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\author{Michael Christ}
\address{
Michael Christ\\
Department of Mathematics\\
University of California \\
Berkeley, CA 94720-3840, USA}
\email{mchrist@math.berkeley.edu}
\thanks{The first author was supported in part by NSF grant DMS-9970660}
\author{Alexander Kiselev}
\address{ Alexander Kiselev\\
Department of Mathematics\\
University of Chicago\\
Chicago, Ill. 60637}
\email{kiselev@math.uchicago.edu}
\thanks{The second author was supported in part by NSF grant DMS-9801530}
\date{\today}
\title[WKB Asymptotics]
{WKB Asymptotics of Generalized Eigenfunctions \\
of One-Dimensional Schr\"odinger Operators}
\begin{abstract}
We prove the WKB asymptotic
behavior of solutions of the differential equation
$-d^2u/dx^2+V(x)u=k^2u$ in two cases. First, for a.e.\ $k^2$ when
$V \in L^p(\reals)$, where $1 \leq p <2$. Second, for a.e.\ $k^2>A$
when $V \in L^{\infty}(\reals)$ and $V' \in L^p(\reals)$,
$1 \leq p <2$, where $A = \limsup_{x \rightarrow \infty}V(x)$.
These results imply that Schr\"odinger operators
with such potentials have absolutely continuous spectrum
on $(0, \infty)$ ($(A, \infty)$ in the second case).
We also establish WKB asymptotic behavior
of solutions for some energy-dependent potentials.
\end{abstract}
\maketitle
\section{Introduction}
Let $D=d/dx$, where $x\in\reals^1$.
In this paper we establish criteria for the solutions of the
ordinary differential equation
\begin{equation}
\label{eq1}
(-D^2+V-k^2)u=0
\end{equation}
to be globally bounded functions for almost every $k$, with
respect to Lebesgue measure, and for the Schr\"odinger
operator $-D^2 +V(x)$ to have absolutely continuous spectrum on the
positive semi-axis. Our main interest here is in potentials $V$
that are either decaying or
have decaying derivative. We say
that a function $f$ belongs to the space
$\ell^p(L^q)(\reals)$ if
\[ \|f\|_{\ell^p(L^q)(\reals)} = \left( \sum\limits_{n=-\infty}^\infty \left(
\int_n^{n+1} |f(x)|^q \,dx \right)^{p/q} \right)^{1/p} < \infty. \]
Denote by $\tilde u_\pm(x,k)$ the unique solutions of (\ref{eq1})
satisfying $\tilde u_\pm(0,k)=1$ and $d\tilde u_\pm/dx\,(0,k)
= \pm i k$.
Our first main result is
\begin{theorem}
\label{thm:main}
Let $1\leq p<2$, and let $V\in \ell^p(L^1)(\reals)$.
Then for
almost every energy $k^2$ with $k\in\reals$, all solutions of
$(-D^2+V-k^2)u=0$ are bounded functions on the real line.
For almost every $k^2,$ there exist solutions
with the WKB asymptotic behavior
\begin{equation}
\label{eq:sol0}
u_{\pm}(x,k) - e^{ \pm ik x \mp \frac{i}{2k}\int_0^x V(y)\,dy}
\to 0 \qquad\text{as } x\to+\infty.
\end{equation}
Moreover
\begin{equation}
\label{logbound}
\int_a^b \log(1+ {\rm sup}_x |\tilde u_{\pm}(x,k)|) \,dk < \infty
\end{equation}
for any $01/2$ as
$|x|\to\infty$, then almost every generalized eigenfunction
is bounded. Moreover,
the same conclusion holds \cite{christkiselev} if there exists
$\delta>0$ such that $(1+|x|)^\delta V\in L^1+L^2$.
The motivation for those investigations was a corollary:
every Schr\"odinger operator with such potential consequently has nonempty
absolutely continuous spectrum,
on any Borel set of positive Lebesgue measure in $(0,\infty)$.
The result is rather sharp, in the sense that there exist potentials
having purely singular spectrum, which are
bounded by $C|x|^{-1/2}$ as $|x|\to\infty$.
Deift and Killip \cite{deiftkillip} subsequently
derived the same conclusion regarding the spectrum under the
still sharper hypothesis $V\in L^1+L^2$, via a remarkable inequality
stemming from inverse scattering
theory. In our context, this inequality can be rephrased in a less
precise form as
\begin{equation}
\label{DK}
{\rm sup}_x \int_a^b \log (1+|\tilde u_\pm(x,k)|)\,dk < \infty \ .
\end{equation}
The bound \eqref{logbound} is interesting because it provides an improvement
of the estimate \eqref{DK} for the potentials we consider.
Although the inequality \eqref{DK} yields more precise information in
one respect,
it fails to imply boundedness of the generalized eigenfunctions or
provide information on their
asymptotic behavior.
It still remains an open question whether almost every eigenfunction
is bounded for every $V\in L^2$.
This, as our analysis will make clear, would be
a nonlinear analogue of Carleson's theorem \cite{carleson}
on the almost everywhere convergence of Fourier series.
Our second main result in this work is devoted to potentials
with decaying derivative.
\begin{theorem}
\label{thm:main1}
Let $V=V_1 +V_2,$ where $V_1 \in L^1(\reals),$ while $dV_2/dx
\in \ell^p(L^1)(\reals)$, with $1 \leq p <2$, and $V_2 \in L^{\infty}
(\reals).$
Then for
almost every energy $k^2>A=\limsup_{|x| \rightarrow \infty}V_2(x),$
all solutions of
$(-D^2+V-k^2)u=0$ are bounded functions on the real line.
Moreover, for almost every $k^2>A$, there exist solutions
with the WKB asymptotic behavior
\begin{equation}
\label{eq:sol1}
u_{\pm}(x,k) -
(k^2-V_2(x))^{-1/4}
e^{\pm i\int_0^x \sqrt{k^2-V_2(y)}\,dy}
\to 0 \qquad\text{as } x\to+\infty.
\end{equation}
\end{theorem}
\it Remark. \rm One scenario where Theorem~\ref{thm:main1} applies is
$V_2 \in L^r(\reals)$ for any $r < \infty,$ $V_2' \in L^p$, $p<2.$
Then $V_2 \rightarrow 0$ as $|x| \rightarrow \infty,$ and there are
solutions with WKB asymptotic behavior for any $k^2>0.$ \\
The proof of Theorems~\ref{thm:main} and \ref{thm:main1} uses
techniques introduced in our earlier work \cite{christkiselev},
but differs markedly from that work in one respect.
In \cite{christkiselev},
solutions of \eqref{eq1} were represented as sums
of infinite series of multilinear operators.
We were able to estimate each term of these series,
but were unable to sum the bounds.
Instead, the extra fractional power $|x|^\delta$
allowed us to show, essentially,
that the remainder after summing sufficiently
many terms of the series belonged to $L^1$, and hence could be
handled in an alternative way, by means of
a classical theorem of Levinson.
In the present paper, we employ a simpler representation
for the solutions and do sum the resulting infinite series.
We also hope that the simplicity of the series will allow
future applications to even larger classes of potentials.
An additional motivation for our work is the desire to have an
approach some of whose elements
might have a chance of being extended to higher
dimensions. While the one-dimensional structure is still essential
for our results, an abolition of the use of Levinson's theorem
appears to be a small step forward in this respect.
Potentially another such small step is the derivation of a
representation for $u_\pm(x,k)$ directly from a
resolvent-based expansion, making no use of reduction to
a first-order system.
We have succeeded in doing this, obtaining
a representation resembling, though more complicated than,
the one employed here. Its derivation was
not completely straightforward; initially we obtained infinitely
many terms which fail to satisfy the desired estimates, but
eventually discovered an algebraic identity which demonstrates
that all the undesirable terms sum to exactly zero.
This derivation has at present no concrete applications,
so is not included in this article.
The existence of solutions with the
asymptotic behavior \eqref{eq:sol0}, \eqref{eq:sol1} for
almost every $k^2$
has direct spectral consequences for Schr\"odinger operators
defined on the half-axis (with some self-adjoint condition at the
boundary) or on the whole axis. It is well known
\cite{Sto, Sim, KL}
that boundedness of solutions of \eqref{eq1}
at some energy implies positivity of the derivative of the spectral
measure. For a more recent alternative approach,
see \cite{christkiselevlast}.
Hence the following theorems are direct corollaries of the above
results.
\begin{theorem}
\label{thm:second}
If $V\in \ell^p(L^1)(\reals)$ for some $1 \leq p<2,$
the positive semi-axis $\reals^+=(0,\infty)$ is an essential support of the
absolutely continuous spectrum of the operator $-D^2+V$.
\end{theorem}
The last conclusion means simply that the absolutely continuous component
of the spectral measure of the operator
gives positive weight to any Borel subset
of $\reals^+$ of positive Lebesgue measure, and zero weight to
the negative semi-axis $\reals^-$.
\begin{theorem}
\label{thm:third}
Assume $V=V_1+V_2$ where $V_1 \in L^1(\reals)$,
$dV_2/dx\in \ell^p(L^1)$, and
$V_2$ tends to zero at infinity.
Then $\reals^+$ is an essential support of the
absolutely continuous spectrum of the operator $-D^2+V$.
If we only assume that $V_2$ is bounded, and
$A=\limsup_{|x|\to\infty} V_2(x),$ then $[A, \infty)$ is part of an
essential support of the absolutely continuous spectrum of
$-D^2+V(x).$
\end{theorem}
\it Remarks. \rm 1. A classical result of Weidmann \cite{Weid} says
that if $V' \in L^1,$ then the spectrum on $\reals^+$ is purely
absolutely continuous.
Our theorem shows that if $V' \in L^p$ with any $p<2,$
and $V \rightarrow 0$ as $x \rightarrow \infty,$ then the
absolutely continuous spectrum still fills $\reals^+$ (but may not be
pure anymore). \\
2. We did not distinguish in the above theorems between
half-line and whole line operators. In the case of whole line operators
we can say in addition that the multiplicity of the absolutely continuous
spectrum is two \cite{Sim}.
In the half-line case, the hypothesis of course only imposed
as $x\to+\infty$.
\\
3. Theorem~\ref{thm:third} is rather sharp in the following sense: \\
a. We can have an example of $V \in L^p,$ $V' \in L^r$ with any $p>2,$ $r>2$
and purely singular spectrum on $\reals^+.$
Existence of such examples follows in a direct manner from the results of
\cite{KLS}. \\
b. The potentials $V(x) = \cos x^{\alpha},$
$0<\alpha <1$ are known to give rise to Schr\"odinger operators with
absolutely continuous
spectrum in $(1, \infty),$
but with pure point spectrum in $(-1,1)$ \cite{Ben, LS, SZ, Sto1}. \\
The last theorem we prove in this paper concerns energy-dependent
potentials. This extension is not directly related to spectral
theory, but is a natural ODE application of our methods.
\begin{theorem}
\label{ODE}
Suppose that $p <2$ and that $W(x,k)$ is real-valued,
and that
\[ \partial^j W(x,k)/\partial k^j\in L^p(\reals) \]
uniformly in $k\in J$ for $j=0,1$. Suppose further that
the derivatives
$\partial^j W(x,k)/\partial k^j \to 0$ as $|x|\to\infty$,
uniformly in $k\in J$, for $j=2,3$.
Then for almost every $k \in J,$
there exist linearly independent, bounded solutions $u_\pm(x,k)$ of
\[ u''(x) = W(x,k) - k^2 u \]
with WKB asymptotic behavior
as $x\to+\infty$.
\end{theorem}
In the hypotheses of this theorem
we have made no effort to economize on derivatives
with respect to $k$.
Related results have been announced
by R.~Killip \cite{killip}, who establishes stability
of the absolutely continuous spectrum under perturbation
of certain classes of background potentials by functions $V$ whose
Fourier transforms are locally in $L^2$, and which satisfy mild
additional decay hypotheses.
\section{The Solution Series}
Let us rewrite the equation \eqref{eq1} as a system
\begin{equation}
\label{sys1}
y' = \left( \begin{array}{cc} 0 & 1 \\ V-k^2 &
0 \end{array} \right)y
\end{equation}
with $y=\begin{pmatrix}u \\ u'\end{pmatrix}$.
Apply a variation of parameters transformation
\[ y = \left( \begin{array}{cc} e^{ikx} & e^{-ikx}
\\ ik e^{ikx} & -ik e^{-ikx} \end{array}
\right) z, \]
to get
\[ z' = \frac{i}{2k}\left(
\begin{array}{cc} -V(x) & -V(x)e^{-2ikx}
\\ V(x)e^{2ikx} & V(x) \end{array} \right)z.
\]
Do one more diagonal transformation
\[ z= \left(
\begin{array}{cc} \exp\left(-\frac{i}{2k} \int_0^x V(t)\,dt \right) & 0 \\
0 & \exp\left(\frac{i}{2k} \int_0^x V(t) \,dt \right)
\end{array} \right)w. \]
Then $w$ satisfies
\begin{equation}
\label{eqsys}
w' = \frac{i}{2k}\left( \begin{array}{cc} 0 & -V(x)
e^{-2ikx+\frac{i}{k}\int_0^x V(t)\,dt} \\ V(x)
e^{2ikx-\frac{i}{k}\int_0^x V(t)\,dt} & 0\end{array} \right)w.
\end{equation}
Let us denote
\begin{equation}
\label{eq:op}
(Tf)(k) = \int_0^\infty e^{2ikx-\frac{i}{k}\int_0^x V(t)\,dt} f(x) \,dx,
\end{equation}
an operator defined initially
on integrable functions of compact support.
We also introduce multilinear operators
\begin{multline}
\label{eq:muop}
T_n (f_1, \dots,f_n)(x,k) \\
=
\left(\frac{i}{2k} \right)^n \int_x^\infty
\int_{t_{1}}^\infty
\cdots
\int_{t_{n-1}}^\infty
\prod_{j=1}^n
\Big[
e^{(-1)^{n-j}(2ikt_j-\frac{i}{k} \int_{0}^{t_j} V(t)\,dt)}
f_j(t_j) dt_j
\Big]\ .
\end{multline}
Iterating system \eqref{eqsys} starting from the vector
$(1,0)$ we obtain the following
formal series expansion for one of
the solutions of equation \eqref{eq1}:
\begin{equation}
\label{keyser}
\begin{split}
u_+(x,k) & = e^{ikx-\frac{i}{2k}\int_0^x V(t)\,dt}
\sum\limits_{n=0}^{\infty}
(-1)^{n} T_{2n}(V, \dots ,V)(x,k) \\
& \ +\
e^{-ikx+\frac{i}{2k}\int_0^x V(t)\,dt} \sum\limits_{n=1}^{\infty}
(-1)^{n} T_{2n-1}(V, \dots,V)(x,k)
\end{split}
\end{equation}
(we stipulate $T_0(V)(x,k)\equiv 1$ in the above formula).
A similar representation can be derived for $u_-(x,k).$
The existence of the individual terms of this expansion,
and convergence of the infinite series,
under the assumptions of Thereom~\ref{thm:main}, will
be proven in later sections. We remark that a similar series
can be derived, relating generalized eigenfunctions of
$-D^2+q+V$ to those of $-D^2+q$,
for various classes of background potentials $q$
(for instance, periodic ones).
We leave the consideration of that situation to a later paper.
Now we derive a different series representation suitable for the
proof of Theorem~\ref{thm:main1}.
Let $\xi(x,k) = \sqrt{k^2-V_2}$.
We can always assume that $k^2-V_2>0$ by
replacing $V_2$ with its restriction to the complement of a suitably
large interval, subsuming the remainder
into an $L^1$ potential $V_1$.
In a system \eqref{sys1} we do a different variation of
parameters-type transformation
\[ y = \left( \begin{array}{cc} e^{i\int_0^x \xi \,dt} & e^{-i\int_0^x \xi
\,dt}
\\ i\xi e^{i\int_0^x \xi \,dt} & -i\xi e^{-i\int_0^x \xi \,dt} \end{array}
\right) z, \]
to get
\[ z' = \frac{\xi'}{2\xi}\left(
\begin{array}{cc} -1 & e^{-2i\int_0^x \xi \,dt} \\ e^{2i\int_0^x \xi
\,dt} &
-1 \end{array} \right)z.
\]
Do one more transformation $z= \xi^{-1/2} w.$
Then $w$ satisfies
\begin{equation}
\label{sys2}
w' = \left( \begin{array}{cc} 0 & -\frac{V_2'(x)}{4 (k^2-V_2(x))}
e^{-2i\int_0^x \xi(t,k) \,dt} \\
-\frac{V_2'(x)}{4 (k^2-V_2(x))} e^{2i\int_0^x \xi(t,k) \,dt} & 0 \end{array}
\right)w.
\end{equation}
Define
\begin{equation}
\label{eq:op1}
(B f)(k) = \int_0^\infty
\tfrac14 (k^2-V_2(x))^{-1}
e^{2i\int_0^x \xi(t,k)
\,dt} f(x) \,dx \ .
\end{equation}
In a similar manner, we introduce multilinear operators
\begin{multline}
\label{eq:muop1}
B_n (f_1, \dots ,f_n)(x,k) \\
=
\int_x^\infty
\int_{t_1}^\infty \cdots
\int_{t_n}^\infty
\prod_{j=1}^n
\Big[
\tfrac14
[ k^2-V(t_j) ]^{-1}\,
e^{2(-1)^{n-j} i\int_{0}^{t_j} \xi(t,k)\,dt}
f_j(t_j)
dt_j
\Big]\ .
\end{multline}
Iterating the system \eqref{sys2} starting from the
vector $(1,0)$, we obtain a series representation for one of the
solutions of equation \eqref{eq1}
with exactly the same structure as in the previous case:
\begin{equation}
\label{keyser1}
\begin{split}
u_+(x,k) =
(k^2-V(x))^{-1/4}
e^{i\int_0^x \xi(t,k)\,dt}
\sum\limits_{n=0}^{\infty}
B_{2n}(V', \dots,V')(x,k) \\
+(k^2-V(x))^{-1/4}
e^{-i\int_0^x \xi(t,k)\,dt}
\sum\limits_{n=1}^{\infty}
B_{2n-1}(V', \dots, V')(x,k).
\end{split}
\end{equation}
We remark that each individual term in the series \eqref{keyser}
(respectively, \eqref{keyser1})
is well-defined for a.e.\ $k$ for any
$V$ (respectively, $V'$)
in $\ell^p(L^1)(\reals)$ according to the results
of \cite{christkiselev},
even though the region of integration is infinite.
We will provide another proof of this fact in Section~\ref{asym}.
Therefore the proof of our
main results reduces to the summation of these infinite series.
This we were unable to accomplish in earlier work.
The estimates for the multilinear transforms which we discuss in the
next section are the essential ingredient of the proof.
\section{The Multiplicative Estimate}
\label{section:estimates}
Here we outline the general
results regarding the estimation of a certain type of multilinear
transforms. Some of the proofs appear in a companion paper \cite{ck2}.
\begin{definition}
A martingale structure on a subinterval $I\subset\reals$ is a
collection of subintervals $\{E^m_j: m\ge 0, 1\le j\le 2^m\}$ of $I$
that satisfy the following conditions, modulo endpoints.
\begin{itemize}
\item
$I = \cup_j E^m_j$ for every $m$.
\item
$E^m_j\cap E^m_{j'}=\emptyset$ for every $j x_1^1 >n$ for some integer $n,$ we have
\begin{multline*}
\|f \|_{\ell^p(L^1)}^p-\|f \cdot \chi_{E^1_1}\|_{\ell^p(L^1)}^p
-\|f \cdot \chi_{E^1_2}\|_{\ell^p(L^1)}^p
\\
=
\left( \int_n^{n+1} |f(x)|\,dx \right)^p - \left( \int_n^{n+1}
|f(x)|\chi_{E^1_1} \,dx \right)^p
-\left( \int_n^{n+1} |f(x)|\chi_{E^1_2} \,dx \right)^p \geq 0,
\end{multline*}
since $(a+b)^p \geq a^p +b^p$ for any positive numbers $a,$ $b$ if $p \geq 1.$
Hence generally
\[ \|f \cdot \chi_{(-\infty, x_1^1]}\|_{\ell^p(L^1)}^p =
\|f \cdot \chi_{[x_1^1, \infty)}\|_{\ell^p(L^1)}^p
\leq 2^{-1}\|f \|_{\ell^p(L^1)}^p. \]
Given any locally integrable functions $h_1,\dots,h_n$, consider a
multilinear expression
\begin{equation}
\label{multi}
M_n(h_1,\dots,h_n)(x,x')
= \int \dots \int_{x \leq x_1 \leq x_2 \dots
\leq x_n \leq x'} \prod_{i=1}^n h_i(x_i) dx_i.
\end{equation}
In the special case
where each $h_i$ is equal either to $h$ or to its complex conjugate
$\overline{h}$, we write simply $M_n(h)$.
(In the latter case,
all estimates stated below are independent of the choices
of $h,\overline{h}$.)
We set
\[ M_n^*(h_1,\dots,h_n)= \sup_{x p$.
For any martingale
structure, any operator $S$ and any function $f$,
$S^*f$ is dominated pointwise by $G_{Sf}$ \cite{ck2}.
By choosing a martingale structure adapted to $f$ in $\ell^p(L^1)$
and applying Proposition~\ref{Gprop} to $G_{f}$,
we conclude that $S^*$ is bounded.
\end{proof}
Actually boundedness of $S$ implies boundedness of $S^*$ whenever
$q>p$, without further restriction on the exponents, as follows
quite directly from the argument in \cite{ck2}. We have proved
only this more restricted statement here
because the required assertion is not explicit in \cite{ck2}.
Finally, we observe that under our hypotheses on the potential
$V$, the operators $T$ and $B$
do actually satisfy $\ell^p(L^1)$ to $L^q$ norm bounds,
which make the above results on multilinear transforms applicable.
\begin{proposition}
\label{l2bound}
Let the operators $T$ and $B$ be defined by
\[ (Tf)(k) = \int_0^\infty e^{2ikx-\frac{i}{k}\int_0^x V(t)\,dt} f(x) \,dx \]
and
\[
(B f)(k)
= \int_0^\infty (4 ( k^2-V_2(x) ))^{-1}\, e^{2i\int_0^x \xi(t,k)
\,dt} f(x) \,dx. \]
Let $p^{-1}+q^{-1}=1,$ and $1 \leq p \leq 2.$
Then $T$, under the conditions of Theorem~\ref{thm:main}, and
$B$, under the conditions of Theorem~\ref{thm:main1},
satisfy
\begin{equation}
\label{TBbound}
\|Tf\|_{L^q(J)} \leq C_J \|f\|_{\ell^p(L^1)(\reals)} \ \text{ and }\
\|Bf\|_{L^q(J)} \leq C_J \|f\|_{\ell^p(L^1)(\reals)}
\end{equation}
for any compact interval $J\subset\reals$
disjoint from zero (or, in the case of
the operator $B$, disjoint from $[-\sqrt{A}, \sqrt{A}]$ with
$A = \limsup_{x \to\infty} V_2(x)$).
\end{proposition}
\begin{proof}[Proof of Proposition]
The operators $T$ and $B$ clearly map $L^1(\reals)$ to $L^\infty
(J),$ so it suffices to show the
$\ell^2(L^1)(\reals)\mapsto L^2(J)$ bounds.
Then \eqref{TBbound} will follow by complex interpolation.
The proof of the $\ell^2(L^1)(\reals)\mapsto L^2(J)$ bound for
$T$ was given in \cite{christkiselev}. Here we will demonstrate
(in a similar way) the bound for $B$. For simplicity, we
will prove the bound on any compact interval $J'\subset (0, \infty)$
for a related operator
\[ (B' f)(E) = \int_0^\infty \tfrac{1}{4} (E-V_2(x))^{-1}
e^{2i\int_0^x \xi(t,\sqrt{E}) \,dt} f(x) \,dx; \]
since $J$ is disjoint from $0,$ the bound for $B$ will follow.
Let $0 \leq \psi \leq 1$ be a
$C_0^\infty((0,\infty))$ function equal to
$1$ on $J.$ Then
\begin{eqnarray*}
\|B'f\|^2_{L^2(J)} & \leq &
\tfrac{1}{16}
\int_\reals \int_\reals f(x)\overline{f(y)}
\int
\frac{\psi(E)}{(E-V_2(x))(E - V_2(y))} e^{2i\int_y^x
\xi(t,\sqrt{E}) \,dt}
\,dE \,dxdy \\
& = & \int_\reals \int_\reals f(x)\overline{f(y)} M(x,y) \,dxdy.
\end{eqnarray*}
It suffices to prove boundedness of $B$ from $\ell^p(L^1)(R,\infty)$
for suitably large $R$, since $B$ maps $L^1$ to $L^\infty$,
and $\ell^p(L^1)(0,R]\subset L^1$.
Under the conditions of Theorem~\ref{thm:main1},
for all $t\ge R$ for sufficiently large $R$, we have
$d\xi(t,\sqrt{E})/dE
=\tfrac12[E-V_2(t)]^{-1/2}>c>0.$ Hence there exists $N$ such that
if $|x-y|>N,$ $|d\int_y^x \xi(t,E)\, dt/dE| > (c/2) |x-y|.$
A double integration by parts in the integral defining $M(x,y)$,
integrating
\[ e^{2i\int_y^x \xi(t,\sqrt{E}) \,dt}
\int_y^x (E-V_2(t))^{-1/2} \,dt \]
and differentiating the other factor, gives
\[ |M(x,y)| \leq C_{N,c} (1+|x-y|)^{-2}). \]
Hence the required bound for $B$ follows from the elementary bound
\[ \int \int |f(x)f(y)|(1+|x-y|)^{-2}\,dxdy \leq
C \|f\|_{\ell^2(L^1)(\reals)}^2\ . \]
\end{proof}
\section{The Asymptotic Behavior of Solutions}
\label{asym}
Proposition~\ref{l2bound} allows us to apply
the maximal estimate that we derived in the previous section
to the multilinear transforms $T_n$ and $B_n$,
and thus to prove an effective
estimate on the terms in the series \eqref{keyser}, \eqref{keyser1}.
However, it must be shown that the definitions of $T_n,B_n$
make sense for general functions in $\ell^p(L^1)$, not merely
those having compact support.
Let us denote $V_y = V \cdot \chi_{(-\infty,y]}.$
\begin{proposition}
\label{T_nbound}
Let $1\le p<2$.
For any $V \in \ell^p(L^1)(\reals)$,
the multilinear transforms
$T_n(V, \dots, V)(x,k)$ defined by \eqref{eq:muop} are well-defined as
the limits
\[ T_n(V, \dots, V)(x,k) = \lim_{y\to\infty}
T_n(V_y, \dots, V_y)(x,k), \]
which exist for almost every $k$.
Moreover,
\begin{equation}
\label{crucialest}
|T_n (V, \dots, V)(x,k)| \leq C^n {G}_{T(V-V_x)}(k)^n/\sqrt{n!}
\end{equation}
for some constant $C<\infty.$
Similarly, for any $V' \in \ell^p(L^1)(\reals)$,
the multilinear transforms
$B_n(V', \dots, V')(x,k)$ are well-defined in the same sense,
and satisfy
\begin{equation}
\label{V'}
| B_n (V', \dots, V')(x,k)| \leq C^n {G}_{B(V' - V'_x)}
(k)^n / \sqrt{n!} \ .
\end{equation}
\end{proposition}
\begin{proof}
It suffices to restrict consideration to an
arbitrary compact interval $J$ disjoint from $0$.
The cases of $T_n$ and $B_n$ are completely parallel,
so we discuss only the former.
Recall the maximal operator
$T^* f(k) = \sup_y |T(f\cdot \chi_{[y,\infty)})(k)|$,
the associated quadratic expression $G_{T^*(\cdot)}$,
and their mapping properties discussed in Lemma~\ref{Gstarlemma}.
We wish to show that
\[ \limsup_{y,z\to\infty} \sup_x|T_n(V_y,\dots,V_y)(x,k)
-T_n(V_z,\dots,V_z)(x,k)|=0\]
for almost every $k\in J$.
The difference is majorized by $\sum_{r=1}^n
|T_n(V_y,\dots,V_y,V_y -V_z,V_z,\dots,V_z)(x,k)|$,
where $V_y - V_z$ appears in the $r$-th position.
Fix any $1\le r\le n$, and assume that $y,z\ge M$.
Then by Proposition~\ref{maximal},
\begin{align*}
\sup_x |T_n(V_y,\dots,V_y,V_y -V_z,V_z,\dots,V_z)(x,k)|
& \le
C^n G_{T(V_y)}^{r-1}(k) G_{T(V_y-V_z)}(k)
G_{T(V_z)}^{n-r}(k)
\\
& \le C^n G_{T^*(V)}^{n-1}(k)
G_{T^*(V\cdot\chi_{[M,\infty)})}(k)\ .
\end{align*}
Here the martingale structure used to define each quantity $G$
is always a structure adapted to $V$, regardless of whether the
subscript is $T^*(V),T(V_y),T(V_z),T(V_y-V_z)$, or $
T^*(V\cdot\chi_{[M,\infty)})$.
To obtain the last line,
we have rewritten $V_y - V_z$ as $ -V\chi_{[y,\infty)}
+ V\chi_{[z,\infty)}$.
The factor $G_{T^*(V\cdot\chi_{[M,\infty)})}(k)$
tends to zero in $L^{q}(J)$, as $M\to\infty$,
by Lemma~\ref{Gstarlemma} and the second conclusion
of Proposition~\ref{Gprop}, while $G_{T^*(V)}\in L^q$.
By summing over $r$, we conclude that the $\limsup$ vanishes
in $L^{q/n}(J)$.
The bounds \eqref{crucialest} and
\eqref{V'} now follow immediately
from the maximal estimate \eqref{max}.
\end{proof}
We are now ready to complete the proofs of our main results.
\begin{proof}[Proof of Theorems~\ref{thm:main} and \ref{thm:main1}.]
It is sufficient to consider $k$ from an arbitrary compact
interval $J=[a,b] \subset (0, \infty).$
Let us recall the series representation \eqref{keyser}:
\begin{eqnarray*}
u_+(x,k)& = & e^{ikx-\frac{i}{2k}\int_0^x V(t)\,dt} +
e^{ikx-\frac{i}{2k}\int_0^x V(t)\,dt}
\sum\limits_{n=1}^{\infty}
(-1)^{n} T_{2n}(V, \dots ,V)(x,k) \\
& & \qquad+\ e^{-ikx+\frac{i}{2k}\int_0^x V(t)\,dt}
\sum\limits_{n=1}^{\infty} (-1)^{n} T_{2n-1}(V, \dots,V)(x,k).
\end{eqnarray*}
Its terms are well-defined by Proposition~\ref{T_nbound},
and it converges uniformly as a function of $x\in\reals$.
An application of Corollary~\ref{maximal1}
shows that this series converges uniformly in $\reals$
as a function of $x$ for almost every $k$,
and leads to the estimate
\begin{equation}
\label{uest}
{\sup}_{x}
|u_+(x,k)| \leq 1+
\sum_{n=1}^{\infty} \frac{B^n
{G}^n_{T(V)}(k)}{\sqrt{n!}}.
\end{equation}
\begin{lemma}
$u_+(x,k)$ satisfies
$(-D^2+V-k^2)u_+=0$, for almost every $k\in\reals$.
\end{lemma}
\begin{proof}
Since the series representation \eqref{keyser}
is obtained from a corresponding series representation for a
solution of the first-order system \eqref{eqsys},
it suffices to prove that the latter series defines
a solution of \eqref{eqsys}.
Write \eqref{eqsys} as $w'=\scriptv(x,k) w$,
where
\begin{equation*}
\scriptv = \frac{i}{2k}
\begin{pmatrix}
0 & -V(x)e^{-2ikx+\frac{i}{k}\int_0^x V}
\\
V(x)e^{2ikx-\frac{i}{k}\int_0^x V} & 0
\end{pmatrix}\ .
\end{equation*}
A formal solution, obtained from the Ansatz $w_0(x) \equiv$
transpose of $(1,\,0)$, is
\begin{equation*}
w(x,k) = \sum_{n=0}^\infty (-1)^n
\scriptt^n w_0(x,k),
\end{equation*}
where $\scriptt w(x,k) = \int_x^\infty \scriptv(t,k)w(t,k)\,dt$
and the formal $n$-fold iterate $\scriptt^n$ of $\scriptt$
is defined rigorously not as an iterate,
but rather by applying Proposition~\ref{T_nbound}
to the $n$-fold multiple integral in the formal expression
for the iterate.
Again, the series converges in $L^\infty(\reals)$ for almost
every $k$, by Proposition~\ref{T_nbound} and Corollary~\ref{maximal1}.
Set $w_N(x,k)$ equal to the partial sum, from $n=0$ to $n=N$.
Then for almost every $k$,
$w_N' - \scriptv w_N = \pm \scriptv\scriptt^N w_0$;
this is obvious for all $k$ for compactly supported potentials, and
follows for general $V\in\ell^p(L^1)$ via approximation,
justified by Proposition~\ref{T_nbound}.
For almost every $k$, the right-hand side of this last equation
converges, as $N\to\infty$, to zero in $\ell^p(L^1)(\reals)$,
while the left-hand side converges to $w'(x,k)-\scriptv w(x,k)$
in the sense of distributions in $x$.
Therefore $w(x,k)$ satisfies the equation
in the sense of distributions, for almost every $k$.
\end{proof}
Notice that
\begin{equation}
\label{aux1}
\frac{B^{2n+1}G^{2n+1}}{\sqrt{(2n+1)!}}
\leq
\sqrt{\frac{B^{2n}G^{2n}}{\sqrt{(2n)!}}
\frac{B^{2n+2}G^{2n+2}}{\sqrt{(2n+2)!}}} \leq
\frac{1}{2} \left( \frac{B^{2n}G^{2n}}{\sqrt{(2n)!}}+
\frac{B^{2n+2}G^{2n+2}}{\sqrt{(2n+2)!}} \right),
\end{equation}
and hence we can simplify the estimate \eqref{uest} to
\[ {\rm sup}_{x} |u_+(x,k)| \leq
2\sum_{n=0}^{\infty} \frac{B^{2n}
{G}^{2n}_{T(V)}(k)}{n!} =
2e^{B^2 {G}^2_{T(V)}(k)}. \]
A similar estimate holds for $u_-.$
Therefore,
\begin{equation}
\label{substitutelogbound}
\int_a^b \log(1+{\rm sup}_{x}|u_\pm (x,k)|)\,dk \leq C
\left(1+ B^2 \int_a^b {G}^2(k) \, dk \right) < \infty,
\end{equation}
since
\[ \|G\|_{L^2([a,b])} \leq C\|G\|_{L^q([a,b])} \leq C_1 \|V\|_
{\ell^p(L^1)(\reals)}. \]
The solution $u_-$ satisfies the same bound by complex conjugation,
and moreover, $\partial_x u_\pm$ also satisfy the same
bounds, since our estimates apply to the solutions of the $2\times 2$
system introduced above.
We have proved a variant of \eqref{logbound}, with
$\tilde u_\pm$ replaced by $u_\pm$.
To estimate $\tilde u_+$,
we now exploit
the asymptotic WKB behavior \eqref{eq:sol0}
of $u_\pm$, which will be established in the next paragraph.
Define coefficients $c_\pm(k)$, independent of $x$, by
\[
\begin{pmatrix}\tilde u_+(x,k) \\ \partial_x\tilde u_+(x,k)
\end{pmatrix}
=
\begin{pmatrix}
u_+(x,k) & u_-(x,k)
\\
\partial_x u_+(x,k) & \partial_x u_-(x,k)
\end{pmatrix}
\cdot
\begin{pmatrix}
c_+ \\c_-
\end{pmatrix} \ .
\]
Now
$\tilde u_+(0,k)=1$, $\partial_x \tilde u_+(0,k)=ik$, and we may
solve for $c_\pm$ by evaluating the matrix equation at $x=0$.
Inverting the $2\times 2$ matrix on the right, we obtain
a cofactor matrix
whose entries satisfy the desired bound, times the reciprocal
of the Wronskian $W(x,k)\equiv W(k)$ of the solutions $u_\pm$.
$W(k)$ may be evaluated
by using the WKB asymptotics \eqref{eq:sol0} in the following
form:
\begin{equation} \label{vectorWKB}
\begin{pmatrix}
u_+(x,k) \\ \partial_x u_+(x,k)
\end{pmatrix}
=
\begin{pmatrix}
e^{ikx-\tfrac{i}{2k}\int_0^x V} \\
ik e^{ikx-\tfrac{i}{2k}\int_0^x V}
\end{pmatrix}
+ \varepsilon(x,k)
\end{equation}
where the vector $\varepsilon(x,k)\to 0$ as $x\to+\infty$,
for almost every $k$.
Granting these asymptotics, it follows that for a.e.\ $k$,
$W(x,k)\to 2ik$ as $x\to +\infty$; but $W$ is independent of $x$,
hence is identically equal to $2ik.$
Thus $\tilde u_+$ is a finite linear combination, with
coefficients locally bounded in $k$,
of products of two factors, each of which satisfies the
desired bound. The logarithm in \eqref{substitutelogbound}
converts these products to sums, so $\tilde u_+$ also
satisfies \eqref{substitutelogbound}.
To analyze asymptotic behavior, we notice that by \eqref{keyser},
\eqref{aux1}, and Proposition~\ref{T_nbound},
\begin{multline*}
{\sup}_{x \leq y}
|u_+(y,k) - e^{iky-\frac{i}{2k}\int_0^y V(t)\,dt}| \leq
\left(B{G}_{T(V - V_x)}(k)+
2\sum_{n=1}^{\infty} \frac{B^{2n}
{G}^{2n}_{T(V - V_x)}(k)}{n!}\right)
\\
\le C \left(e^{B^2 {G}^2_{T(V - V_x)}(k)+
B {G}_{T(V - V_x)}(k)}-1 \right).
\end{multline*}
>From this estimate we infer that for a.e.\ $k \in \reals$
\[ u_+(x,k) - e^{ikx-\frac{i}{2k}\int_0^x V(t)\,dt}
\rightarrow 0 \]
as $x \to +\infty.$
The solution $u_-(x,k)$ with WKB asymptotic behavior is
simply the complex conjugate of $u_+$.
To obtain the vector asymptotics \eqref{vectorWKB}, it
suffices to observe that with the notation of Section~2,
\begin{equation*}
\begin{pmatrix}
u \\ \partial_x u
\end{pmatrix}
= y =
\begin{pmatrix}
e^{ikx} & e^{-ikx}
\\
ike^{ikx} & -ike^{-ikx}
\end{pmatrix}
\begin{pmatrix}
\exp(\tfrac{-i}{2k}\int_0^x V) & 0
\\
0 & \exp(\tfrac{i}{2k}\int_0^x V)
\end{pmatrix}
w,
\end{equation*}
where the above analysis proves that
$w(x,k)$ equals the transpose of $(1,\,0)$ plus
a vector-valued function that tends to zero as $x\to+\infty$,
for almost every $k$.
The proof of Theorem~\ref{thm:main1} follows along the same lines,
when $V_1\equiv 0$.
The addition of the $L^1$ potential $V_1$ constitutes a further
perturbation, which again preserves the WKB asymptotic
behavior, by a classical theorem of Levinson.
This can also be shown
by elementary estimates employing a variation of parameters
transformation and integral equation for the solution.
\end{proof}
\section{A more general ODE application}
It remains to prove Theorem~\ref{ODE}.
Observe first that the operator
\[
T f(k) = \int_0^\infty e^{i(2kx-k^{-1}\int_0^x W(t,k)\,dt)}
f(x)\,dx
\]
maps $L^p(\reals)$ boundedly to $L^q(J)$, where
$q = p/(p-1)$, for all $1\le p\le 2$.
Indeed, the proof of Proposition~\ref{l2bound} still applies;
to obtain the bound $|M(x,y)|\le C|x-y|^{-2}$ we require
control of three derivatives of $W$ with respect to $k$.
Let
\[
\Omega =
\sum_{r=0}^1
\iint_{\reals\times J}
|\partial_\rho^r W(x,\rho)|^p\,dx\,d\rho\ .
\]
Choose a martingale structure $\{E_j^m\}$ so that
for all $m,j$,
\begin{equation}
\label{martin}
\iint_{E^m_j\times J}
\sum_{r=0}^1
|\partial_\rho^r W(x,\rho)|^p\,dx\,d\rho
= 2^{-m} \Omega.
\end{equation}
The pointwise in $k$ estimate \eqref{max} still applies, and gives
\[
{\rm sup}_{y}
|T_n(W(\cdot,k), \dots, W(\cdot,k))(y,k)|
\leq C^n {G}_{T(W(\cdot ,k))}(k)^n/ \sqrt{n!}\ ,
\]
where
\begin{equation}
\label{newG}
{G}_{T(W(\cdot,k))}(k) = \sum\limits_{m=1}^\infty m
\left( \sum\limits_{j=1}^{2^m}
| T(W(x,k) \cdot \chi_j^m)(k)|^2 \right)^{1/2}.
\end{equation}
Here $T(W(x,k)\cdot\chi^m_j)(\rho)$
denotes the function obtained by letting $T$ act on the
function $x\mapsto W(x,k)\chi^m_j$, restricted to $\rho=k$.
All proofs from the previous sections would extend to this case if we could
apply Proposition~\ref{Gprop} to estimate ${G}$ in terms of $W$.
However, the bound \eqref{TBbound} is not applicable because of the
$k$--dependence of $W$.
Instead, we will bound ${G}_{T(W(\cdot,k))}(k)$
directly. The proof of Theorem~\ref{ODE} reduces to the following
result.
\begin{proposition}
Let $1\le p<2$. Assume that
$W(x,k)$ satisfies the hypotheses of Theorem~{\rm \ref{ODE}},
and let $E_j^m$ be a martingale structure satisfying
\eqref{martin}.
Then ${G}_{T(W(\cdot,k))}(k)$,
defined by \eqref{newG}, satisfies
\begin{equation}
\label{Gbound2}
\|{G}_{T(W(\cdot,k))}(k)\|_{L^q(J)} \leq
C\Omega^{1/p},
\end{equation}
where $q^{-1}+p^{-1}=1.$
\end{proposition}
\begin{proof}
Clearly,
\[ {G}_{T(W(\cdot,k))}(k) \leq \sup_{\rho \in J}
{G}_{T(W(\cdot,\rho))}(k), \]
so it suffices to bound the latter expression,
\[ \sup_{\rho \in J}
{G}_{T(W(\cdot,\rho))}(k)=
\sup_{\rho \in J} \sum\limits_{m=1}^\infty m
\left( \sum\limits_{j=1}^{2^m} | T(W(x,\rho)
\cdot \chi_j^m)(k)|^2 \right)^{1/2}. \]
Notice that
\begin{align*}
\int_J \sup_{\rho \in J} &
\left( \sum\limits_{j=1}^{2^m} | T(W(x, \rho) \cdot \chi_j^m)(k)|^2
\right)^{q/2}\, dk
\\
& \leq
C 2^{m(\frac{q}{2}-1)}
\sum_j
\int_J
\sup_{\rho \in J}
| T(W(x,\rho) \cdot \chi_j^m(x))(k)|^q \, dk
\\
& \leq
C 2^{m(\frac{q}{2}-1)}
\sum_j
\int_J
(\int_J \sum_{r=0}^1
|\partial_\rho^r T(W(x,\rho)\chi^m_j(x))(k)|^p
\,d\rho
)^{q/p}\,dk, \\
\intertext{by
the Sobolev embedding theorem applied in $L^p(J,d\rho)$.
Since $\partial_\rho$ commutes with $T$, we may continue
by writing}
& =
C 2^{m(\frac{q}{2}-1)}
\sum_j
\int_J
(\int_J \sum_{r=0}^1
|T(\partial_\rho^r W(x,\rho)\chi^m_j(x))(k)|^p
\,d\rho
)^{q/p}\,dk
\\
& \le
C 2^{m(\frac{q}{2}-1)}
\sum_{r=0}^1
\sum_j
(\int_J
(\int_J
|T(\partial_\rho^r W(x,\rho)\chi^m_j(x))(k)|^q
\,dk
)^{p/q}
\,d\rho)^{q/p}
\\
& \le
C 2^{m(\frac{q}{2}-1)}
\sum_{r=0}^1
\sum_j
(\int_J \int_\reals
|\partial_\rho^r W(x,\rho)\chi^m_j(x)|^p\,dx\,d\rho
)^{q/p}
\\
& \le
C 2^{m(\frac{q}{2}-1)}
\sum_{r=0}^1
\sum_j
2^{-m(\frac{q-p}{p})}\Omega^{\frac{q-p}{p}}
\iint_{J\times\reals}
|\partial_\rho^r W(x,\rho)\chi^m_j(x)|^p
\,dx\,d\rho
\\
& \le C 2^{mq(\frac{1}{2}-\frac{1}{p})}\Omega^{q/p}.
\end{align*}
We have invoked first Minkowski's integral inequality,
then the $L^p\mapsto L^q$ bound for $T$.
Finally, the triangle inequality implies that
\[ \|{G}_{T(W(\cdot,k))}(k)\|_{L^q(J)} \leq C\Omega^{1/p}
\sum\limits_{m=1}^\infty m 2^{(1/2- 1/p)m} \leq C\Omega^{1/p}. \]
\end{proof}
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\end{document}