TITLE: Derivative of fraction of vectors
QUESTION [0 upvotes]: Hi is there anyone that can help me with this calculation?
So, I don’t know the exact way to differentiate the fraction of vector function. I tried to use the formal way but the components of the numerator had different size of dimension. Btw I actually solved the question by Cauchy-Schwarts inequality but am just curious with the technique the solution suggested. Thanks.

REPLY [1 votes]: $
\newcommand\R{\mathbb R}
$
I assume that $x'$ means the transpose of $x$.
By differentiate, I assume they mean take gradient $\nabla_\alpha$. The facts we need are

The product rule holds. If $f(\alpha)$ and $g(\alpha)$ are scalar-valued functions then
$$
  \nabla_\alpha f(\alpha)g(\alpha)
    = (\nabla_\alpha f(\alpha))g(\alpha) + f(\alpha)(\nabla_\alpha g(\alpha)).
$$
If $f : \R \to \R$ and $g : \R^n \to \R$ then the chain rule is
$$
  \nabla_\alpha f(g(\alpha))
  = \frac{\partial f}{\partial g}\nabla g(\alpha).
$$
If $x$ is a vector independent of $\alpha$ then
$$
  \nabla_\alpha(\alpha'x) = \nabla_\alpha(x'\alpha) = x.
$$

All together, we see
$$\begin{aligned}
  \nabla_\alpha\rho_{yw}^2
&= \frac{2(\alpha'\sigma_{yx})[\nabla_\alpha(\alpha'\sigma_{yx})]}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)}
  - \frac{(\alpha'\sigma_{yx})^2}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)^2}[\nabla_\alpha(\alpha'\Sigma_{xx}\alpha)]
\\
&= \frac{2\alpha'\sigma_{yx}}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)}\sigma_{yx} - \frac{(\alpha'\sigma_{yx})^2}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)^2}[\nabla_\beta(\beta'\Sigma_{xx}\alpha) + \nabla_\beta(\alpha'\Sigma_{xx}\beta)]
\\
&= \frac{2\alpha'\sigma_{yx}}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)}\sigma_{yx} - \frac{(\alpha'\sigma_{yx})^2}{\sigma_{yy}(\alpha'\Sigma_{xx}\alpha)^2}[\Sigma_{xx} + \Sigma_{xx}']\alpha.
\end{aligned}$$
The first equality is application of the product rule followed by application of the chain rule in each term. The second equality is application of fact (3.) in the first term and application of the product rule in the second term. Finally, the third equality follows application of fact (3.) in the second term, noting that $\alpha'\Sigma_{xx}\beta = (\Sigma_{xx}'\alpha)'\beta$.
I assume now that $\Sigma_{xx}$ is symmetric so that $\Sigma_{xx} + \Sigma_{xx}' = 2\Sigma_{xx}$. Setting $\nabla_\alpha\rho_{yw}^2 = 0$ and making the obvious cancellations gives
$$
  \sigma_{yx} = \frac{\alpha'\sigma_{yx}}{\alpha'\Sigma_{xx}\alpha}\Sigma_{xx}\alpha
  \implies
  \alpha = \frac{\alpha'\Sigma_{xx}\alpha}{\alpha'\sigma_{yx}}\Sigma_{xx}^{-1}\sigma_{yx}.
$$


TITLE: Is there a bijection between uncountable sets?
QUESTION [3 upvotes]: I know that there is a bijection between naturals and rationals. I also know that there is no bijection between naturals and reals (diagonal argument).
But, I have never heard of the existence of a bijection between uncountable sets (ex aleph-one). Is there a way to create a (computable ?) function that takes an element from an uncountable set and outputs (in infinite time ?) an element from another uncountable set ?
(I do not have a strong mathematical background, so please keep it simple or use terms of computer science)
[EDIT]
It seems that my question was very trivial. An answer would be y = f(R) where f is just one-to-one. I was hoping for something more sophosticated :( . Sorry for the inconvenience.
[EDIT2]
How we would construct a bijection between these sets ?
A = reals
B = reals without naturals
C = reals without primes

REPLY [0 votes]: Since you are asking about computable bijections, and the other answers (at the time of writing this) do not address this point, let me weigh in on this.
1) There is no computable bijection $f : \mathbb{R} \to \mathbb{R}\setminus\mathbb{Q}$.
In fact, whenever $f : \mathbb{R} \to \mathbb{R}\setminus \mathbb{Q}$ is a computable function, then it already is a constant function. The reason is that computable functions are always continuous, and the image of a connected space such as $\mathbb{R}$ under a continuous function has to be connected again.
2) There is no computable bijection $g : \mathbb{R}\setminus\mathbb{Q} \to \mathbb{R}$.
This one is slightly trickier, but still follows from the continuity considerations. Here, however, it is demanding the injectivity that gets us. We can have a computable surjection from $\mathbb{R}\setminus \mathbb{Q}$ to $\mathbb{R}$.
3) As for 1), we cannot have a computable surjection from $\mathbb{R}$ to $\mathbb{R} \setminus \mathbb{P}$. The most we can get is the interval between two primes.
4) We don't get a computable bijection from $\mathbb{R}\setminus\mathbb{P}$ to $\mathbb{R}$ either, but a computable surjection works.
5) There is a computable surjection from $\mathbb{R}\setminus\mathbb{Q}$ to $\mathbb{R}\setminus\mathbb{P}$, but not the other way around.
The bijection between $2^\mathbb{N}$ and the Cantor middle third set inside $\mathbb{R}$ mentioned by celtschk is computable in both directions.
The standard (but a bit out-dated) textbook in this area is Weihrauch: Computable Analysis (2000). A briefer intro in a semilar style is Brattka, Hertling & Weihrauch: A tutorial on computable analysis (2008). More general, but maybe also less newcomer friendly is Pauly: On the topological aspects of the theory of represented spaces (2016).


text:
\chapter{Truncation in $\T$}

\section{Introduction}\label{intro}

\noindent
In this chapter we recollect from \cite{ADH} how to construct the field $\T$ of logarithmic-exponential transseries from $\Texp$, and extend the notion of splitting from sets of exponential transmonomials to sets of logarithmic-exponential transmonomials in order to prove our main result.

As to $\T$, this is basically $\Texp$ extended with logarithms. For a detailed description, see
Section~\ref{consT} below. For now, we just mention the following basic facts about $\T$ in order to
 state the main result of this dissertation in a way that can be easily grasped. \begin{enumerate}
\item $\T$ is a truncation closed subfield of a Hahn field $\R[[G^{\LE}]]$, with $\R(G^{\LE})\subseteq \T$.
Here $G^{\LE}$ is a monomial group whose elements are the \textbf{logarithmic-exponential transmonomials}.
It contains the group $G^{\E}$ of exponential transmonomials as a subgroup, and the subfield $\Texp$ of the Hahn field $\R[[G^{\E}]]\subseteq \R[[G^{\LE}]]$, defined as in Section~\ref{METEXP},

 is a subfield of $\T$.
\item We regard $\T$ as an {\em ordered\/} subfield of the ordered Hahn field $\R[[G^{\LE}]]$. The map $f\mapsto \ex^f$ on $\Texp$ extends naturally to an isomorphism, called \textbf{exponentiation}, of the ordered additive group of $\T$ onto its multiplicative group $\T^{>}$ of positive elements; we denote it also by $f\mapsto \ex^f$.
\item There is given a natural automorphism $f\mapsto f{\uparrow}$ of the ordered field $\T$ over $\R$, to be thought of as $f(x) \mapsto f(\ex^x)$. Its $n$th iterate is denoted by $f\mapsto f{\uparrow}^n$.
Its inverse is $f\mapsto f{\downarrow}$, to be thought of as $f(x) \mapsto f(\log x)$, and the $n$th iterate of this inverse is denoted by $f\mapsto f{\downarrow}_n$.
\item With $G^{\E,n}:= G^{\E}{\downarrow}_n$ we  have

an increasing sequence $G^{\E}\subseteq G^{\E,1}\subseteq G^{\E,2}\subseteq \cdots \subseteq G^{\LE}$, and
\[G^{\LE}\ =\ \bigcup_n G^{\E,n}.\]
In particular, $\uparrow$ and $\downarrow$ restrict to automorphisms of the (ordered) monomial group $G^{\LE}$.
\item The derivation $\der$ on $\Texp$ extends naturally to a derivation on $\T$, also denoted by $\der$, such that $\der(\ex^f)=\der(f)\ex^f$ for all $f\in \T$. As an ordered field equipped with this extended derivation, $\T$ is a Liouville closed $H$-field with constant field $\R$.
\end{enumerate}

\medskip\noindent
Let $K$ be a subfield of $\T$. In view of (2) we can define $K$ to be {\bf closed under exponentiation\/} if $\ex^K\subseteq K$, and we define the {\bf closure of $K$ under exponentiation\/} to be the smallest subfield
$L\supseteq K$ of $\T$ that is closed under exponentiation; we denote this $L$ by $K^{\exp}$. (For subfields of $\Texp$ this agrees with the previous chapter.) Note that if $\der(K)\subseteq K$, then $\der(K^{\exp})\subseteq K^{\exp}$. Call a set $S\subseteq \T$ {\bf truncation closed\/} if it is truncation closed as a subset of $\R[[G^{\LE}]]$. We can now state one of our results; it is proved in the next section and does not involve the derivation:

\begin{proposition}\label{trexp} If the subfield $K\supseteq \R$ of $\T$ is truncation closed, then so is $K^{\exp}$.
\end{proposition}

\medskip\noindent
In the next section we also show that the property for sets $S\subseteq G^{\E}$ to split can be extended uniquely to sets $S\subseteq G^{\LE}$ in such a way that the following conditions are satisfied: \begin{enumerate}
\item[{(Sp1)}] for $S\subseteq G^{\E}$ to split as defined earlier agrees with $S$ to split as a subset of $G^{\LE}$;
\item[{(Sp2)}] for $S\subseteq G^{\LE}$, $S$ splits iff $S\cap G^{\E,n}$ splits for all $n$;
\item[{(Sp3)}] for $S\subseteq G^{\LE}$, $S$ splits iff $S{\uparrow}$ splits.
\end{enumerate}

\noindent
In the next section we also prove the following variant of Lemma~\ref{expsplit}:

\begin{proposition}\label{trexpsplit} Suppose $K\supseteq \R$ is a truncation closed subfield of $\T$ such that $K\cap G^{\LE}$ splits.  Then $K^{\exp}\cap G^{\LE}$ splits.
\end{proposition}

\noindent
Any differential subfield of $\T$ containing $\R$ is an $H$-field, where the subfield is given the ordering induced by $\T$. Let $K\supseteq \R$ be a differential subfield of $\T$. We define $K^{\Li}$ to be the smallest real closed subfield of $\T$ containing $K$ that is closed under exponentiation and integration, where the latter means that for all $f\in K$ there exists $g\in K$ with $f=\der(g)$. Then $K^{\Li}$ is also a differential subfield of $\T$, and is Liouville closed as an $H$-field. In fact, $K^{\Li}$ is the smallest differential subfield of $\T$ that contains $K$ and is Liouville closed; we call $K^{\Li}$ the {\bf Liouville closure of $K$ in $\T$}.

 We can now state our main result, proved in the last section:

\begin{theorem}\label{thC} \label{THC} Let $K\supseteq \R$ be a truncation closed differential subfield of $\T$ such that $K\cap G^{\LE}$ splits.
Then $K^{\Li}$ is truncation closed and $K^{\Li}\cap G^{\LE}$ splits.
\end{theorem}

\noindent
For example, $K:=\R(x)$ is a truncation closed differential subfield of $\T$ with $K\cap G^{\LE}=x^{\Z}$. Thus
the Liouville closure $\R(x)^{\Li}$ of $\R(x)$ in $\T$ is truncation closed, and $\R(x)^{\Li}\cap G^{\LE}$ splits.

\subsection*{Changing the derivation}
The derivation $\der$ that we use here is not the derivation $\frac{d}{dx}$ on $\T$ that is used systematically in \cite{ADH}, but it is closely related: $\der=x\cdot\frac{d}{dx}$ and we explain here why it doesn't matter which of these derivations we use.
First, $\T$ with the derivation $\der$ is isomorphic via $\uparrow$ to $\T$ with the derivation $\frac{d}{dx}$, since for $f\in \T$ we have a chain rule for $f{\uparrow}$
thought of as $f(\ex^x)$:
 \[ \frac{df{\uparrow}}{dx}\ =\ \ex^x\cdot\big(\frac{df}{dx}\big){\uparrow}\ =\ \big(x\frac{df}{dx}\big){\uparrow}\ =\ \der(f){\uparrow}.\]
Next, $\uparrow$ preserves truncation: for $f,g\in \T$,
$$f\truncof g\ \Longleftrightarrow\ f{\uparrow}\truncof g{\uparrow}. $$
See the next section for these facts. Let $(\T,\frac{d}{dx})$ refer to
the $H$-field $\T$ with derivation $\frac{d}{dx}$ instead of
$\der$. Thus $(\T, \frac{d}{dx})$ is also Liouville closed. Let $K$ be a subfield of $\T$. If $x\in K$, then $K$ is a differential subfield of $\T$ iff $K$ is a differential subfield of $(\T, \frac{d}{dx})$. Note that if $K$ is a Liouville closed subfield of
$\T$, then we do have $x\in K$, and so $K$ is also a Liouville
closed subfield of $(\T, \frac{d}{dx})$; likewise with the roles of $\T$ and $(\T, \frac{d}{dx})$ interchanged. For differential subfields of $(\T,\frac{d}{dx})$ we define its Liouville closure in
$(\T,\frac{d}{dx})$ in the same way as for $\T$ (with $\frac{d}{dx}$ instead of $\der$). It should be clear now that Theorem \ref{thC} goes through with $(\T,\frac{d}{dx})$ instead of
$\T$, and with $K^{\Li}$ replaced by the smallest Liouville closed
subfield of $(\T,\frac{d}{dx})$ containing $K$.

\section{Constructing $\T$ and Splitting}\label{consT}

\noindent
The introduction to this chapter described basic facts about $\T$, but not how $\T$ is obtained from $\Texp$. In this section we take care of that. Most of it
is taken from \cite[Appendix A]{ADH} to which we refer for proofs omitted here.  In this section we also extend the notion of splitting to subsets of $G^{\LE}$
and obtain Propositions~\ref{trexp} and ~\ref{trexpsplit} on the exponential closure of subfields of $\T$.

We start by using the alternative notation $\R[[x^\R]]^{\E}$ for $\Texp$ to indicate $\R[[x^{\R}]]$ as
the first step in its construction and to suggest the presence of the formal variable $x$.
The idea is to
replace $x$ here by $\log x$, $\log \log x$, and so on. Formally, we introduce distinct symbols $\ell_0, \ell_1,\ell_2,\ldots$, with $\ell_0$ standing for $x$, and
$\ell_1$, $\ell_2, \dots$ to be interpreted later as $\log x$, $ \log \log x,\cdots$ so that
 for each $f(x) \in  \R[[x^\R]]^{\E}$ we have a corresponding $f(\ell_n)\in \R[[\ell_n^\R]]^{\E}$. More precisely,
we take for each $n$  a copy $G^{\E,n}$ of the ordered abelian group $G^{\E}$ with an isomorphism
\[\fm\mapsto \fm\downer_n\ :\  G^{\E}\rightarrow G^{\E,n}\]
of ordered abelian groups, where $x^r\downer_n = \ell_n^r \in G^{\E,n}$ for $r\in \R$; by convention, $x:= x^1$ and $\ell_n:= \ell_n^1$.  For $n=0$ we take $G^{\E,0}=G^{\E}$ and let ${\downarrow}_0$ be the identity map.
 The above map $\fm \mapsto \fm{\downarrow}_n$ extends uniquely to a strongly additive $\R$-linear bijection
\[f\mapsto f\downer_n\ :\  \R[[G^{\E}]]\rightarrow \R[[G^{\E,n}]],\]
between Hahn spaces over $\R$. This map $f\mapsto f\downer_n$ is an isomorphism of ordered Hahn fields over $\R$.
We denote the image of $\R[[x^\R]]^{\E}\subseteq \R[[G^{\E}]]$ under this map by $\R[[\ell_n^\R]]^{\E}$ and make the latter into an {\em exponential\/} ordered field so that
\[f\mapsto f\downer_n\ :\  \R[[x^\R]]^{\E}\rightarrow \R[[\ell_n^\R]]^{\E}\]
is an isomorphism of exponential ordered fields. We denote the exponentiation on $\R[[\ell_n^\R]]^{\E}$ by $\exp$ as well.
For $n=0$ we have of course $f\downer_0 = f$ for $f\in \R[[G^{\E}]]$.
Recall that $G^{\E}$ is the union of an increasing sequence $(G_m)_m$ of convex subgroups. Thus we have an increasing sequence $(G_m\downer_n)_m$ of convex subgroups  of $G^{\E,n}$ with $\bigcup_m G_m\downer_n = G^{\E,n}$ and  $\R[[\ell_n^\R]]^{\E}= \bigcup_m \R[[G_m\downer_n]]$.

So far these are just notational conventions, but now a fact that goes beyond mere notation: for each $n$ there is a unique $\R$-linear embedding
$$ \R[[\ell_n^\R]]^{\E}\to \R[[\ell_{n+1}^\R]]^{\E}$$
of exponential ordered fields with the following properties: \begin{itemize}
\item it sends $\ell_n^r$ to $\exp(r\ell_{n+1})$ for all $r\in \R$;
\item for every $m$ it maps $G_m\downer_n$ into $G_{m+1}\downer_{n+1}$ and $\R[[G_m\downer_n]]$ into $\R[[G_{m+1}\downer_{n+1}]]$;
\item for every $m$ its restriction to a map $\R[[G_m\downer_n]]\to \R[[G_{m+1}\downer_{n+1}]]$ between Hahn spaces over $\R$ is strongly additive.
\end{itemize}
Identifying $\R[[\ell_n^\R]]^{\E}$ with its image in  $\R[[\ell_{n+1}^\R]]^{\E}$ under this map we have $\ell_n^r=\exp(r\ell_{n+1})$ for $r\in \R$ and $G_m\downer_n\subseteq G_{m+1}\downer_{n+1}$ for all $n$, and so we obtain a chain
\[G^{\E}\ =\ G^{\E,0}\ \subseteq\ G^{\E,1}\ \subseteq\ G^{\E,2}\ \subseteq\  \cdots\]
of ordered groups and a corresponding chain
\[\Texp\ =\ \R[[\ell_0^\R]]^{\E}\ \subseteq\ \R[[\ell_1^\R]]^{\E}\ \subseteq\  \R[[\ell_2^\R]]^{\E}\ \subseteq\ \cdots,\]
of exponential ordered fields.
 We set
\[ G^{\LE}\ := \bigcup_n G^{\E,n},\qquad \T\  := \ \bigcup_n \R[[\ell_n^\R]]^{\E},\]
making $G^{\LE}$ into an ordered group extension of all $G^{\E,n}$, and $\T$ into an exponential ordered field extension of all $\R[[\ell_n^\R]]^{\E}$.
It also makes $\T$ a truncation closed subfield of $\R[[G^{\LE}]]$.
We refer to the elements of $\T$ as \textbf{transseries}, or $\LE$-series, and to the elements of $G^{\LE}$ as \textbf{transmonomials}, or $\LE$-monomials.
Note that by virtue of this construction $\T$ and $\R[[G^{\LE}]]$ are real closed.

\subsection*{Exponentiation in $\T$}
We obtained $\T$ as an ordered exponential field, and denote its exponential map by $\exp$. For $f\in \T$ we also write $\ex^f$ in place of $\exp(f)$. We have $\exp(\T)=\T^{>}$. (For $\Texp$ we do not have $\exp(\Texp)=\Texp^{>}$.) The inverse of $\exp$ is denoted by $\log : \T^{>}\to \T$, so $\log(\ell_n)=\ell_{n+1}$
for all $n$.  For $f= c\fm(1-\epsilon)$ for $c\in \R^{>}$, $\fm\in G^{\LE}$, $\epsilon \in \T^{\prec 1}$,
$$\log(f)\ =\ \log(\fm) + \log(c) - \sum_{i=1}^{\infty} \frac{\epsilon^i}{i}\qquad  \text{(with the usual real value of $\log c$)}.$$
A key fact is that $\log(G^{\LE})=\{f\in \T:\ \supp f\succ 1\}$.
(This was used in Section 4.3.) At this point we have justified items (1) and (2) about $\T$ in the introduction to this chapter.

\subsection*{Proof of Proposition~\ref{trexp}} This goes just as for $\Texp$: First, for a subfield $E$ of $\T$ its small exponential closure  (in $\R[[G^{\LE}]]$)
is contained in $\T$, and is truncation closed if $E\supseteq \R$ is truncation closed. Next, we can basically copy the proof of Lemma~\ref{exptrunc}, using  the fact
that $\ex^a\in G^{\LE}$ if $a\in \T$ satisfies $\supp a \succ 1$.

\subsection*{The upward shift operator} As to items (3) and (4) in the introduction to this chapter, we just refer to \cite[Appendix A]{ADH}
for the construction of the automorphism $f\mapsto f{\uparrow}$
(called the {\bf upward shift}) of the ordered field $\T$
such that (3) and (4) hold. The restriction of the $n$th iterate of its inverse $f\mapsto f{\downarrow}$ to $\R[[x^{\R}]]$ agrees with the map $f\mapsto f{\downarrow}_n: \R[[x^{\R}]]\to \R[[\ell_n^{\R}]]$ in the above construction of $\T$, so
the various ways we used the notation ${\downarrow}_n$ are in agreement. Thus we have an ordered field isomorphism $f\mapsto f{\downarrow}: \R[[G^{\E,n}]]\to \R[[G^{\E,n+1}]]$ induced by the isomorphism $$\fm\mapsto \fm{\downarrow}\ :\  G^{\E,n} \to G^{\E,n+1}$$
of ordered abelian groups. Hence for $f,g\in \R[[G^{\E,n}]]$ we have: $f\truncof g \Leftrightarrow f{\downarrow}\truncof g{\downarrow}$. Since this equivalence holds for any $n$, it holds for all $f,g\in \T$, which justifies a fact stated at the end of the introduction to this chapter.

It is also important that the upward shift is not just an isomorphism of $\T$ as an ordered field, but also of the {\em exponential} ordered field $\T$, that is, $\exp(f{\uparrow})=\exp(f){\uparrow}$ for $f\in \T$.

\subsection*{Splitting}
Here we define what it means for a set $S\subseteq G^{\LE}$ to
split so as to satisfy conditions (Sp1), (Sp2), (Sp3) from the
introduction to this chapter. First we show in the next lemma that for subsets of $G^{\E}$ to split is invariant under certain operations. The
proof uses a key fact about the automorphism $\uparrow$, namely that $G_n{\uparrow}\subseteq G_{n+1}$ for all $n$.

\begin{lemma}\label{sp1} Let $S\subseteq G^{\E}$. Then \begin{enumerate}
\item[\rm{(i)}]  $S$ splits $\ \Longleftrightarrow\ $ $S{\uparrow}$ splits;
\item[\rm{(ii)}] $S$ splits $\ \Longrightarrow\ $  $S{\downarrow}\cap G^{\E}$ splits.
\end{enumerate}
\end{lemma}
\begin{proof} We use here notation from the last chapter for $\smallk_{*}=\Texp$, so $G_n= \fM^{(n)}=\fM_0\cdots \fM_n$.  For $\fm=\fm_0\cdots\fm_n$, with $\fm_i\in \fM_i$ for $i=0,\dots,n$, we have $\fm{\uparrow}=1\cdot\fm_0{\uparrow} \cdots \fm_n{\uparrow}$
with $1\in \fM_0$ and $\fm_i{\uparrow}\in \fM_{i+1}$ for $i=0,\dots,n$. Applying this to $\fm\in S$ yields (i).  As to (ii), assume $S$  splits, and let
$\fm\in S{\downarrow}\cap G^{\E}$ be as above. Then $\fm{\uparrow}\in S$, and so by the above, $\fm_i{\uparrow}\in S$ for $i=0,\dots,n$,
hence $\fm_i\in S{\downarrow}\cap G^{\E}$ for those $i$. Thus $S{\downarrow}\cap G^{\E}$ splits.
\end{proof}

\noindent
Let $S\subseteq G^{\E,n}$. Then $S=R{\downarrow}_n$ for a unique
$R\subseteq G^{\E}$, namely $R=S{\uparrow}^n$. Let us say that
$S$ \textbf{splits in $G^{\E,n}$} if $R$ as a subset of $G^{\E}$ splits. Now we also have $S\subseteq G^{\E,n+1}$, and:

\begin{lemma}\label{sp2} $S$ splits in $G^{\E,n}$ iff $S$ splits in $G^{\E,n+1}$.
\end{lemma}
\begin{proof} With $R$ as above, we have $R{\uparrow}=S{\uparrow}^{n+1}$. Apply Lemma~\ref{sp1}(i)  to $R$ in the role of $S$.
\end{proof}

\noindent
Next we consider sets $S\subseteq G^{\LE}$, and we say that such a set $S$ {\bf splits} if $S\cap G^{\E,n}$ splits in $G^{\E,n}$ for all $n$. In order to show that
this notion has the properties (Sp1), (Sp2), (Sp3) stated in the introduction to this chapter, we need one more fact:

\begin{lemma}\label{sp3} Suppose $S\subseteq G^{\E,n+1}$ splits in $G^{\E,n+1}$. Then $S\cap G^{\E,n}$ splits in $G^{\E,n}$.
\end{lemma}
\begin{proof} Let $R:= S{\uparrow}^{n+1}$. Then $R\subseteq G^{\E}$  splits, so $R{\downarrow}\cap G^{\E}$ splits by Lemma~\ref{sp1}(ii). Now use that $R{\downarrow}\cap G^{\E}=\big(S\cap G^{\E,n}\big){\uparrow}^n$.
\end{proof}

\noindent
It is now routine to show that (Sp1), (Sp2), (Sp3) hold: for (Sp2), first observe as a consequence of Lemmas~\ref{sp2} and ~\ref{sp3} that for $S\subseteq G^{\E,n}$, $S$  splits iff $S$ splits in $G^{\E,n}$.  It is also easy to check that there is no other way to extend the notion of splitting for
subsets of $G^{\E}$ to subsets of $G^{\LE}$ so that (Sp1), (Sp2), (Sp3) hold.

\medskip\noindent
We also extend some facts from Section~\ref{neat} to the present setting:

\begin{lemma}\label{sp4} Let $R\supseteq \R$ be a truncation closed subring of $\R[[G^{\LE}]]$ such that $R\cap G^{\LE}$ splits. Let $V$ be a truncation closed $\R$-linear subspace of $\R[[G^{\LE}]]$ such that $V\cap G^{\LE}$ splits. Then the $($truncation closed$)$ subring
$R[V]$ of $\R[[G^{\LE}]]$ has the property that $R[V] \cap G^{\LE}$ splits. The $($truncation closed$)$ fraction field $F$ of $R$ in $\R[[G^{\LE}]]$ has the property that $F\cap G^{\LE}$ splits.
\end{lemma}

\noindent
The proofs consist of routine reductions to Lemmas~\ref{split1} and ~\ref{split2}.

\begin{lemma}\label{sp5} Let $E\supseteq \R$ be a truncation closed ordered subfield of the $($real closed$)$ Hahn field $\R[[G^{\LE}]]$ such that $E\cap G^{\LE}$ splits.
Let $F$ be the real closure of $E$ in $\R[[G^{\LE}]]$. Then $F\cap G^{\LE}$ splits.
\end{lemma}
\begin{proof}  Use that by Lemma~\ref{actr} we have $F\cap G^{\LE}=\{\fg\in G^{\LE}:\ \fg^n\in E\cap G^{\LE}\}$.
\end{proof}

\subsection*{Proof of Proposition~\ref{trexpsplit}} Let $K\supseteq \R$ be a truncation closed subfield of $\T$ such that $K\cap G^{\LE}$ splits. Our job is to show that $K^{\exp}\cap G^{\LE}$
splits. We set $K_n:= K\cap \R[[\ell_n^{\R}]]^{\E}$, so
$K_n$ is a truncation closed subfield of $\R[[G^{\E,n}]]$ such that
$K_n\cap G^{\E,n}$ splits. The exponential closure $K_n^{\exp}$ of
$K_n$ in $\T$ is also its exponential closure in
$\R[[\ell_n^{\R}]]^{\E}$, so $K_n^{\exp}\cap G^{\E,n}$ splits
in $G^{\E,n}$ by Lemma~\ref{expsplit}. Taking the union over all $n$ now gives the desired result.

\subsection*{The derivation of $\T$}  We take this from \cite[Appendix A]{ADH}, but alert the reader that the derivation on $\T$ we denote here by
$\der$ equals the derivation $x\cdot \frac{d}{dx}$  where $\frac{d}{dx}$ is the derivation on $\T$ constructed in that appendix (and which, unfortunately for us, is denoted
there by $\der$). Keeping that in mind we easily verify item (5) of the introduction to this chapter. Moreover, if $m\ge n$, then
$$\der \R[[G_m{\downarrow}_n]]\ \subseteq\
\R[[G_m{\downarrow}_n]],$$ and the restriction of $\der$ to $\R[[G_m{\downarrow}_n]]$ is a strong operator on that Hahn field over $\R$. Fixing $n$ and taking the
union over all $m\ge n$ shows that $\R[[\ell_n^{\R}]]^{\E}$ is a differential subfield of $\T$.

We define the elements $e_n\in G_n\subseteq \Texp$ by
$e_0 := x$ and $e_{n+1}:= \exp(e_n)$. An easy induction on $n$ gives $e_n=x{\uparrow}^n$. We also set
$\fe_n:= e_0e_1\cdots e_{n-1}$ (so $\fe_0=1$ and $\fe_1=x$).

The interaction of $\der$ and $\uparrow$ is described in the  commutative diagram

\[\xymatrixcolsep{8pc}\xymatrix{
\T \ar[d]_{\upper^n} \ar[r]^{\der} & \T \ar[d]^{\ \upper^n} \\
\T \ar[r]^{\frac{1}{\fe_n}\der} & \T
}\]
which by restriction yields the commutative diagram
\[\xymatrixcolsep{8pc}\xymatrix{
\R[[\ell_n^{\R}]]^{\E} \ar[d]_{\upper^n} \ar[r]^{\der} & \R[[\ell_n^{\R}]]^{\E} \ar[d]^{\ \upper^n} \\
\Texp \ar[r]^{\frac{1}{\fe_n}\der} & \Texp
}\]
With this last diagram we routinely reduce questions about $\R[[\ell_n^{\R}]]^{\E}$ involving $\der$ to corresponding questions about $\Texp$.
We are now finally ready for the proof of our main result.

\section{Proof of Theorem~\ref{thC}}

\noindent
Let $K\supseteq \R$ be a truncation closed differential subfield of $\T$ such that
$K\cap G^{\LE}$ splits. Our job is to show that then the Liouville
closure $K^{\Li}$ of $K$ in $\T$ is truncation closed, and that
$K^{\Li}\cap G^{\LE}$ splits. First we observe that $x\in K^{\Li}$, and thus the iterated logarithms
$\ell_n$ are also in $K^{\Li}$.
Moreover, $L:=K(\ell_n:\ n=0,1,2,\dots)$ is still a truncation closed
differential subfield of $\T$ and $L\cap G^{\LE}$ splits, so replacing $K$ by $L$ we arrange that $\ell_n\in K$ for all $n$. We can also assume $K$ is not Liouville closed. For such $K$ it is enough to show: there exists a differential field extension $L\subseteq K^{\Li}$ of $K$ such that $K\ne L$, $L$ is truncation closed, and $L\cap G^{\LE}$ splits. We now treat separately the
various cases that may occur because of the $H$-field $K$ not being Liouville closed:

\medskip\noindent
{\em Case 1:\ $K$ is not real closed.} Then we can take for $L$ the real closure of $K$ in $\T$,  by Lemma~\ref{sp5}.

\medskip\noindent
{\em Case 2:\ $K$ is not closed under exponentiation.}
Then we can take for $L$ the exponential closure of $K$ in $\T$,
by Propositions~\ref{trexp} and ~\ref{trexpsplit}.

\medskip\noindent
{\em Case 3:\ $f\in K$ is such that $f\notin \der(K)$.}
We begin by observing that $\fe_n{\downarrow}_n=\ell_1\cdots \ell_n$, and that if $m < n$, then no element of $\R[[\ell_m^{\R}]]^{\E}$ has $\frac{1}{\ell_1\cdots \ell_n}$ in its support. We set
$K_n:= K\cap \R[[\ell_n^{\R}]]^{\E}$, and take $n$ so high that
$f\in K_n$ and $\frac{1}{\ell_1\cdots \ell_n}\notin\supp(f)$. Note that $K_n$ is truncation closed and
$K_n\cap G^{\E,n}$ splits. Since $K$ contains all $\ell_m$, we also have
$\frac{1}{\ell_1\cdots \ell_n}\in K_n\cap G^{\E,n}$.  Thus $F:=K_n{\uparrow}^n$ is a truncation closed subfield of $\Texp$ such that $F\supseteq \R$, $f{\uparrow}^n\in F$, $\frac{1}{\fe_n}\in (F\cap G^{\E})\setminus \supp(f{\uparrow}^n)$, and
$F\cap G^{\E}$ splits. We now use the second commutative diagram at the end of the previous section, and let $\derdelta:=\frac{1}{\fe_n}\der$ be the derivation on $\Texp$ corresponding to the bottom horizontal line of that diagram.

Then Theorem \ref{thmB} and the remark following its proof yields a truncation closed differential subfield $F_{\infty}$ of $F^{\Li}$ such that
$f{\uparrow}^n\in \derdelta(F_{\infty})$ and $F_{\infty}\cap G^{\E}$ splits. Applying ${\downarrow}_n$ we obtain a truncation closed
differential subfield $H:= F_{\infty}{\downarrow}_n$ of $K_n^{\Li}\subseteq K^{\Li}$ such that $H\supseteq \R$, $f\in \der(H)$ and $H\cap G^{\E,n}$ splits. Then it follows from Lemma~\ref{sp4} that $K(H)$ is a truncation closed
differential subfield of $K^{\Li}$ such that
$f\in \der K(H)$
and $K(H)\cap G^{\LE}$ splits. So $L:=K(H)$ does the job.

\medskip\noindent
This concludes the proof of Theorem~\ref{thC}.

\section{Future Directions}

\noindent
In this section we discuss possible ways of extending the results in this dissertation. First, there are other natural extension procedures for differential subfields
$K\supseteq \R$ of $\T$  besides taking the Liouville closure in $\T$. For example, $\T$ is {\em linearly surjective}, and every $K$ as above has a {\em linearly surjective closure} in $\T$. (A differential field $E$ is said to be {\bf linearly surjective\/}  if every linear differential equation $y^{(n)} + a_1y^{(n-1)} + \cdots + a_ny=b$ with $n\ge 1$ and $a_1,\dots,a_n, b\in E$ has a solution in $E$.  For $K$ as above there is a smallest linearly surjective differential subfield of $\T$ containing $K$,  the
\textbf{linearly surjective closure of $K$ in $\T$}.) The obvious question is whether Theorem~\ref{thC} goes through for the
linearly surjective closure of $K$ in $\T$ instead of its Liouville closure  in $\T$.  Going beyond linear differential equations, the most powerful
elementary property of $\T$ is that it is {\em newtonian}. (We refer to \cite[Chapter 14]{ADH} for a precise definition.)  If $K$ as above has {\em asymptotic integration} (as defined in \cite[Section 9.1]{ADH}), then $K$ has a {\em newtonization}
in $\T$ (a smallest newtonian differential subfield of $\T$ that contains $K$), and one can ask if Theorem~\ref{thC} goes through for the newtonization
instead of the Liouville closure.

Another natural direction: try to connect the material above to surreal numbers. We refer to \cite{ADHNo} for the description of a canonical embedding of
$\T$ as an exponential ordered field into the exponential ordered field $\No$ of surreal numbers, with $x\in \T$ corresponding to $\omega\in \No$.  For subsets of $\No$, the notion of being {\em initial} plays a key role, and is similar to being truncation closed for subsets of Hahn fields. Elliot Kaplan has shown that the image of $\T$ in $\No$ under the canonical embedding is initial. This raises natural questions for truncation closed subfields $K\supseteq \R$ of $\T$:

\begin{itemize}
\item if $K\cap G^{\LE}$ splits, does it follow that the image of $K$ in $\No$ is initial?
\item if the image of $K$ in $\No$ is initial, does it follow that $K\cap G^{\LE}$ splits?
\end{itemize}

config: arxiv


TITLE: 3D rotation of coordinates axis
QUESTION [0 upvotes]: I am struggling to understand the 3D rotation.
The initial object position is $A(x_1, y_1, z_1).$ We need rotate it into $A'(x_2,y_2,z_2)$ about $z$-axis by angle $\angle A'PA=\theta.$
And we take $P$ be the any point in $z$-axis. It makes $\angle APS=\phi.$
Now we find $cos \phi= \frac{PS}{OP}=\frac{x_1}{r}\Rightarrow x_1=rcos \phi.$ But we know $OQ =x_1$ but here slides take $PS =x_1$ and they forcefully match the answer which is wrong.
My question is slides has wrong or my concepts is wrong? If both are wrong so how we find 3D rotation about any coordinate axis?

REPLY [1 votes]: The following is a note on $3D$ rotation that I wrote a while ago.  Click on the image and ZOOM +.


text:
\begin{document}
\pagestyle{plain}

\title[]{\Large The massive Feynman propagator \\ on asymptotically  Minkowski spacetimes II}

\address{Universit\'e Paris-Sud XI, D\'epartement de Math\'ematiques, 91405 Orsay Cedex, France}
\email{christian.gerard@math.u-psud.fr}
\author{\large Christian G\'erard  \& Micha{\l} Wrochna}
\thanks{\emph{Acknowledgments.} We would like to warmly thank Christian B\"{a}r, Jan Derezi\'nski, Alexander Strohmaier and Andr\'as Vasy for useful conversations, and in particular for pointing out to us the link between invertibility and general features of the Klein-Gordon operator. Support from the grant ANR-16-CE40-0012-01 is gratefully acknowledged.}
\address{Universit\'e Grenoble Alpes, Institut Fourier, UMR 5582 CNRS, CS 40700, 38058 Grenoble \textsc{Cedex} 09, France}
\email{michal.wrochna@univ-grenoble-alpes.fr}
\keywords{pseudodifferential calculus, scattering theory, Quantum Field Theory on curved spacetimes, Feynman propagators}
\subjclass[2010]{81T13, 81T20, 35S05, 35S35}
\begin{abstract}
We consider the massive Klein-Gordon equation on short-range asymptotically Minkowski spacetimes. Extending our results in \cite{GW1}, we show that the Klein-Gordon operator with Feynman type boundary conditions at infinite times is invertible and that its inverse, called the {\em Feynman inverse},  satisfies the microlocal conditions of Feynman parametrices
 in the sense of Duistermaat and H\"ormander. This supplements the recent work of Vasy \cite{Va} with more explicit techniques.
\end{abstract}

\maketitle

\section{Introduction}\label{sec0}\init
The present paper is a continuation of \cite{GW1}, which was devoted to the existence of the  {Feynman propagator} for Klein-Gordon fields on asymptotically Minkowski spacetimes. Let us first briefly recall the motivation of \cite{GW1}.

On Minkowski spacetime $(\rr^{1+d}, \alteta)$, the Klein-Gordon operator $P_{\free}= \p_{t}^{2}- \Delta_{\rx}+\altm^{2}$ has four distinguished inverses, the {\em retarded/advanced} inverses $G_{\rm ret/adv}$ which are Fourier multipliers by $((\tau\pm\i 0)^{2}- (k^{2}+\altm^{2}))^{-1}$, and the {\em Feynman/anti-Feynman} inverses $G_{{\rm F}/\overline{\rm F}}$ which are Fourier multipliers by $((\tau)^{2}- (k^{2}+\altm^{2})\pm \i 0)^{-1}$.

The retarded/advanced inverses exist on any globally hyperbolic spacetime $(M, g)$ and are characterized as the unique solutions of
\[
\begin{array}{l}
PG_{\rm ret/adv}= G_{\rm ret/adv}P= \one, \\[2mm]
 \supp G_{\rm ret/adv}v\subset J^{\pm}\left(\supp v\right), \ v\in \coinf(M),
\end{array}
\]
where $P= - \Box_{g}+ V$, $V\in\cinf(M; \rr)$ and $J^{\pm}(K)$ is the future/past causal shadow of a set $K\subset M$.  Their difference $G= G_{\rm ret}- G_{\rm adv}$, called the {\em causal} or {\em Pauli-Jordan propagator} is used in  the algebraic quantization of free Klein-Gordon fields on $(M, g)$.

The Feynman/anti-Feynman inverses on Minkowski spacetime play a  fundamental role in the perturbative renormalization of interacting Klein-Gordon fields.

It is not a priori clear  how the  Feynman/anti-Feynman inverses generalize to an arbitrary globally hyperbolic spacetime $(M, g)$, but some of the properties they need to satisfy  are understood since the work of Duistermaat and H\"ormander \cite{DH}, who proved the existence and uniqueness modulo smooth terms of \emph{Feynman parametrices}.

Namely, if  $\Phi_t$ is the Hamilton flow of $p(x,\xi)=\xi\cdot \altg^{-1}(x)\xi$ restricted to the \emph{characteristic set} $\cN=p^{-1}(\{0\})$ (understood as a subset of $T^*M\setminus\zero$, where $\,\zero$ is the zero section of the cotangent bundle), one says that
 $\tilde{G}_\F$ is a {\em Feynman parametrix} if the operators $\one- \tilde{G}_\F P$ and $\one-P \tilde{G}_\F$ have smooth Schwartz kernels and
\beq\label{eq:fewf}
\WF'(\tilde{G}_{\F})= (\diag_{T^*M})\cup\textstyle\bigcup_{s\leq 0}(\Phi_s(\diag_{T^*M})\cap \pi^{-1}\cN).
\eeq
Here  $\Phi_s$ is the bicharacteristic flow acting on the left component of $\diag_{T^*M}$ (i.e., the diagonal in $(T^*M\times T^*M)\setminus\zero$), and $\pi:\cN\times\cN\to\cN$ is the projection to the left component.

This leaves open the question of the existence of a {\em canonical Feynman inverse} $G_{\F}$ satisfying \eqref{eq:fewf} on a globally hyperbolic spacetime $(M, g)$, see \cite{DS2} for a discussion.

If $(M, g)$ is {\em stationary}, i.e.~admits a global, complete, time-like Killing vector field $K$ and if  the potential  $V$ is invariant under $K$, it is well known that the {\em vacuum state} (with respect to the group of isometries generated by $K$) generates a canonical Feynman inverse.

In \cite{GW1} we considered this problem for {\em asymptotically Minkowski} spacetimes. Inspired by  works by Gell-Redman, Haber and Vasy \cite{GHV,positive} and B\"{a}r and Strohmaier \cite{BS} on closely related non-elliptic Fredholm problems, we introduced Hilbert spaces $\cY^{m}$, $\cX^{m}_{\F}$, where $m\in \rr$ is an order of Sobolev regularity and the subscript $\F$ refers to what one can call {\em Feynman boundary conditions}, which are imposed at $t= \pm \infty$.  We refer the reader to \ref{sec0.1.1} and \ref{sec0.2.1} for precise definitions.

We proved in \cite[Thm.~1.2]{GW1} in the massive case that $P:\cX^{m}_{\F}\to \cY^{m}$ is a {\em Fredholm operator} and we related  its index  to the index of  some `Feynman' wave operator $W_{\F}^{\dag}$.  We also showed that $P$ has a pseudo inverse $\tilde{G}_{\F}: \cY^{m}\to \cX^{m}_{\F}$ (i.e., an inverse modulo compact errors) which is at the same time a Feynman parametrix. Though obtained with different methods, these results are analogous to the outcome of \cite{GHV,VW} in the massless case, and are consistent with the general program outlined in Vasy's work \cite{positive}.

Our method of proof consisted in writing the Klein-Gordon equation as a first order system, which can be diagonalized globally in time modulo smoothing and decaying in time errors. After these reductions the problem can be handled by rather explicit methods.

Recently, Vasy \cite{Va} considered the same problem by working directly on the scalar operator $P$ using microlocal methods, motivated by the issue of essential self-adjointness of $P$ (see \cite{DS1}). Solving in this setting a conjecture by Derezi\'nski and Siemssen \cite{DS2,D}, he constructed the Feynman inverse $G_{\F}$ between microlocal Sobolev spaces as the boundary value $(P-\i 0)^{-1}$ of the resolvent of $P$.

In the present paper we  will  recover Vasy's result in our framework by showing  that $P: \cX_{\F}^{m}\to \cY^{m}$ is indeed {\em invertible} rather than merely Fredholm, and that its inverse $G_{\F}$ is a {\em Feynman inverse}, i.e.~the wavefront set of its kernel equals the r.h.s.~in \eqref{eq:fewf},  see Thm.~\ref{thm1} below.

The fact that $P$ is of index zero could actually be concluded directly from \cite{GW1} (see the proof of Prop.~\ref{prop1.1} in the present paper). Its injectivity from $\cX^{m}_{\F}$ to $\cY^{m}$ is less evident, the proof turns out however to be very easy, by adapting arguments of \cite{Va} to our framework. The advantage of having such a result in our framework is that this provides a rather explicit parametrix for $G_{\rm F}$.

\subsection{Klein-Gordon operators on asymptotically Minkowski spacetimes}\label{sec0.1}
In this subsection we recall the framework considered in \cite{GW1}.
\subsubsection{Asymptotically Minkowski spacetimes}\label{sec0.1.1}
We consider    $M=\rr^{1+d}$ equipped with  a Lorentzian metric $\altg$ such that
\[
 \begin{array}{rl}
({\rm aM} i)&\altg_{\mu\nu}(y)- \alteta_{\mu\nu} \in S^{-\delta}_{\std}(\rr^{1+d}), \ \delta>1,\\[2mm]
({\rm aM} ii)&(\rr^{1+d}, \altg) \hbox{ is globally hyperbolic},\\[2mm]
({\rm aM} iii)&(\rr^{1+d}, \altg) \hbox{ has a  time function }\tilde{t}\hbox{ with  }\tilde{t}- t\in S^{1-\epsilon}_{\std}(\rr^{1+d})\hbox{ for }\epsilon>0.
\end{array}
\]
where $\alteta_{\mu\nu}$ is the Minkowski metric and $S_{\std}^{\delta}(\rr^{1+d})$ stands for the class of smooth functions $f$ such that for  $\langle x \rangle  = (1+|x|)^{\12}$,
\[
\p^{\alpha}_{x}f\in O(\langle x\rangle^{\delta- |\alpha|}), \ \alpha\in \nn^{1+d}.
\]
We recall $\tilde{t}$ is a {\em time function} if $\nabla \tilde{t}$ is a time-like vector field.  It is a {\em Cauchy time function} if in addition its level sets  are Cauchy surfaces for $(M, \altg)$.

It is shown in \cite{GW1} that if $({\rm aM} i)$ holds, then $({\rm aM} ii)$ is equivalent to the familiar {\em non trapping condition} for null geodesics of $\altg$, and if $({\rm aM} i), ii) , iii)$ hold there exists a  Cauchy time function $\tilde{t}$ such that  $\tilde{t}- t\in \coinf(M)$.

Replacing $t$ by $t-c$, $\tilde{t}$ by $\tilde{t}-c$ for $c\gg 1$ we can also assume that $\Sigma\defeq \{t=0\}= \{\tilde{t}=0\}$ is a Cauchy surface for $(M, \altg)$, which can be canonically identified with $\rr^{d}$. In the sequel we will fix such a time function $\tilde{t}$.

\subsubsection{Klein-Gordon operator}\label{sec0.1.2}
We  fix a real function $V\in \cinf(M; \rr^{d})$ such that
\[
({\rm aM} iv) \quad V(x)- \altm^{2} \in S^{-\delta}_{\std}(\rr^{1+d}),\hbox{ for some }\altm>0,\ \delta>1,
\]
and consider the Klein-Gordon operator
\[
P= - \Box_{\altg}+V.
\]
\subsection{The Feynman inverse of $P$}\label{sec0.2}
We now introduce the Hilbert spaces  $\cX^{m}_{\F}$, $\cY^{m}$ between which $P$ will be  invertible. The spaces $\cY^{m}$ are standard spaces of right hand sides for the Klein-Gordon equations, their essential property being that they are $L^{1}$ in $t$, with values in some Sobolev spaces of order $m$ (not to be confused with the Klein-Gordon mass parameter $\altm$). The spaces $\cX^{m}_{\F}$ incorporate the {\em Feynman boundary conditions}, which are imposed at $t= \pm \infty$.

\subsubsection{Hilbert spaces}\label{sec0.2.1}
Using the Cauchy time function $\tilde{t}$ we can  identify $M$ with $\rr\times \Sigma$, using the flow $\phi_{t}$ of the vector field $v= \frac{\altg^{-1}d\tilde{t}}{d\tilde{t}\cdot \altg^{-1} d\tilde{t}}$ and obtain  the diffeomorphism:
\beq\label{e0.10}
\chi: \rr\times \Sigma\ni (t, \rx)\to \phi_{t}(\rx)\in M,
\eeq
such that
\[
\chi^{*}\altg= - c^{2}(t, \rx)dt^{2}+ h(t, \rx)d\rx^{2}.
\]
For $m\in \rr$ we denote by $H^{m}(\rr^{d})$ the usual Sobolev spaces on $\rr^{d}$. We set for $\12<\gamma<\12+ \delta$:
\[
\cY^{m}\defeq \{u\in \cD'(M): \chi^{*}u\in \langle t\rangle^{-\gamma}L^{2}(\rr; H^{m}(\rr^{d}))\}
\]
with norm $\|v\|_{\cY^{m}}= \| \langle t\rangle^{\gamma}\chi^{*}u\|_{L^{2}(\rr; H^{m}(\rr^{d}))}$.  The exponent $\gamma$ is chosen such that $ \langle t\rangle^{-\gamma}L^{2}(\rr)\subset L^{1}(\rr)$. Similarly we set
\[
\cX^{m}\defeq \{u\in \cD'(M): \chi^{*}u\in C^{0}(\rr; H^{m+1}(\rr^{d}))\cap C^{1}(\rr; H^{m}(\rr^{d})), \ Pu\in \cY^{m}\}.
\]
We equip  $\cX^{m}$ with the norm
\[
\| u\|_{\cX^{m}}= \| \varrho_{0} u\|_{\cE^{m}}+ \| Pu\|_{\cY^{m}},
\]
where $\varrho_{s}u= \col{u\traa{\Sigma_{s}}}{\i^{-1}\p_{n}u\traa{\Sigma_{s}}}$ is the Cauchy data map on $\Sigma_{s}\defeq \tilde{t}^{-1}(\{s\})$  and $\cE^{m}\defeq H^{m+1}(\rr^{d})\oplus H^{m}(\rr^{d})$ is the  energy space of order $m$. From the well-posedness of the inhomogeneous Cauchy problem for $P$ one easily deduces that $\cX^{m}$ is a Hilbert space.

\subsubsection{Feynman boundary conditions}\label{sec0.2.2}
Let us set
\[
c^{\pm}_{\free}=\12\begin{pmatrix}1 & \pm \sqrt{-\Delta_\rx+\altm^2} \\ \pm \sqrt{-\Delta_\rx+\altm^2} & 1\end{pmatrix},
\]
which are the spectral projections on $\rr^{\pm}$ for the generator
\[
H_{\free}= \mat{0}{1}{-\Delta_\rx+\altm^2}{0}
\]
of the Cauchy evolution for the free Klein-Gordon operator $P_{\free}= \p_{t}^{2}- \Delta_{\rx}+ \altm^{2}$, to which $P$  is asymptotic when $t\to \pm \infty$.

We then set
\[
\cX^{m}_{\rm F}\defeq \{u\in \cX^{m}: \lim_{t\to \mp\infty}c^{\pm}_{\free}\varrho_{t}u=0\hbox{ in }\cE^{m}\}.
\]
It is easy to see that $\cX^{m}_{\F}$ is a closed subspace of $\cX^{m}$.

In this paper we will prove the following theorem:
\begin{theoreme}\label{thm1}
 Assume $({\rm aM})$. Then $P: \cX^{m}_{\rm F}\to \cY^{m}$ is boundedly invertible for all $m\in\rr$. Its inverse $G_{\rm F}$ is called the {\em Feynman inverse} of $P$. It satisfies:
 \[
  \WF(G_{{\rm F}})'= ({\rm diag}_{T^*M})\cup\textstyle\bigcup_{t\leq 0}(\Phi_t({\rm diag}_{T^*M})\cap \pi^{-1}\cN).
  \]
\end{theoreme}
\section{Proof of Thm.~\ref{thm1}}\label{sec1}\init
We now give the proof of Thm.~\ref{thm1}. The first step consists in replacing $P$ by $\chi^{*}P$, where $\chi$ is the diffeomorphism in \eqref{e0.10}. After an additional conformal transformation,  we can reduce $\chi^{*}P$ to a model Klein-Gordon operator of the type introduced in Subsect.~\ref{sec1.1}. The reduction procedure is explained in details in \cite[Sect. 4]{GW1}. Theorem \ref{thm1} is then reduced to Thm.~\ref{thm1.2} below, whose proof will be explained in this section.
 \subsection{Model Klein-Gordon operators}\label{sec1.1}
Let us recall the model Klein-Gordon operators introduced in \cite[Sect. 2]{GW1}.
We work on $\rr^{1+d}$ with elements $x= (t, \rx)$ equipped
with the Lorentzian metric
\[
g= - dt^{2}+ \alth_{ij}(t, \rx)d\rx^{i}d\rx^{j},
\]
where $t\mapsto \alth_{t}= \alth_{ij}(t, \rx)d\rx^{i}d\rx^{j}$ is a smooth family of Riemannian metrics on $\rr^{d}$.  Fixing a real potential $V\in \cinf(\rr^{1+d}; \rr)$ the Klein-Gordon operator $P = -\Box_{g}+V$ takes the form
\begin{equation}
\label{e1.0}
P= \p_{t}^{2}+ r(t, \rx)\p_{t}+ a(t, \rx, \p_{\rx}),
\end{equation}
where
\[
\begin{array}{l}
a(t)= a(t, \rx, \pe_{\rx})= - |\alth|^{-\12}\pe_{i}\alth^{ij}|\alth|^{\12}\pe_{j}+ \altV(t,\rx),\\[2mm]
 r(t)= r(t,\rx )= |\alth|^{-\12}\p_{t}(|\alth|^{\12})(t,\rx ).
\end{array}
\]
The operator $a(t)$ is formally selfadjoint for the  time-dependent scalar product
\[
(u|v)_{t}= \int_{\Sigma}\bar{u}v|h_{t}|^{\12}d\rx,
\]
and $P$ is formally selfadjoint for the scalar product
\[
(u|v)=  \int_{\rr\times \Sigma}\bar{u}v|h_{t}|^{\12}d\rx dt.
\]
Conditions $({\rm aM})$ on the original metric $\altg$ and potential $V$ imply similar asymptotic conditions on $a(t, \rx, \p_{\rx})$ and $r(t, \rx)$ when $t\to \pm \infty$. More precisely one has:
\[
({\rm Hstd}) \  \ \beal
a(t, \rx, \p_{\rx})= a_{\outin}(\rx, \p_{\rx})+ \Psi_{\std}^{2, - \delta}(\rr; \rr^{d})  \hbox{ on }\rr^{\pm}\times \rr^d,\\[2mm]
r(t)\in \Psi_{\std}^{0, -1-\delta}(\rr; \rr^{d}),\\[2mm]
a_{\outin}(\rx, \p_{\rx})\in \Psi_{\scc}^{2,0}(\rr^{d})\hbox{\ is  elliptic},\\[2mm]
a_{\outin}(\rx, \p_{\rx})= a_{\outin}(\rx, \p_{\rx})^{*}\geq C_{\infty}>0,
\eeal
\]
where $\Psi_{\std}^{m, \delta}(\rr; \rr^{d})$ is the class of time-dependent pseudodifferential operators on $\rr^{d}$ associated to symbols $m(t, \rx, k)$ such that
\[
 \p_{t}^{\gamma}\p_{\rx}^{\alpha}\p_{\spexi}^{\beta}m(t, \rx, \spexi)\in O((\langle t\rangle+ \langle\rx\rangle)^{\delta- \gamma- |\alpha|}\langle \spexi\rangle^{m-|\beta|}),  \ \gamma\in \nn, \ \alpha, \beta\in \nn^{d}.
\]
Similarly $\Psi_{\scc}^{m, \delta}(\rr^{d})$ is the class of pseudodifferential operators on $\rr^{d}$ associated to symbols $m( \rx, k)$ such that
\[
\p_{\rx}^{\alpha}\p_{\spexi}^{\beta}m(\rx, \spexi)\in O( \langle\rx\rangle^{\delta-|\alpha|}\langle \spexi\rangle^{m-|\beta|}),  \ \alpha, \beta\in \nn^{d}.
\]
We refer the reader to \cite[Subsect.~2.3]{GW1} for more details.

\subsubsection{A further reduction}\label{sec1.1.1}
It is convenient to perform a further reduction to the case $r=0$. Namely setting $R= |h_{0}|^{\frac{1}{4}}|h_{t}|^{- \frac{1}{4}}$, we see that
\[
L^{2}(\Sigma, |h_{0}|^{\12}d\rx)\ni\tilde{u}\mapsto R\tilde{u}\in L^{2}(\Sigma, |h_{t}|^{\12}d\rx)
\]
is unitary and that
\[
R^{-1}PR\eqdef\tilde{P}= \p_{t}^{2}+ \tilde{a}(t, \rx, \p_{\rx})
\]
where
\[
\tilde{a}(t)= rR^{-1}(\p_{t}R)+ R^{-1}(\p_{t}^{2}R)+ R^{-1}a(t)R
\]
is formally selfadjoint for $(\cdot| \cdot)_{0}$. Clearly $\tilde{a}(t, \rx, \p_{\rx})$ satisfies also $({\rm Hstd})$, with the same asymptotic $a_{\outin}(\rx, \p_{\rx})$. It is also immediate that the Hilbert spaces $\cY^{m}$, $\cX^{m}$, $\cX^{m}_{\F}$ introduced in Subsect.~\ref{sec1.3} are invariant under the map $u\mapsto Ru$ and hence  it suffices to prove Thm.~\ref{thm1.2} for $P$ replaced by $\tilde{P}$.

To simplify notation we will denote again $\tilde{P}$ by $P$. Summarizing we have $P= \p_{t}^{2}+ a(t, \rx, \p_{\rx})$, conditions $({\rm Hstd})$ are satisfied (with $r= 0$) and $a(t, \rx, \p_{\rx})$ is formally selfadjoint on
$L^{2}(\Sigma, |h_{0}|^{\12}d\rx)$ for all $t\in \rr$.

\subsection{Approximate diagonalization}\label{sec1.2}
Setting $\varrho u= \col{u}{\i^{-1}\p_{t}u}$, the equation $Pu=v$ is equivalent to
\[
(D_{t}- H(t))\varrho u= - \pi_{1}^{*}v, \ \ H(t)= \mat{0}{1}{a(t)}{0}
\]
where $\pi_{i}f= f_{i}$ for $f=\col{f_{0}}{f_{1}}$.

 One can  then construct an operator $T$ with $Tf(t)= T(t)f(t)$, and $t\mapsto T(t)$ is a smooth family of matrix-valued pseudodifferential operators on $\rr^{d}$ such that
\[
T^{-1}(D_{t}- H(t))T= D_{t}- H^{\rm ad}(t)\eqdef P^{\adg},
\]
where $H^{\rm ad}(t)$ is almost diagonal ie
\[
H^{\rm ad}(t)= H^{\rm d}(t)+ V^{\rm ad}_{-\infty}(t),
\]
\beq\label{e1.-1}
H^{\rm d}(t)= \mat{\epsilon^{+}(t)}{0}{0}{- \epsilon^{-}(t)},
\eeq
where $\epsilon^{\pm}(t)$ are time-dependent pseudodifferential operators on $\rr^{d}$, with principal symbols equal to $\pm (k\dual h^{-1}(t, \rx)k)^{\12}$, and $V^{\rm ad}_{-\infty}(t)$ is an off-diagonal matrix of time-dependent operators on $\rr^{d}$ such that
 \[
 (\langle \rx\rangle + \langle t\rangle)^{m}V^{\rm ad}_{-\infty}(t)(\langle \rx\rangle + \langle t\rangle)^{-m+\delta}: H^{-p}(\rr^{d})\to H^{p}(\rr^{d})
 \]
  is uniformly bounded in $t$ for all $m, p\in \rr$.
Let us denote by $\cU^{\adg}(t, s)$ for $t,s\in \rr$ the Cauchy evolution generated by $H^{\adg}(t)$, i.e.~the solution of
\[
(D_{t}- H^{\adg}(t)) \cU^{\adg}(t,s)= 0, \ \ \cU^{\adg}(s, s)= \one.
\]
Then $\cU^{\adg}(t, s)$ is {\em symplectic}, which translates into the identity:
 \beq\label{e1.1}
H^{\rm ad}(t)^{*}q^{\rm ad}= q^{\rm ad}H^{\adg}(t), \ \ q^{\rm ad}\defeq\mat{1}{0}{0}{-1},
\eeq
where the adjoint is computed w.r.t.~the scalar product of $\cH^{0}= L^{2}(\rr^{d}, |h_{0}|^{\12}d\rx;  \cc^{2})$.

\subsection{Hilbert spaces}\label{sec1.3}
We set for $m\in \rr$
 \[
\cE^{\sobo}\defeq H^{\sobo+1}(\rr^{d})\oplus H^{\sobo}(\rr^{d}), \ \ \cH^{\sobo}\defeq H^{\sobo}(\rr^{d})\oplus H^{\sobo}(\rr^{d}), \ \sobo\in \rr,
\]
where $H^{m}(\rr^{d})$ are the usual Sobolev spaces. Fixing  $\gamma$ with $\12<\gamma<\12 + \delta$, we set:
\[
\cY^{\sobo}\defeq \langle t\rangle^{-\gamma}L^{2}(\rr; H^{\sobo}), \ \ \cY^{\adg, \sobo}\defeq   \langle t\rangle^{-\gamma}L^{2}(\rr; \cH^{\sobo}),
\]
and denote by $\cX^{m}$ the space of $u\in C^{1}(\rr; H^{m})\cap C^{0}(\rr; H^{m+1})$ such that $Pu\in \cY^{m}$, by $\cX^{\adg, \sobo}$ the space of $u^{\adg}\in C^{0}(\rr; \cH^{m})$ such that $P^{\adg}u^{\adg}\in \cY^{\adg}$. These spaces are equipped with the Hilbert space norms
 \beq\label{defdenorme}\begin{array}{l}
 \|u^{\adg}\|^{2}_{\sobo}\defeq \| \varrho^{\adg}_{0}u^{\adg}\|^{2}_{\cH^{\sobo}}+ \| P^{\adg}u^{\adg}\|^{2}_{\cY^{\adg, \sobo}},\\[2mm]
  \|u\|^{2}_{\sobo}\defeq \| \varrho_{0}u\|^{2}_{\cE^{\sobo}}+ \| Pu\|^{2}_{\cY^{\sobo}},
\end{array}
 \eeq
 where $\varrho_{t}u= \col{u(t)}{\i^{-1}\p_{t}u(t)}$, $\varrho^{\adg}_{t}u^{\adg}= u^{\adg}(t)$ are the Cauchy data maps at $t=0$. The well-posedness of the inhomogeneous Cauchy problems for $P$ and $P^{\adg}$, see \cite[Lemma 3.5]{GW1} implies that $\cX^{(\adg),m}$ are Hilbert spaces.

\subsubsection{Feynman boundary conditions}\label{sec1.3.1}
Let us set
\[
\pi^{+}= \mat{1}{0}{0}{0}, \ \ \pi^{-}= \mat{0}{0}{0}{1}.
\]
It is straightforward to show that
\beq\label{e1.4}
\cX^{\adg, m}_{\rm F}\defeq\{u^{\adg}\in \cX^{\adg, m}: \lim_{t\to -\infty}\pi^{+}\varrho^{\adg}_{t}u^{\adg}= \lim_{t\to +\infty}\pi^{-}\varrho^{\adg}_{t}u^{\adg}=0\hbox{ in }\cH^{m}\}.
\eeq
is a closed subspace of $\cX^{\adg, m}$.
In \cite[Sect. 3.7]{GW1} $\cX^{\adg, m}_{\rm F}$  is defined using scattering data maps, see \cite[Def.~3.7]{GW1}:  one sets
\[
\epsilon_{\outin}= a_{\outin}(\rx, \p_{\rx})^{\12}, \ \ H_{\outin}^{\adg}= \mat{\epsilon_{\outin}}{0}{0}{-\epsilon_{\outin}}
\]
 and $\cU_{\outin}^{\adg}(t,s)= \e^{\i(t-s)H^{\adg}_{\outin}}$. Note that $H_{\outin}^{\adg}$ are actually diagonal \emph{exactly}, hence $\cU_{\outin}^{\adg}(t,s)$ commute with $\pi^+$ and $\pi^-$. The scattering data maps are
 \[
\varrho^{\adg}_{\outin}u^{\adg}= \lim_{t\to \pm \infty}\cU_{\outin}^{\adg}(0, t)u^{\adg}(t)\hbox{ in }\cH^{m},
\]
in terms of which $\cX^{\adg, m}_{\rm F}$ can be defined as
\[
\cX^{\adg, m}_{\rm F}= \{u^{\adg}\in \cX^{\adg, m}: \pi^{+}\varrho_{\rm in}^{\adg}u^{\adg}= \pi^{-}\varrho_{\rm out}^{\adg}u^{\adg}=0\}.
\]
 Both definitions are the same using that $\cU^{\adg}(t, s)$ and $\cU^{\adg}_{\rm out/in}(t,s)$  are uniformly bounded in $B(\cH^{m})$, as shown in \cite[Prop.~5.6]{GW2}.

Similarly we define $\cX^{m}_{\F}$ as
\[
\cX^{m}_{\F}\defeq\{u\in \cX^{m}: \lim_{t\to-\infty}c^{-}_{\rm out}\varrho_{t}u= \lim_{t\to+\infty}c^{+}_{\rm in}\varrho_{t}u=0\hbox{ in }\cE^{m}\}
\]
where
$c^{\pm}_{\outin}= \12\mat{1}{\pm a_{\outin}^{\12}}{\pm a_{\outin}^{\12}}{1}$
 are the spectral projections on $\rr^{\pm}$ of $H_{\outin}= \mat{0}{1}{a_{\outin}}{0}$. Again, $\cX^{m}_{\F}$ is a closed subspace of $\cX^{m}$.

\subsection{Invertibility of $P$}
We now prove the following theorem:
\begin{theoreme}\label{thm1.2}
 $P: \cX^{m}_{\rm F}\to \cY^{m}$ is boundedly invertible with inverse
 \[
G_{\rm F}= - \pi_{0} T G^{\adg}_{\rm F}T^{-1}\pi_{1}^{*}.
\]
Moreover $G_{\rm F}$ is a {\em Feynman inverse} of $P$, i.e.,
\beq\label{e1.6}
 \WF(G_{{\rm F}})'= ({\rm diag}_{T^*M})\cup\textstyle\bigcup_{t\leq 0}(\Phi_t({\rm diag}_{T^*M})\cap \pi^{-1}\cN).
\eeq
\end{theoreme}

In view of the results in \cite{GW1}, the only part that deserves special attention is the proof that $P$ is injective, which is reduced to a similar statement about $P^{\adg}$.
The proof of Lemma \ref{lemma1.1} below is inspired by  the work of Vasy \cite[Prop.~7]{Va}, which in turn relies on  arguments of Isozaki \cite{I} from $N$-body scattering. In our framework it turns out to be very simple.
\begin{lemma}\label{lemma1.1}
 One has:
 \[
\Ker P^{\rm ad}|_{\cX^{\adg, m}_{\rm F}}= \{0\}\hbox{ for all }m\geq 1.
\]
\end{lemma}
\proof
 Let us set $\chi_{\epsilon}(t)= \int_{|t|}^{+\infty}\one_{[1, 2]}(\epsilon s)s^{-r}ds$ for some $0<r<1$. Note that $\supp \chi_{\epsilon}\subset \{|t|\leq 2 \epsilon^{-1}\}$. Let us still denote by $\chi_{\epsilon}$ the operator $\chi_{\epsilon}\otimes\one_{\cc^{2}}$.  Recalling that $q^{\adg}$ is defined in \eqref{e1.1},  we compute for $u^{\adg}\in \cX^{\adg, m}_{\rm F}$:
 \[
 \bea
&\int_{\rr}(P^{\adg}u^{\adg}(t)| q^{\adg}\chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}dt-(\chi_{\epsilon}(t)u^{\adg}(t)| q^{\adg} P^{\adg}u^{\adg}(t))_{\cH^{0}}dt\\[2mm]
&= \int_{\rr} (D_{t}u^{\adg}(t)| q^{\adg} \chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}- (u^{\adg}(t)| q^{\adg}\chi_{\epsilon}(t)D_{t}u^{\adg}(t))_{\cH^{0}}\\[2mm]
&\phantom{=}\, + \int_{\rr}  (u^{\adg}(t)| q^{\adg}[H^{\adg}(t), \chi_{\epsilon}(t)]u^{\adg}(t))_{\cH^{0}}dt,
\eea
\]
using that $H^{\adg *}(t)q^{\adg}= q^{\adg}H^{\adg}(t)$, $\chi_{\epsilon}(t)^{*}q^{\adg}= q^{\adg}\chi_{\epsilon}(t)$ and $u^{\adg}(t)\in \Dom H^{\adg}(t)$ since $m\geq 1$.
We have $[H^{\adg}(t), \chi_{\epsilon}(t)]=0$,  and using  that $\chi_{\epsilon}$ is compactly supported in $t$ we can integrate by parts in $t$ in the second line and obtain
\begin{equation}
\label{e1.2}
\bea
&\int_{\rr}(P^{\adg}u^{\adg}(t)| q^{\adg}\chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}dt-(\chi_{\epsilon}(t)u^{\adg}(t)| q^{\adg} P^{\adg}u^{\adg}(t))_{\cH^{0}}dt\\[2mm]
&=-\i\int_{\rr}(u^{\adg}(t)| q^{\adg}\p_{t}\chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}dt.
\eea
\end{equation}
Note that we used here that the scalar product in $\cH^{0}$ does not depend on $t$, which is the reason for the reduction to $r=0$ in \ref{sec1.1.1}.

Since $P^{\adg}u^{\adg}=0$ this yields
\beq\label{e1.5}
\int_{\rr}(u^{\adg}(t)| q^{\adg}\p_{t}\chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}dt=0.
\eeq
We claim that:
\beq\label{e1.3}
\begin{array}{rl}
i)&\| \pi^{\pm}u^{\adg}(t)\|^{2}_{\cH^{0}}\in O(t^{1- \delta}) \hbox{ when }t\to \mp \infty,\\[2mm]
ii)&\| \pi^{\pm}u^{\adg}(t)\|^{2}_{\cH^{0}}= c^{\pm}+ O(t^{1-\delta})\hbox{ when }t\to \pm \infty.
\end{array}
\eeq
for $c^{\pm}= \lim_{t\to \pm\infty}\|\pi^{\pm}u^{\adg}(t)\|^{2}_{\cH^{0}}$. The proof of \eqref{e1.3}  is elementary:  we have $H^{\adg}(t)- H_{\outin}^{\adg}\in O(t^{-\delta})$ in $B(\cH^{0})$ when $t\to \pm \infty$, see e.g.~\cite[Subsect.~2.5]{GW1}, which  using $\delta>1$ and  the Cook argument (i.e., estimating first the derivative in time, and then integrating) yields:
\[
\begin{array}{l}
W_{\rm out/in}^{\dag}u^{\rm ad}= \lim_{t\to \pm\infty}\cU_{\rm out/in}^{\adg}(0, t)u^{\rm ad}(t)\hbox{ exists in }\cH^{0},\\[2mm]
\|W_{\rm out/in}^{\dag}u^{\rm ad}- \cU_{\rm out/in}^{\adg}(0, t)u^{\rm ad}(t)\|_{\cH^{0}}\in O(t^{1-\delta}).
\end{array}
\]
 Since $\cU_{\rm out/in}^{\adg}(0, t)$ is unitary on $\cH^{0}$ and $u^{\rm ad}(t)\in\cX^{m}_{\F}$, this implies \eqref{e1.3}.

Next, we compute
\[
\bea
&\int_{\rr}(u^{\adg}(t)| q^{\adg}\p_{t}\chi_{\epsilon}(t)u^{\adg}(t))_{\cH^{0}}dt\\[2mm]
& = \int_{\rr}\p_{t}\chi_{\epsilon}(t)\|\pi^{+}u^{\rm ad}(t)\|_{\cH^{0}}^{2}dt- \int_{\rr}\p_{t}\chi_{\epsilon}(t)\|\pi^{-}u^{\rm ad}(t)\|_{\cH^{0}}^{2}dt.
\eea
\]
Since  $\p_{t}\chi_{\epsilon}(t)= -{\rm sgn}(t)\one_{[\epsilon^{-1}, 2\epsilon^{-1}]}(|t|)|t|^{-r}$ we have using \eqref{e1.3}:
\[
0\leq \int_{\rr^{\mp}}|\p_{t}\chi_{\epsilon}(t)|\|\pi^{\pm}u^{\rm ad}(t)\|_{\cH^{0}}^{2}dt\leq C\int\one_{[\epsilon^{-1}, 2\epsilon^{-1}]}(|t|) |t|^{-r- \delta+1}dt\in O(\epsilon^{r+ \delta-2}),
\]
and
\[
\bea
\int_{\rr^{\pm}}\p_{t}\chi_{\epsilon}(t)\|\pi^{\pm}u^{\rm ad}(t)\|_{\cH^{0}}^{2}dt&= \mp\int_{\rr^{\pm}} \one_{[\epsilon^{-1}, 2\epsilon^{-1}]}(|t|)c^{+}|t|^{-r} dt+ O(\epsilon^{r+ \delta-2})\\[2mm]
&=\mp Cc^{\pm}\epsilon^{r-1}+ O(\epsilon^{r+ \delta-2}).
\eea
\]
Using \eqref{e1.5} this yields $C\epsilon^{r-1}(c^{+}+ c^{-})\in O(\epsilon^{r+ \delta-2})$ hence $c^{+}= c^{-}=0$ since $\delta>1$. Therefore  by \eqref{e1.3} we have $\lim_{t\to \pm \infty}\| u^{\adg}(t)\|_{\cH^{0}}=0$. Since  the Cauchy evolution $\cU^{\adg}(t, s)$ is uniformly bounded in $B(\cH^{0})$ we have $u^{\adg}(0)=0$ hence $u=0$. \qed

\begin{proposition}\label{prop1.1}
  The operator $P^{\adg}: \cX_{\rm F}^{\adg, m}\to \cY^{\adg,m}$ is boundedly invertible for all $m\in \rr$.
  \end{proposition}
\proof
Recall that the  diagonal operator $H^{\d}(t)$  was introduced in \eqref{e1.-1}. In \cite[Thm.~3.16]{GW1} it is shown that $P^{\d}= D_{t}- H^{\d}(t):  \cX_{\rm F}^{\adg, m}\to \cY^{\adg,m}$ is boundedly invertible, with inverse $G^{\d}_{\rm F}$ given by:
 \[
\bea
G^{\d}_{\rm F} v^{\adg}(t)&\defeq  \i\int_{-\infty}^{t}\cU^{\d}(t, 0)\pi^{+}\cU^{\d}(0,s)v^{\adg}(s)ds\\ &\phantom{=\,}-\i\int_{t}^{+\infty}\cU^{\d}(t, 0)\pi^{-}\cU^{\d}(0,s)v^{\adg}(s)ds.
\eea
\]
Moreover by \cite[Lemma 3.7]{GW1} $V^{\adg}_{-\infty}$ is compact from $\cX^{\adg, m}$ to $\cY^{m}$  hence also from $\cX_{\rm F}^{\adg, m}$ to $\cY^{m}$ since $\cX_{\rm F}^{\adg, m}$ is closed in $\cX^{\adg, m}$.

Therefore $P^{\adg}= P^{\d}+ V^{\adg}_{-\infty}: \cX_{\rm F}^{\adg, m}\to \cY^{m}$ is Fredholm of index $0$. If $u^{\adg}\in \Ker P^{\rm ad}|_{\cX^{\adg, m}_{\rm F}}$, we have $u^{\adg}= - G^{\d}_{\rm F}V^{\adg}_{-\infty}u^{\adg}$, from which we deduce that $u^{\adg}\in \cX^{\adg, m'}_{\rm F}$ for any $m'$, using that $V^{\adg}_{-\infty}$ is smoothing in $\rx$. By Lemma \ref{lemma1.1} we have  $\Ker P^{\rm ad}|_{\cX^{\adg, m}_{\rm F}}= \{0\}$ hence $P^{\adg}:\cX_{\rm F}^{\adg, m}\to \cY^{m}$  is boundedly invertible. \qed

\subsubsection{Proof of Thm.~\ref{thm1.2}}
 By \cite[(3.20)]{GW1} we know that  \[
 \pi_{0}T\in B(\cX^{\adg, \sobo+ \12}, \cX^{\sobo}), \ \ T^{-1}\pi_{1}^{*}\in B(\cY^{\sobo}, \cY^{\adg, \sobo+\12})
 \]
  hence $G_{\rm F}: \cY^{m}\to \cX^{m}$.  Since $(D_{t}- H(t))TG^{\adg}_{\rm F}T^{-1}=TG^{\adg}_{\rm F}T^{-1}(D_{t}- H(t))= \one$, we obtain that $P G_{\rm F}= G_{\rm F}P= \one$.
 We also have $\varrho\pi_{0}T G^{\adg}_{\rm F}T^{-1}\pi_{1}^{*}= T G^{\adg}_{\rm F}T^{-1}\pi_{1}^{*}v$.
From \cite[ equ. (3.25)]{GW1} we know that  $\varrho_{\overline{\rm F}}= \varrho^{\adg}_{\overline{\rm F}}T^{-1}\varrho$, hence $\varrho_{\overline{\rm F}}G_{\rm F}= 0$, i.e.~$G_{\rm F}: \cY^{m}\to \cX_{\rm F}^{m}$.

To prove the second statement,  let  $\tilde{G}_{\rm F}= - \pi_{0} T G^{\d}_{\rm F}T^{-1}\pi_{1}^{*}$.  We have $G_{\rm F}^{\d}- G_{\rm F}^{\adg}\eqdef R= G^{\d}_{\rm F}V^{\adg}_{-\infty}G^{\adg}_{\F}$ by the resolvent identity.  It is shown in \cite[Lemma 3.7]{GW1} that $V^{\adg}_{-\infty}: \cX^{\adg, m}\to \cY^{m'}$ is bounded for all $m'>m$, hence $R: \cY^{\adg, m}\to \cX^{\adg, m'}$ for all $m'>m$ i.e.~it is smoothing in the $\rx$ variables. We use then that
$D_{t}R= H^{\d}(t)R+ V^{\adg}_{-\infty}G^{\adg}_{\F}$, $RD_{t}= R H^{\adg}(t)+ G^{\d}_{\rm F}V^{\adg}_{-\infty}$ to gain regularity in the $t$ variable and obtain that $R: \cE'(\rr^{1+d}; \cc^{2})\to \cinf(\rr^{1+d}; \cc^{2})$.  Therefore, $G_{\rm F}- \tilde{G}_{\rm F}$ is a smoothing operator and it is shown in \cite[Thm.~3.18]{GW1} that $\WF(\tilde{G}_{{\rm F}})'$ equals the r.h.s.~of \eqref{e1.6}, which completes the proof.
 \qed