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\begin{document}
\title[Random Schr\"odinger Operators]
{ Absolutely Continuous Spectrum in Random Schr\"odinger Operators}
\author{ Abel Klein}
\address{Department of Mathematics, University of California, Irvine, CA 92717-3875}
\email{aklein@@math.uci.edu}
% Don't type a period at the end; it will be supplied.
\thanks{The author was supported in part by NSF Grant DMS-9208029.}
\keywords{ Random Schr\"odinger operators, Anderson model,
extended states, absolutely continuous spectrum, localization}
% Math Subject Classifications
\subjclass{Primary 82B44; Secondary 81Q10}
\date{}
\begin{abstract}
The spectrum of the Anderson Hamiltonian $\;H_\lb=-\De +\lb V$ on the Bethe
Lattice is absolutely continuous inside the spectrum of the
Laplacian, if the disorder $\lb$ is sufficiently small. More precisely, given any closed interval $I$
contained in the interior of the spectrum of the (centered) Laplacian
$\De$ on the Bethe lattice, for small disorder $H_\lb$ has
purely absolutely continuous spectrum in $I$ with probability one (i.e.,
$\si_{ac}( H_\lb) \cap I = I$ and $\si_{pp}( H_\lb) \cap I =\si_{sc}( H_\lb) \cap I= \emptyset$
with probability one). The proof is discussed and regularity properties are proven
for the spectral measures restricted to such intervals of absolute continuity.
\end{abstract}
\maketitle
\section{Introduction}
The Anderson Hamiltonian \ci{And}
is the random Schr\"odinger operator
\begin{equation}
H_\lb\;\;=\;\;{\textstyle\frac{1}{2}}\De\, +\,\lb V\;\;\;\mbox{on}\;\;\; \ell^2({\bbz^d})\,. \la{ha}
\end{equation}
Here
the (centered) Laplacian $\De$ is defined by
\begin{equation}
(\De u)(x)\;\;=\;\;\sum_{y} {u(y)}\;,
\end{equation}
where the sum runs over all nearest neighbors of $x\,$, and $V$ is a
random potential, with $V(x)$, $x \in {{\bbz^d}}$, being independent, identically distributed random
variables with common probability distribution $\mu$. The real parameter $\lb$ is
called the {\em disorder}.
The spectrum of the Hamiltonian $\, H_\lb$ is given by
\begin{equation}
\sigma(H_\lb) = \sigma({\textstyle\frac{1}{2}}\Delta) + \lb\, \mbox{supp} \; \mu =
[-d,d] +\lb\, \mbox{supp} \; \mu \la{erg}
\end{equation}
with probability one, due to ergodic considerations \ci{P,CL}. For each choice of $V$ the spectrum of $H_\lb$ can be decomposed
into pure point spectrum, $\sigma_{pp}(H_\lb)$, absolutely continuous
spectrum, $\sigma_{ac}(H_\lb)$, and singular continuous spectrum,
$\sigma_{sc}(H_\lb)$. Ergodicity also gives the existence of sets $\Sigma_{\lb,pp}\; , \; \Sigma_{\lb,ac}\; ,
\; \Sigma_{\lb,sc} \subset {{\bbr}}$ such that $\sigma_{pp}(H_\lb) = \Sigma_{\lb,pp}
\; , \; \sigma_{ac}(H_\lb) = \Sigma_{\lb,ac}$ and $\sigma_{sc}(H_\lb) = \Sigma_{\lb,sc}$
with probability one \ci{KS1,CL}.
In the physics literature (\ci{And,MT,T,AAT,AT,MS} and others) the following picture is given:
In one and two dimensions, as long as the potential is random (i.e., $\lb \not= 0$), the model
shows {\em exponential localization} (i.e., pure point spectrum with exponentially decaying
eigenfunctions ). In three and more dimensions
both localized and {\em extended states} (i.e., absolutely continuous spectrum)
are expected for small disorder, with the energies of extended and localized states being
separated by a {\em mobility edge}.
In one dimension there are now mathematical proofs of exponential localization for any disorder
(e.g., \ci{GMP,KS1,CKM,DK} and others). In the multidimensional case exponential localization
has been proven
for large disorder or low energy (e.g., \ci{FS,FMSS,DLS,SW,DK,AM,A,K3,Gf} and others);
for small disorder there exist energies
$E_{\lambda, loc}^\pm$, with $ |E_{\lambda, loc}^\pm| > d$ and
\mbox{$\lim_{\lambda \to 0} E_{\lambda, loc}^\pm= \pm d$},
such that $H_{\lambda}$ has pure point spectrum in
\mbox{$(-\infty, E_{\lambda, loc}^- ) \cup ( E_{\lambda, loc}^+,\infty)$ \cite{Sp,FK,A,Sp2}.}
Only localization in two dimensions for small disorder is still an open problem.
But up to now there is no mathematical
proof of the occurrence of absolutely continuous spectrum in the Anderson model on $\bbz^d$.
The Bethe lattice (or Cayley tree), ${{\B}}$, is an infinite connected graph with no closed loops and a
fixed degree (number of nearest neighbors) at each vertex (site or point). The degree is called
the coordination number and the connectivity, $K$, is one less the coordination number
($K \ge 2$, so $\B$ is not the line $\bbr$) .
The distance between two sites $x$ and $y$ will be denoted by $d(x,y)$
and is equal to the length of the shortest path connecting $x$ and $y$. The Anderson Hamiltonian
on the Bethe lattice is given by (\ref{ha}) with ${{\B}}$ substituted for ${\bbz^d}$.
It was first studied by Abou-Chacra, Anderson and Thouless
\ci{AAT}; their self-consistent approximation for the study of localization
becomes exact in the Bethe lattice. The resulting equations were further studied by
Abou-Chacra and Thouless \ci{AT}, who argued that, on the Bethe lattice, for sufficiently
low disorder there should be an energy at which localization breaks
down, which converges to $\frac{K +1}{2}$ in the zero disorder limit.
An outline of expected results was given by Kunz and Souillard \cite{KS}.
Miller and Derrida \ci{MD} performed a weak disorder expansion inside the
spectrum of the zero disorder Hamiltonian, and
computed perturbatively the density of states and conducting properties corresponding to
extended states. They also found the existence of an energy, which converges (from outside)
to the edge $\sqrt{K}$ of the spectrum of ${1 \over 2}\De$ in the zero disorder limit, above
which the density of states and conducting properties vanish to all orders in perturbation
theory. The Bethe lattice model was also discussed by Mirlin and Fyodorov \ci{MF}.
Ergodicity considerations still apply (see Acosta and Klein \cite[Appendix]{AK}), so the first
equality in
(\ref{erg}) and the following statements are still valid in the Bethe lattice, where we have
$ \sigma(\Delta) \,=\, [-2\sqrt{ K},2\sqrt{ K}]\,$. In particular,
\begin{equation}
\sigma(H_\lb)\;\; =\;\;[-\sqrt{ K},\sqrt{ K}] +\lb\, \mbox{supp} \; \mu
\end{equation}
with probability one.
The nature of the spectrum for
energies $E$ with $\sqrt{K} \le |E| \le \frac{K +1}{2}$, at low disorder, does not seem to have been
properly discussed in the physics literature. The Abou-Chacra and Thouless calculations
indicate that we do not have localization in these intervals for
small disorder; Miller and Derrida's weak disorder expansions suggest that the corresponding
states cannot be ``too extended", since they should not have conducting properties.
So there should be continuous spectrum, but of which type? It could be absolutely
continuous, but it would be of a different nature then the one inside the free spectrum. It is
plausible that near $E=\pm \frac{K +1}{2}$ there is a transition from pure point spectum
to either singular continuous spectrum or absolutely continuous spectrum ``without conducting
properties", and that near $E=\pm \sqrt{K}$ there is another
transition where the spectrum changes to absolutely
continuous ``with conducting properties". If so, there is a {\it mobility interval} instead of
a {\it mobility edge}!
Localization for large disorder or low energies has only recently been proven by Aizenman
and Molchanov \ci{AM}, and Aizenman \ci{A}
proved localization for energies beyond $\frac{K +1}{2}$
at weak disorder, confirming half of Abou-Chacra and Thouless' prediction \ci{AT}.
On the Bethe lattice we proved that the Anderson Hamiltonian has
``extended states'' for small disorder \cite{K1,K2}. More precisely, given any closed
interval $I$ contained in
the interior of the spectrum of ${1 \over 2}\De$ on the Bethe lattice, we proved that
for small disorder $H_\lb$ has
purely absolutely continuous spectrum in $I$ with probability one, and its integrated density of
states is continuously differentiable on the interval. These results agree with Miller and Derrida's
conclusions \ci{MD}. We have also studied the spreading of wave packets evolving under the
Anderson Hamiltonian on the Bethe Lattice: for
small disorder we showed that the averaged mean square distance travelled by a particle in a
time $t$ \ grows as $t^2$ for large $t$ \cite{K4}.
The precise results requires hypotheses about the single site
potential probability distribution $\mu$: the localization results require $\mu$ to
have a bounded density with respect to Lebesgue measure, while the extended states
results call for $\mu$ to have a characteristic function $h(t)=\int {{\e}^{-itv}d\mu (v)}$
which is twice differentiable with bounded first and second derivatives on $(0,\infty)$.
(True for any any probability distribution $\mu$ with a
finite second moment (e.g., uniform, Gaussian or Bernoulli distributions) and for the
Cauchy distribution).
In this article we show that by requiring more differentiability of the characteristic function
$h(t)$ we can get absolutely continuous spectrum with certain
regularity properties, which will be expressed in terms of the spectral measures $\nu_{\lb,x},\;
x \in \B$, given by
$$d\nu_{\lb,x}(E) = \left\langle {\de_x,dP_\lb (E)\de_x} \right\rangle\;,$$
where $dP_\lb (E)$ is the spectral measure of the operator $H_\lb$.
\bd Let $p \in \bbn$. A probability measure $\mu$ will be said to be $p$-admissible
if its characteristic function $h(t)$ is $2p$-times differentiable on $(0,\infty)$, with
all $2p$ derivatives bounded on $(0,\infty)$.
(True for any any probability distribution $\mu$ with $\int |v|^{2p} d\mu (v) < \infty$).
\ed
\bt \la{main} Let $\;H_\lb$ be the Anderson Hamiltonian on the Bethe lattice with $\mu$
$p$-admissible, $p \in \bbn$. Then for any $E \in (0 , \sqrt K)$ there exists $\lb(E) > 0$,
such that given any $\lb$ with$ |\lb| < \lb(E) $, with probability one the spectrum of $\;H_\lb$
in $ [-E, E]$ is purely absolutely continuous (i.e., $\Sigma_{\lb,ac} \cap [-E, E] = [-E, E] $ and
$\Sigma_{\lb,pp} \cap [-E, E] = \Sigma_{\lb,sc} \cap [-E, E]=\emptyset $),
the spectral measures $\nu_{\lb,x}$ being absolutely continuous in this interval
with $\frac{d\nu_{\lb,x}}{dE} \in L^{2p}([-E, E])$ for all $x \in \B$.
\et
The $p=1$ case was proved in \cite{K1,K2}.
The Green's function of $H_\lb$ is given by
\begin{equation}
G_{\lb}\, (x,y;z)\;\;=\;\;\left\langle {\de_x,(H_{\lb} -z)^{-1}\de_y} \right\rangle
\end{equation}
for $x,y \in {\B}$ and $z = E + i\eta$ with $E \in \bbr$, $\eta> 0$.
For any $x \in \B$ and any potential $V$,
$ G_{\lb}\, (x,x; E +i\eta) $ is a continuous function of
$ (\lb, E,\eta) \in \bbr \times \bbr \times (0, \infty)$, so
$ \E (|G_{\lb}\, (x,x; E +i\eta)|^{2p}) $ is also a
continuous function of $ (\lb, E,\eta) \in \bbr \times \bbr \times (0, \infty)$ for any
$p \in \bbn$.
Theorem \ref{main} will follow from the fact that we can let $\eta \downarrow 0$ inside the
spectrum of $ \frac{1}{2} \De$.
\bt \la{cor} Let $\;H_\lb$ be the Anderson Hamiltonian on the Bethe lattice with $\mu$
$p$-admissible, $p \in \bbn$. Then for any $E \in (0 , \sqrt K)$ there exists $\lb(E) > 0$,
such that for all $x \in \B$
the continuous function
\begin{equation}
(\lb, E',\eta) \in (-\lb(E),\lb(E)) \times [-E,E] \times (0, \infty) \;\longrightarrow\;
\E (|G_{\lb}\, (x,x; E' +i\eta)|^{2p} )
\end{equation}
has a continuous extension to $ (-\lb(E),\lb(E)) \times [-E,E] \times [0, \infty) $.
In particular,
\begin{equation}
\sup_{\lb;\, |\lb| < \lb(E)}\, \sup_{E';\,|E' | \le E} \, \sup_{\eta;\, 0 <\eta }\,
\E (|G_{\lb}\, (x,x; E' +i\eta)|^{2p} )\;\;<\;\; \infty\;. \la{sup}
\end{equation}
\et
Theorem \ref{main} follows from (\ref{sup}). Let
$E \in (0 , \sqrt K)$ and $\lb(E) > 0$ as in Theorem
\ref{cor}, so (\ref{sup}) holds. For $|\lb| < \lb(E) $ and any $x \in \B$ we use
Fubini's Theorem and Fatou's Lemma to obtain
\beq
\lefteqn{
\E \left( \liminf_{\eta \downarrow 0}\int_{-E}^E |G_{\lb}\, (x,x; E' +i\eta)|^{2p}\,dE' \right)
\;\;\le \quad \quad \quad \quad \quad } \\
&&\quad \quad \liminf_{\eta \downarrow 0} \int_{-E}^E \E (|G_{\lb}\, (x,x; E' +i\eta)|^{2p} )\,dE'\;\;<
\;\; \infty\;. \nonumber
\eeq
Thus we must have
\beq
\lefteqn{
\liminf_{\eta \downarrow 0}\int_{-E}^E ({\rm Im}\,G_{\lb}\, (x,x; E' +i\eta))^{2p}\,dE' \;\;\le
\qquad \qquad \qquad} \\
&& \quad \quad \quad \quad \liminf_{\eta \downarrow 0}\int_{-E}^E |G_{\lb}\, (x,x; E' +i\eta)|^{2p}\,dE' \;\;<\;\;\infty
\nonumber
\eeq
with probability one. Since $G_{\lb}\, (x,x; E +i\eta)$ is the Stieltjes transform of the
measure $\nu_{\lb,x}$, it follows (see \ci[Theorem 2.1]{simon}) that, with probability one,
the spectral measures $\nu_{\lb,x}$ are absolutely continuous in the interval $(-E, E)$
with $\frac{d\nu_{\lb,x}}{dE} \in L^{2p}(-E, E)$ for all $x \in \B$. Theorem \ref{main} now follows.
\section{Formulas}
Let $\;H_\lb$ be the Anderson Hamiltonian on the Bethe lattice with $\mu$
$p$-admissible.
We fix an arbitrary site in $\B$ which we will call the origin and denote by $0$. Given two nearest neighbors
sites $x,y \in \B$, we will denote by ${\B}^{(x|y)}$ the lattice obtained by removing from $\B$
the branch emanating from $x$ that passes through $y$; if we do not specify which
branch was removed we will simply write ${\B}^{(x)}$. Each vertex in ${\B}^{(x)}$ has degree
$K +1$, with the single exception of $x$ which has degree $K$.
Given $\La \subset \B$, we will use $\,H_{\lb, \La}$ to denote the operator $\,H_{\lb}$
restricted to $\ell^2 (\Lambda)$ with Dirichlet boundary conditions.
The Green's function corresponding to $\,H_{\lb, \La}$ will be denoted by
\begin{equation}
G_{\lb, \La}\, (x,y;z)\;\;=\;\;\left\langle {\de_x,(H_{\lb, \La} -z)^{-1}\de_y} \right\rangle
\end{equation}
for $x,y \in {\La}$ and $z = E + i\eta$ with $E \in \bbr$, $\eta> 0$.
We will write $H_{\lb}$, $H_{\lb}^{(x|y)}$ and $H_{\lb}^{(x)}$ for
$H_{\lb, \Bs}$, $H_{\lb, {\Bs}^{(x|y)}}$ and $H_{\lb, {\Bs}^{(x)}}$, respectively. Similarly,
we will use $G_{\lb}\, (x,y;z)$ for
$G_{\lb, \Bs}\, (x,y;z)$ and $G_{\lb}\, (z)$, $G_{\lb}^{(x|y)}\, (z)$,
$G_{\lb}^{(x)}\, (z)\;$ for $G_{\lb}\, (0,0;z)$, $G_{\lb,{\Bs}^{(x|y)}}\, (x,x;z)$,
$G_{\lb,{\Bs}^{(x)}}\,(x,x;z)\;$, respectively.
We start by stating some useful formulas (see \ci{K2} for more details). The next proposition
is a consequence of the resolvent equation.
\bp \la{res} For any $\lb \in \bbr$, $E \in \bbr$ and $\eta > 0$ we have
\beq
G_{\lb}\, (z)\;\;=\;\; -\,\left(z -\lb V(0) +{\textstyle{\frac{1}{4}}} \!\sum_{x:\,d(x,0)=1}
G_{\lb}^{(x|0)}\, (z) \right)^{-1}\; \la{G3}
\eeq
and, for any two nearest neighbors sites $x,y \in \B$,
\beq
G_{\lb}^{(x|y)}\, (z)\;\;=\;\; -\,\left(z - \lb V(x) +{\textstyle{\frac{1}{4}}}
\!\sum_{x':\,d(x',x)=1,\,x'\not=y} G_{\lb}^{(x'|x)}\, (z) \right)^{-1}\;. \la{G4}
\eeq
\ep
\bp For any $\lb \in \bbr$, $E \in \bbr$ and $\eta > 0$ we have
\beq
G_{\lb}\, (z)\;\;=\;\; \frac{ i}{\pi} \int_{\bbrs^2} \e^{i(z -\lb V(0))\vp ^2}
\exp \left\{{\textstyle{\frac{i}{4}}} \!\sum_{x:\,d(x,0)=1} G_{\lb}^{(x|0)}\, (z)\, \vp^2\right\}\, d^2\vp
\la{G2}\;,
\eeq
and, for any two nearest neighbors sites $x,y \in \B$,
\beq
\lefteqn{\qquad \quad \e^{ \frac{i}{4} G_{\lb}^{(x|y)}\, (z) \vp^2}\;\;=} \la{G22} \\
&&-\frac{1}{\pi}\int_{\bbrs^2} \e^ { - i \vp\cdot\vp'}\, \partial\left\{\e^{i(z -\lb V(x)) {{\vp'}^{2}}}
\exp \left\{{\textstyle{\frac{i}{4}}}
\!\sum_{x':\,d(x',x)=1,\,x'\not=y}\!\! G_{\lb}^{(x'|x)}\, (z) \,{{\vp'}^{2}}\right \}
\right\}\, d^2\vp'\;, \nonumber
\eeq
where $ \vp^2=\vp\cdot \vp$ and $\partial f(\vp^2)= f'(\vp^2)$.
\ep
\bpf If we perform the integration in (\ref{G2}) and
(\ref{G22}) we obtain (\ref{G3}) and (\ref{G4}). \epf
\bd
For any $\lb \in \bbr$, $E \in \bbr$ and $\eta > 0$ let
\beq
\xi_{\lb,z} ( t, s)\;\;&=&\;\;
\E \left(\exp{ \left\{\frac{i}{4} \left ( G_{\lb}^{(0)}\, (z) \,t\, - \,\overline{ G_{\lb}^{(0)}\, (z) }\,
s\right)\right\}}\right) \la{xia} \\
&=&\;\;\E \left(\exp{ \left\{\frac{1}{4} \left[i{\R}_{\lb}^{(0)} (z) \,(t-s)\, -\,
{\I}_{\lb}^{(0)} (z)\, (t+s)\right]\right\}}\right) \la{xi}
\eeq
for $t,s >0$, where ${\R}_{\lb}^{(x|0)} (z) + i{\I}_{\lb}^{(x|0)} (z)$ is the decomposition of
$ G_{\lb}^{(x|0)}\, (z)$ into its real and imaginary parts.
\ed
If $\lb =0$ we can calculate $G_{0}^{(0)}\, (z) $ \ci{AK} obtaining
\beq
\xi_{0,z} (t, s)\;\;=\;\;
\e^ { \frac{ i }{2K } \{(-z +\sqrt{z^2 - K})t\, - \, \overline{(-z +\sqrt{z^2 - K})}s \} } \;,\la{ze0}
\eeq
where we always make the choice Im $\sqrt{ \ } >0$. If $ |E| <\sqrt{ K}$, we have the
pointwise limit %@xi00
\beq
\xi_{0,E} (t, s)\;\;\equiv
\;\;\lim_{\eta \downarrow 0}\xi_{0,z} (t, s)\;\;=\;\;
\e^ { \frac{ 1 }{2K }\{-iE(t-s) -\sqrt{ K- E^2}(t+s) \} } \;.\la{xi00}
\eeq
\bl \la{eqs} For any $\lb \in \bbr$, $E \in \bbr$ and $\eta > 0$ we have
\beq
\lefteqn{\qquad \qquad\E(| G_{\lb}\, (z)|^{2p}) \; \;=} \la{EGG}\\
&&{\textstyle\frac{1}{\pi^{2p}}}{\displaystyle \int_{\bbrs^{2p} \times\bbrs^{2p} } } \e^{iE (\vp_+^2-\vp_- ^2) -\eta(\vp_+^2+\vp_- ^2)}
h(\lb (\vp_+^2-\vp_- ^2)) [\xi_{\lb,z} ( \vp_+^2, \vp_- ^2)]^{K+1}
d^{2p}\vp_+ d^{2p}\vp_- \;, \nonumber
\eeq
and
\beq
\lefteqn{\qquad \qquad \quad \xi_{\lb,z} ( \vp_+^2, \vp_- ^2)\;\;=\;\;
{\textstyle\frac{1}{\pi^{2p}}}{\displaystyle \int_{\bbrs^{2p} \times\bbrs^{2p} } }
\e^ { - i (\vp_+\cdot\vp_+' - i \vp_-\cdot\vp_-')} \;\times } \la{xiK} \\
&& \partial_+^p \partial_-^p
\left\{\e^{iE ({\vp_+'}{^2}-{\vp_-'}{^2}) -\eta({\vp_+'}{^2}+{\vp_-'}{^2})}
h(\lb ({\vp_+'}{^2}-{\vp_-'}{^2}))
[\xi_{\lb,z} ( {\vp_+'}{^2}, {\vp_-'}{^2})]^K \right \} d^{2p}{\vp}_+' d^{2p}{\vp}_-' , \nonumber
\eeq
with
\beq
\partial_\pm g(\vp_+^2,\vp_-^2 )\;\;= \frac{\partial}{\partial \vp_{\pm}^2} g(\vp_+^2,\vp_-^2 )\;.
\eeq
\el
\bpf Equations (\ref{EGG}) and (\ref{xiK}) follow from (\ref{G2}) and (\ref{G22}) by taking the
$p$th power of each side, multiplying it by its complex conjugate, and taking expectations
of each side, using the independence of the potential at different sites. \epf
\section{Nonlinear analysis}
To handle the nonlinear equations (\ref{EGG}) and (\ref{xiK}), we
introduce the Banach spaces ${\K}_r$, $1 \le r \le \infty$, given by the completion of
$$
\{g\!:\;[0,\infty) \times [0,\infty) \to \bbc \;\;\mbox{of class $C^\infty$ }\,;\;\;
\|g\|_{{\K}_r} \equiv|\!|\!|\!|g|\!|\!|\!|_2 + |\!|\!|\!|g|\!|\!|\!|_r < \infty \}\;,
$$
where
\beq
|\!|\!|\!|g|\!|\!|\!|_r ^2 \;\;= \;\;
\sum_{a,b=0}^p
\| (2\partial_+)^a (2\partial_-)^b
g(\vp_+^2,\vp_-^2 )\|_{L^r(\bbrs^{2p} \times\bbrs^{2p} ,d^{2p}\vp_+ d^{2p}\vp_-)}^2\;\;.
\nonumber
\eeq
We define linear operators
\beq
\lefteqn{({\T} g)( \vp_+^2, \vp_- ^2)\;\;=} \\
&&{\textstyle\frac{1}{\pi^{2p}}}
{\displaystyle \int_{\bbrs^{2p} \times\bbrs^{2p} } }
\e^ { - i (\vp_+\cdot\vp_+' - i \vp_-\cdot\vp_-')}\,
\partial_+^p \partial_-^p
\left\{g ( {\vp_+'}{^2}, {\vp_-'}{^2}) \right \} d^{2p}{\vp}_+' d^{2p}{\vp}_-' \nonumber
\eeq
and
\beq
{\cB} (\lb,z) =
M(\e^{iE (\vp_+^2-\vp_- ^2) -\eta(\vp_+^2+\vp_- ^2)} h(\lb (\vp_+^2-\vp_- ^2)) )\;,
\eeq
where $M(g(\vp_+^2,\vp_-^2 ))$ denotes multiplication by the function $g(\vp_+^2,\vp_-^2 ) $.
It turns out that $\T$ is unitary on ${\K}_2$ \ci{CK} and is a bounded linear operator from
${\K}_1$ to ${\K}_\infty$, ${\cB}_{\lb,z}$ is a bounded linear operator on all ${\K}_r$, and
$g \to g^n$ is a continuous map from ${\K}_\infty$ to ${\K}_1$
for any $n=2,3,\ldots$. We can prove the following lemma.
\bl \la{xita}
\begin{enumerate}
\item[(I)] $\xi_{\lb,z} \in {\K}_\infty$ for all $\lb \in \bbr$ and $z=E+i\eta$ with
$\eta >0$. The map $(\lb,E, \eta) \to \xi_{\lb,E +i\eta}$ is continuous from
$\bbr \times \bbr \times (0,\infty)$ to $ {\K}_\infty$.
\item[(II)] If $|E| < \sqrt{K}$ we have $\xi_{0,E} \in {\K}_\infty$ and
\beq
\lim_{\eta \downarrow 0} \xi_{0,E+i\eta}\;\;=\;\;\xi_{0,E} \;\;\;\mbox{in}\;\; {\K}_\infty \;.
\eeq
\item[(III)] The integral equation(\ref{xiK}) can be rewritten as a fixed point equation in
${\K}_\infty$: %@fpx
\beq \la{fpx}
\xi_{\lb,z} \;\;=\;\;{\T}{\cB}_{\lb,z} \xi_{\lb,z}^K \;,
\eeq
valid for all $\lb \in \bbr$ and $z=E+i\eta$ with $\eta >0$, and also valid for $\lb=0$ and
$z=E$ with $|E| < \sqrt{K}$ .
\end{enumerate}
\el
The next step is a fixed point analysis.
\bl \la{Q} The map
$Q:\,\bbr \times \bbr \times [0,\infty) \times {\K}_\infty \to {\K}_\infty$, defined by
\beq
Q(\lb, E,\eta,g)\;\;=\;\;{\T} {\cB}_{\lb,E + i \eta} g^K \;-g \;\;,
\eeq
is continuous. $Q$ is continuously Frechet differentiable with respect to $g$, the partial
derivative being
\beq
Q_g(\lb, E,\eta,g)\;\;=\;\;K {\T} {\cB}_{\lb,E + i \eta} M( g^{K-1}) \;-\; I \;.
\eeq
Moreover, for any $E$ such that $|E| < \sqrt{K}$ we have $ Q(0, E,0,\xi_{0,E}) =0$ and
\beq
0 \notin \si( Q_g(0, E,0,\xi_{0,E})) \;. \la{000}
\eeq
\el
\bpf The proof of this lemma is routine except for (\ref{000}). We have
$ Q_g(0, E,0,\xi_{0,E}) = K{\A}_{0,E} - I$ where ${\A}_{0,E}= {\T}{\cB}_{0,E } M( \xi_{0,E}^{K-1}) $.
As in \ci{K2},
following Acosta and Klein \ci{AK} (see their Propositions 3.2 and 3.3 and Theorem 3.5), we can
show that ${\A}_{0,E}^2$ is a compact operator on ${\K}_\infty$ and
\beq
\si({\A}_{0,E}) \;\;=\;\;\{ {\cal E}_{i,j} =E_i\bar{E}_j \,;\;\; i,j= 0,1,2,\ldots \} \cup \{0\}\;,
\eeq
with
\beq
E_n\;\;=\;\; \left(\frac{-E +i\sqrt{ K- E^2}}{K} \right)^{2n}\;\;\;\mbox{for}\;\; n= 0,1,2,\ldots \;. \la{E}
\eeq
Since ${\cal E}_{i,j} \not= {1 \over K}$ for any $ i,j=0, 1,2,\ldots$, (\ref{000}) follows. \epf
Lemma \ref{Q} tells us that the hypotheses of the Implicit Function Theorem
(see \ci[2.7.2]{N}) are satisfied by the function
$Q(\lb, E,\eta,g)$ at $(0,E,0,\xi_{0,E} )$, if $ |E| < \sqrt{K} $. It follows
that for each $E$ such that $|E| < \sqrt{K}$ there exist $\lb_E > 0$,
$\varepsilon_E >0$, $\eta_E >0$ and $\de_E >0$, such that for each
$$ (\lb, E',\eta) \in (-\lb_E,\lb_E)\times (E - \ve_E,E + \ve_E) \times [0, \eta_E) $$
there is a unique $\, \omega_{\lb,E',\eta} \in {\K}_\infty$ with
$\| \omega_{\lb,E',\eta}- \xi_{0,E} \|_{ {\K}_\infty}< \de_E$,
such that $ Q(\lb, E',\eta,\omega_{\lb,E',\eta}) =0$. Moreover, the map
$$
(\lb, E',\eta) \in (-\lb_E,\lb_E)\times (E - \ve_E,E + \ve_E) \times [0, \eta_E) \;\longrightarrow\;
\omega_{\lb,E',\eta} \in {\K}_\infty
$$
is continuous. Combining with Lemma \ref{xita}, and using the uniqueness of
$ \omega_{\lb,E',\eta}$ as above for each $ (\lb, E',\eta)$, we get
\bt \la{xize}
For any $E$ such that $|E| < \sqrt{K}$ there exist $\lb_E > 0$, $\varepsilon_E >0$
and $\de_E >0$, such that the map
$$
(\lb, E',\eta) \in (-\lb_E,\lb_E)\times (E - \ve_E,E + \ve_E) \times (0, \infty) \;\longrightarrow\;
\xi_{\lb,E'+i\eta} \in {\K}_\infty \la{xizexi}
$$
has a continuous extension to $ (-\lb_E,\lb_E)\times (E - \ve_E,E + \ve_E) \times [0, \infty)$
satisfying (\ref{fpx}).
\et
Theorem \ref{cor} now follows from (\ref{EGG}), Theorem \ref{xize}, the
translation invariance of expectations, and a simple compactness argument.
Full details for the $p=1$ case can be found in Klein \ci{K2}.
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\end{document}