psi}$. Then, in Cartesian coordinates on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$, taking into account Theorem 3.29, we
have then 3 symplectomorphisms $C_{e},C_{\psi},C_{\psi e}$ defined for $x\neq
0,\xi\neq 0$ and $x,\xi\neq 0$, respectively. Their components are parts of
principal $SG$-symbols:

(3.39) $\displaystyle C_{e}(x,\xi)$
$\displaystyle=(Y^{1}_{e}(x,\xi),\dots,Y^{n}_{e}(x,\xi),H^{e}_{1}(x,\xi),\dots,H^{e}_{n}(x,\xi)),$
$\displaystyle C_{\psi}(x,\xi)$
$\displaystyle=(Y^{1}_{\psi}(x,\xi),\dots,Y^{n}_{\psi}(x,\xi),H^{\psi}_{1}(x,\xi),\dots,H^{\psi}_{n}(x,\xi)),$
$\displaystyle C_{\psi e}(x,\xi)$ $\displaystyle=(Y^{1}_{\psi
e}(x,\xi),\dots,Y^{n}_{\psi e}(x,\xi),H^{\psi e}_{1}(x,\xi),\dots,H^{\psi
e}_{n}(x,\xi)),$ $\displaystyle Y_{e}^{j}$
$\displaystyle\in\mathit{SG}^{(1),0},\quad
Y_{\psi}^{j}\in\mathit{SG}^{1,(0)},\sigma_{e}(Y^{j}_{\psi})=\sigma_{\psi}(Y^{j}_{e})=Y^{j}_{\psi
e}$ $\displaystyle H^{e}_{k}$ $\displaystyle\in\mathit{SG}^{(0),1},\quad
H^{\psi}_{k}\in\mathit{SG}^{0,(1)},\sigma_{e}(H_{k}^{\psi})=\sigma_{\psi}(H_{k}^{e})=H_{k}^{\psi
e}.$

The $SG$ estimates for these functions follow directly from the previous
considerations, Chapter 6 in [And], and our particular choice of coordinates.
Now, the twisted graphs
$\Lambda_{\bullet}=\operatorname{\mathrm{gr}}^{\prime}(C_{\bullet})$ of these
maps are conic Lagrangians (either in $x,\xi$ or both). As in the classical
theory of canonical graphs, we can find (possibly after rearranging) a
partition $I=(1,\dots,d),J=(d+1,\dots,n)$ so that $(x^{J},\xi_{I},\eta)$ can
be taken as coordinates on $\Lambda_{\bullet}$.

Notice that, in principle, in what follows we should choose different sets of
coordinates for each map. However, the boundary hyper-surfaces intersect
cleanly and all the changes of coordinates just defined are either
diffeomorphism or homogeneous extensions outside a compact neighbourhood of
$0$, so they preserve this structure. Hence, near the corner we can always
take the same partitions $I,J$.

From here to the end of the proof, we employ a modified Einstein convention,
namely: the indices $i$ belong to $I$, $j$ to $J$, $k$ to $\\{1,\dots,n\\}$,
and repeated $i$ or $j$ means summing only over $I$ and $J$. The other
coordinates are defined implicitly on $\Lambda_{\bullet}$ as

(3.40)
$x^{i}=X^{i}_{\bullet}(x^{J},\xi_{I},\eta),\quad\xi_{j}=\Xi^{\bullet}_{j}(x^{J},\xi_{I},\eta),\quad\eta_{k}=H_{k}^{\bullet}(x^{J},\xi_{I},\eta).$

In view of Chapter 6 of [And], we see that these function $X,\Xi,H$ must
satisfy $SG$-estimates. In particular, they belong to the following classes
(the classicality is implied by the fact that we are pulling-back smooth
functions on the compactified space along $R$):

(3.41) $\displaystyle X^{i}_{e}$
$\displaystyle\in\mathit{SG}^{(1),0}(\mathbb{R}^{n+d}\times\mathbb{R}^{n-d}),\quad
X^{i}_{\psi}\in\mathit{SG}^{1,(0)}(\mathbb{R}^{n+d}\times\mathbb{R}^{n-d}),$
$\displaystyle\Xi_{j}^{e}$
$\displaystyle\in\mathit{SG}^{(0),1}(\mathbb{R}^{n+d}\times\mathbb{R}^{n-d}),\quad\Xi_{j}^{\psi}\in\mathit{SG}^{0,(1)}(\mathbb{R}^{n+d}\times\mathbb{R}^{n-d}),$
$\displaystyle H_{k}^{e}$
$\displaystyle\in\mathit{SG}^{(0),1}(\mathbb{R}^{n+d}\times\mathbb{R}^{n-d}),\quad
H_{k}^{\psi}\in\mathit{SG}^{0,(1)}(\mathbb{R}^{n+d},\mathbb{R}^{n-d}).$

Still following the ideas of [CS17], we now define directly homogeneous phase
functions that parametrise locally the graphs of these diffeomorphisms and
show they can be patched together to a single $SG$-phase function. To begin
with, we look at the condition that $C_{e}$ be an $e-$homogeneous canonical
transformation. This amounts to $C_{e}$ preserving the 1-form $\alpha_{e}$,
namely $C_{e}^{\ast}\alpha_{e}=\alpha_{e}$. In the above coordinate patches,
this is expressed as

(3.42) $\displaystyle 0$
$\displaystyle=(x\operatorname{\mathrm{d}\\!}\xi-y\operatorname{\mathrm{d}\\!}\eta)|_{\Lambda_{e}}$
$\displaystyle=X^{i}_{e}\operatorname{\mathrm{d}\\!}\xi_{i}+x^{j}\left(\frac{\partial\Xi^{e}_{j}}{\partial
x^{j_{1}}}\operatorname{\mathrm{d}\\!}x^{j_{1}}+\frac{\partial\Xi^{e}_{j}}{\partial\xi_{i}}\operatorname{\mathrm{d}\\!}\xi_{i}+\frac{\partial\Xi^{e}_{j}}{\partial
y^{k}}\operatorname{\mathrm{d}\\!}y^{k}\right)-Y_{e}^{k}\operatorname{\mathrm{d}\\!}\eta_{k}$
$\displaystyle=\left(X_{e}^{i}-x^{j}\frac{\partial\Xi^{e}_{j}}{\partial\xi_{i}}\right)\operatorname{\mathrm{d}\\!}\xi_{i}+x^{j_{1}}\frac{\partial\Xi^{e}_{j_{1}}}{\partial
x^{j}}\operatorname{\mathrm{d}\\!}x^{j}+\left(x^{j}\frac{\partial\Xi^{e}_{j}}{\partial\eta_{k}}-Y_{e}^{k}\right)\operatorname{\mathrm{d}\\!}\eta_{k}.$

Therefore, all expressions in parenthesis must vanish on the graph. Very
similar relations hold true for $C_{\psi}$, which we give explicitly
hereafter:

(3.43) $\displaystyle\Xi^{\psi}_{j}+\xi_{i}\frac{\partial
X^{i}_{\psi}}{\partial x^{j}}-\eta_{k}\frac{\partial Y^{k}_{\psi}}{\partial
x^{j}}$ $\displaystyle=0,$ $\displaystyle\xi_{i}\frac{\partial
X^{i}_{\psi}}{\partial\xi_{I}}-\eta_{k}\frac{\partial
Y_{\psi}^{k}}{\partial\xi_{I}}$ $\displaystyle=0,$
$\displaystyle\xi_{i}\frac{\partial
X^{i}_{\psi}}{\partial\eta_{l}}-\eta_{k}\frac{\partial
Y^{k}_{\psi}}{\partial\eta_{l}}$ $\displaystyle=0.$

Let us consider first the functions $S_{\bullet}$ defined by

(3.44) $\displaystyle S_{e}(x^{J},\xi_{I},\eta)$
$\displaystyle=x^{j}\Xi_{j}^{e}(x^{J},\xi_{I},\eta),$ $\displaystyle
S_{\psi}(x^{J},\xi_{I},\eta)$
$\displaystyle=-X_{\psi}^{i}(x^{J},\xi_{I},\eta)\xi_{i}+Y^{k}_{\psi}\eta_{k}.$

We prove that they are generating functions for the canonical transformations
$C_{\bullet}$. First looking at $S_{e}$ we have, using (3.42), that

(3.45) $\displaystyle\frac{\partial S_{e}}{\partial x^{j}}$
$\displaystyle=\Xi^{e}_{j}+x^{j_{1}}\frac{\partial\Xi^{e}_{j_{1}}}{\partial
x^{j}}=\Xi^{e}_{j},$ $\displaystyle\frac{\partial S_{e}}{\partial\xi_{i}}$
$\displaystyle=x^{j}\frac{\partial\Xi^{e}_{j}}{\partial\xi_{i}}=X^{i}_{e},$
$\displaystyle\frac{\partial S_{e}}{\partial\eta^{k}}$
$\displaystyle=x^{j}\frac{\partial\Xi^{e}_{j}}{\partial\eta_{k}}=Y_{e}^{k};$

hence $S_{e}$ generates $\Lambda_{e}$. The computation for $S_{\psi}$ using
(3.43) is very similar and gives

(3.46) $\frac{\partial S_{\psi}}{\partial
x^{j}}=\Xi_{j}^{\psi},\quad\frac{\partial
S_{\psi}}{\partial\xi_{i}}=-X^{i}_{\psi},\quad\frac{\partial
S_{\psi}}{\partial\eta_{k}}=Y^{k}_{\psi}.$

We have then established that $S_{\bullet}$ is a generating function for
$\Lambda_{\bullet}$, so we now consider the phase functions

(3.47) $\displaystyle\varphi_{e}(x^{I},x^{J},y,\xi_{I},\eta)$
$\displaystyle\equiv x^{i}\xi_{i}+y^{k}\eta_{k}+x^{j}\Xi^{e}_{j}$
$\displaystyle=x^{i}\xi_{i}+y^{k}\eta_{k}+S_{e}(x^{J},\xi_{I},\eta)\in\mathit{SG}^{(1),1}(\mathbb{R}^{2n}\times\mathbb{R}^{n+d}),$
$\displaystyle\varphi_{\psi}(x^{I},x^{J},y,\xi_{I},\eta)$ $\displaystyle\equiv
x^{i}\xi_{i}-y^{k}\eta_{k}-\xi_{i}X^{i}_{\psi}+\eta_{k}Y^{k}_{\psi}$
$\displaystyle=x^{i}\xi_{i}-y^{k}\eta_{k}+S_{\psi}(x^{J},\xi_{I},\eta)\in\mathit{SG}^{1,(1)}(\mathbb{R}^{2n}\times\mathbb{R}^{n+d}).$

Then $\operatorname{\mathrm{d}\\!}_{\,(\xi_{I},\eta)}\varphi_{\bullet}=0$ if
and only if (3.42) and (3.43) hold true. Then, computing the other derivatives
gives the desired parametrisation.

It remains to show that the functions $\varphi_{\bullet}$ can be realised as
the principal symbol of an $SG$-function. To this end, the methods of [CS17]
and [CDS19] still prove viable: using (2.18) and keeping in mind that the
tuples $(X^{i}_{e},X^{i}_{\psi},X^{i}_{\psi
e}),(\Xi^{e}_{j},\Xi_{j}^{\psi},\Xi_{j}^{\psi e})$ and
$(H_{k}^{e},H_{k}^{\psi},H_{k}^{\psi e})$ are principal symbols, we compute
$\sigma_{e}(\varphi_{\psi})-\sigma_{\psi}(\varphi_{e})$ restricted to the
graph of $C_{\psi e}$:

(3.48) $\displaystyle\sigma_{e}(\varphi_{\psi})$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}\varphi_{\psi}(\lambda
x,\xi_{I},\lambda y)$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}(\lambda
x^{i}\xi_{i}-\lambda y^{k}\eta_{k}-X^{i}_{\psi}(\lambda
x^{J},\xi_{I},\eta)\xi_{i}+Y^{k}_{\psi}(\lambda x^{J},\xi_{I},\eta))$
$\displaystyle=x^{i}\xi_{i}-y^{k}\eta_{k}-X^{i}_{\psi e}\xi_{i}+Y^{k}_{\psi
e}\eta_{k},$ $\displaystyle\sigma_{\psi}(\varphi_{e})$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}\varphi_{e}(x,\lambda\xi_{I},y)$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}(\lambda
x^{i}\xi_{i}+\lambda y^{k}\eta_{k}+x^{j}\Xi^{e}_{j}(x^{J},\lambda\xi_{I},y))$
$\displaystyle=x^{i}\xi_{i}+y^{k}\eta_{k}+x^{j}\Xi^{\psi e}_{j}$
$\displaystyle\implies\quad(\sigma_{\psi}(\varphi_{e})-\sigma_{e}(\varphi_{\psi}))|_{\Lambda_{\psi
e}}$ $\displaystyle=(2y^{k}\eta_{k}+x^{j}\Xi^{\psi e}_{j}+X^{i}_{\psi
e}\xi_{i}-Y^{k}_{\psi e}\eta_{k})|_{\Lambda_{\psi e}}$
$\displaystyle=X^{i}_{\psi e}\xi_{i}+x^{j}\Xi_{j}^{\psi e}+Y^{k}_{\psi
e}\eta_{k}=\left((x,y),(\xi,\eta)\right)|_{\Lambda_{\psi e}}$

However, recall that $\Lambda_{\psi e}$ is bi-conic, so the phase
$\varphi_{\psi e}$ parametrising it is bi-homogeneous of degree $\mathbbm{1}$
in $(x,y)$ and $(\xi_{I},\eta)$. Since on the graph we have
$(\xi,\eta)=\operatorname{\mathrm{d}\\!}_{(x,y)}\varphi_{\psi
e}(x,y,\xi_{I},\eta)$, we can apply Euler’s equation for homogeneous functions
twice to obtain

(3.49) $\displaystyle\left((x,y),(\xi,\eta)\right)|_{\Lambda_{\psi e}}$
$\displaystyle=\left((x,y),\operatorname{\mathrm{d}\\!}_{\,(x,y)}\varphi_{\psi
e}(x,y,\xi_{I},\eta)\right)$ $\displaystyle=\varphi_{\psi
e}(x,y,\xi_{I},\eta)|_{\Lambda_{\psi
e}}=\left((\xi_{I},\eta),\operatorname{\mathrm{d}\\!}_{\,(\xi_{I},\eta)}\varphi_{\psi
e}(x,y,\xi_{I},\eta)\right)$ $\displaystyle=0,$

where for the last equality we noticed that
$\operatorname{\mathrm{d}\\!}_{\,(\xi_{I},\eta)}\varphi_{\psi e}=0$ is exactly
the relation defining the set in $\mathbb{R}^{3n+d}$ which parametrises the
graph of $C_{\psi e}$. Therefore, on the graph of $C_{\psi e}$ we have that
$\sigma_{e}(\varphi_{\psi})=\sigma_{\psi}(\varphi_{e})$, which is the
compatibility condition for $SG$-principal symbols. This proves that
$(\varphi_{e},\varphi_{\psi})$ can be realised as the principal symbol of a
function
$\varphi\in\mathit{SG}^{\mathbbm{1}}(\mathbb{R}^{2n}\times\mathbb{R}^{n+d})$.
This concludes the proof.

∎

######  Remark 3.43.

Looking at the phase functions (3.47), it is clear that, in bi-conic
neighbourhoods of infinity, they actually belong to the class $\mathcal{Q}$.
Namely, they are given as a sum of two terms $f(x,\theta)+g(y,\theta)$ for
$\theta=(\xi_{I},\eta)$ satisfying appropriate $SG$ estimates.

######  Remark 3.44.

We remark that, while our discussion above was limited to the model case
$\mathbb{B}^{n}\times\mathbb{B}^{n}$, our definition of SCTs and the related
results all have a local character (in the sense that they can be checked in
coordinates near the corners). Therefore, they apply mutatis mutandis to
general scattering manifolds, their scattering cotangent bundles, and the
fibre-wise compactifications thereof. However, the theory of FIOs on
scattering manifolds has not yet reached a completely satisfactory status. In
particular, the concept of elliptic FIO in this setting has not yet been
defined and analysed to the extent that we need. So, for our purposes in the
coming Chapter 4 we will stick to the model case.

##  4\. Order-preserving isomorphisms

###  4.1. Preliminary definitions and auxiliary results

In this chapter, we work exclusively in the model case of $\mathbb{R}^{n}$ and
its compactification $\mathbb{S}^{n}_{+}$. Although we believe that most of
what follows should hold true in general for operators defined on an
asymptotically Euclidian manifold $X$ (or even scattering), the theory of FIOs
in this setting has not been studied in the required depth to allow us to
formulate certain results below. On the other hand the nature of the argument
is such that, given the existence of a sufficiently precise calculus
structure, the computations need only to be performed locally, thus reducing
them to the model case. Therefore our choice of fixing $X=\mathbb{S}^{n}_{+}$
and working with $SG$-classes below.

Recall that, until now, we have specialised to the sub-classes of classical
symbols, in order to study the analytical and geometrical properties of the
principal symbol. Here, we specialise further by assuming that the order of
the involved operators is $(m_{e},m_{\psi})\in\mathbb{Z}^{2}$. There will be
only a single exception to this rule, which will be mentioned explicitly.
Again, we always omit to write $cl,cl(e),cl(\psi)$ in the corresponding
notations in Definition 2.8.

We want to address the question of the order-preserving isomorphisms in the
$SG$-setting. Our main object of investigation is the following.

######  Definition 4.1.

Consider the algebra $\mathit{LG}$ and an algebra isomorphism (not necessarily
topological nor a *-isomorphism)
$\imath\colon\mathit{LG}\rightarrow\mathit{LG}$. We say that $\imath$ is an
SG-order preserving isomorphism (SGOPI) if for any $m\in\mathbb{Z}^{2}$ it
holds true

(4.1) $\imath(\mathit{LG}^{m})\subset\mathit{LG}^{m},$

that is, $\imath$ preserves the double filtration on $\mathit{LG}$.

The approach for this result is very much in line with the original paper
[DS76], but a number of differences arise, due to the introduction of the
second filtration. In particular, we have to work with products of manifolds
with boundary, and many of the ideas in [CDS19] and [MZ96], as we developed
further in the previous chapters, are useful.

For later reference, we list various, easy properties of SGOPIs in the
following lemma. These are direct algebraic consequences of Definition 4.1

######  Lemma 4.2.

Let $\imath$ be a SGOPI. Then:

  1. (1)

$\imath$ maps ideals to ideals and in particular maximal ideals to maximal
ideals;

  2. (2)

$\imath(\mathcal{R}G)=\mathcal{R}G$.

We will need to employ, in the course of the proof of Theorem 4.19, the
principle which has come to be known as Milnor’s exercise (namely Problem 1-C
in Section 1 of [MS74]). It is generally presented in the following form:

> For a compact smooth manifold $X$, the maximal ideals in
> $\mathcal{C}^{\infty}(X)$ are given by functions vanishing at a point.
> Namely, $I\triangleleft\mathcal{C}^{\infty}(X)$ is maximal if and only if
> $I=I_{p}\equiv\\{f\in\mathcal{C}^{\infty}(X)\text{ s.t. }f(p)=0\\}$ for some
> $p\in X$.

A direct corollary is that any algebra isomorphism
$F\colon\mathcal{C}^{\infty}(X)\rightarrow\mathcal{C}^{\infty}(Y)$ is induced
by a diffeomorphism $C\colon X\rightarrow Y$ via pull-back and, hence,
automatically continuous. For our purposes, we need to consider $X,Y$
manifolds with corners and an algebra isomorphism
$F\colon\mathcal{C}^{\infty}(\partial
X)\rightarrow\mathcal{C}^{\infty}(\partial Y)$, and ask ourselves the question
whether $F$ is also induced by a diffeomorphism $C\colon\partial
X\rightarrow\partial Y$. We state here a slight generalisation of this
principle, applicable to certain sub-algebras of continuous functions on a
compact topological space $X$. This version has arisen in a discussion with
Philipp Schmitt, concerning the minimal conditions which such a subalgebra has
to satisfy in order to be able to characterise the maximal ideals.

######  Proposition 4.3 (Milnor’s exercise).

Let $X$ be a compact topological space and $\mathcal{A}\subset\mathcal{C}(X)$
a sub-algebra having the same unit as $\mathcal{C}(X)$. Assume the following:

  1. (1)

$\mathcal{A}$ is spectrally invariant in $\mathcal{C}(X)$, namely,
$\mathcal{C}(X)^{-1}\cap\mathcal{A}=\mathcal{A}^{-1}$, where the superscript
$-1$ denotes the group of invertibles;

  2. (2)

$\mathcal{A}$ is closed under complex conjugation (or simply $\mathcal{A}$
consists of real-valued functions).

Then, every maximal ideal in $\mathcal{A}$ is of the form $I_{p}$ for some
$p\in X$. In particular it has codimension 1.

###### Proof.

Let $I\triangleleft\mathcal{A}$. We claim that there exists $p\in X$ such that
$f(p)=0$ for every $f\in I$. Arguing by contradiction, assume that for each
point $x\in X$ we can find a function $f_{x}\in I$ with $f_{x}(x)\neq 0$. In
particular, by continuity of the elements in $\mathcal{A}$ there exists an
open cover $\\{U_{x}\\}_{x\in X}$ of $X$ where $f_{x}(y)\neq 0$ for all $y\in
U_{x}$. By compactness, we can look at a finite sub-cover
$\\{U_{0},\dots,U_{n}\\}$ associated to the points $x_{0},\dots,x_{n}$ and the
functions $f_{0},\dots,f_{n}$. Then for all $i$,
$\left|f_{i}\right|^{2}=\overline{f_{i}}f_{i}$ are non-negative elements of
$I$ which only vanish (if anywhere) outside $U_{i}$. The pointwise sum
$f=\sum_{i=0}^{n}\left|f_{i}\right|^{2}$ is therefore everywhere positive and
belongs to $I$. By spectral invariance, $f$ is invertible in $\mathcal{A}$ so
$I=\mathcal{A}$, contradicting our assumption of maximality. The proof is
complete. ∎

Applying this to $\mathcal{C}^{\infty}(B_{sc}X)$ and
$\mathcal{C}^{\infty}(B_{sc}Y)$ (algebras which clearly satisfy the conditions
above), for two scattering manifolds, $X,Y$ gives that any algebraic
isomorphism is induced by a diffeomorphism $B_{sc}X\rightarrow B_{sc}Y$.
Notice that there is a little extra structure hidden here: smooth functions of
$B_{sc}X$ are actually pairs of smooth functions on manifolds with boundary
together with an identification of the boundaries, so this really means that
we obtain a triple of compatible diffeomorphisms. In the model case
$\mathbb{B}^{n}\times\mathbb{B}^{n}$, this excludes directly the possibility
of the symplectic rotation of Remark 3.41. For the sake of clarity and to push
the analogy between scattering and $SG$ as far as possible, we give in Lemma
4.5 an argument adapted to this situation.

Next, we give a complete proof of (an adaptation of) the spectral argument
used in [MM17] to exclude the possibility of a skew-symplectic diffeomorphism.

######  Lemma 4.4.

Let $\imath$ be an SGOPI. Then the following holds true:

  1. (1)

If $A\in\mathit{LG}^{m}$ is elliptic, $\imath(A)$ is elliptic as well;

  2. (2)

Let $A_{\bullet}$ be a $\bullet$-order reduction, that is
$A_{\bullet}\in\mathit{LG}^{\mathbbm{1}_{\bullet}}$ is elliptic and self-
adjoint, and let $B_{\bullet}=\imath(A_{\bullet})$. If $a_{\bullet}>0$ and
$\mathcal{I}\mathit{m}\,b_{\bullet}=0$, we also have $b_{\bullet}>0$.

###### Proof.

  1. (1)

Consider a parametrix $R$ of $A$. Then there exist operators
$K_{1},K_{2}\in\mathcal{R}G$ such that $AR-\delta=K_{1},RA-\delta=K_{2}$.
Applying $\imath$ to these relations gives immediately that $\imath(R)$ is a
parametrix of $\imath(A)$, therefore $\imath(A)$ is elliptic.

  2. (2)

First notice that the assumption of self-adjointness is not really
restrictive, since any elliptic operator with positive symbol is equal to a
self-adjoint one modulo lower order operators. Therefore, assume
$A_{\bullet}\in\mathit{LG}^{\mathbbm{1}_{\bullet}}$ has the required
properties and let $B_{\bullet}=\imath(A_{\bullet})$. By the previous point,
$B_{\bullet}$ is elliptic as well, so its symbol can be either positive or
negative everywhere in view of the assumption $b_{\bullet}\in\mathbb{R}$.
Assume, arguing by contradiction, that $b_{\bullet}<0$. We can then find an
operator $N_{\bullet}\in\mathit{LG}^{0}$ so that
$B_{\bullet}=B_{\bullet}^{W}+N_{\bullet}$ where $B_{\bullet}^{W}$ is the Weyl
operator associated with $b_{\bullet}$. Notice, in particular, that
$B_{\bullet}$ is a bounded perturbation of its Weyl counterpart. By
assumption, $B_{\bullet}^{W}$ is unbounded self-adjoint and has real spectrum
bounded from above. We can also estimate

(4.2) $\mathrm{spec}(B_{\bullet})\subset\\{\lambda\in\mathbb{C}\text{ s.t.
}\mathrm{dist}(\lambda,\mathrm{spec}(B_{\bullet}^{W}+N_{\bullet}))\leq\|N_{\bullet}\|\\}.$

We conclude that there exists a constant $K_{\bullet}\in\mathbb{R}$ such that
$B_{\bullet}^{W}+N_{\bullet}-t$ is invertible for all
$t\notin(-\infty,K_{\bullet}]\times[-\|N_{\bullet}\|,\|N_{\bullet}\|]$, with
inverse being an operator lying in $\mathit{LG}^{-\mathbbm{1}_{\bullet}}$. Let
now $M_{\bullet}=\imath^{-1}(N_{\bullet})$ and consider
$A_{\bullet}+M_{\bullet}-t$, which has to be invertible with inverse in
$\mathit{LG}^{-\mathbbm{1}_{\bullet}}$ for the same $t$’s. The spectrum of
$A_{\bullet}$ is real and bounded from below since $A_{\bullet}$ is positive,
moreover $M_{\bullet}$ is bounded, so that, just like in (4.2), the spectrum
of $A_{\bullet}+M_{\bullet}$ is unbounded but contained in a tube
$[D_{\bullet},+\infty)\times[-\|M_{\bullet}\|,\|M_{\bullet}\|]$ for some
$D_{\bullet}\in\mathbb{R}$. Then, $A_{\bullet}+M_{\bullet}-t$ is invertible
for all $t$ outside this set, but, at the same time, there exists at least one
$\tilde{t}_{\bullet}\in\operatorname{\mathrm{spec}}(B_{\bullet}^{W}+N_{\bullet})$
for which $A_{\bullet}+M_{\bullet}-\tilde{t}_{\bullet}$ cannot be invertible,
since the spectra are unbounded. This is a contradiction. We conclude that
$b_{\bullet}$ has to be positive as well, completing the proof.

∎

###  4.2. The case of the formal symbol algebra

We begin our investigation with the formal symbol algebra
$\mathcal{B}G={}^{\mathit{LG}}\\!\\!\diagup_{\mathcal{R}G}\cong{}^{\mathit{SG}}\\!\\!\diagup_{\mathit{SG}^{-\infty\mathbbm{1}}}$.
At this level, we can exploit the explicit relation between (asymptotic
expansions of) symbols and operators.

######  Lemma 4.5.

Given an SGOPI $\imath$, there exists a scattering canonical transformation
$C\colon\partial(\mathbb{B}^{n}\times\mathbb{B}^{n})\rightarrow\partial(\mathbb{B}^{n}\times\mathbb{B}^{n})$
such that for all $(a_{\bullet})\in\Sigma G^{m}$ it holds true
$\imath(a_{\bullet})=a_{\bullet}\circ C_{\bullet}^{-1}$.

###### Proof.

The algebraic properties of $\imath$ guarantee that

$\imath\left(\mathit{LG}^{m_{1},m_{2}}\diagup\mathcal{R}G\right)=\mathit{LG}^{m_{1},m_{2}}\diagup\mathcal{R}G.$

$\imath$ also acts on the space of principal symbols. Indeed, it preserves
$\mathit{SG}^{-\mathbbm{1}_{e}}$, $\mathit{SG}^{-\mathbbm{1}_{\psi}}$,
$\mathit{SG}^{-\mathbbm{1}_{e}}\oplus\mathit{SG}^{-\mathbbm{1}_{\psi}}$ as
ideals in $\mathit{SG}^{0}$, so it descends to a map $\imath_{pr}$ on $\Sigma
G^{0}\cong\mathcal{C}^{\infty}(\partial(\mathbb{S}^{n}_{+}\times\mathbb{S}^{n}_{+}))$,
whose elements can be identified with pairs of functions $(a_{e},a_{\psi})$ on
the respective (open) boundary face
$\widetilde{\mathcal{W}}_{e}=\mathbb{S}^{n-1}\times\mathbb{R}^{n},\widetilde{\mathcal{W}}_{\psi}=\mathbb{R}^{n}\times\mathbb{S}^{n-1}$,
having the same “limit” in the corner $\widetilde{\mathcal{W}}_{\psi
e}=\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}$. That is, they extends smoothly to
the whole $\partial(\mathbb{S}_{+}^{n}\times\mathbb{S}_{+}^{n})$. We have then
the commutative diagram

(4.3) $\begin{CD}\mathit{LG}^{0}@>{\imath}>{}>\mathit{LG}^{0}\\\
@V{\sigma_{pr}}V{}V@V{\sigma_{pr}}V{}V\\\ \Sigma
G^{0}@>{\imath_{pr}}>{}>\Sigma G^{0},\end{CD}$

meaning that we have maps $\imath_{\bullet}$ satisfying

(4.4)
$\sigma_{pr}(\imath(A))=(\imath_{e}a_{e},\imath_{\psi}a_{\psi},\imath_{\psi
e}a_{\psi e}).$

In view of the multiplicative properties of $\sigma_{pr}$ in Proposition 2.16
we see that the maps $\imath_{\bullet}$ are multiplicative on the respective
spaces. We can then apply a Milnor-type argument to obtain bijections of
$\widetilde{\mathcal{W}}_{\bullet}$. Let $I^{\bullet}$ be a maximal ideal in
$\mathcal{C}^{\infty}(\widetilde{\mathcal{W}}_{\bullet})$. This is given by
the set $I^{\bullet}_{p^{\bullet}}$ of those functions on
$\widetilde{\mathcal{W}}_{\bullet}$ which vanish at
$p^{\bullet}\in\widetilde{\mathcal{W}}_{\bullet}$. Then, $\imath_{\bullet}$
gives a correspondence
$\chi_{\bullet}\colon\widetilde{\mathcal{W}}_{\bullet}\rightarrow\widetilde{\mathcal{W}}_{\bullet}$
defined by
$\imath_{\bullet}(I^{\bullet}_{p^{\bullet}})=I^{\bullet}_{\chi_{\bullet}(p^{\bullet})}$.

We may repeat the same argument with $\imath^{-1}$ to obtain another triple of
bijections
$\zeta_{\bullet}\colon\widetilde{\mathcal{W}}_{\bullet}\rightarrow\widetilde{\mathcal{W}}_{\bullet}$.
By writing
$I^{\bullet}_{p}=\imath_{\bullet}^{-1}\imath_{\bullet}(I^{\bullet}_{p})$ we
find then that $\zeta_{\bullet}=\chi_{\bullet}^{-1}$. Furthermore, we see that
it holds true

(4.5) $\imath_{\bullet}a_{\bullet}=a_{\bullet}\circ\chi_{\bullet}^{-1};$

indeed for
$a_{\bullet}\in\mathcal{C}^{\infty}(\widetilde{\mathcal{W}}_{\bullet})$ we
have $a_{\bullet}-a_{\bullet}(p^{\bullet})1\in I^{\bullet}_{p_{\bullet}}$,
hence
$\imath_{\bullet}(a_{\bullet})(\chi_{\bullet}(p^{\bullet}))-a_{\bullet}(p^{\bullet})1=0$,
as claimed. We remark here in addition that the identification of principal
symbols of order $0$ with smooth (in the sense of Remark 3.11) functions on
$\partial(\mathbb{B}^{n}\times\mathbb{B}^{n})$ is canonical, since it does not
depend on the choice of a boundary defining function.

These maps must be smooth. To see this, for example, for $\bullet=e$, it
suffices to choose local coordinates $(\theta^{i},\xi_{j})$ on
$\widetilde{\mathcal{W}}_{e}$ near a point $p_{e}$. By definition, these
coordinates are smooth function on $\mathbb{S}^{n-1}\times\mathbb{R}^{n}$, so
they can be identified canonically with homogeneous symbols
$\tilde{\theta}^{i},\tilde{\xi}_{j}$ of order $0$, namely, with elements of
$\mathit{SG}^{(0),1}$. We can then apply $\imath$ to obtain

(4.6) $\displaystyle\theta^{i}\circ\chi_{e}^{-1}$
$\displaystyle=\imath(\tilde{\theta}_{i})\in\mathit{SG}^{(0),1},$
$\displaystyle\xi_{j}\circ\chi_{e}^{-1}$
$\displaystyle=\imath(\tilde{\xi}_{j})\in\mathit{SG}^{(0),1}.$

It follows that the components of $\chi_{e}^{-1}$ are smooth functions by
composition, so $\chi_{e}^{-1}$ is smooth itself. The same argument, using
$\imath_{e}^{-1}$, gives that $\chi_{e}$ is actually a diffeomorphism.

Having determined the action of $\imath$ on principal symbols of order
$(0,0)$, we use order reductions to extend it to $\Sigma G^{m}$ for any
$m\in\mathbb{Z}^{2}$. Namely, recalling Lemma 2.32 we write
$A\in\mathit{LG}^{m}$ as

(4.7) $A=P^{m_{e}}Q^{m_{\psi}}B$

for $B\in\mathit{LG}^{0}$ and $P$, resp. $Q$, an $e$-order reduction, resp. a
$\psi$-order reduction. Thus the image of $A$ via $\imath$ can be computed as

(4.8) $\imath(A)=\tilde{P}^{m_{e}}\tilde{Q}^{m_{\psi}}\imath(B),$

where
$\tilde{P}=\imath(P)\in\mathit{LG}^{\mathbbm{1}_{e}},\tilde{Q}=\imath(Q)\in\mathit{LG}^{\mathbbm{1}_{\psi}}$
are elliptic and $\imath(B)$ is an operator of order $(0,0)$. To determine the
action of $\imath$ on principal symbols of any order it suffices then to
describe it on the order reductions. Looking at the pairs picture of principal
symbols (cf. Proposition 2.16), we see that
$\sigma_{pr}(\imath(B))=(b_{e}\circ\chi_{e}^{-1},b_{\psi}\circ\chi_{\psi}^{-1})$
and

(4.9)
$\sigma_{pr}(\imath(A))=(\tilde{p}_{e}^{m_{e}}\tilde{q}_{e}^{m_{\psi}}b_{e}\circ\chi_{e}^{-1},\tilde{p}_{\psi}^{m_{e}}\tilde{q}_{\psi}^{m_{\psi}}b_{\psi}\circ\chi_{\psi}^{-1})$

for $(\tilde{p}_{e},\tilde{p}_{\psi})$ and $(\tilde{q}_{e},\tilde{q}_{\psi})$
the principal symbols of $\tilde{P}$ and $\tilde{Q}$, respectively. In
particular we analyse closely the case $m=\mathbbm{1}$, for which we know that
$\Sigma G^{\mathbbm{1}}$ is a Lie algebra by Lemma 2.34.

For any $a,\alpha\in\mathit{SG}^{\mathbbm{1}}$ consider the relation
$\imath\\{\sigma_{pr}(\alpha),\sigma_{pr}(a)\\}=\sigma_{pr}(\\{\imath(\alpha),\imath(a)\\})$
and write $a$ and $\alpha$ as in (4.7), namely

(4.10) $\displaystyle a$ $\displaystyle=pqb,$ $\displaystyle\alpha$
$\displaystyle=pq\beta.$

Denoting $r=pq,\tilde{r}=\imath(r)$, we pass to principal symbols and look at
the single components. Let us work out in detail what happens for $\bullet=e$,
since for the case $\bullet=\psi$ the same proof suffices up to an exchange in
the homogeneities, and the exit behaviour is the main novelty here. The above
relation for the Poisson brackets reads

(4.11)
$\displaystyle\\{\tilde{r}_{e}b_{e}\circ{\chi_{e}^{-1}},\tilde{r}_{e}\beta_{e}\circ\chi_{e}^{-1}\\}$
$\displaystyle=\imath_{e}(\\{r_{e}b_{e},r_{e}\beta_{e}\\})$
$\displaystyle=\imath_{e}(r_{e})\imath_{e}(r_{e})^{-1}\imath(\\{r_{e}b_{e},r_{e}\beta_{e}\\})$
$\displaystyle=\tilde{r}_{e}\imath(r_{e}^{-1}\\{r_{e}b_{e},r_{e}\beta_{e}\\}).$

In particular, if we choose $\beta_{e}=1$, we obtain that, for any $b_{e}$,

(4.12)
$\\{\tilde{r}_{e},b_{e}\circ\chi^{-1}_{e}\\}=\\{r_{e},b_{e}\\}\circ\chi^{-1}_{e}.$

We claim now that we can extend the map $\chi_{e}$ homogeneously to a map
$C_{e}$ so that $\tilde{r}_{e}=r_{e}\circ C_{e}^{-1}.$ This choice will give
that $\\{r_{e}\circ C_{e}^{-1},b_{e}\circ C_{e}^{-1}\\}=\\{r_{e},b_{e}\\}\circ
C_{e}^{-1}$, so that going back to (4.11) we find that for any two symbols
$a,\alpha$ of order $\mathbbm{1}$ it holds true

(4.13) $\\{\alpha_{e}\circ C_{e}^{-1},a_{e}\circ
C_{e}^{-1}\\}=\\{\alpha_{e},a_{e}\\}\circ C_{e}^{-1}.$

This is equivalent to $C_{e}$ being a canonical transformation.

To see that we can indeed make such a choice, recall that homogeneous maps in
$\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}$ are written as in (3.31) for some
$f_{e}$ real-valued and smooth on $\mathbb{S}^{n-1}\times\mathbb{R}^{n}$. We
now define $C_{e}$ to be the 1-homogeneous extension of $\chi_{e}$ given by

(4.14)
$C_{e}(\rho_{1},\theta,\xi)=\left(\frac{{p}_{e}(1,\theta,\xi)}{\tilde{p}_{e}(1,\chi_{e}(\theta,\xi))}\rho_{1},\chi_{e}(\theta,\xi)\right).$

Recall that here $\theta$ are coordinates on $\mathbb{S}^{n-1}$,
$\rho_{1}\in\mathbb{R}^{+}$ and $\xi\in\mathbb{R}^{n}$. This choice satisfies
$\tilde{r}_{e}=r_{e}\circ C_{e}^{-1}$. Indeed, notice that both $q_{e}$ and
$\tilde{q}_{e}$ do not play a rôle here since they are $e$-homogeneous of
degree $0$. More precisely,
$q_{e}(1,\theta,\xi)=\tilde{q}_{e}(1,\chi_{e}(\theta,\eta))$. Keeping this in
mind, and denoting $\chi_{e}(\theta,\xi)=(\varphi,\eta)$ with
$\sigma_{1}\in\mathbb{R}^{+}$ the newly introduced coordinate in the target
space, the check is immediate, using the $e$-homogeneity of $p_{e}$ and
$\tilde{p}_{e}$:

(4.15) $\displaystyle\tilde{r}_{e}(\sigma_{1},\varphi,\eta)$
$\displaystyle=\tilde{p}_{e}(\sigma_{1},\varphi,\eta)\tilde{q_{e}}(1,\varphi,\eta)$
$\displaystyle r_{e}\circ C_{e}^{-1}(\sigma_{1},\varphi,\eta)$
$\displaystyle=r_{e}\left(\frac{{p}_{e}(1,\chi_{e}^{-1}(\varphi,\eta)}{\tilde{p}_{e}(1,\varphi,\eta)}\sigma_{1},\varphi,\eta\right)$
$\displaystyle=\frac{{p}_{e}(1,\chi_{e}^{-1}(\varphi,\eta)}{\tilde{p}_{e}(1,\varphi,\eta)}\sigma_{1}\cdot
p_{e}(1,\chi_{e}^{-1}(\varphi,\eta))q_{e}(1,\chi_{e}^{-1}(\varphi,\eta))$
$\displaystyle=\tilde{p}(\sigma_{1},\varphi,\eta)q_{e}(1,\chi_{e}^{-1}(\varphi,\eta)).$

With this choice, $C_{e}$ is then a homogeneous diffeomorphism
$\mathcal{W}_{e}\rightarrow\mathcal{W}_{e}$, preserving the Poisson bracket
for $e$-principal symbols.

Having constructed the required extensions, it follows that $\chi$ is a
scattering canonical transformation. The proof is complete. ∎

By Theorem 3.42, we can locally parametrise the graph of $\chi$ via $SG$-phase
functions of order $\mathbbm{1}$ and type $\mathcal{Q}_{gen}$. Covering
$\operatorname{\mathrm{graph}}\chi$ with these coordinate patches and picking
local amplitudes, we can construct an elliptic FIO $F$ of type
$\mathcal{Q}_{gen}$ associated with $\chi$, in the sense that its principal
symbol can be identified with a function on the graph. If we denote by
$F^{\\#}$ the parametrix of $F$, which is again elliptic of type
$\mathcal{Q}_{gen}$ and is associated with the inverse map $\chi^{-1}$, we
have then that $\jmath(P)\equiv FPF^{\\#}$ is an automorphism of
${}^{\mathit{LG}}\\!\\!\diagup_{\mathcal{R}G}$ preserving principal symbols,
in view of Theorem 2.47. Our next goal is to analyse $\jmath$ more closely by
mirroring the argument of [DS76] and refining it to $SG$-classes. Before we
start with that task, we need to prove an auxiliary result, adapted from
Theorem 2.2.10 in [AM08].

######  Lemma 4.6.

Consider $X=\mathbb{R}^{n}_{0}\times\mathbb{R}^{k}$ and the space
$\mathcal{H}^{m,l}(X)$ of functions of $(x,y)\in X$ being positively
homogeneous of degree $m$ in $x$ and $y-$classical of degree $l$.
$\mathcal{H}=\bigcup\mathcal{H}^{m,l}$ is a filtered algebra. Consider a
derivation $\theta\colon\mathcal{H}\rightarrow\mathcal{H}$. Then there exists
$V\in\mathfrak{X}(X)$ such that $\theta=\mathcal{L}_{V}$ as a derivation.

###### Proof.

Let us make a few remarks to begin with. First, given a vector field
$V\in\mathfrak{X}(X)$, the Lie derivative $\mathcal{L}_{V}$ is well defined
for functions in $\mathcal{H}\subset\mathcal{C}^{\infty}(X)$. In particular,
it is a derivation. However, for a general $V$, there is no guarantee that
$V(\mathcal{H})\subset\mathcal{H}$ since the local expressions of the
coefficients of $V$ need not be $y-$classical.

Second, $\theta$ is a local operator on $\mathcal{H}$, i.e. if
$a\in\mathcal{H},(x_{0},y_{0})\dot{\in}U$ and $a|_{U}=0$, then
$\theta(a)(x_{0},y_{0}))=0$. To see this, pick a smooth cut-off function $g$
such that $g=1$ on $U\dot{\ni}(x_{0},y_{0}),U\subset V$ and $g=0$ outside $V$.
It follows that $a=(1-g)a$ everywhere, so by the derivation property and the
assumptions we have

(4.16) $\displaystyle\theta(a)(x_{0},y_{0})$
$\displaystyle=\theta(a)(x_{0},y_{0})(1-g(x_{0},y_{0}))-\theta(g)(x_{0},y_{0})a(x_{0},y_{0})$
$\displaystyle=0,$

as required.

We can now proceed to the main part of the proof. The locality property
implies that we can define restrictions of $\theta$ to open subsets
$(x,y)\dot{\in}V\subset X$ by

(4.17) $\theta|_{V}(a)(x,y)\equiv\theta(ga)(x,y),$

where $g$ is a smooth cut-off such that $g=0$ outside $V$ and $g=1$ on some
$(x,y)\dot{\in}U\subset V$. Again by locality, it follows that $\theta|_{V}$
doesn’t actually depend on the choice of such a cut-off. We keep denoting by
$\theta$ the restrictions to open subsets.

Pick a chart $(U,\rho)$ on $X$ with coordinates $(x^{i},y^{\alpha})$, let
$p\in U$ and $a\in\mathcal{H}$. Assume $\rho(p)=q=(x_{0}^{i},y_{0}^{\alpha})$.
Then, we can write, in a sufficiently small $W\dot{\ni}q$ and denoting
$c_{q}(t)\equiv q+t(x^{i}-x^{i}_{0},y^{\alpha}-y^{\alpha}_{0})$

(4.18) $\displaystyle({\rho}_{\ast}a)(x^{i},y^{\alpha})$
$\displaystyle=({\rho}_{\ast}a)(q)+\int_{0}^{1}\frac{\partial}{\partial
t}[({\rho}_{\ast}a)(c_{q}(t))]\operatorname{\mathrm{d}\\!}t$
$\displaystyle=({\rho}_{\ast}a)(q)+(x^{i}-x^{i}_{0})\int_{0}^{1}\frac{\partial{\rho}_{\ast}a}{\partial
x^{i}}(c_{q}(t))\operatorname{\mathrm{d}\\!}t+(y^{\alpha}-y^{\alpha}_{0})\int_{0}^{1}\frac{\partial{\rho}_{\ast}a}{\partial
y^{\alpha}}(c_{q}(t))\operatorname{\mathrm{d}\\!}t.$

Let $U^{\prime}=\rho^{-1}(W)\dot{\ni}p$ and $u\in U^{\prime}$. There exist
then functions $g_{i},g_{\alpha}\in\mathcal{C}^{\infty}(U)$ such that

(4.19) $g_{i}(p)=\frac{\partial{\rho}_{\ast}a}{\partial x^{i}}(q),\quad
g_{\alpha}(p)=\frac{\partial{\rho}_{\ast}a}{\partial y^{\alpha}}(q)$

and

(4.20)
$a(u)=a(p)+(x^{i}-x^{i}_{0})g_{i}(u)+(y^{\alpha}-y^{\alpha}_{0})g_{\alpha}(u).$

We apply $\theta$ to (4.20) to obtain

$\theta(a)(u)=\theta(x^{i})(u)g_{i}(u)+(x^{i}-x^{i}_{0})\theta(g_{i})(u)+\theta(y^{\alpha})(u)g_{\alpha}(u)+(y^{\alpha}-y^{\alpha}_{0})\theta(g_{\alpha})(u),$

so that, evaluating this expression at $p$ and using (4.19), we have

(4.21) $\theta(a)(p)=\theta(x^{i})(p)\frac{\partial{\rho}_{\ast}a}{\partial
x^{i}}(p)+\theta(y^{\alpha})(p)\frac{\partial{\rho}_{\ast}a}{\partial
y^{\alpha}}(p).$

It is readily seen that a change of coordinates does not affect this
expression. We define then a vector field $V_{\rho}$ on $U$ by setting

(4.22)
$V_{\rho}(x^{i},y^{\alpha})\equiv\left((x^{i},y^{\alpha}),(\theta(x^{i})(u),\theta(y^{\alpha})(u))\right)$

for $\rho(u)=(x^{i},y^{\alpha})$. It follows that $V_{\rho}|_{U}$ is
independent of the chart, so that the collection of these objects defines a
vector field $V\in\mathfrak{X}(X)$. Now, by definition, the Lie derivative
with respect to $V$ in a local chart $(U,\rho)$ of $a\in\mathcal{H}$ is

(4.23) $\displaystyle\mathcal{L}_{V}a|_{U}$
$\displaystyle=D(a\circ\rho^{-1})(x^{i},y^{\alpha})V_{\rho}(x^{i},y^{\alpha})$
$\displaystyle=\frac{\partial}{\partial
x^{i}}(a\circ\rho^{-1})(x^{i},y^{\alpha})\theta(x^{i})(x^{i},y^{\alpha})+\frac{\partial}{\partial
y^{\alpha}}(a\circ\varphi^{-1})(x^{i},y^{\alpha})\theta(y^{\alpha})(x^{i},y^{\alpha})$
$\displaystyle=\theta(a)(u),$

and that the claim follows. The proof is complete. ∎

######  Remark 4.7.

This lemma shows that it suffices to have a derivation on (certain)
subalgebras of $\mathcal{C}^{\infty}(\mathbb{R}^{n}_{0}\times\mathbb{R}^{n})$
to determine a vector field on $\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}$. We
will find use for this fact in the proof of Lemma 4.10 below. In addition, we
will see that the properties of the subalgebra (in this case, homogeneity in
$x$ and classicality of order $0$ in $y$) are reflected in the properties of
the coefficients of the obtained vector field.

######  Remark 4.8.

Notice that Duistermaat and Singer do not need this specialised result. Indeed
in their setting they obtain derivations on the whole
$\mathcal{C}^{\infty}(\mathbb{S}^{\ast}X)$, which are given by vector fields
by the standard theory.

We can now begin our analysis of the map $\jmath$, which will lead to the
following first main result.

######  Theorem 4.9.

Assume given an automorphism
$\jmath\colon{}^{\mathit{LG}}\\!\\!\diagup_{\mathcal{R}G}\rightarrow{}^{\mathit{LG}}\\!\\!\diagup_{\mathcal{R}G}$
preserving principal symbols, namely
$\jmath(P)-P\in{}^{\mathit{LG}^{m-\mathbbm{1}}}\\!\\!\diagup_{\mathcal{R}G}$
whenever $P\in\mathit{LG}^{m}$. Then, $\jmath$ is given by conjugation with
some elliptic $B\in\mathit{LG}^{s}$ for some $s\in\mathbb{C}^{2}$.

For the sake of clarity, we split the proof into a series of lemmata.

######  Lemma 4.10.

Assume that, for some $l\geq 1$, we have
$\jmath(P)-P\in\mathit{LG}^{m-l\mathbbm{1}}$ for any $P\in\mathit{LG}^{m}$
with principal symbol $(p_{e},p_{\psi})$. Then $\sigma_{pr}(\jmath(P)-P)$ only
depends on the principal symbol of $P$ and it is obtained as
$(\beta_{e}p_{e},\beta_{\psi}p_{\psi})$ for two vector fields
$\beta_{e},\beta_{\psi}$. Moreover these vector fields are Hamiltonian, that
is, there exist functions $f_{e},f_{\psi}$ such that
$\beta_{\bullet}p_{\bullet}=H_{f_{\bullet}}p_{\bullet}\equiv\\{f_{\bullet},p_{\bullet}\\}$.

###### Proof.

For a fixed $l\geq 1$ consider, for all $m\in\mathbb{Z}$, the map
$Z^{m}\colon\mathit{LG}^{m}\rightarrow{}^{\mathit{LG}^{m-l\mathbbm{1}}}\\!\\!\diagup_{\mathit{LG}^{m-(l+1)\mathbbm{1}}}$,
$Z^{m}(P)\equiv\jmath(P)-P\mod\mathit{LG}^{m-(l+1)\mathbbm{1}}$. $Z^{m}$ only
depends on the principal symbol of $P$. Indeed, if $Q=P+W$ for some
$W\in\mathit{LG}^{m-\mathbbm{1}}$, it follows that

(4.24) $\displaystyle Z^{m}(Q)$ $\displaystyle=Z^{m}(P)+Z^{m}(W)$
$\displaystyle=\jmath(P)-P+\jmath(W)-W\mod\mathit{LG}^{m-(l+1)\mathbbm{1}}$
$\displaystyle=Z^{m}(P)$

since, by assumption, $\jmath(W)-W\in\mathit{LG}^{m-(l+1)\mathbbm{1}}$. Hence,
by composition with the principal symbol map, $Z^{m}$ descends to a map
$\beta^{m}\colon\Sigma G^{m}\rightarrow\Sigma G^{m-le}$. We show that
$\beta^{m}$ (or rather the direct sum
$\beta=\bigoplus_{m\in\mathbb{Z}^{2}}\beta^{m}$) is a bi-derivation of the bi-
algebra $\Sigma G$. To this end, consider $Z(PQ)$ for some operators
$P\in\mathit{LG}^{m},Q\in\mathit{LG}^{k}$. Recalling the algebraic properties
of $\jmath$, we have by definition,

$\displaystyle Z^{m+k}(PQ)$
$\displaystyle=\jmath(PQ)-PQ\mod\mathit{LG}^{m+k-l\mathbbm{1}}$
$\displaystyle=\jmath(P)\jmath(Q)-\jmath(P)Q+\jmath(P)Q-PQ\mod\mathit{LG}^{m+k-l\mathbbm{1}}$
$\displaystyle=\jmath(P)Z^{k}(Q)+Z^{m}(P)Q\mod\mathit{LG}^{m+k-l\mathbbm{1}}$
$\displaystyle=Z^{m}(P)Z^{k}(Q)+PZ^{k}(Q)+Z^{m}(P)Q\mod\mathit{LG}^{m+k-l\mathbbm{1}}$
$\displaystyle=PZ^{k}(Q)+Z^{m}(P)Q\mod\mathit{LG}^{m+k-l\mathbbm{1}},$

where we noticed that
$Z^{m}(P)Z^{k}(Q)\in\mathit{LG}^{m-l\mathbbm{1}}\cdot\mathit{LG}^{k-l\mathbbm{1}}\subset\mathit{LG}^{m+k-2l\mathbbm{1}}\subset\mathit{LG}^{m+k-l\mathbbm{1}}$.
Taking principal symbols gives the Leibniz rule. Similarly, for $Z[P,Q]$ we
obtain

$Z^{m+k-\mathbbm{1}}[P,Q]=[Z^{m}(P),Q]+[P,Z^{k}(Q)]\mod\mathit{LG}^{m+k-(l+1)\mathbbm{1}},$

so that $\beta$ is a derivation with respect to the Poisson bracket.
Therefore, $\beta$ acts as a bi-derivation on the space of principal symbols
$\Sigma G$. Keeping in mind the pairs picture of Proposition 2.16, denote the
action of $\beta$ as

(4.25) $\beta(p_{\psi},p_{e})=(\beta_{\psi}p_{\psi},\beta_{e}p_{e}).$

Similarly to (4.24), one sees that $\beta_{\psi}p_{\psi}$, respectively
$\beta_{e}p_{e}$, only depends on the component $p_{\psi}$, respectively
$p_{e}$, of the principal symbol, and it holds true that
$\sigma_{\psi}^{m_{\psi}-l}(\beta_{e}p_{e})=\sigma_{e}^{m_{e}-l}(\beta_{\psi}p_{\psi})$.
We can write, more explicitly,

(4.26) $\displaystyle\beta_{\psi}p_{\psi}$
$\displaystyle=\sigma_{\psi}^{m_{\psi}-l}(\beta\check{p})$
$\displaystyle\beta_{e}p_{e}$
$\displaystyle=\sigma_{e}^{m_{e}-l}(\beta\check{p}).$

Applying Lemma 4.6, we obtain that both $\beta_{\psi}$ and $\beta_{e}$ are
given by vector fields on $\mathbb{R}^{n}\times\mathbb{R}^{n}_{0}$ and
$\mathbb{R}^{n}_{0}\times\mathbb{R}^{n}$, respectively. We have then

(4.27) $\displaystyle\beta_{\psi}$
$\displaystyle=\gamma^{i}_{\psi}\frac{\partial}{\partial
x^{i}}+\rho^{\psi}_{k}\frac{\partial}{\partial\xi_{k}},$
$\displaystyle\beta_{e}$ $\displaystyle=\gamma^{i}_{e}\frac{\partial}{\partial
x^{i}}+\rho^{e}_{k}\frac{\partial}{\partial\xi_{k}},$

where, by definition,
$\gamma_{\bullet}^{i}=\beta_{\bullet}x^{i},\rho^{\bullet}_{k}=\beta_{\bullet}\xi_{k}$.
In particular, by definition of $\beta$, it holds true that

(4.28) $\displaystyle\gamma^{i}_{\psi}(x,\xi)$
$\displaystyle=\beta_{\psi}x^{i}\in\mathit{SG}^{(-l),1-l}_{cl(x)},$
$\displaystyle\rho_{k}^{\psi}(x,\xi)$
$\displaystyle=\beta_{\psi}\xi_{k}\in\mathit{SG}^{(1-l),-l}_{cl(x)},$
$\displaystyle\gamma_{e}^{i}(x,\xi)$
$\displaystyle=\beta_{e}x^{i}\in\mathit{SG}^{-l,(1-l)}_{cl(\xi)},$
$\displaystyle\rho^{e}_{k}(x,\xi)$
$\displaystyle=\beta_{e}\xi_{k}\in\mathit{SG}^{1-l,(-l)}_{cl(\xi)},$

so that the components of the obtained vector fields mirror the extra
properties of the algebra of functions from which they are derived. Recall
also that $\\{x^{i},x^{j}\\},\\{x^{i},\xi_{j}\\}$ and $\\{\xi_{i},\xi_{j}\\}$
are all constant, hence, if we apply $\beta_{\bullet}$, to them we obtain 0.
On the other hand, by using the derivation property, we see that it must hold
true that

(4.29)
$\frac{\partial\gamma^{i}_{\bullet}}{\partial\xi_{j}}=\frac{\partial\gamma^{j}_{\bullet}}{\partial\xi_{i}},\quad\frac{\partial\rho^{\bullet}_{i}}{\partial
x^{j}}=\frac{\partial\rho^{\bullet}_{j}}{\partial
x^{i}},\quad\frac{\partial\gamma_{\bullet}^{i}}{\partial
x^{j}}=-\frac{\partial\rho^{\bullet}_{j}}{\partial\xi^{i}}.$

But this is the same as saying that the (symplectic) dual 1-form to
$\beta_{\bullet}$ is closed, hence locally equal to
$\operatorname{\mathrm{d}\\!}f_{\bullet}$ for some smooth $f_{\bullet}$.
Locally, we have then shown $\beta_{\bullet}=H_{f_{\bullet}}$, the Hamiltonian
vector field defined by $f_{\bullet}$. The proof is complete. ∎

The next step consists in establishing under which conditions $\jmath$ is
given by conjugation with an $SG\Psi$DO at the level of principal symbols.

######  Lemma 4.11.

There exist $B\in\mathit{LG}^{s}$, $s\in\mathbb{C}^{2}$, such that
$\sigma_{pr}(\jmath(P))=\sigma_{pr}(BPB^{-1})$ if and only if
$\sigma_{pr}(B)=(e^{-\operatorname{\mathrm{i}}f_{e}},e^{-\operatorname{\mathrm{i}}f_{\psi}})$
for functions $f_{\bullet}$ such that $\beta_{\bullet}=H_{f_{\bullet}}$.

###### Proof.

Notice first that $BPB^{-1}-P=[B,P]B^{-1}$, so that taking principal symbols
yields

(4.30) $\displaystyle\sigma_{pr}(BPB^{-1}-P)$
$\displaystyle=\sigma_{pr}([B,P])\sigma_{pr}(B^{-1})$
$\displaystyle=\left(\frac{1}{\operatorname{\mathrm{i}}b_{e}}\\{b_{e},p_{e}\\},\frac{1}{\operatorname{\mathrm{i}}b_{\psi}}\\{b_{\psi},p_{\psi}\\}\right)$
$\displaystyle=\left(H_{\operatorname{\mathrm{i}}\log
b_{e}}(p_{e}),H_{\operatorname{\mathrm{i}}\log b_{\psi}}(p_{\psi})\right).$

Remark that, for an invertible $b$, both logarithms exist. Then applying Lemma
4.10 with $l=1$ gives that
$\sigma_{pr}(BPB^{-1}-P)=\sigma_{pr}(\jmath(P)-P)=(H_{f_{e}}(p_{e}),H_{f_{\psi}}(p_{\psi}))$
for some smooth $f_{e},f_{\psi}$ if and only if

(4.31) $\left\\{\begin{aligned}
b_{\psi}&=e^{-\operatorname{\mathrm{i}}f_{\psi}}\\\
b_{e}&=e^{-\operatorname{\mathrm{i}}f_{e}}\end{aligned}\right.$

are homogeneous in $\xi$, respectively $x$, of degree $s_{\psi}$, respectively
$s_{e}$. Using Euler’s equation, we can rewrite this as

(4.32) $\left\\{\begin{aligned} \xi_{k}\frac{\partial
f_{\psi}}{\partial\xi_{k}}&=\operatorname{\mathrm{i}}s_{\psi},\\\
x^{j}\frac{\partial f_{e}}{\partial
x^{j}}&=\operatorname{\mathrm{i}}s_{e}.\end{aligned}\right.$

Recalling that $\partial_{\xi_{k}}{f_{\bullet}}$ and $\partial_{x^{j}}{f_{e}}$
are, respectively, the components $\gamma_{\psi}^{k}$ and $\rho_{j}^{e}$ of
the vector fields $\beta_{\bullet}$, we see that the claim is equivalent to
$\xi_{k}\gamma^{k}_{\psi}$ and $x^{j}\rho_{j}^{e}$ being constant. We check
this directly by computing the derivatives of these expressions w.r.t. $x^{r}$
and $\xi_{r}$. Considering, for instance, the vector field $\beta_{\psi}$, we
have
$\partial_{x^{j}}{\gamma_{\psi}^{i}}=-\partial_{\xi^{i}}{\rho^{\psi}_{j}}$ and
$\partial_{\xi_{j}}{\gamma^{i}_{\psi}}=\partial_{\xi_{i}}{\gamma^{j}_{\psi}}$,
in view of the symmetry relations of (4.29). Then, recalling the homogeneities
of (4.28) and that $l=1$, we obtain

(4.33) $\displaystyle\frac{\partial(\xi_{k}\gamma^{k}_{\psi})}{\partial
x^{r}}$ $\displaystyle=\xi_{k}\frac{\partial\gamma^{k}_{\psi}}{\partial
x^{r}}=-\xi_{k}\frac{\partial\rho^{\psi}_{r}}{\partial\xi_{k}}=0,$
$\displaystyle\frac{\partial(\xi_{k}\gamma^{k}_{\psi})}{\partial\xi_{r}}$
$\displaystyle=\gamma^{r}_{\psi}+\xi_{k}\frac{\partial\gamma^{k}_{\psi}}{\partial\xi_{r}}=\gamma^{r}_{\psi}+\xi_{k}\frac{\partial\gamma^{r}_{\psi}}{\partial\xi_{k}}=0.$

In a completely analogous fashion one proves the corresponding results for
$\beta_{e}$. This concludes the proof. ∎

######  Lemma 4.12.

Assume that for some $l>1$ we have
$\jmath(P)-P\in\mathit{LG}^{m-l\mathbbm{1}}$ for any $P\in\mathit{LG}^{m}$.
Then there exist $C\in\mathit{LG}^{(1-l)\mathbbm{1}}$ such that
$(I-C)\circ\jmath(P)\circ(I-C)^{-1}-P\in\mathit{LG}^{m-(l+1)\mathbbm{1}}$.

###### Proof.

We apply Lemma 4.10 again by defining directly the Hamiltonian functions of
the vector fields. Recall that $\gamma^{j}_{\bullet}$ and $\rho_{k}^{\bullet}$
are the components of the vector field $\beta_{\bullet}$ determining the
action of $\jmath$ on principal symbols. Using these functions, set

(4.34) $\left\\{\begin{aligned}
c_{\psi}&\equiv\frac{1}{1-l}\xi_{j}\gamma^{j}_{\psi}\in\mathit{SG}^{(1-l),1-l}_{cl(x)},\\\
c_{e}&\equiv\frac{1}{l-1}x^{j}\rho^{e}_{j}\in\mathit{SG}^{1-l,(1-l)}_{cl(\xi)}.\end{aligned}\right.$

With these definitions, we see that $H_{c_{\bullet}}=\beta_{\bullet}$. Indeed,
for instance

(4.35) $\displaystyle\frac{\partial c_{\psi}}{\partial x^{k}}$
$\displaystyle=\frac{1}{1-l}\xi_{j}\frac{\partial\gamma^{j}_{\psi}}{\partial
x^{k}}=\frac{1}{l-1}\xi_{j}\frac{\partial\rho^{\psi}_{k}}{\partial\xi_{j}}$
$\displaystyle=\frac{1}{l-1}(1-l)\rho^{\psi}_{k}=-\rho^{\psi}_{k},$
$\displaystyle\frac{\partial c_{\psi}}{\partial\xi_{k}}$
$\displaystyle=\frac{1}{1-l}\left[\gamma^{k}_{\psi}+\xi_{j}\frac{\partial\gamma^{j}_{\psi}}{\partial\xi_{k}}\right]$
$\displaystyle=\frac{1}{1-l}\left[\gamma^{k}_{\psi}+\xi_{j}\frac{\partial\gamma_{\psi}^{k}}{\partial\xi_{j}}\right]=\frac{1}{1-l}\left[\gamma^{k}_{\psi}-l\gamma^{k}_{\psi}\right]$
$\displaystyle=\gamma^{k}_{\psi},$

where we have used again (4.28) and (4.29). Wit the aim of assiociating an
operator with $c=(c_{e},c_{\psi})$, we verify that, indeed, $c\in\Sigma
G^{(1-l)\mathbbm{1}}$, namely, that
$\sigma^{1-l}_{e}(c_{\psi})=\sigma_{\psi}^{1-l}(c_{e})$. Computing these
symbols, we have to prove that

(4.36) $\xi_{j}\gamma^{j}_{\psi,1-l}=-x^{j}\rho_{j}^{e,1-l},$

where $\gamma^{j}_{\psi,1-l}$, respectively $\rho_{j}^{e,1-l}$, is the
$(1-l)$-homogeneous component in the asymptotic expansion of
$\gamma^{j}_{\psi}$, respectively $\rho_{j}^{e}$. To this end, consider the
third relation in (4.29), multiply it by $\xi_{k}$ and take the trace to
obtain

(4.37) $\xi_{k}\frac{\partial\gamma^{k}_{\psi}}{\partial
x^{j}}=-\xi_{k}\frac{\partial\rho^{\psi}_{k}}{\partial\xi_{j}}=(l-1)\rho^{\psi}_{k}.$

Here we took again advantage of the homogeneity properties in (4.28). The
first and last sides of (4.37) are $x-$classical symbols, so we can expand
them in $x-$homogeneous functions. Since asymptotic expansions are uniquely
determined, we find that the two must be equal term by term, so the top order
relation reads

(4.38) $\xi_{k}\frac{\partial\gamma^{k}_{\psi,1-l}}{\partial
x^{j}}=(l-1)\rho^{\psi,-l}_{j}.$

Multiplying by $x^{j}$ (namely, taking the trace of the matrix
$(x^{r}\xi_{k}\partial_{x^{j}}\gamma^{k}_{\psi,1-l})$), we obtain by
homogeneity

(4.39) $(l-1)x^{j}\rho_{j}^{\psi,-l}=(1-l)\xi_{j}\gamma^{j}_{\psi,1-l}.$

On the other hand, considering $p=\xi_{r}$ as a symbol and applying $\beta$
gives

(4.40) $\left.\begin{aligned} (\beta
p)_{\psi}&=\rho^{\psi}_{r}\in\mathit{SG}^{(1-l),-l}_{cl(x)}\\\ (\beta
p)_{e}&=\rho^{e}_{r}\in\mathit{SG}^{1-l,(-l)}_{cl(\xi)}\end{aligned}\right\\}\implies\rho^{\psi,-l}_{r}=\rho^{e,1-l}_{r},$

where we computed the principal symbol of $\beta p$. Since (4.40) holds true
for any $r$, it follows that we can substitute it in (4.39) to obtain (4.36),
as required.

Let then $C\in\mathit{LG}^{-l\mathbbm{1}}$ be an operator whose principal
symbol is $\operatorname{\mathrm{i}}c$ and let $(I-C)^{\\#}$ be a parametrix
of $(I-C)$, that is

$(I-C)(I-C)^{\\#}=(I-C)^{\\#}(I-C)=I+R$

for some $R$ smoothing. Recall here that $l>1$ so $C$ has negative integer
order and thus $I-C\in\mathit{LG}^{0}$ with $\sigma^{0}_{pr}(I-C)=1$.
Therefore, the parametrix of $I-C$ exists and has a compact remainder since
$R$ has kernel in the Schwartz class. Moreover, $(I-C)^{\\#}=I-C^{\prime}$ for
some $C^{\prime}\in\mathit{LG}^{-\mathbbm{1}}$. Let then
$P\in\mathit{LG}^{m}$. By commuting $P$ and $I-C$ and noticing that $[I,P]=0$,
we obtain

(4.41) $\displaystyle(I-C)^{\\#}P(I-C)$
$\displaystyle=(I-C)^{\\#}\left((I-C)P+[P,I-C]\right)$
$\displaystyle=(I-C)^{\\#}(I-C)P+(I-C)^{\\#}[C,P]$
$\displaystyle=P+[C,P]\mod\mathit{LG}^{m-(l+1)\mathbbm{1}}.$

Thence, the principal symbol of $(I-C)^{\\#}P(I-C)-P$ equals the principal
symbol of the commutator, namely, the Poisson bracket of the principal
symbols. Specifically, we have

(4.42) $\displaystyle\sigma_{pr}\left((I-C)^{\\#}P(I-C)-P\right)$
$\displaystyle=-\operatorname{\mathrm{i}}H_{\operatorname{\mathrm{i}}c}(p)=H_{c}(p)$
$\displaystyle=\\{c,p\\}=\beta(p)$ $\displaystyle=\sigma_{pr}(\jmath(P)-P).$

It follows that
$\jmath(P)-P=(I-C)^{\\#}P(I-C)-P\mod\mathit{LG}^{m-(l+1)\mathbbm{1}}$, so in
particular there exists $Q\in\mathit{LG}^{m-(l+1)\mathbbm{1}}$ such that
$\jmath(P)=(I-C)^{\\#}P(I-C)+Q$. Conjugating with the parametrix $(I-C)^{\\#}$
gives now

(4.43) $\displaystyle(I-C)\jmath(P)(I-C)^{\\#}$
$\displaystyle=(I-C)(I-C)^{\\#}P(I-C)(I-C)^{\\#}+(I-C)Q(I-C)^{\\#}$
$\displaystyle=P+RP+PR+RPR+(I-C)Q(I-C)^{\\#}.$

Now, on the one hand $PR$ and $RP$ are smoothing, since $R\in\mathcal{R}G$, on
the other hand we have, thanks to Theorem 2.47,

(4.44) $(I-C)Q(I-C)^{\\#}\in\mathit{LG}^{m-(l+1)\mathbbm{1}}.$

Thence, it holds true that
$(I-C)^{\\#}\jmath(P)(I-C)-P\in\mathit{LG}^{m-(l+1)\mathbbm{1}}$. The proof is
complete. ∎

###### Proof of Theorem 4.9.

Exploiting Lemmas 4.11 and 4.12 we can set up an inductive procedure which
constructs a sequence of pseudo-differential operators
$B_{0},C_{1},C_{2},\dots$, where:

  1. (1)

$B_{0}$ is elliptic of some order $s\in\mathbb{C}$ and
$C_{j}\in\mathit{LG}^{-j\mathbbm{1}}$;

  2. (2)

conjugation with $B_{l}=(I-C_{l})\dots(I-C_{1})B_{0}$ gives an automorphism of
${}^{\mathit{LG}}\\!\\!\diagup_{\mathcal{R}G}$, approximating $\jmath$ up to
order $m-(l+2)\mathbbm{1}$.

A computation of the asymptotic expansion of the symbol of $B_{l}$ shows that
$(I-C_{l+1})B_{l}$ only changes the symbol up to $s-(l+1)\mathbbm{1}$. There
exists, then, an elliptic operator $B\in\mathit{LG}^{s}$, such that, for each
$l$, we have $B-B_{l}\in\mathit{LG}^{s-(l+1)\mathbbm{1}}.$ Thus, the
difference $B\jmath(P)B^{-1}-P$ is smoothing, in view of said asymptotic
expansion. This proves the claim. ∎

The preceding results proven thus far enable us to prove the main result of
this section, namely the characterisation of the SGOPS of the formal symbol
algebra $\mathcal{B}G$.

######  Theorem 4.13 (OPIs of the formal $SG$-algebra).

Let $\imath\colon\mathcal{B}G\rightarrow\mathcal{B}G$ be an $SG$OPI on the
formal symbol algebra $\mathcal{B}G$. There exists then an elliptic SGFIO $A$,
of type $\mathcal{Q}_{gen}$, such that
$\imath(P+\mathcal{R}G)=A^{\\#}PA+\mathcal{R}G$ for any $P\in\mathit{LG}^{m}$.

###### Proof.

On the one hand, we know that there exists an elliptic SGFIO $F$ of type
$\mathcal{Q}_{gen}$ such that $F\imath(P)F^{\\#}-P$ is an automorphism of
$\mathcal{B}G$ preserving principal symbols. On the other hand, Theorem 4.9
guarantees that every such automorphism is given by conjugation with an
elliptic $B\in\mathit{LG}^{s}$ for some $s\in\mathbb{C}^{2}$. Therefore, we
see that, mod $\mathcal{R}G$, $F\imath(P)F^{\\#}=BPB^{\\#}$, so that setting
$A\equiv B^{\\#}F$ gives that $\imath(P)=A^{\\#}PA$. This concludes the proof.
∎

###  4.3. Lifting the characterisation to $\mathit{LG}$

We now turn to the problem of lifting this characterisation to the whole
algebra. We notice first that we are able to take advantage of the Eidelheit
Lemma of [DS76] without any further hassle.

######  Lemma 4.14 (Eidelheit-type Lemma).

Given an algebra isomorphism
$\varphi\colon\mathcal{R}G\rightarrow\mathcal{R}G$ there exists a topological
isomorphism $V\colon\mathcal{S}\rightarrow\mathcal{S}$ such that
$\varphi(P)=VPV^{-1}$ for any $P\in\mathcal{R}G$.

###### Proof.

We show that the assumptions of Lemma 3 in [DS76] hold true for
$\mathcal{E}=\widetilde{\mathcal{E}}=\mathcal{S},\mathcal{U}=\tilde{\mathcal{U}}=\mathcal{R}G$.
Indeed, $\mathcal{S}$ is an infinite dimensional Fréchet space and
$\mathcal{R}G$ comprises linear bounded operators on $\mathcal{S}$. Moreover,
for $u,v\in\mathcal{S}$, the rank 1 operator $u\otimes v$ lies in
$\mathcal{R}G$. Then, picking a sequence $v_{j}$ converging to $v$ in the weak
topology, we see that $u\otimes v_{j}$ converges to $u\otimes v$ in the
operator topology, so that
$\mathcal{S}^{\prime}=\mathcal{F}=\widetilde{\mathcal{F}}$ in the notation of
[DS76]. The claim follows then directly from the quoted result. ∎

On the other hand, Lemma 4 of [DS76] is not as straightforward to generalise
to $SG$-operators. Indeed, when looking at the proof there, one is confronted
with the possible incapability of choosing a function
$u\in\mathcal{S}(\mathbb{R}^{n})$ with the property that $u(x)\neq u(y)$ for
each $x,y\in\mathbb{R}^{n}$. Accordingly, it is not clear whether such a claim
is at all true. We set out then to prove directly that the composition of the
Eidelheit isomorphism with the FIO coming from the formal algebra is a
multiple of the identity up to some operator with Schwartz kernel. Consider to
this end the composition $E=VA$ of the Eidelheit isomorphism $V$ with the FIO
$A$ coming from Theorem 4.13.

######  Lemma 4.15.

$E\colon\mathcal{S}\rightarrow\mathcal{S}$ is bounded and extends to a bounded
operator $E\colon\mathit{HG}_{k\mathbbm{1}}\rightarrow\mathit{L}^{2}$ for some
$k\in\mathbb{N}$.

###### Proof.

$E$ is clearly bounded as an operator $\mathcal{S}\rightarrow\mathit{L}^{2}$.
Then, there is a finite set of semi-norms $\\{p_{0},\dots,p_{n}\\}$ on
$\mathcal{S}$ which estimate $\|Eu\|_{\mathit{L}^{2}}$, namely,
$\|Eu\|_{\mathit{L}^{2}}\leq\max_{i\in\\{0,\dots,n\\}}p_{i}(u)$. Thus, there
is an integer $k$ so that $\|E\|_{\mathit{L}^{2}}\leq\|u\|_{\mathit{H}_{k}}$.
This shows that $E$ extends as a bounded operator
$\mathit{HG}_{k\mathbbm{1}}\rightarrow\mathit{L}^{2}$. The proof is complete.
∎

We choose now order reductions $P,Q$, as in the proof of Lemma 4.5 and
consider $K\equiv ER^{-k}$ as an operator
$\mathit{L}^{2}\rightarrow\mathit{L}^{2}$, where we denote $R=PQ$. Our goal is
to prove that $K$ is an $SG$-pseudo-differential operator of order $(0,0)$.
For this, we look at commutators and use the characterisation of Schrohe
[Sch87] of $SG$-pseudo-differential operators on the weighted Sobolev spaces
$\mathit{HG}_{(l,k)}$. Here and later we write $\operatorname{\mathrm{ad}}K$
for the operator on $\mathit{LG}$ acting by commutation with $K$, namely
$(\operatorname{\mathrm{ad}}K)(P)=[K,P]$. We owe the idea of the following
strategy to Ryszard Nest, whom we thank for the helpful suggestion. We start
with the following easy lemma.

######  Lemma 4.16.

$\operatorname{\mathrm{ad}}{K}$ preserves the double filtration and $K$
extends to a bounded operator
$K\colon\mathit{HG}_{r\mathbbm{1}}\rightarrow\mathit{HG}_{r\mathbbm{1}}$ for
every $r\in\mathbb{R}^{2}$.

###### Proof.

Fix $r\in\mathbb{R}^{2}$ and consider $v\in\mathit{HG}_{r\mathbbm{1}}$.
Setting $v_{0}\equiv\Lambda^{r\mathbbm{1}}v$ for $\Lambda$ an elliptic
$SG\Psi$DO of order $\mathbbm{1}$, we have that $v_{0}\in\mathit{L}^{2}$ and

$Kv=K\Lambda^{-r\mathbbm{1}}v_{0}=K\Lambda^{-r\mathbbm{1}}K^{-1}Kv_{0}.$

Remarking that $\operatorname{\mathrm{ad}}{E}$ preserves the double filtration
of $\mathit{LG}$, we see that

$KPK^{-1}=\Lambda^{-l\mathbbm{1}}E\Lambda^{-k\mathbbm{1}}P\Lambda^{k\mathbbm{1}}E^{-1}\Lambda^{l\mathbbm{1}},$

so that also $\operatorname{\mathrm{ad}}{K}$ preserves the double filtration.
It follows directly that
$(\operatorname{\mathrm{ad}}{K})(\Lambda^{-r\mathbbm{1}})$ has order
$-r\mathbbm{1}$, and since $Kv_{0}\in\mathit{L}^{2}$ by assumption, we have
$Kv\in\mathit{HG}_{r\mathbbm{1}}$. ∎

######  Proposition 4.17.

$K$ is a (non necessarily classical) $SG$-pseudo-differential operator of
order $(0,0)$.

###### Proof.

We prove that for every $\alpha,\beta\in\mathbb{N}^{n}$ there exists an
operator
$R^{\alpha}_{\beta}\in\mathit{LG}^{-\left|\alpha\right|,-\left|\beta\right|}$
such that, continuously,

(4.45)
$(\operatorname{\mathrm{ad}}M_{x})^{\alpha}(\operatorname{\mathrm{ad}}\partial)^{\beta}K=R^{\alpha}_{\beta}K\colon\mathit{HG}_{r}\rightarrow\mathit{HG}_{r+(\left|\alpha\right|,\left|\beta\right|)},$

where $M_{x^{j}}$ is the multiplication operator by $x^{j}$. This is known by
[Sch88] to be equivalent to $K\in\mathit{LG}^{0}$. We show first that, for
every $\beta\in\mathbb{N}^{n}$, there exists
$Q_{\beta}\in\mathit{LG}^{0,-\left|\beta\right|}$ such that

(4.46) $(\operatorname{\mathrm{ad}}\partial)^{\beta}K=Q_{\beta}K.$

We argue by induction on $\left|\beta\right|$.

For $\left|\beta\right|=1$ we have

$\displaystyle(\operatorname{\mathrm{ad}}\partial_{x^{j}})K$
$\displaystyle=\left[\partial_{x^{j}},K\right]=(\partial_{x^{j}}-K\partial_{x^{j}}K^{-1})K$
$\displaystyle=Q_{j}K.$

Here, $Q_{j}\in\mathit{LG}^{0,-1}$, since $\operatorname{\mathrm{ad}}K$ is an
automorphism preserving the principal symbol. So the base step holds true.

Assume the claim holds true for every $\left|\beta\right|\leq r$ and consider
then $\gamma\in\mathbb{N}^{n}$ with $\left|\gamma\right|=r+1$. Then
$\gamma=\beta+\mathbbm{1}_{j}$ for some $j\in\\{1,\dots n\\}$ and some $\beta$
with $\left|\beta\right|=n$. We write

$\displaystyle(\operatorname{\mathrm{ad}}\partial)^{\gamma}K$
$\displaystyle=(\operatorname{\mathrm{ad}}\partial_{x^{j}})\left[(\operatorname{\mathrm{ad}}\partial)^{\beta}K\right]$
$\displaystyle=(\operatorname{\mathrm{ad}}\partial_{x^{j}})(Q_{\beta}K)=(\operatorname{\mathrm{ad}}\partial_{x^{j}})(Q_{\beta})K+Q_{\beta}(\operatorname{\mathrm{ad}}\partial_{x^{j}})K$
$\displaystyle=\left[\partial_{x^{j}},Q_{\beta}\right]K+Q_{\beta}Q_{j}K,$

having used the inductive hypothesis twice and the properties of
$\operatorname{\mathrm{ad}}$. Now, $Q_{\beta}Q_{j}=\widetilde{Q}_{\beta
j}\in\mathit{LG}^{0,-\left|\beta\right|-1}$ by composition, on the other hand
$\partial_{x^{j}}Q_{\beta},Q_{\beta}\partial_{x^{j}}\in\mathit{LG}^{1,-\left|\beta\right|}$.
However, they have the same principal symbol, so that in view of 5. in
Proposition 2.16 we have $[\partial_{x^{j}},Q_{\beta}]=\check{Q}_{\beta
j}\in\mathit{LG}^{0,-\left|\beta\right|-1}$. It follows now that

(4.47) $(\operatorname{\mathrm{ad}}\partial)^{\gamma}K=(\check{Q}_{\beta
j}+\widetilde{Q}_{\beta j})K\equiv Q_{\gamma}K,$

with
$Q_{\gamma}\in\mathit{LG}^{0,-\left|\beta\right|-1}=\mathit{LG}^{0,-\left|\gamma\right|}$.
By induction, then, (4.46) holds true for any $\beta\in\mathbb{N}^{n}$, as
claimed.

We now prove (4.45) for any $\alpha,\beta\in\mathbb{N}^{n}$. For
$\left|\alpha\right|=0$ there is nothing to prove. For clarity’s sake, we
spell out the case $\alpha=\mathbbm{1}_{j}$. Then,

$\operatorname{\mathrm{ad}}x^{j}(\operatorname{\mathrm{ad}}\partial)^{\beta}K=\operatorname{\mathrm{ad}}x^{j}(Q_{\beta}K)=\left[x^{j},Q_{\beta}\right]K+Q_{\beta}\left[x^{j},K\right].$

Similarly as above $[x^{j},K]=(x^{j}-Kx^{j}K^{-1})K=\widetilde{P}_{j}K$ with
$\widetilde{P}_{j}\in\mathit{LG}^{-1,0},$ so that
$Q_{\beta}\widetilde{P}_{j}=\widetilde{R}^{j}_{\beta}\in\mathit{LG}^{-1,-\left|\beta\right|}$
by composition. On the other hand,
$[x^{j},Q_{\beta}]=\overline{R}^{j}_{\beta}\in\mathit{LG}^{-1,-\left|\beta\right|}$
in view of the observation above about the order of commutators in the
$SG$-calculus.

Assume now that (4.47) holds true for any $\alpha\in\mathbb{N}^{n}$ such that
$\left|\alpha\right|\leq n$, and let $\gamma=\alpha+\mathbbm{1}_{j}$ for some
$j$. Then, using the properties of $\operatorname{\mathrm{ad}}$ as a
derivation,

(4.48)
$\displaystyle(\operatorname{\mathrm{ad}}x)^{\gamma}(\operatorname{\mathrm{ad}}\partial)^{\beta}(K)$
$\displaystyle=\operatorname{\mathrm{ad}}x^{j}(\operatorname{\mathrm{ad}}x)^{\alpha}(\operatorname{\mathrm{ad}}\partial)^{\beta}(K)$
$\displaystyle=\operatorname{\mathrm{ad}}x^{j}(P^{\alpha}_{\beta}K)=\left[x^{j},P^{\alpha}_{\beta}\right]K+P^{\alpha}_{\beta}\left[x^{j},K\right]$
$\displaystyle=\left(\left[x^{j},P^{\alpha}_{\beta}\right]+P^{\alpha}_{\beta}P^{j}\right)K,$

with $P^{\alpha}_{\beta}$ given by inductive hypothesis and $P^{j}$ given by
the previous step with $\left|\beta\right|=0$. Now, $\widetilde{P}^{\alpha
j}_{\beta}\equiv\left[x^{j},P^{\alpha}_{\beta}\right]\in\mathit{LG}^{-\left|\alpha\right|-1,-\left|\beta\right|}$
by assumption and the properties of $[\,,]$. On the other hand, by
composition, $\overline{P}^{\alpha j}_{\beta}\equiv
P^{\alpha}_{\beta}P^{j}\in\mathit{LG}^{-\left|\alpha\right|-1,-\left|\beta\right|}$
as well. Thus, setting

(4.49) $P^{\alpha+\mathbbm{1}_{j}}_{\beta}\equiv\widetilde{P}^{\alpha
j}_{\beta}+\overline{P}^{\alpha j}_{\beta}$

it follows
$P^{\gamma}_{\beta}\in\mathit{LG}^{-\left|\gamma\right|,-\left|\beta\right|}$.
The induction is complete.

Armed with this relation, it is now easy to show that the required mapping
properties hold true. Indeed, $K$ maps
$\mathit{HG}_{r}\rightarrow\mathit{HG}_{r}$ continuously for each
$r\in\mathbb{R}^{2}$ by Lemma 4.16. On the other hand,
$P^{\alpha}_{\beta}\in\mathit{LG}^{-\left|\alpha\right|,-\left|\beta\right|}$
gives exactly that
$P^{\alpha}_{\beta}\colon\mathit{HG}_{r}\rightarrow\mathit{HG}_{r+(\left|\alpha\right|,\left|\beta\right|)}$
continuously for each $r\in\mathbb{R}^{2}$. The composition
$P^{\alpha}_{\beta}K$ satisfies then the same properties. Thus, we have proven
the characterization (4.45) and $K$ is a pseudo-differential operator of
$SG$-type of order $0,0$. ∎

Notice, in addition, that, while Lemmas 4.15 and 4.16, together with
Proposition 4.17, imply that $E$ is a pseudo-differential operator as well,
its order is not necessarily integral. Therefore,
$\operatorname{\mathrm{ad}}E$ is not an inner automorphism, cf. also [MM17].
This is the reason why, in general, we cannot expect to obtain an FIO of
integer order.

If we start with the inverse of the Eidelheit isomorphism, $V^{-1}$, we
obtain, by the same argument, another pseudo-differential operator,
$\tilde{E}$. They satisfy $EP-
PE\in\mathcal{R}G,\tilde{E}P-P\tilde{E}\in\mathcal{R}G$, for any
$P\in\mathit{LG}^{m}$. We notice that this means, in particular, that $E$
almost commutes with Shubin operators since
$\Gamma^{m}(\mathbb{R}^{n})\subset\mathit{SG}^{m,m}$. Notice that here we are
disregarding classicality, notion which has different meanings for $\Gamma$
and $\mathit{SG}$. However, Lemma 4.18 below, suggested in a private
communication by Elmar Schrohe, is a statement about smooth, bounded functions
on $\mathbb{R}^{2n}$ which does not need classicality in any sense. Therefore,
it can be proven almost exactly as in the Master thesis of Robert Hesse [Hes].
We reproduce here the proof since the aforementioned work may not be readily
available.

######  Lemma 4.18.

Let $E,\tilde{E}\colon\mathcal{S}\rightarrow\mathcal{S}$ be $\Psi$DOs of
$SG$-type, parametrices of each other, such that
$[E,P],[\tilde{E},P]\in\mathcal{R}G$ for each $P\in\mathit{LG}$. Then,
$E=cI+R$ for some $c\in\mathbb{C},R\in\mathcal{R}G$.

###### Proof.

First, notice that the conditions on $E$ and $\tilde{E}$ imply that their
symbols $e,\tilde{e}$ are of order at most 0 and their derivatives are rapidly
decreasing. Indeed, $\\{e,p\\}\in\mathcal{S}\,\forall
p\in\mathit{SG}\iff\partial_{x}e,\partial_{\xi}e\in\mathcal{S}$, and it
follows that $e$ is bounded. Moreover, $\nabla e$ is a conservative vector
field with potential $e$. Namely, for each path
$\gamma\colon[a,b]\rightarrow\mathbb{R}^{2n}$ we have

(4.50)
$e(\gamma(a))-e(\gamma(b))=\int_{a}^{b}\nabla(e)(\gamma(s))\cdot\dot{\gamma}(s)\operatorname{\mathrm{d}\\!}s.$

Fix now some point $z$ in $\mathbb{S}^{2n-1}$ (an oriented direction in
$\mathbb{R}^{2n}$) so that, for $1<t_{1}<t_{2}$, it holds true

(4.51) $e(t_{2}z)-e(t_{1}z)=\int_{t_{1}}^{t_{2}}\nabla(e)(sz)\cdot
z\operatorname{\mathrm{d}\\!}s.$

By assumption, $\nabla e$ has rapidly decaying component. It follows that, for
any $v\in\mathbb{R}^{2n}$, we have $\left|\nabla
e(v)\right|\lesssim_{N}\braket{v}^{-N}$ and, for each $M\geq 2$, we can
estimate the integral as

(4.52) $\left|\int_{t_{1}}^{t_{2}}\nabla e(sz)\cdot
z\operatorname{\mathrm{d}\\!}s\right|\lesssim_{M}\braket{s}^{2-2M}|_{t_{1}}^{t_{2}}.$

So, the integral converges to 0 uniformly in $t_{2}$ as we take the limit
$t_{1}\rightarrow\infty$. We can, on the other hand, also estimate
$\left|e(t_{2}z)-e(t_{1}z)\right|\lesssim_{k}\braket{t_{1}}^{k}$. In
particular then, we can pass to the limit $t_{2}\rightarrow\infty$ in this
expression to obtain, for the radial limit
$l(z)\equiv\lim_{t\rightarrow\infty}e(tz)$, the bound

(4.53) $\left|l(z)-e(tz)\right|\lesssim_{k}\braket{t}^{-k}.$

We claim now that $l(z)$ does not depend on $z$, tht is, the radial limit is
constant on the sphere $\mathbb{S}^{2n-1}$. To see this, choose another $w\neq
z$ on $\mathbb{S}^{2n-1}$ and assume, possibly after having applied an
orthogonal transformation, that $z=(1,0,\dots,0)$ and
$w=(\cos\alpha,\sin\alpha,0,\dots,0)$. We consider a family of paths
$\gamma_{t}\colon[0,\alpha]\rightarrow\mathbb{R}^{2n}$ given by
$\gamma_{t}(\theta)=(t\cos\theta,t\sin\theta,0,\dots,0)$. Pick an
$\varepsilon>0$. Using (4.53) for both directions $z,w$ we know that there
exists a $T>1$ such that for all $t\geq T$ we have
$\left|l(z)-e(tz)\right|<\varepsilon$ and
$\left|l(w)-e(tw)\right|<\varepsilon$. On the other hand, for each fixed
$t\geq T$, we have

(4.54) $e(tz)-e(tw)=\int_{\gamma_{t}}\nabla
e(z)\cdot\operatorname{\mathrm{d}\\!}z=\int_{0}^{\alpha}\nabla
e(\gamma_{t}(\theta))\cdot\dot{\gamma}_{t}(\theta)\operatorname{\mathrm{d}\\!}\theta,$

so that taking absolute values and noticing that
$\left|\gamma_{t}\right|=\left|\dot{\gamma}_{t}\right|=t$, we estimate, for
each $M\geq 2$,

(4.55) $\left|e(tz)-e(tw)\right|\leq\int_{0}^{\alpha}\left|\nabla
e(\gamma_{t}(\theta))\cdot\dot{\gamma}_{t}(\theta)\right|\operatorname{\mathrm{d}\\!}\theta\lesssim_{M}\frac{\alpha
t}{\braket{t}^{M}},$

where we again used the fact that $\nabla e$ has rapidly decaying components.
Clearly, for each $M$, the right-hand side decays to zero. We collect then

(4.56)
$\left|l(w)-l(z)\right|\leq\left|l(w)-e(tw)\right|+\left|e(tz)-l(z)\right|+\left|e(tw)-e(tz)\right|\leq
3\varepsilon,$

which shows that $l(z)=l(w)$ as claimed. We let $c$ be the constant value of
$l$ on $\mathbb{S}^{2n-1}$ and look at the function $f=e-c$. Clearly, $f$ has
rapidly decreasing derivatives. On the other hand, we can compute, for
$z\in\mathbb{S}^{2n-1}$, that

(4.57) $f(tz)=e(tz)-l(z)=-\int_{t}^{+\infty}\nabla e(sz)\cdot
z\operatorname{\mathrm{d}\\!}s$

and conclude, as before, that $f(tz)$ is rapidly decreasing as a function of
$t$. As above, the convergence to zero is also uniform in $\mathbb{S}^{2n-1}$
and we discover that $f$ is rapidly decreasing itself. Summing up, we have
proven that $f\in\mathcal{S}(\mathbb{R}^{2n})$.

Now, set $R=\operatorname{\mathrm{Op}}(f)\in\mathcal{R}G$. Then, $e=c+f$
implies $E=cI+R$, and, since $\tilde{E}$ is a parametrix for $E$, we have, for
some $R^{\prime}\in\mathcal{R}G$,

(4.58) $(cI+R)\tilde{E}=I+R^{\prime},\implies c\tilde{E}=I\mod\mathcal{R}G.$

Therefore, $c$ must be different from $0$, and we deduce that also
$\tilde{E}=\frac{1}{c}I+S$ for some $S\in\mathcal{R}G$. This concludes the
proof. ∎

With all the above pieces in place, we can now state and prove our third main
result.

######  Theorem 4.19 (Characterisation of $SG$-order-preserving isomorphisms).

Let $\imath\colon\mathit{LG}\rightarrow\mathit{LG}$ be an SGOPI. Then, there
exists an invertible, classical SGFIO $A$ of type $\mathcal{Q}_{gen}$ such
that, for all $P\in\mathit{LG}$, we have

(4.59) $\imath(P)=A^{-1}PA.$

###### Proof.

Consider $E=FV^{-1}$ where $V$ is the Eidelheit isomorphism of Lemma 4.14 and
$F$ the SGFIO obtained from Theorem 4.9 by considering the induced isomorphism
on the formal symbol algebra $\mathcal{B}G$. Then, by the above discussion, we
have that
$E\colon\mathcal{S}(\mathbb{R}^{n})\rightarrow\mathcal{S}(\mathbb{R}^{n})$ is
continuous. Moreover, it is an elliptic $SG\Psi$DO of order $(0,0)$, with
parametrix $\tilde{E}=VF^{\\#}$. Furthermore, both $E$ and $\tilde{E}$ commute
$\mod\mathcal{R}G$ with every $P\in\mathit{LG}$. By Lemma 4.18, it follows
that $E=cI+R$ with $R\in\mathcal{R}G$, so that $F=cV+RV$. But then
$V=c^{-1}(F-RV)$ and $V^{-1}=c(F-RV)^{\\#}+S$ with some $S\in\mathcal{R}G$, so
that $V$ is an invertible SGFIO, as claimed. ∎

The following corollary is now completely straightforward.

######  Corollary 4.20.

Let $\imath\colon\mathit{LG}\rightarrow\mathit{LG}$ be an algebra isomorphism
satisfying the condition

(4.60)
$\imath(\mathit{LG}^{m_{1},m_{2}})\subset\mathit{LG}^{m_{2},m_{1}}\quad\forall(m_{1},m_{2})\in\mathbb{Z}^{2}.$

Then,
$\imath(P)=(\operatorname{\mathcal{F}}A)^{-1}P\operatorname{\mathcal{F}}A$,
where $A$ is an invertible $\mathcal{Q}$-FIO and $\operatorname{\mathcal{F}}$
is the Fourier transform.

###### Proof.

This is obtained by combining Theorem 4.19 with Proposition 3.28. Namely,
consider the isomorphism
$\jmath(P)=\operatorname{\mathcal{F}}\imath(P)\operatorname{\mathcal{F}}^{-1}$,
which, by assumption, is now an SGOPI. It follows that $\jmath(P)=A^{-1}PA$
for some invertible $\mathcal{Q}$-FIO. Then
$\imath(P)=\operatorname{\mathcal{F}}A^{-1}PA\operatorname{\mathcal{F}}$, as
claimed. ∎

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