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\maketitle

\begin{abstract}
\noindent We review  geometrical properties of a static spacetime $(M,g)$, including geodesic completeness, causality, standard splittings, compact $M$, closed geodesics and geodesic connectedness. We pay special attention to the critical quadratic behavior at infinity of the coefficients $\beta$, $\beta^{-1}$  ($\beta = -g(K,K)$, being $K$ a timelike irrotational Killing vector field), which essentially control completeness, causality and geodesic connectedness. Recent references are specially discussed.

\smallskip

\noindent {\em Keywords}: Static spacetime, geodesic, completeness, connectedness, closed geodesic, variational methods, causality, critical quadratic case.
\end{abstract}

\section{Introduction}

\noindent Static spacetimes
are one of the simplest classes of Lorentzian manifolds. They  admit well-known physical interpretations related to the synchronizability and invariance of the metric for a field of priviledged observers $K$ (see, for example, \cite{SW}), and
 include some classical spacetimes, as outer Schwarzschild and Reissner Nordstr\"om. Thus, many of their geometric properties have been studied from different viewpoints and, recently, there has been renewed progress made \cite{AU}, \cite{BCFS}, \cite{CMP},   \cite{Sa-St}.
Our purpose is to review and discuss some of the geometrical properties and techniques, sketching some of the proofs (at least when the results have not been explicitly stated in the references).

This paper is organised as follows. In Section \ref{s2} both, geodesic completeness  and the possibility to split the static spacetime as a standard one,
are characterized  (Theorems \ref{t2a}, \ref{t2b}). In Section \ref{s3},  the causal ladder of a standard static spacetime is shown to be  causally continuous but not  causally simple (Theorem \ref{t3a}, Remark \ref{r3a}), and possible standard splittings in the globally hyperbolic case are discussed, Remark \ref{r3b}. Causality Theory and different geometrical tools apply for the problem of closed geodesics and connectedness by causal geodesics. In Section \ref{s4}, which is based on \cite{Sa-St}, this is shown explicitly when $M$ is compact. In fact, some general causal and geometrical properties (Theorem \ref{t4a}, Proposition \ref{p4a}) are shown to imply the connectivity by  timelike geodesics (Theorem \ref{t4c}) and the existence of a closed timelike geodesic (Theorem \ref{t4d}); this  improves widely the analogous variational results in \cite{CMP}. In Section \ref{s5}, general properties of geodesics are discussed, including related problems as periodic trajectories or boundaries. Finally, in Section \ref{s6}, we show that the variational techniques applied in \cite{BCFS} (firstly introduced for the Riemannian case in \cite{CFS}) yield accurate results on geodesic connectedness, improving previous results and techniques.

\section{Completeness and standard static spacetimes} \label{s2}

In what follows, $(M,g)$ denotes a  connected $n(\geq 2)$-Lorentzian manifold $(-,+,$ $\cdots , +)$, which is smooth enough ($C^\infty$ as a simplification, but we only need $C^1$ for most purposes).  Our notation and conventions will be standard, as in the books \cite{BEE}, \cite{O}, \cite{SW}. A Lorentzian manifold will be called a  stationary spacetime if it admits a timelike Killing vector field $K$, and {\em static} if, additionally, $K$ is  irrotational (the orthogonal distribution to $K$ is involutive). In this case, we will refer to $K$ as the  static vector field, which will be assumed to time-orient the spacetime. $K$ is not determined univocally; nevertheless, if another static $K'$ exists and it satisfies

$K'= f K$ for some $f\in C^\infty (M)$, then $f$ is a constant \cite[p. 224]{SW}. Some properties of Lorentzian manifolds admitting a Killing vector field are reviewed in \cite{Sa-grec}.
A {\em standard} static spacetime is a product
$\R\times S$ endowed with the metric
\be \label{ems}
 g[(t,x)] = -\beta(x) dt^2 + g_S[x]
\ee
where $g_S$ is a Riemannian metric on $S$. Locally, any static spacetime looks like a standard one, with $K$ identifiable to $\partial_t$. Note that function $\beta = -g(K,K) \in C^\infty(M)$  makes sense also in the non-standard case (as well as the integral hypersurfaces of the orthogonal foliation $K^\perp$).

An auxiliary Riemannian metric $g_R$ can be constructed by reversing
the sign of $g(K,K)$, that is,
$ g_R(u,v) = g(u,v)+ 2 g(u,K)g(v,K)/\beta , $ for all $u,v \in TM. $
The qualitative behavior of $\beta$ at infinity for this Riemannian metric (or, eventually, $g_S$ in the standard case) will be especially relevant. A function $V$ on a complete Riemannian manifold behaves {\em at most quadratically} (at infinity) if, for some $A >0$:
\be \label{e2} V(x) \leq A\cdot d(x)^2
\ee
for all $x$ outside a compact subset, where $d$ is the distance (canonically associated to the Riemannian metric) to some fixed point $x_0$.

The (geodesic) completeness of a static spacetime is interesting not only in its own right, but also in relation to the structure of the spacetime.

\begin{teor} \label{t2a} Let $(M,g)$ be a static spacetime:

(1) If $g$ is complete, then both the static vector field $K$ and the integral hypersurfaces of
$K^\perp$ are complete with the restriction of the metric $g$.

(2) If the auxiliary Riemannian metric $g_R$ is complete and $\beta^{-1}$ behaves at most quadratically,  then $g$ is complete. In particular, the following are complete: (a) compact static spacetimes, and (b) standard static spacetimes with $g_S$  complete and $\beta^{-1}$ at most quadratic on $(S,g_S)$.

\end{teor}
{\em Sketch of proof.} For {\it (1)}, see \cite[Theorem 2.1(2)]{Sa-St}. For {\it (2)}, following \cite[Proposition 2.1]{RS2}, let $\gamma: [0,1)\rightarrow M$ be a $g$-geodesic. It is enough to prove that the $g_R$-length of $\gamma$ is bounded, (see \cite[Lemma 5.8]{O}). As $K$ is Killing, $g(\dot \gamma,K)= \lambda$ (constant) and $|\dot \gamma|^2_R (:=g_R(\dot \gamma,\dot \gamma))= C+2\lambda^2/\beta\circ \gamma $, with $C=g(\dot \gamma,\dot \gamma)$. Thus, choosing $x_0=\gamma(0)$ in the definition of $d$ below (\ref{e2}), as well as appropiate constants $B,D>0$:
$$ |\dot \gamma|_R(s) \leq B\cdot d(\gamma(s)) + D \leq B\cdot  \int_0^s
|\dot \gamma|_R (\bar s)d\bar s + D , $$
 and, thus,
$$ \log \left(B\cdot  \int_0^s
|\dot \gamma|_R (\bar s)d\bar s + D \right) - \log D \leq  B s \leq B .$$
Therefore, the length of $\gamma$ is bounded, as required.
Finally, {\it (a)} is obvious and, for {\it (b)}, recall \cite[Proposition 7.40]{O}. $\quad \quad $
\cvd

\noindent {\bf Note.} (1) The proof of the first assertion in {\it (2)} holds in the stationary case (and {\it (a)} holds even for conformally stationary spacetimes). (2) The optimality of the at most quadratic behaviour for $\beta^{-1}$ can be explicitly checked in the standard case with $S=\R$.

\smallskip

\noindent Completeness of $K$ is an obvious necessary condition to ensure that a static spacetime is standard with $K=\partial_t$. Notice that this condition is very mild and automatically satisfied when $M$ is compact.
In the simply connected case, it is also sufficient \cite[Theorem 2.1(1)]{Sa-St}:

\begin{teor} \label{t2b}
A simply connected static spacetime is standard if and only if at least one of its static vector fields $K$ is complete.
\end{teor}

\noindent {\bf Note.} No analogous result holds in the stationary case. In fact, there exist simply connected compact stationary manifolds of any dimension $n\geq 5$ and $n=3$ (the exceptions $n=2,4$ appear only by topological reasons: no Lorentzian manifolds of such dimensions can be compact and simply connected).

\smallskip

\noindent Then, as a consequence of Theorems \ref{t2a}, \ref{t2b} we have (for further study in the non-simply connected case, see \cite{GO}):

\begin{coro} \label{c2a}
The universal covering of any geodesically complete static spacetime is a standard static spacetime.
\end{coro}

\section{Causality} \label{s3}

Causality of static spacetimes is simplified by considering the conformal metric
$g^* = g/\beta$. In the standard case, $t$ is a time function (which implies causal stability) and $g^*$ is globally a product metric. Moreover:

\begin{teor} \label{t3a}
Any standard static spacetime $(M,g)$ as in {\rm (\ref{ems}) }
is causally continuous, and they are equivalent:

(A) $(M,g)$ is globally hyperbolic.

(B) The conformal metric $g_S^* = g_S/\beta$ is complete.

(C) Each slice $t=$constant is a Cauchy hypersurface.

In particular, this holds if $g_S$ is complete and $\beta$ at most quadratic.
\end{teor}
{\em Sketch of proof.} For the first assertion,  by \cite[Th. 3.25, Prop. 3.21]{BEE}, it is enough to prove past and future reflectivity. Let us show past reflectivity  $I^+(p) \supseteq I^+(q) \Rightarrow I^-(p) \subseteq I^-(q)$ (future is analogous). Put $p=(t_p,x_p), q=(t_q,x_q)$. Assuming the first inclusion, it is enough to prove $p_{-\epsilon} := (t_p-\epsilon,x_p) \in I^-(q)$, for all $\epsilon >0$. As $q_\epsilon := (t_q + \epsilon , x_q) \in I^+(p)$, there exists a future-directed timelike curve $\gamma(s)=(s,x(s)), s\in [t_p,t_q + \epsilon]$ joining $p$ and $q_\epsilon$. Then, the future-directed timelike curve $\gamma_{-\epsilon}(s)=(s-\epsilon  ,x(s))$ connects $p_{-\epsilon}$ and $q$, as required.

For the remainder, use   \cite[Theorems 3.67, 3.69]{BEE}. $\triangle$

\begin{remas}\label{r3a} {\rm
(1) When $g_S^*$ is not complete, the spacetime may be non-causally simple. In fact,  Take  $(S,g^*_S) = \R^2\backslash \{(1,y) : y \leq 1\}$, and choose $p=(0,0,0), q=(\sqrt 8, 2, 2) \in \R \times S$. Clearly
$q\in  \overline{J^+(p)} \backslash J^+(p)$, that is, $J^+(p)$ is not closed  as required.

(2) The required behaviour at infinity for $\beta$ is different in Theorems \ref{t3a} and \ref{t2a}, which shows the independence between completeness and global hyperbolicity. More striking, even if $g_S$ is incomplete (and, thus, so is $g$ for any $\beta$) a conformal factor $\Omega $ exists such that  $g_S^* = \Omega g_S$ is complete \cite{NO}. Thus, the corresponding $g=-\beta dt^2+g_S$  will be globally hyperbolic choosing $\beta = 1/\Omega$.

}\end{remas}
\noindent For further results  on causality in the standard case, see \cite{Sa-ba}.

In the non-standard one, recall first that a stationary spacetime is called {\em standard} when its metric can be written as in (\ref{ems}) plus, eventually, cross terms between $\R$ and $S$ independent of $t$.

\begin{propo}\label{t3c}
A globally hyperbolic spacetime $(M,g)$ which admits a complete stationary vector field $K$ is standard stationary.
\end{propo}
{\em Proof.} By using \cite{BS}, a smooth spacelike Cauchy hypersurface $S$ can be found. Then, recall that $\R \times S \rightarrow M, (t,x) \rightarrow \Phi_t(x)$ is an isometry, where $\Phi$ is the flow of $K$. $\quad $
\cvd

\begin{remas} \label{r3b} {\em (1) In the proof each hypersurface $t=$ constant is  Cauchy and, again, the completeness of $K$ is essential (consider an open diamond $I^+(p) \cap I^-(q)$ in Lorentz-Minkowski space).

(2)  Proposition  \ref{t3c} holds in particular in the static case, but:

{\em in general,
a complete static globally hyperbolic spacetime $(M,g)$ is not a standard static one.}

To construct an explicit counterexample, take the cylinder $\R\times S^1$,
$g= -E(\theta )dt^2 + 2 F(\theta ) dt d\theta + G(\theta) d\theta^2$, $E,  G >0$, which is complete and admits the Cauchy hypersurface $t=0$. Clearly,  $K=\partial_t$ is a static vector field (the distribution $K^\perp$ is one-dimensional), and it is unique (up to a constant)
for choices of  $E, F, G$ with non-constant curvature (no two independent Killing vector fields can exist on a neighborhood with non-constant curvature,
see the proof of \cite[Theorem 4.2(1)]{Sa-trans}). But, for most choices of $E,F,G$, the integral curves of $K^\perp$ are not circles and, thus, neither Cauchy hypersurfaces (see \cite[Example 2.4]{Sa-St} for related computations).
}\end{remas}

\section{Compact static spacetimes} \label{s4}

In the bidimensional case, compact static spacetimes are essentially globally conformally flat Lorentzian tori, exhaustively studied in \cite{Sa-trans}. For the higher dimensional case, causality plays an important role.

\begin{teor}\label{t4a} Let $(M,g)$ be a compact static spacetime. Then:

(1) Its universal Lorentzian covering $(\bar M,\bar g)$ is standard static and admits a (possibly non-compact) Cauchy hypersurface.

(2) $(M,g)$ is totally vicious, i.e., $I^+(p)=I^-(p)=M, \forall p \in M$.

\end{teor}
{\em Sketch of proof.} {\it (1)} Straightforward from Theorem \ref{t2a}(2), Corollary \ref{c2a}, Theorem \ref{t3a} and the example in Remark \ref{r3b}(2). {\it (2)} The following general result (which includes the stationary case) holds \cite[Theorem 1.1]{Sa-St}:
{\em any compact Lorentzian manifold $(M,g)$ which admits a timelike conformal vector field  is totally vicious}. $\quad \quad $
\cvd

\noindent Even more,  deck transformations of $(\bar M,\bar g)$ satisfy:
\begin{propo} \label{p4a}
Let $(M,g)$ be a compact static manifold, and $(\bar M,\bar g)$ its universal Lorentzian covering. Then any deck transformation $\phi: \bar M \rightarrow \bar M$ can be written as
$$
\phi(t,x)= (t+T_\phi , \phi^S(x)),
$$
for some diffeomorphism  $\phi^S$ of $S$ and $T_\phi \in \R$.

\end{propo}

The striking difference between the causality of $M$ and its universal covering $\bar M$, yields interesting consequences for geodesic connectedness and existence of closed geodesics. Concretely, for the first question, one can ensure not only geodesic connectedness but also the existence of  timelike connecting geodesics:
\begin{teor} \label{t4c}
Any pair of points $p,q $ in a compact static spacetime $(M,g)$
 can be joined by means of a timelike geodesic.
\end{teor}
{\em Sketch of proof.} Join $p, q$ by means of a timelike curve $\gamma$ (Theorem \ref{t4a}(2)), and lift this curve to $\bar \gamma$ in the universal covering. Because of global hyperbolicity (Theorem \ref{t4a}(1)), its extremes $\bar p, \bar q$ can be joined by a timelike geodesic (Avez-Seifert theorem), which projects onto the required one. $\quad \quad$
\cvd

\noindent For the second question, it is known (Tipler \cite{Ti}, see also \cite[Theorem 4.15]{BEE}): {\em any compact spacetime regularly covered by a spacetime which admits a {\em compact} Cauchy hypersurface, contains a closed timelike geodesic.} An extension of Tipler's technique yields:

\begin{propo} \label{p4b}
Let $(M,g)$ be a compact Lorentzian manifold which admits a regular Lorentzian covering $\Pi: \bar M\rightarrow M$ such that $\bar M$ is globally hyperbolic. Assume that a
conjugacy class ${\cal C}$ of the group of deck transformations of $\bar M$ is finite
and contains a closed timelike curve $\alpha$.
Then there exists at least one closed timelike geodesic in ${\cal C}$.
\end{propo}
A particular case of this result is \cite[Corollary 4.7]{CMP}.
Even more, Proposition \ref{p4b} admits further extensions, which are applicable whenever the deck transformations satisfy the conclusion of Proposition \ref{p4a}. Then \cite{Sa-St}:

\begin{teor} \label{t4d}
For a compact static spacetime $(M,g)$,
any conjugacy class ${\cal C}$ of the group of deck transformations, containing a closed timelike curve,  admits a closed timelike geodesic. In particular, there exists at least one closed timelike geodesic in $M$.

\end{teor}

\section{Geodesic equations} \label{s5}

For simplicity we will consider in this section the standard case $\R \times S$, but extensions to the non-standard one are obvious. From a straightforward computation, $\gamma(s) =(t(s),x(s))$ is a geodesic if and only if:
\be \label{e5a}
D_s \dot{ x} = -\frac{1}{2}\dot{t}^2 \nabla \beta (x) \, , \quad \quad \quad  \beta (x) \dot t = \lambda \, ,
\ee
for some $\lambda \in \R$.
When $\lambda =0$ then $\gamma(s) =(t_0, x(s))$, for some $t_0\in \R$, is essentially a geodesic on $S$. Otherwise, we can normalize the value of $\lambda$, and the following result holds (see \cite{Sa-nonlin} for details):

\begin{propo} \label{p5a}
Let $\gamma(s) =(t(s),x(s))$ be a  geodesic of $S$ satisfying {\rm (\ref{e5a})} with $\lambda = \sqrt{2}$ and $E=g(\dot \gamma,\dot \gamma)/2$.  Let $V=-1/\beta$, then:

(1) $x(s)$ is a solution of classical equation
$ D_s \dot{ x} = -\nabla V(x)$, with ``total energy'' $E=g_S(\dot x, \dot x)/2+ V(x)$.

(2) $x(s)$ is a pregeodesic (i.e., geodesic  up to a reparameterization) for the Jacobi metric
$g_E = (E - V) g_S$, whenever $E>V$.
\end{propo}
Conversely, from any curve satisfying {\it (1)} or {\it (2)}, a solution to (\ref{e5a}) with $\lambda= \sqrt{2}$ (i.e., a normalized geodesic) can be constructed.
This yields an equivalence between the set of geodesics (non-orthogonal to $K$)
in the static spacetime and the trajectories for the potential $V$, which can be applied to some problems \cite{Sa-nonlin}. In particular, to look for geodesics $\gamma(s)=(t(s),x(s))$ with ``energy'' (rest mass, if non-spacelike) $-2E=-g(\dot \gamma,\dot \gamma)$, which are ``spacelike closed'' in the following sense: for some $b>0$ (proper period) and $ T>0$ (universal period), $ x(b)=x(0), \dot x(b)=\dot x(0), t(b)=t(0)+T$. Proposition \ref{p5a} applies directly to find such geodesics with fixed value of $E$ or $b$; for fixed $T$ (``T-periodic trajectories''), see \cite{Sa-pams} and references therein.

Finally, we point out that, starting with Proposition \ref{p5a}, the equivalence between two  notions of {\em convexity} for incomplete $g_S$ can be proven \cite{BGS}. When $(S,g_S)$ can be regarded as an open subset with smooth boundary $\partial S$ of a bigger complete Riemannian manifold, there are some natural Riemannian notions of convexity, essentially equivalent. Nevertheless, their equivalence for a Lorentzian manifold, as a static spacetime $(\R \times S, g)$, has to be proven (see \cite{Sa-cata} for a review). Even more, convexity for each causal character (timelike, lightlike, spacelike) must be taken into account. In the static case, the problem is solved in \cite{BGS}, proving in particular the equivalence between the geometrical and  variational notions of convexity, which are useful for the problem of geodesic connectedness (see also \cite{BCFS} for non-smooth boundaries).

\section{Variational approach. Geodesic connectedness} \label{s6}

A different approach is required for geodesic connectedness  (see \cite{Sa-cata} for a survey on this problem). This  will be discussed only in the standard case $M=\R\times S$ because, essentially, the natural hypotheses for connectedness are applicable to the universal covering, and Theorem \ref{t2b} would be also applicable.
A fruitful idea is the variational characterization of connecting geodesics by Benci, Fortunato and Giannoni \cite{BFG} (see also \cite{Ma}):  given two points $p_0=(t_0,x_0), p_1=(t_1,x_1)\in \R\times S$, a (absolutely continuous) curve $\gamma: [0,1] \rightarrow \R\times S, $ $\gamma(s)=(t(s),x(s))$ with $p_0=\gamma(0), p_1=\gamma(1)$ is a geodesic if and only if the $x(s)$ component is a critical point of the functional:
\begin{equation}
\label{e6a}
J(x) =\ {1\over 2} \int_0^1 g_S(\dot x(s),\dot x(s)) ds\
 -\ \frac{\Delta_t^2}{2}\ \left(\int_0^1 {1\over \beta(x(s))}\ ds\right)^{-1},
\end{equation}
where $\Delta_t^2 = (t_1-t_0)^2$. Thus, the problem is reduced to consider the functional $J$ on curves $x:[0,1]\rightarrow S$ connecting $x_0, x_1$. Even more, these authors studied directly the functional $J$, ensuring the existence of critical points if: (i) $g_S$ is complete, and (ii)  $\beta$ is (upper) bounded. Recently, Caponio, Masiello and Piccione \cite{CMP} showed that  assumption (ii) can be weakened in: (ii') $\beta$ is subquadratic (i.e., inequality (\ref{e2}) holds replacing $d^2$ by $d^{2-\epsilon}$, for some $\epsilon >0$). On the other hand, Allison and Unal \cite{AU} showed that, when (i) and (ii) holds, if $S$ is pseudoconvex and non-returning, then so is $M$; in this particular case, an alternative non-variational proof of geodesic connectedness is obtained.

Recently, we \cite{BCFS} have obtained a fully satisfactory approach to this problem, which can be summarized in the following points:
\begin{enumerate}
\item By using Cauchy-Schwarz inequality, $J$ is lower bounded by
\begin{equation}\label{e6b}
J(x)  \geq  \frac{1}{2}\int_0^1 g_S(\dot x(s),\dot x(s))\ ds\
 -\ \frac{\Delta_t^2}{2}\ \int_0^1  \beta(x) \ ds ,
\end{equation}
which can be seen as the action $A= \int L$ associated to a typical Lagrangian type $L=T-V$, with $T=g_S(\dot x,\dot x)/2$ (kinetic energy) and $V= \Delta_t^2 \beta /2$ (potential energy). General properties which imply the existence of critical points for a functional (coercitivity, boundedness from below), if are fulfilled by $A$, then are also fulfilled by $J$.

\item These general properties for action $A$ were known to hold when $g_S$ is complete and $V$ subquadratic (this re-proves Caponio et al. result). Even more, an optimal result for $V$ at most quadratic was obtained in \cite{CFS}. Nevertheless, this result does not guarantees the existence of critical points for $V$ quadratic (in fact, there are simple counterexamples) and, thus, neither the geodesic connectedness of $M$.

\item Studying directly the functional $J$ for $\beta$ quadratic (without the loss of information in Cauchy-Schwarz inequality), by using the accurate technique in \cite{CFS},  both coercitivity and lower boundedness for $J$, are ensured, i.e.:

\end{enumerate}

\begin{teor} \label{t6a}
A standard static spacetime with $g_S$ complete and $\beta$ at most quadratic
 is geodesically connected.
\end{teor}

\begin{remas} {\rm (1)
The assumption on completeness for $g_S$ is unavoidable (a strip of the
universal covering of two--dimensional anti--de Sitter spacetime
such as $M= \R \times (-\pi/4, \pi/4) $, $g = (dx^2-dt^2)/\cos^2x$ is not geodesically connected). Nevertheless, completeness can be weakened by the existence of a con\-vex boundary.
The ``at most quadratic'' behaviour for $\beta$ is optimal, as shown by a family of explicit counterexamples \cite[Section 7]{BCFS} (inspired by \cite[Section 6.1]{FS}).

(2) Causally  related points will be connectable by (maximizing) causal geodesics because the assumptions imply global hyperbolicity (Theorem \ref{t3a}). But causal connectability will also hold if the completeness of $g_S$ is weakened by the time and lightlike convexities of the boundary.

(3) A related variational problem is the connectability by means of trajectories under a potential vector (as charged particles), see for example, \cite{Min}. In particular, when applied to static spacetimes, this also yields implications on connectability for stationary ones.
}
\end{remas}

config: arxiv


TITLE: Multivariable Limit.
QUESTION [4 upvotes]: I'm trying to show that a multivariable function is continuous at the point $(0,0)$. To do so, I'd like to show the following limit:
$\lim_{(x_1, x_2) \to (0,0)} \frac{(x_1)^3(x_2) - (x_1)(x_2)^3}{(x_1)^2 +(x_2)^2}=0$
Unfortunately, I don't have much experience with limits of multiple variables. Can anyone help?

REPLY [3 votes]: Changing to polar coordinates works nicely here:
$$\eqalign{
\lim_{(x_1,y_1)\rightarrow(0,0)}\   {x_1^3x_2-x_1x_2^3\over x_1^2+x_2^2}
&=\lim_{r\rightarrow 0^+}\   {r^4\cos^3\theta\sin\theta -r^4\cos\theta\sin^3\theta\over r^2}\,  \cr
&=\lim_{r\rightarrow 0^+}\ \bigl[\,r^2(\cos^3\theta\sin\theta - \cos\theta\sin^3\theta)\,\bigr] \cr
&=0.
}
$$


TITLE: Let (Y,d) be a complete metric space, and let G be a family of continuous functions from Y to $\mathbb R$...
QUESTION [0 upvotes]: Let $(X,d)$ be a complete metric space, and $F$ be a family of continuous functions from X to $\mathbb R$.  Suppose that for each $x\in X$ there exists $M_x\in$$\mathbb R$ such that $f(x)\le M_x$ for all $f \in F$.  Prove there exists an open ball $B=$ { $x\in X|d(x,x_o)<\epsilon$ } and a constant $M\in \mathbb R$ such that $f(x)\le M$ for $f\in F$ and all $x\in B$.

REPLY [2 votes]: HINT: For $n\in\Bbb Z^+$ let $A_n=\{x\in X:M_x\le n\}$; clearly $X=\bigcup_{n\in\Bbb Z^+}A_n$. $X$ is complete, so by the Baire category theorem there are an $m\in\Bbb Z^+$ and a non-empty open $U\subseteq X$ such that $U\subseteq\operatorname{cl}A_m$.


TITLE: What is the covalent radius of different molecules of different atoms?
QUESTION [0 upvotes]: I mean the covalent radius of different molecules of same atoms  is one half of the distance between the nuclei. Then what is the covalent radius of molecules of different atoms.

REPLY [1 votes]: Molecules don't have a covalent radius, only atoms that make up molecules through covalent bonds have a covalent radius. Even then it's a slightly tortured concept, in my view.
Atoms however do not possess a single covalent radius. The bond length (inter-nuclear distances of bonded atoms) varies considerably from one bond type to the next.
In particular triple bonds ($\sigma$ + 2 $\pi$) are shorter than double bonds ($\sigma$ + 1 $\pi$), which in turn are shorter than single bonds ($\sigma$ only). This is due to increased inter-nuclear electron shielding for triple and double bonds. The higher the shielding, the closer the nuclei can be, because their electrostatic repulsion is reduced.
In addition, in single bonds, the participating Atomic Orbitals also matter: $\sigma_{ss}$ are not exactly the same as $\sigma_{sp}$ or $\sigma_{pp}$ bonds.
Difference in Pauling electronegativity also influences bond length somewhat.
Despite that, average covalent radii for different atoms have been computed, see here.


text:
\begin{document}

\title[Reflection coefficients]{Generalized reflection coefficients}

\author{Christian Remling}

\address{Mathematics Department\\
University of Oklahoma\\
Norman, OK 73019}

\email{cremling@math.ou.edu}

\urladdr{www.math.ou.edu/$\sim$cremling}

\date{June 4, 2014}

\thanks{2010 {\it Mathematics Subject Classification.} Primary 34L40 47B36 81Q10}

\keywords{Reflection coefficient, absolutely continuous spectrum}

\thanks{CR's work has been supported
by NSF grant DMS 1200553}
\begin{abstract}
I consider general reflection coefficients for arbitrary one-dimensional whole line
differential or difference operators of order $2$. These reflection coefficients
are semicontinuous functions of the operator:
their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity
result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps
(thus recovering and generalizing a result of Last-Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an
attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more
specialized situations.
\end{abstract}
\maketitle
\section{Introduction}
A \textit{Herglotz function }is a holomorphic map $F:\C^+\to\C^+$ on the upper half plane $\C^+=\{ z\in\C : \textrm{Im}\, z>0\}$.
Given a pair of Herglotz functions $m_{\pm}$, we define generalized \textit{reflection coefficients }as follows:
\begin{equation}
\label{defR}
R_+(z) = \frac{\overline{m_+(z)}+m_-(z)}{m_+(z)+m_-(z)}, \quad\quad
R_-(z) = \frac{m_+(z)+\overline{m_-(z)}}{m_+(z)+m_-(z)}
\end{equation}
While, on a formal level, it is convenient to make Herglotz functions the basic objects, I'm really motivated
by the spectral theory of whole line differential or difference operators.
We thus like to think of $m_{\pm}$ as representing the half line Titchmarsh-Weyl $m$ functions of, for example, a Jacobi matrix
\[
(Ju)_n = a_n u_{n+1} + a_{n-1}u_{n-1} + b_n u_n
\]
on $\ell^2(\Z)$, or perhaps of a Schr\"odinger operator
\[
(Hu)(x) = -u''(x) + V(x) u(x)
\]
on $L^2(\R)$, or of more general operators. This abstract and general framework
provides the natural setting for what I want to do here.

It is in fact \textit{always }possible
to interpret a given pair of Herglotz functions as the half line $m$ functions
of a sufficiently general whole line spectral problem. Namely, the Herglotz functions are in one-to-one correspondence to trace-normed
(that is, $\textrm{tr }H(x)=1$) canonical systems
\[
Ju'(x) = zH(x) u(x)
\]
on the half line $0\le x < \infty$. Here, $J=\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$, and $H(x)\in\R^{2\times 2}$ is positive semidefinite.
See \cite{Win} for this fundamental result, which is based on important earlier work by de~Branges \cite{dB} and Potapov \cite{Pota}.

Thus \textit{pairs }of Herglotz functions can be parametrized by \textit{whole line }trace-normed canonical systems
(one half line for each Herglotz function). In particular, Jacobi and Schr\"odinger equations can be rewritten as canonical systems.
We will often find it convenient to write $H(x)$ (with $x\in\R$) to refer in this way to a pair of Herglotz functions. This is mainly a notational device.

As we will see, the reflection coefficients $R_{\pm}$ have very remarkable properties. Their use was first advertised by Rybkin in \cite{Ryb},
and Breuer-Ryckman-Simon \cite{BRSim} work with closely related general reflection coefficients.
Here, we want to use $R_{\pm}$ to analyze the absolutely continuous spectrum in completely general situations.

We will be especially interested in the set (more precisely, equivalence class of sets, defined up to sets of Lebesgue measure zero)
\[
\Sigma_{ac} = \{x\in\R : \textrm{\rm Im }m_{\pm}(x) > 0 \} .
\]
This is the set where the underlying whole line operator has
absolutely continuous spectrum of multiplicity $2$.

Another important notion in the study of the absolutely continuous spectrum is the following:
\begin{Definition}
We call the canonical system $H(x)$ \textit{reflectionless }on a Borel set $A\subset\R$ of positive Lebesgue measure if
\[
m_+(x) = -\overline{m_-(x)} \:\: \textrm{for almost every }x\in A .
\]
\end{Definition}
If $H(x)$ is reflectionless on $A$, then $A\subset\Sigma_{ac}$. Conversely, however, a general canonical system can not
be expected to be reflectionless on $\Sigma_{ac}$; quite on the contrary, reflectionless operators are very rare. These points are also illuminated
by Proposition \ref{P1.1} below.

Reflectionless operators are important because they can be thought of as the basic building blocks
of arbitrary operators with absolutely continuous spectrum; see \cite{Remac} for more on this.

We can read off both $\Sigma_{ac}$ and the set where $H$ is reflectionless from the reflection coefficients.
This immediately recommends them for further use if one is interested in the absolutely continuous spectrum.
Observe first of all that we may define
$R_{\pm}(z)$ by the formulae given for all $z\in\C^+$. Since Herglotz functions have boundary values
$F(x)\equiv\lim_{y\to 0+} F(x+iy)$ for (Lebesgue-)almost all $x\in\R$, we can also consider the almost everywhere defined functions
$R_{\pm}(x)$. The boundary value of the Herglotz function $m_++m_-$ cannot be equal to zero on a positive measure set,
so there are no problems with the denominators here.

It is also helpful to observe that $|R_+|=|R_-|$, so we can unambiguously write $|R|$ if we are only interested
in the \textit{absolute values }of the reflection coefficients. More generally, we will frequently use the simplified notation
$R$ to refer to an unspecified reflection coefficient ($R_+$ or $R_-$).
\begin{Proposition}
\label{P1.1}
(a) Up to sets of measure zero,
\[
\Sigma_{ac} = \{ x\in\R : |R(x)|< 1 \} .
\]
(b) $H$ is reflectionless precisely on the set where $R(x)=0$.
\end{Proposition}
\begin{proof}
This is immediate from the definition of $R$ and the fact that $\textrm{Im }m_{\pm}(x)\ge 0$.
\end{proof}
We could similarly recover $\Sigma^{(\sigma)}_{ac}$, defined as the set where
$\textrm{Im }m_{\sigma}>0$, for $\sigma=+$ or $-$: indeed, $x\in\Sigma^{(\sigma)}_{ac}$ precisely if
$R_{\sigma}(x)\not= 1$. However, we will make no use of these sets here.

The following is the main result of this paper; recall our notational convention to write
$H$ (thought of as the coefficient of a canonical system) to refer to a pair of Herglotz functions $m_{\pm}$.
\begin{Theorem}[Semicontinuity of $R(x)$ and $\Sigma_{ac}$]
\label{Tsemicon}
If $m^{(n)}_{\pm}(z)\to m_{\pm}(z)$ locally uniformly on $z\in\C^+$,
then the reflection coefficients satisfy
\[
\left| R(x; H)\right| \le \limsup_{n\to\infty} \left| R(x; H_n) \right|
\]
for almost every $x\in\R$.

In particular, if $A$ denotes the set where the $\limsup$ is $<1$, then $\Sigma_{ac}(H)\supset A$.
\end{Theorem}
It is obvious from the definition that the reflection coefficients are continuous off the real line, that is,
$R(z;H_n)\to R(z;H)$ locally uniformly on $z\in\C^+$. It is also clear that $R(x;H)$ for $x\in\R$ will not,
in general, be \textit{continuous }(rather
than just semicontinuous) in the second argument. For example, an arbitrary pair $m_{\pm}$ of Herglotz functions
can be approximated by Herglotz functions with purely singular measures, and then $R(x;H_n)=1$
for almost every $x\in\R$ and all $n$. Or, as will become clear in a moment (see Proposition \ref{P1.2} below),
we can have $|R(x; H_n)|\ge c>0$ and $R(x;H)=0$ on a positive measure set $x\in A$ even if the approximating
problems have purely absolutely continuous spectra. In fact, the results on reflectionless limit points \cite{Remac}
say that something of this sort will always happen in certain situations.

Theorem \ref{Tsemicon} deals with completely general sequences of operators $H_n$, and then the $\limsup$ could easily
equal one and we are not getting a non-trivial statement. It is therefore worth pointing out that in the situations
of greatest interest, $|R(x; H_n)|$ will actually be constant along the whole sequence, and then the
$\limsup$ just equals the reflection coefficient $|R(x; H_1)|$ of the first operator of the sequence.

Let me elaborate on this in more detail.
We are often interested in sequences that are generated by iterating a map on the associated operators.
Of particular interest are the shift map and flows of an integrable hierarchy such as the Toda (Jacobi operators)
or KdV (Schr\"odinger operators) hierarchies. These maps are induced by an associated transfer matrix,
and this will imply that they preserve $|R|$.

Let me state this more formally.
Let $T(z)$ be an entire, $SL(2,\C)$ valued
function with $T(x)\in SL(2,\R)$ for $x\in\R$.
If $m_{\pm}^{(1)}$ are the half line $m$ functions of the
canonical system $H_1(x)$, define two new functions $m_{\pm}^{(2)}$ on $\C^+$
as follows:
\begin{equation}
\label{defphi}
m_-^{(2)}(z) = T(z)m_-^{(1)}(z) , \quad\quad m_+^{(2)}(z) = IT(z)I\,m_+^{(1)}(z)
\end{equation}
Here $I= \left( \begin{smallmatrix} 1 & 0\\ 0 & -1 \end{smallmatrix} \right)$, and a matrix acts on a complex number
as a linear fractional transformation, that is,
\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix} z = \frac{az+b}{cz+d} .
\]
(So $I$ implements multiplication by $-1$.)
The new functions $m_{\pm}^{(2)}$ are not guaranteed to be Herglotz functions, but if they are, we obtain a transformation
$m_{\pm}^{(1)}\mapsto m_{\pm}^{(2)}$.

\begin{Definition}
\label{D1.1}
We say that a transformation of canonical systems $H_1\mapsto H_2$ is of type TM (as in transfer matrix) if
the $m$ functions of the new system $H_2$ are obtained as in \eqref{defphi}, for some $T(z)$ as above.
\end{Definition}
This definition may not, at first sight, look especially natural. The next two Propositions will, I hope, show that it is useful here.
\begin{Proposition}
\label{P1.3}
The following maps are of type TM:
\begin{enumerate}
\item shift maps;
\item (time $t$ maps of) flows from the Toda hierarchy;
\item (time $t$ maps of) flows from the KdV hierarchy
\end{enumerate}
\end{Proposition}
By a \textit{shift map, }we of course mean a map that sends $H(x)$ to $H(x-a)$, for some $a\in\R$.
This includes (as restrictions) shifts $V(x)\mapsto V(x-a)$ of Schr\"odinger operators and shifts
$(a_n,b_n)\mapsto (a_{n-k},b_{n-k})$ of Jacobi matrices.

The list from Proposition \ref{P1.3} is, of course, not exhaustive. For example, we do not go beyond
polynomial dependence on $z$ here in the sense that either $T(z)$ itself is a polynomial (Jacobi shift)
or is obtained by solving a differential equation with coefficients that are polynomials in $z$.
\begin{proof}[Sketch of proof]
This follows from the fact that the $m$ functions can be represented by formulae of the type
$m_{\pm}(z) =\pm u_2(0,z)/u_1(0,z)$, where $u$ solves the canonical system (or other associated
equations, in cases (2), (3)) $Ju'=zHu$. If we now shift $H$ by $a$, then $u$ is updated by the transfer matrix
$T(a,z)$, which indeed leads to \eqref{defphi}.

We obtain the same conclusion for
Toda and KdV type flows because these also have associated $SL(2,\C)$/$SL(2,\R)$ cocycles that update solutions.
See, for example, \cite{GeHol1,GeHol2,Teschl} for a discussion of this.
\end{proof}
\begin{Proposition}
\label{P1.2}
If $H_1\mapsto H_2$ is of type TM, then
\[
\left| R(x;H_2)\right| = \left| R(x; H_1) \right|
\]
for almost all $x\in\R$.
\end{Proposition}
\begin{proof}
Since $T(z)$ is assumed to be continuous, \eqref{defphi} will also hold for almost all $z=x\in\R$. Then we
can obtain $R(x;H_2)$ directly from $R(x; H_1)$ by using formula \eqref{defR} for $z=x$ and updating
$m_{\pm}(x)$ according to \eqref{defphi}.
We can then verify by a direct computation that this will not change $|R(x)|$. More explicitly, if $T(x) =\left( \begin{smallmatrix}
a & b \\ c & d \end{smallmatrix} \right)$, then we find that
\[
R_+(x; H_2) = \frac{c m -d}{c\overline{m}-d}\, R_+(x; H_1) , \quad m = m_+^{(1)}(x) .
\]
\end{proof}
In the next Corollary, I'll focus on the
version for Jacobi operators, but analogous results hold in other settings, with the same proof.
\begin{Corollary}
\label{C1.1}
Suppose that $J_n\to J$ in the weak operator topology (equivalently, assume pointwise convergence of the
coefficients $a, b$). If $J_{n+1}$ is obtained from $J_n$ by:
\begin{enumerate}
\item a shift, or
\item a (time $t$ map of a) flow from the Toda hierarchy,
\end{enumerate}
then $\Sigma_{ac}(J)\supset \Sigma_{ac}(J_1)$.
\end{Corollary}
\begin{proof}
These transformations are of type TM by Proposition \ref{P1.3}.
By Proposition \ref{P1.2}, $|R(x; J_n)|$ is constant along the whole sequence, so the result is immediate from Theorem \ref{Tsemicon}
if we also recall that pointwise convergence of the coefficients of a Jacobi matrix
is equivalent to the locally uniform convergence of the $m$ functions.
\end{proof}

The pure shift version of the Corollary was obtained earlier by Last-Simon \cite{LS}, and this is now also
an immediate consequence of the fact that limit points under the shift map are reflectionless on
$\Sigma_{ac}(J_1)$ (see \cite{Remac} for this); the result for Toda flows is new. Here we can in particular
consider $\omega$~limit points $J=\lim\phi_{t_n}(J_1)$ under a fixed Toda flow $\phi_t$; of course, much
more general limits are possible, too.

If we only allow shifts and consistently shift in one direction only (say $(a_n,b_n)\mapsto (a_{n-k},b_{n-k})$ with $k\ge 0$),
then we have the much stronger statement available that all limit points are reflectionless
on $\Sigma_{ac}(J_1)$ \cite[Theorem 1.4]{Remac}. Such a statement cannot possibly hold in the
general setup of Theorem \ref{Tsemicon} or Corollary \ref{C1.1}; for example, we could have that $J_n=J_1$ for all $n\ge 1$.
We must impose an irreversibility condition on the evolution to have any right to expect such a result. It does not seem
ridiculous, however, to hope that if $J=\lim \phi_{t_n}(J_1)$ with $t_n\to\infty$ and $\phi$ a Toda type flow,
then $J$ will be reflectionless on $\Sigma_{ac}$. Kotani \cite{KotKdV} has such a result
for the KdV flow with special initial conditions. In this work, Kotani runs a localized (in energy) version of the proof from
\cite{Remac}, making use of the associated cocycle of the KdV flow.

Inspired by this and Corollary \ref{C1.1}, it is now tempting to ask what class of evolutions of this type lets one
run the original proof and obtain reflectionless limit points. I'll give in to this temptation in the final section of
this paper. The answer we obtain here will be satisfying and disappointing at the same time:
we essentially recover the class of shift maps, but now in the generality of canonical systems.

Theorem \ref{Tsemicon} is proved in Section 3, and Section 2 discusses
a criterion for convergence of Herglotz functions that will be a crucial ingredient to this proof.
\section{Convergence of Herglotz functions}
Our proof of Theorem \ref{Tsemicon} will be based on the following criterion for convergence of Herglotz functions.
Recall again that the boundary value $F(x)$ ($x\in\R$) of a Herglotz
function cannot be equal to zero on a positive measure set, so we can
certainly define $\ln F(x)$ almost everywhere. We will always use the value with $0\le \textrm{Im}\,\ln F(x)\le\pi$.

In fact, we will be working with (complex) logarithms a lot in this section and the next, and much ink can be saved
if at this point already we agree once and for all that
\begin{equation}
\label{convlog}
0\le \textrm{\rm Im}\, \ln w < 2\pi
\end{equation}
is the value we want whenever a logarithm makes an appearance somewhere.
\begin{Theorem}
\label{THergl}
Let $F_n, F$ be Herglotz functions, and fix $R>0$. Then $\ln F_n(x), \ln F(x) \in L^2(-R,R)$.
Moreover, if $F_n(z)\to F(z)$ locally uniformly on $z\in\C^+$, then
\[
\ln F_n(x)\to \ln F(x) \:\: \textrm{\rm weakly in } L^2(-R,R) .
\]
\end{Theorem}
The converse is also true and follows automatically from this, by compactness. More precisely,
if $F_n,F$ are Herglotz functions and $\ln F_n\to\ln F$ weakly in $L^2(A)$ for some bounded $A\subset\R$
of positive Lebesgue measure, then $F_n(z)\to F(z)$ locally uniformly on $\C^+$. Indeed, if this were
false, then $F_n(z)\to G(z)$ on a subsequence for some $G$ not identically equal to $F$, by normal families.
Since Herglotz functions are determined by their boundary values on a positive measure set, this
contradicts the Theorem. (To tell the truth, we really need a slightly more general version here because in this
argument, we could have $G\equiv a\in\R_{\infty}$.)
\begin{proof}
Any Herglotz function $F$ has a holomorphic logarithm $\ln F(z)$, which is a Herglotz function itself (recall \eqref{convlog}!).
Since $\textrm{Im}\, \ln F <\pi$, the representing measure of $\ln F$ is purely absolutely continuous; its density, up to a factor of $\pi$,
is called the \textit{Krein function }of $F$:
\[
\xi(x) = \frac{1}{\pi}\lim_{y\to 0+} \textrm{Im }\ln F(x+iy) .
\]
So we have that $0\le\xi\le 1$.

The Herglotz representation of $\ln F$ then reads
\begin{align}
\label{lnF}
\ln F(z) & = C + \int_{-\infty}^{\infty} \left( \frac{1}{t-z} - \frac{t}{t^2+1}\right)\xi(t)\, dt\\
\nonumber
& = C + \int_{-\infty}^{\infty} \frac{1+tz}{t-z}\, \zeta(t)\, dt ;
\end{align}
here we have introduced $\zeta(t)=\xi(t)/(t^2+1)$.

It is well known that locally uniform convergence
of Herglotz functions implies (in fact, is essentially equivalent to) weak $*$ convergence of the associated measures. See,
for example, \cite[Sect.\ 2]{Remac} for a more detailed discussion.
If now $F_n\to F$ locally uniformly, then also $\ln F_n(z)\to\ln F(z)$,
so we obtain that $\zeta_n\, dt\to \zeta\, dt$ in weak $*$ sense, as Borel measures on the one-point compactification
$\R_{\infty}$ of $\R$. The absence of a term linear in $z$ in the Herglotz representation of $\ln F$ implies
that our measures don't give weight to the added point $\infty$, so it is enough
to extend the integrations over $\R$. We then have that
\[
\int_{-\infty}^{\infty}  f(t)\zeta_n(t)\, dt \to \int_{-\infty}^{\infty} f(t)\zeta(t)\, dt
\]
for all test function $f$ that have a continuous extension to $\R_{\infty}$.

Our assignment is to analyze the sequence of functions
\[
\ln F_n(x)\equiv \lim_{y\to 0+}\ln F_n(x+iy) ,
\]
as $n\to\infty$.
We will do this by looking separately at the various contributions from \eqref{lnF}.

Since $C=\ln |F(i)|$, it is certainly clear that $C_n\to C$. Next, the weak $*$ convergence $\zeta_n\, dt\to\zeta\, dt$ together
with the uniform bound on the $\zeta_n$ give that
\[
\int_{|t|>R+1} \frac{1+tx}{t-x}\zeta_n(t)\, dt - \int_{|t|>R+1} \frac{1+tx}{t-x}\zeta(t)\, dt \to 0 ,
\]
uniformly on $|x|\le R$. So let's now focus on the integrals from the right-hand side of \eqref{lnF}, cut off at $|t|=R+1$. Then both summands
from the first line of \eqref{lnF}
become integrable separately, and
\[
\int_{-R-1}^{R+1} \frac{t}{t^2+1}\xi_n(t)\, dt \to \int_{-R-1}^{R+1} \frac{t}{t^2+1}\xi(t)\, dt ,
\]
by the weak $*$ convergence again. It remains to analyze
\[
f_n(x) \equiv \lim_{y\to 0+} \int_{-R-1}^{R+1} \frac{1}{t-x-iy}\, \xi_n(t)\, dt ,
\]
for $|x|<R$ and as $n\to\infty$. It is clear that $\textrm{Im }f_n = \pi\xi_n$ converges to $\textrm{Im }f=\pi\xi$ weakly on
$L^2(-R,R)$: this follows from the weak $*$ convergence $\zeta_n\, dx\to\zeta\, dx$ and the fact that $0\le \xi_n\le 1$.

The real part of $f_n$ can alternatively be computed as a Hilbert transform:
\begin{equation}
\label{2.1}
\textrm{\rm Re}\; f_n(x) = \lim_{y\to 0+} \int_{\{ t\in\R : |t-x|>y\} } \chi_A(t) \frac{\xi_n(t)}{t-x}\, dt \equiv (H(\chi_A\xi_n))(x)
\end{equation}
This holds for almost all $x\in (-R,R)$, and we have abbreviated $A=(-R-1,R+1)$. The Hilbert transform defines
a bounded operator on $L^2(\R)$, and it satisfies $H^*=-H$. This shows, first of all, that indeed $\ln F\in L^2(-R,R)$
for any Herglotz function $F$: since $\chi_A\xi\in L^2(\R)$, we have that $\textrm{Re}\, f\in L^2(-R,R)$ by \eqref{2.1}, and the other contributions
from \eqref{lnF} are bounded on $x\in (-R,R)$. Next, the desired weak convergence in $L^2(-R,R)$ also follows: write $B=(-R,R)$
for notational convenience, and let
$g\in L^2(B)$ be a test function. Then
\begin{align*}
\langle g, \textrm{Re}\, f_n \rangle_{L^2(B)} & = \langle \chi_B g, H(\chi_A\xi_n) \rangle_{L^2(\R)} = - \langle H(\chi_B g), \xi_n\rangle_{L^2(A)}\\
& \to - \langle H(\chi_B g), \xi \rangle_{L^2(A)} = \langle g, \textrm{Re}\, f \rangle_{L^2(B)} .
\end{align*}
Here, the convergence follows from the weak convergence $\xi_n\to\xi$ in $L^2(A)$ that was observed earlier, and the last equality is established
by reversing the first two steps.
\end{proof}
\section{Proof of Theorem \ref{Tsemicon}}
If $C<1$ and $L>0$ are given and we define
\[
B=\{ x\in [-L,L] :\limsup_{n\to\infty} |R(x;H_n)|< C \} ,
\]
then we can find an $A\subset B$ of almost full measure inside $B$ and $N\ge 1$,
such that $|R(x; H_n)|\le C$ for all $x\in A$ and $n\ge N$.
It thus suffices to show that $|R(x; H)|\le C$ almost everywhere on $A$ for such an $A$.

We consider the functions
\[
L^{(n)}_{\sigma}(x) = \ln \left( R_{\sigma}(x; H_n) - 1\right) ,
\]
for $x\in A$ (and $n\ge N$, $\sigma=\pm$).
The argument of the logarithm lies in the left half plane;
so $L^{(n)}$ has imaginary part in $(\pi/2, 3\pi/2)$ by \eqref{convlog}.
Since $R$ is bounded away from $1$, the $L^{(n)}$ are uniformly bounded on $A$. Thus they converge weakly in $L^2(A)$
on a suitable subsequence, which, for notational convenience, we assume to be the original sequence. The weak limits will take
values in the closed convex hull of the union of the ranges of the $L^{(n)}$. We now make use of the following elementary
fact; this will be proved after we have completed the proof of the theorem.
\begin{Lemma}
\label{L3.1}
Let $C<1$. Then $\{ \ln (z-1) : |z|\le C \}$ is a convex subset of $\C$.
\end{Lemma}
So we can now introduce two new functions $r_{\sigma}(x)$ on $x\in A$ to describe the weak limits, as follows:
\begin{equation}
\label{3.1}
\ln \left( R_{\sigma}(x; H_n) - 1\right) \xrightarrow{w} \ln \left( r_{\sigma}(x) - 1\right)
\end{equation}
By the Lemma, these may be required to take values in $|r|\le C$.

From the definition \eqref{defR}, we have that
\begin{equation}
\label{3.31}
R_{\sigma}-1 = -\frac{2i\,\textrm{\rm Im}\, m_{\sigma}}{m_++m_-} = 2i g\, \textrm{\rm Im}\, m_{\sigma} ;
\end{equation}
Here we have abbreviated $g=-1/(m_++m_-)$. Note that $g$ is a Herglotz function, too.
We would now like to take logarithms on both sides, but to be able to do this, we also need to make sure
that $\textrm{Im}\, m>0$ or, equivalently, $R\not = 1$. This is of course clear for $R=R(x;H_n)$, from the definition
of the set $A$, but we will also want to use these formulae for $R=R(x;H)$. This issue will be addressed in a moment.

Let's first use \eqref{3.31} for $R=R(x;H_n)$. As planned, take logarithms to obtain that
\begin{equation}
\label{3.3}
\ln \left( R_{\sigma}(x;H_n)-1\right) = \ln 2ig_n(x) + \ln\textrm{Im}\, m_{\sigma}^{(n)}(x) .
\end{equation}
The hypothesis of Theorem \ref{Tsemicon}
implies that also $g_n(z)\to g(z)$ locally uniformly. Thus $\ln 2ig_n(x)\xrightarrow{w} \ln 2ig(x)$ weakly in $L^2(A)$ by
Theorem \ref{THergl}. This shows that the weak limit
\begin{equation}
\label{defP}
P_{\sigma}(x) \equiv \textrm{\rm w--}\!\!\lim_{n\to\infty} \ln\textrm{Im}\, m^{(n)}_{\sigma}(x) ;
\end{equation}
exists, at least if we take it along the subsequence that makes \eqref{3.1} happen. We obtain the following limiting version of \eqref{3.3}:
\begin{equation}
\label{3.32}
\ln \left(r_{\sigma}(x)-1 \right) = \ln 2ig(x) + P_{\sigma}(x)
\end{equation}

We have the following fundamental estimate on $P_{\sigma}$.
\begin{Lemma}
\label{L3.2}
For almost every $x\in A$, we have that
\[
\textrm{\rm Im}\, m_{\sigma}(x) \ge e^{P_{\sigma}(x)} .
\]
\end{Lemma}
\begin{proof}[Proof of Lemma \ref{L3.2}]
Drop $\sigma$ in the notation, and let $I=(a,b)$ be an interval with
$\rho(\{a\} )=\rho(\{ b\} )=0$, where $\rho$ is the measure from the Herglotz representation of $m$. Then
\begin{align*}
\frac{\pi\rho(I)}{|I|} & = \lim_{n\to\infty} \frac{\pi\rho_n(I)}{|I|} \ge \limsup_{n\to\infty} \frac{1}{|I|}
\int_{I\cap A} \textrm{\rm Im}\, m_n(x)\, dx\\
& \ge \frac{|I\cap A|}{|I|}\lim_{n\to\infty} \exp \left( \frac{1}{|I\cap A|}\int_{I\cap A} \ln\textrm{\rm Im}\, m_n(x)\, dx \right) \\
& = \frac{|I\cap A|}{|I|} \exp \left( \frac{1}{|I\cap A|}\int_{I\cap A} P(x) \, dx\right) .
\end{align*}
We have used Jensen's inequality to pass to the second line, and the limit in this formula exists by \eqref{defP}.
Now take $I=(x-h,x+h)$, fix $x$ and send $h\to 0+$. For almost every $x\in A$,
the following statements will be true:
\[
\frac{|I\cap A|}{|I|}\to 1, \quad \frac{1}{|I|} \int_{I\cap A} P(t)\, dt \to P(x), \quad \frac{\pi \rho(I)}{|I|} \to \textrm{\rm Im}\, m(x)
\]
Thus, if we also avoid those $h>0$ for which $\rho(\{ x\pm h\})>0$, we obtain that $\textrm{Im}\, m(x)\ge e^{P(x)}$ for these
$x$, as claimed.
\end{proof}

In particular, this implies that $\textrm{Im}\, m_{\sigma}(x)>0$ for almost all $x\in A$, so we may indeed take
logarithms in \eqref{3.31} also for $R=R(x;H)$. Comparison of this formula with \eqref{3.32} then yields
\begin{equation}
\label{3.2}
\ln \left( R_{\sigma}(x; H) - 1\right) = \ln \left( r_{\sigma}(x) - 1\right) + \ln\textrm{\rm Im}\, m_{\sigma}(x) - P_{\sigma}(x) .
\end{equation}
(Note that it would of course have been too naive to expect, based on \eqref{3.1} perhaps, that $R(x;H)=r(x)$.)
We reformulate this one more time: For almost all $x\in A$, we have that
\begin{align}
\label{3.6a}
\textrm{\rm Im}\, \ln \left( R_{\sigma}(x)-1\right) & = \textrm{\rm Im}\, \ln \left( r_{\sigma}(x)-1\right) = \textrm{\rm Im}\,
\ln 2ig(x), \\
\label{3.6b}
\textrm{\rm Re}\, \ln \left( R_{\sigma}(x)-1\right) & \ge \textrm{\rm Re}\, \ln \left( r_{\sigma}(x)-1\right) .
\end{align}
To obtain this, we just take real and imaginary parts of \eqref{3.2}; the inequality in \eqref{3.6b} is obtained from Lemma \ref{L3.2}.

To finish the proof, we now discuss conditions \eqref{3.6a}, \eqref{3.6b} for fixed $x$. We know that $|r_+(x)|, |r_-(x)|\le C$,
and we wish to show that then $|R_{\pm}(x)|\le C$ also. Recall that we of course have that $|R_+|=|R_-|$, but $r_{\pm}$ were obtained
from certain weak limits, so we cannot be sure if these quantities satisfy the same identity.

Write $R_{\pm}-1=\rho_{\pm} e^{i\varphi}$, with $\pi/2<\varphi<3\pi/2$. Note that both numbers indeed have the same
phase $\varphi$ by \eqref{3.6a}. From \eqref{3.6b}, we know that $\rho_{\pm}\ge\rho_0$, where
$\rho_0=|z_0|$ is defined as the absolute value of the smaller of the (usually two) points of intersection of the ray
$te^{i\varphi}$, $t\ge 0$ with the circle $\{ z-1: |z|=C\}$. Also, let $\rho_1=|z_1|\ge \rho_0$ be the absolute value of the other
point of intersection.
We are trying to show that $|R_+|\le C$ or, equivalently, $\rho_+\le\rho_1$.

By elementary geometry, if we compute the points where $\{te^{i\varphi}: t\ge 0\}$ and $\{ z-1 : |z|=C\}$ intersect, we find that
\begin{equation}
\label{3.7}
\rho_j = -\cos\varphi \pm \sqrt{C^2-\sin^2\varphi} \quad\quad (j=0,1);
\end{equation}
more precisely, we obtain $\rho_1$ if we take the plus sign (notice that $-\cos\varphi>0$ for our range
of $\varphi$'s). Notice also that only those values of $\varphi$ for which the ray $te^{i\varphi}$, $t\ge 0$,
intersects the circle $\{ z-1: |z|=C\}$ are compatible with \eqref{3.6a}, \eqref{3.6b}, and this implies that $|\sin\varphi|\le C$.

Next, observe that
\[
-\frac{\overline{R_+}}{R_-} = -\frac{\overline{g}}{g} = \frac{\overline{2ig}}{2ig} = e^{-2i\varphi} .
\]
The last step is by \eqref{3.6a}. It follows that
\[
R_- = -\overline{R_+}e^{2i\varphi} = -\left( 1+\rho_+ e^{-i\varphi} \right) e^{2i\varphi} ,
\]
so $R_- -1=(-\rho_+-2\cos\varphi)e^{i\varphi}$, and this yields
\[
\rho_- = -2\cos\varphi -\rho_+\equiv f(\rho_+) .
\]
From \eqref{3.7}, we see that $f(\rho_1)=\rho_0$,
so we would obtain smaller values $f(\rho_+)<\rho_0$ if we had $\rho_+>\rho_1$, but we also know that $\rho_-\ge\rho_0$, so this
can't happen. We have proved that $\rho_+\le\rho_1$, as desired. \hfill $\square$
\begin{proof}[Proof of Lemma \ref{L3.1}]
The set under consideration can be described as
\[
\{ w=x+iy: 1-C\le e^x \le 1+C, \varphi(x)\le y \le 2\pi - \varphi(x) \} ,
\]
where $\varphi(x)\in (\pi/2,3\pi/2)$ is the angle determined by that ray $te^{i\varphi(x)}$, $t\ge 0$, that intersects the
circle $\{ z-1 :|z|=C\}$ for the first time at $t=e^x$. We want to show that the complementary angle $\psi(x)=\pi-\varphi(x)$
is a concave function of $x$ in the range indicated.

Elementary geometry shows that
\[
\cos\psi(x) = \frac{1}{2} (e^x + Ae^{-x}) ,
\]
with $A=1-C^2>0$. From this, we learn that $(\cos\psi)''=\cos\psi$, so
\[
-\psi'' \sin\psi =(1+\psi'^2)\cos\psi .
\]
Since $0\le\psi<\pi/2$, this implies that $\psi''\le 0$, as desired.
\end{proof}
\section{Reflectionless limits}
As mentioned in the introduction, if we only apply (let's say: left) shifts to (let's say) a Jacobi matrix,
then any limit not only has a $\Sigma_{ac}$ that contains the original $\Sigma_{ac}$ (which is constant
along the approximating sequence, by Propositions \ref{P1.1}, \ref{P1.3}, \ref{P1.2}), but it satisfies the
much stronger condition of being reflectionless on this set \cite[Theorem 1.4]{Remac}.

We now ask the following perhaps somewhat artificial, but not easy to resist question: What is the most
general version of this result that can be obtained by running a sufficiently abstract version of the original proof,
using the setup of the present paper? To make this more concrete, let's define:
\begin{Definition}
\label{D4.1}
A transformation $H_1\mapsto H_2$ of type TM is said to be of type $\mathcal S$ if the associated $T(z)$ can be chosen so that
we have that $T(z)w\in\C^+$ whenever $w,z\in\C^+$. If, in addition, $T(0)=1$, then we say that the transformation is of type $\mathcal S_0$.
\end{Definition}
In other words, in addition to the properties from Definition \ref{D1.1}, we now also demand that the linear fractional
transformation $w\mapsto T(z)w$ is a Herglotz function for all $z\in\C^+$ (equivalently, $z\mapsto T(z)w$ is Herglotz
for all $w\in\C^+$). This condition played a key role in the proof of
\cite[Theorem 1.4]{Remac}, and this is our reason for introducing it here. Abusing notation, we will also write $T\in\mathcal S$.

The relation of the restricted class $\mathcal S_0$ to $\mathcal S$ is easily clarified: if $T(z)\in\mathcal S$ and we set
$A=T(0)\in SL(2,\R)$, then both $T(z)A^{-1}$ and $A^{-1}T(z)$ will be in $\mathcal S_0$.
Conversely, if $A\in SL(2,\R)$ and $T_0(z)\in\mathcal S_0$, then $AT_0, T_0A\in\mathcal S$.
All this is immediate from the definitions if we recall that an $A\in SL(2,\R)$ induces an automorphism of $\C^+$.

We will indeed be able to generalize \cite[Theorem 1.4]{Remac} to this setting. However, nothing essentially new is obtained.
Rather, the transformations of type $\mathcal S_0$ are precisely the left shifts, and if we work with $\mathcal S$, then this
is perhaps followed (or preceded, if we prefer) by
an $SL(2,\R)$ transformation.
\begin{Theorem}
\label{T4.1}
$T(z)\in\mathcal S_0$ precisely if there exist $L\ge 0$ and a trace-normed coefficient $H(x)$ on $[0,L]$ so that
$T(z)=T(L,z)$ where $T(x,z)$ solves
\[
JT'(x,z) = zH(x)T(x,z), \quad T(0,z)=1 .
\]
\end{Theorem}
So this says that the suspense built at the beginning of this section was completely unjustified:
we are not getting beyond the class of shift maps with this attempt at increasing the scope of \cite[Theorem 1.4]{Remac}.
We will obtain a rather general version of this result, though, which we'll state and prove below.

Theorem \ref{T4.1} follows from the characterization of the transfer matrices of canonical systems
that is due to Potapov \cite{Pota} and de~Branges \cite[Theorem 37]{dB}.
Namely, $T(z)=T(L,z)$ for a $T(x,z)$ as above precisely if $T(z)$ is entire with values in
$SL(2,\C)$ and $T(x)\in SL(2,\R)$ if $x\in\R$, $T(0)=1$ and
\begin{equation}
\label{Jin}
-i \left( T^*(z)J T(z) - J \right) \ge 0
\end{equation}
for $z\in\C^+$. If we compare these conditions with the ones from Definition \ref{D4.1}, we see that in order to establish
Theorem \ref{T4.1}, we must show that in the presence of the other conditions just listed, \eqref{Jin} is equivalent to
the Herglotz property from Definition \ref{D4.1}. This is known, but we sketch a proof anyway for the reader's convenience.
The claim boils down to showing the following:
\begin{Lemma}
Let $M\in SL(2,\C)$. Then $w\mapsto Mw$ is a Herglotz function precisely if
\begin{equation}
\label{Min}
-i (M^* JM - J)\ge 0 .
\end{equation}
\end{Lemma}
\begin{proof}
If \eqref{Min} is assumed, then $\textrm{Im}\, Mw>0$ follows at once from the observation that
\begin{equation}
\label{4.2}
\frac{1}{2i} (\overline{w}, 1) J \begin{pmatrix} w \\ 1 \end{pmatrix} = \textrm{\rm Im}\, w .
\end{equation}
So assume now that $Mw$ is Herglotz. Then $M$ maps $\R_{\infty}$ to a circle (possibly a line parallel to $\R_{\infty}$ or even
$\R_{\infty}$ itself, if $M\in SL(2,\R)$) in $\C^+$. We shift so that
the bottom point of this circle touches the real line at $0$ after shifting. In other words, we choose $z\in\C^+\cup \{ 0\}$ so that
$M_1=\left( \begin{smallmatrix} 1 & -z \\ 0 & 1 \end{smallmatrix} \right) M$ is still a Herglotz function, but now $M_1 t=0$
for some $t\in\R_{\infty}$. We also move $t$ to the origin, that is, we introduce
\[
P = \begin{pmatrix} 1 & -z \\ 0 & 1 \end{pmatrix} M\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} .
\]
This of course assumes that $t\not=\infty$; the case $t=\infty$ is similar and is left to the reader. So we have that
\[
M= \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} P \begin{pmatrix} 1 & -t \\ 0 & 1 \end{pmatrix} \equiv APB .
\]
It suffices to show that $-iC^*JC\ge -iJ$ for the individual factors $C=A,P,B$. This is obvious for $B$ (with equality,
since $B\in SL(2,\R)$) and $A$ (by a calculation), so we can focus on $P$.

Since $P0=0$, we have that
\[
P = \begin{pmatrix} a & 0 \\ b & a^{-1} \end{pmatrix} ,
\]
with $a\not= 0$. Write $w=x+iy$ and compute
\begin{equation}
\label{4.1}
\frac{1}{2i} (\overline{w}, 1) P^*JP \begin{pmatrix} w \\ 1 \end{pmatrix} = Ay^2 + By + Ax^2 + Cx \equiv F(x,y)
\end{equation}
with
\[
A = \textrm{\rm Im}\, a\overline{b}, \quad B = \textrm{Re}\, a/\overline{a}, \quad C = \textrm{\rm Im}\, a/\overline{a} .
\]
We are interested in the range $x\in\R$, $y\ge 0$. By \eqref{4.2}, the left-hand side of \eqref{4.1} computes a positive multiple of $\textrm{Im}\, Pw$,
so we know that $F(x,y)\ge 0$, and we wish to show that in fact $F(x,y)\ge y$. From the condition that $F\ge 0$, we infer that
$A\ge 0$, $C=0$, so $a/\overline{a}=\pm 1$, that is, either $a=id$ or $a=d$ for some $d\in\R$. The first
case would make $B=-1$, but then $F(0,y)<0$ for small $y>0$, so this is impossible. Hence $a\in\R$ and $B=1$, so
\[
F(x,y) = Ay^2 + y + Ax^2 \ge y ,
\]
as desired.
\end{proof}
\begin{Theorem}
\label{Trefl}
Suppose that $H_n\mapsto H_{n+1}$ is of type $\mathcal S$ and the sequence $H_n$ approaches a
limit $H$ in the sense that $m_{\pm}(z; H_n)\to m_{\pm}(z;H)$
locally uniformly on $\C^+$. Then either:\\
(1) $H$ is reflectionless on $\Sigma_{ac}(H_1)$, or\\
(2) $T_n(z)\cdots T_1(z)\to T(z)$ locally uniformly on $\C$ for some $T\in\mathcal S$ (see comments below).
\end{Theorem}
An equivalent way of stating (1) is to say that $R(x;H)=0$ on $\Sigma_{ac}(H_1)$. This is the conclusion we were hoping for;
in case (2), the whole sequence $H_n$ just approaches the result of mapping $H_1$ by a single shift type map.
Actually, a few comments on the precise formulation of part (2) are in order: Observe that if $H_1\mapsto H_2$
is a given transformation of type $\mathcal S$, then the corresponding matrix $T(z)$ is determined up to a sign. Indeed,
if we again write $T=AT_0$, $T_0\in\mathcal S_0$, $A\in SL(2,\R)$, then $A$ as a linear fractional transformation can be
found from \eqref{defphi}. This gives $A$ as a matrix, up to a sign ($A$ and $-A$ generate the same linear fractional transformation).
We can then see from Theorem \ref{T4.1} that (the matrix) $T_0(z)$ is uniquely determined.
Thus the $T_j(z)$ from case (2) are well defined, up to signs, and the claim really is that after suitably choosing
these signs, we will have the asserted convergence of the matrices.

So in case (2),
we obtain that $H(x)$ is $H_1(x-a)$, possibly followed by an $SL(2,\R)$ transformation. In particular $|R(x,H)|=|R(x,H_1)|$.
Obviously, this second case cannot be ruled out in general; for instance, we could be dealing with the sequence
of shifts $H(x)\mapsto H(x-2^{-n})$.

To prove Theorem \ref{Trefl}, we follow the proof of \cite[Theorem 1.4]{Remac} that was given in this reference as closely as
possible. I will not try to summarize the whole argument here. Rather, I'll focus right away on the one place where new ingredients are needed.
Namely, Lemma A.2 and Corollary A.3 of \cite{Remac} were established by a direct calculation that depended on the specific form
of transfer matrices of Jacobi operators. In our more general setting, we will use a function theoretic version of the Weyl theory
of nested disks for $m$ functions of problems with variable endpoints.

Let me introduce some notation.
Let $T_n(z)\in\mathcal S$. We abbreviate
\[
P_n(z) = T_n(z)\cdots T_1(z) .
\]
Observe that $P_n\in\mathcal S$ also. For fixed $z\in\C^+$ and $n\ge 1$, the set $\{ P_n(z)w : w\in\C^+ \}$ is a disk in the upper half plane
(possibly degenerate, that is, of the form $\{x+iy: y>y_0\}$),
and we let $R_n(z)$ be its hyperbolic diameter:
\[
R_n(z) = \sup_{w,w' \in\C^+} \gamma \left( P_n(z)w, P_n(z)w' \right) \in (0,\infty]
\]
As explained, $\gamma$ denotes hyperbolic distance in the upper half plane. $R_n(z)=\infty$ is possible if the
disk $\{ P_n(z)w\}$ touches $\R_{\infty}$ or is a half plane. Since holomorphic self-maps of $\C^+$ (aka Herglotz functions)
decrease hyperbolic distance, $R_n(z)$ is decreasing in $n$ for fixed $z\in\C^+$. Thus
\[
R(z) = \lim_{n\to\infty} R_n(z) \in [0,\infty]
\]
exists for all $z\in\C^+$.

By normal families, given a sequence
of Herglotz functions, we can always pass to a locally uniform limit on a subsequence. In particular, we can do
this for a sequence of the form $P_n(z)w_n$, with $w_n\in\C^+$. Any such limit will either
be a Herglotz function itself, or it will be identically equal to $a$ for some $a\in\R_{\infty}$.
\begin{Theorem}
\label{TWeyl}
Suppose that for some $w_0,z_0\in\C^+$, the sequence $P_n(z_0)w_0$ has a subsequence that converges to a $\zeta\in\C^+$. Then either:\\
(1) $R(z)\equiv 0$ on $z\in\C^+$, or\\
(2) $R(z)>0$ for all $z\in\C^+$.\\
In the first case, $R_n(z)\to 0$ locally uniformly on $\C^+$.
\end{Theorem}
This can be viewed as a function theoretic version of Weyl's limit point/limit circle alternative. Note, however,
that our disks $\{ P_n(z)w: w\in\C^+\}$ move around and are not nested.
\begin{proof}
I will, as expected, work on the upper half plane in this proof, but our intuition is actually better served
if we imagine our functions taking values in the disk model instead, with its hyperbolic metric.

For $w_n,w'_n\in\C^+$, consider the Herglotz functions $P_n(z)w_n$, $P_n(z)w'_n$.
As explained above, these will approach limits $F(z), G(z)$ along suitable subsequences.
If we also take $w_n\not= w'_n$, then $P_n(z)w_n\not=
P_n(z)w'_n$ for all $z\in\C^+$, so by Hurwitz's Theorem, either $F-G$ is zero free or $F\equiv G$.

Now suppose that
$R(z_0)=0$ for some $z_0\in\C^+$. Then $F(z_0)=G(z_0)$, so $F\equiv G$ for any two limits $F$, $G$.
If we had $R(z_1)>0$ for some $z_1\in\C^+$, then we could pick $w_n,w'_n$ so that $\gamma(P_n(z_1)w_n,P_n(z_1)w'_n)\ge c >0$,
but we also just saw that these two sequences must have the same limit if we make them both converge by passing to a subsequence.
This is only possible if in fact $P_{n_j}(z_1)w\to a\in\R_{\infty}$ for all $w\in\C^+$; here, $a$ is independent of $w$.
However, this condition implies that $P_{n_j}(z)w\to a$ for all $z\in\C^+$. In this whole argument, we are free to
choose $n_j$ as a subsequence of the sequence from the hypothesis of the Theorem. So this scenario is not possible, and
thus indeed $R\equiv 0$ as soon as $R(z_0)=0$.

The locally uniform convergence of $R_n$ in this case follows by essentially the same argument.
Fix a sequence $n_j\to\infty$ as in the hypothesis, that is, $P_{n_j}(z_0)w_0\to\zeta\in\C^+$.
Now if we had $R_n(z'_n)\ge 2\epsilon >0$ for arbitrarily large $n$ and certain $z'_n$'s drawn from a compact
set $K\subset\C^+$, then, since $R_n$ is decreasing, we could also arrange that
$R_{n_j}(z_j)\ge 2\epsilon$ for certain arbitrarily large $j$. We could then find $w_j,w'_j\in\C^+$
so that $\gamma(P_{n_j}(z_j)w_j, P_{n_j}(z_j)w'_j)\ge \epsilon>0$,
even though both sequences approach the same limit, uniformly on $K$ (after passing to another subsequence).
This leaves only one possible excuse: the common limit must lie in $\R_{\infty}$, but this contradicts the choice of the sequence $n_j$,
which makes sure that all limits are genuine Herglotz functions.
\end{proof}
Let us now return to Theorem \ref{Trefl}. First of all, the hypothesis of Theorem \ref{TWeyl} is clearly satisfied here:
we know that $P_n(z)m^{(1)}_-(z)\to m_-(z)\in\C^+$.

We wish to show that cases (1), (2) of Theorem \ref{TWeyl} lead
to (1) and (2) of Theorem \ref{Trefl}, respectively.

If we are in case (1) of Theorem \ref{TWeyl}, then the additional statement that $R_n(z)\to 0$ \textit{uniformly }on compact
subsets of $\C^+$ can now take the role of Corollary A.3 of \cite{Remac} (which was used in the original proof).
The argument now proceeds exactly as in \cite{Remac}, with no further modifications necessary. I'll leave the matter at that.

Now consider case (2) (of Theorem \ref{TWeyl}). By the observations made above, a $T\in\mathcal S$ can be written as
either $T=AT_0$ or $T=T_1A$, with $A=T(0)\in SL(2,\R)$ and $T_j\in\mathcal S_0$. This means that in the product $P_n$, we can
move the constant $SL(2,\R)$ factors all the way to the left and write
\[
P_n(z) = B_n S_n(z)\cdots S_1(z) ,
\]
with $B_n\in SL(2,\R)$ and $S_j\in\mathcal S_0$. Observe also that the $S_j$, $1\le j\le n$, don't change if we add new factors on the left and
consider $P_k$ with $k>n$. By Theorem \ref{T4.1}, each $S_j$ is a transfer matrix of a canonical system on an interval $I_j$.
The product $S_n\cdots S_1$ then is the transfer matrix of the concatenation of these canonical systems. More specifically, arrange this so that
$S_j$ is the transfer matrix across $(x_{j-1},x_j)$. I now claim that $\lim_{j\to\infty} x_j<\infty$. This will follow from the important result
of de~Branges \cite[Theorem 42]{dB} that a trace-normed canonical system on a half line is in the limit point case at the singular endpoint.
See also \cite{AchaLPC} for a transparent alternative proof.

So assume, to obtain a contradiction, that $x_j\to\infty$. We shift the intervals so that we now have canonical systems on $[-x_j,0]$, and
we want to consider limits of these problems; these will be canonical systems on a left half line $(-\infty,0]$.
We are taking these limits with respect to a metric that is built in a similar way as the metric
on (Schr\"odinger operator) potentials that was used in \cite{Remcont}. Two canonical systems are close if their coefficient functions
almost agree (in a weak sense, not pointwise) on a sufficiently long initial interval $[-L,0]$; the metric pays little attention to what happens far out.
We will not spell out the details here; the important features
are that this gives a compact space of canonical systems, and convergence with respect to this metric is equivalent to the
locally uniform convergence of the $m$ functions. (In fact, this property could almost serve as the definition.)

Fix any limit point $H_0(x)$, $x<0$. By standard Weyl theory and what we just reviewed, any limiting function of the type
\begin{equation}
\label{4.11}
M(z)=\lim P_{n_j}(z)w_{n_j}
\end{equation}
is a half line $m$ function of this problem. By the result just quoted, there can be only one
such function. However, we are currently assuming that $R(z)>0$ on $\C^+$, and this makes sure
that we can get more than just one function from \eqref{4.11}. Indeed it suffices to fix $z_0\in\C^+$,
then pick $w_n,w'_n$ so that $\gamma(P_n(z_0)w_n,P_n(z_0)w'_n)\ge c>0$ and pass to limits along
a subsequence.

This contradiction shows that the collection of intervals $(x_{j-1},x_j)$ is of bounded total length. So there exist $L\ge 0$ and
a trace-normed $H(x)$ on $(0,L)$ so that $P_n(z)=B_n T(x_n,z)$, where $T(x,z)$ is the transfer matrix of this system across $(0,x)$,
and $0=x_0\le x_1 \le \ldots$, $x_n\to L$. So in particular, we have that $T(x_n,z)\to T(L,z)$ locally uniformly on $\C$,
and this matrix is in $\mathcal S_0$.

It remains to show that with suitable choices of ($n$ dependent) signs, we will also obtain that $\pm B_n\to B$ for some
$B\in SL(2,\R)$. Recall that
\begin{equation}
\label{4.21}
B_nT(x_n,z)m_-^{(1)}(z)\to m_-(z) .
\end{equation}
By normal families again, $B_n$ (as a sequence of linear fractional
transformations) converges locally uniformly on $\C^+$ along suitable subsequences. The limit cannot be identically equal
to an $a\in\R_{\infty}$, by \eqref{4.21}, so it is a linear fractional transformation $B\in SL(2,\R)$ itself. Moreover,
\eqref{4.21} identifies $B$ (as a map) uniquely, so there is no need to pass to a subsequence. For the matrices, this means
that $\pm B_n\to B$, as desired (note that we originally derived the convergence of the \textit{maps, }not the matrices).


TITLE: Need help finding the limit of this sequence.
QUESTION [2 upvotes]: If $x_1< x_2$ are real numbers, and $x_n = \frac{1}{2}(x_{n-2} + x_{n-1})$ for every $n > 2$, how do I show $\lim x_n = \frac{1}{3}x_1 + \frac{2}{3}x_2$?

REPLY [0 votes]: Due to the form of the recursion, you can always write
$$
x_n = a_k x_{n-k} + b_k x_{n-k-1},
$$
where $a_k$ and $b_k$ are some appropriate coefficients. Now, assume $n$ is very large and solve $a_k$, $b_k$. Using the recursion we get
$$
x_n = (a_k/2+b_k) x_{n-k-1} + a_k/2 \cdot x_{n-k-2},
$$
so $a_{k+1} = a_k/2+b_k$ and $b_{k+1} = a_k/2$. Using these we hence obtain the recursion $a_{k+1} = \frac{1}{2}(a_k+a_{k-1})$. With standard techniques (assuming $a_k = u^k$ and solving a quadratic equation that follows), we get the full solution as $a_k = C_1 + C_2(-1/2)^k$. From the initial values $a_1 = 1/2$ and $a_2 = 3/4$ we get the solution that
$$
a_k = 2/3 + (-1/2)^k \cdot 1/3
$$
and
$$
b_k = a_{k-1}/2 = 1/3 + (-1/2)^{k-1} \cdot 1/6.
$$
Now, we assumed $n \rightarrow \infty$, so we can take $k \rightarrow \infty$ as well and get that $a_k \rightarrow 2/3$ and $b_k \rightarrow 1/3$. At the limit the coefficient $a_k$ is the coefficient of $x_2$ and $b_k$ the coefficient of $x_1$, so the claim follows.