Content-Type: multipart/mixed; boundary="-------------0110161419415" This is a multi-part message in MIME format. ---------------0110161419415 Content-Type: text/plain; name="01-379.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-379.keywords" Schr\"odinger operators, scattering theory, slowly decaying potentials ---------------0110161419415 Content-Type: application/x-tex; name="scat6.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="scat6.tex" % modified 8/21/2001 by AK. Tiny changes made after that by MC. % this is file for entire paper on scattering and wave operators \documentclass[12pt]{amsart} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \usepackage{amssymb} %\usepackage{times} \usepackage{mathtime} \begin{document} \numberwithin{equation}{section} % define theorem environments %%% \newtheorem{theorem}{Theorem}[section] %\def\thetheorem{\unskip} \newtheorem{proposition}[theorem]{Proposition} %\def\theproposition{\unskip} \newtheorem{conjecture}[theorem]{Conjecture} \def\theconjecture{\unskip} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{observation}[theorem]{Observation} %\def\thelemma{\unskip} \theoremstyle{definition} \newtheorem{definition}{Definition} \def\thedefinition{\unskip} \newtheorem{remark}{Remarks} \def\theremark{\unskip} \newtheorem{question}{Question} \def\thequestion{\unskip} \newtheorem{example}{Example} \def\theexample{\unskip} \newtheorem{problem}{Problem} \def\intprod{\mathbin{\lr54}} \def\reals{{\mathbb R}} \def\integers{{\mathbb Z}} \def\complex{{\mathbb C}\/} \def\distance{\operatorname{distance}\,} \def\ZZ{ {\mathbb Z} } \def\e{\varepsilon} \def\p{\partial} \def\rp{{ ^{-1} }} \def\Re{\operatorname{Re\,} } \def\Im{\operatorname{Im\,} } \def\intprod{\mathbin{\lr54}} \def\reals{{\mathbb R}} \def\integers{{\mathbb Z}} \def\complex{{\mathbb C}\/} \def\distance{\operatorname{distance}\,} \def\ZZ{ {\mathbb Z} } \def\e{\varepsilon} \def\p{\partial} \def\rp{{ ^{-1} }} \def\Re{\operatorname{Re\,} } \def\Im{\operatorname{Im\,} } \def\scriptx{{\mathcal X}} \def\scripti{{\mathcal I}} \def\scripth{{\mathcal H}} \def\scriptm{{\mathcal M}} \def\scripte{{\mathcal E}} \def\scriptt{{\mathcal T}} \def\scriptb{{\mathcal B}} \def\frakg{{\mathfrak g}} \def\frakG{{\mathfrak G}} \def\ov{\overline} \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 by NSF grant DMS-9970660, and by the Miller Institute for Basic Research in Science, while he was on appointment as a Miller Research Professor} \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 by NSF grant DMS-0102554 and by an Alfred P. Sloan Research Fellowship} \title[Scattering and wave operators] {Scattering and wave operators for one-dimensional Schr\"odinger operators with slowly decaying nonsmooth potentials} \begin{abstract} We prove existence of modified wave operators for one-dimensional Schr\"odinger equations with potential in $L^p(\reals)$, $p<2$. If in addition the potential is conditionally integrable, then the usual M\"oller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential. \end{abstract} \date{\today} \maketitle % \begin{center} \fbox{\bf Draft August 15, 2001} \end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $H_V$ be a one-dimensional Schr\"odinger operator defined by \begin{equation}\label{start} H_V = -\frac{d^2}{dx^2} + V(x). \end{equation} Let us discuss the case where the operator is defined on a half-axis, with some self-adjoint boundary condition at zero. We are interested in potentials decaying at infinity, for which we may expect that asymptotically as time tends to infinity, motion of the associated perturbed quantum system resembles the free evolution. What is the critical rate of decay of the potential for which the dynamics remains close to free for large times? The mathematical framework for studying these questions is provided by scattering theory. Recall that the wave operators associated with $H_V$ and $H_0$ are defined by \begin{equation}\label{wo} \Omega_\pm f = \lim_{t \rightarrow \mp \infty} e^{itH_V} e^{-itH_0} f, \end{equation} where the limit is understood in the strong $L^2$ sense. In particular, existence of the wave operators implies that the absolutely continuous spectrum of $H_V$ fills all of $\reals^+$ (see, e.g. \cite{RS3}) and moreover provides significant information on large time dynamics $e^{-itH_V}.$ The wave operators will be called asymptotically complete if the range of $\Omega_\pm$ coincides with the orthogonal complement of the subspace spanned by eigenvectors of the operator $H_V$. An alternative equivalent characterization is that the range of the wave operators is equal to the absolutely continuous subspace $\scripth_{\text{ac}}(H_V)$ of the operator $H_V$, and that the singular continuous spectrum $\sigma_{{\rm sc}}(H_V)$ is empty. %\cite{RS3}. We note that the intended intuitive meaning of asymptotic completeness is that the dynamics of the perturbed operator can be divided into two well-understood parts: scattering states traveling to infinity in a way similar to the free evolution, and bound states which remain confined in a certain sense for all times. %From this perspective, the definition given above is %somewhat outdated in the view %of recent research showing that point spectrum can in some cases lead %to very nontrivial dynamics. %However, it is still useful; We postpone more detailed discussion of the notion of asymptotic completeness to Section \ref{section:completeness}. A well-known result, going back to the very first years of the rigorous scattering theory, says that if $V \in L^1,$ then the wave operators exist and are asymptotically complete. Moreover, in this case the spectrum on the positive semi-axis is purely absolutely continuous, and there can be only discrete spectrum below zero, possibly accumulating at zero. There has been much work extending the existence of wave operators to wider classes of potentials, with some additional conditions on derivatives, or for potentials of certain particular forms. Generally, for more slowly decaying potentials one needs to modify the free dynamics in the definition of wave operators in order for the limits to exist. The first work of this type was that of Dollard \cite{Dol}, who studied the Coulomb potential. Further developments used computation of asymptotic classical trajectories by means of a Hamilton-Jacobi equation to build an appropriate phase correction to the free dynamics, which was used to prove existence of modified wave operators (in any dimension). See, for instance, Buslaev and Matveev \cite{BM}, Ahsholm and Kato \cite{AK}; the strongest results are contained in H\"ormander \cite{hormanderwaveoperators}. For example, H\"ormander \cite{hormanderwaveoperators} gives existence of wave operators if $|V(x)| \leq C(1+|x|)^{-1/2-\epsilon}$ and $|D^{\alpha}V(x)| \leq C(1+|x|)^{-3/2-\epsilon}$ for any $|\alpha|=1.$ One needs more conditions on derivatives to compensate for slower decay. Another series of works (see \cite{BA,BAD,D1,W1} for further references) studied oscillating potentials of Wigner-von Neumann type, for example $V(x) = \sin (cx^{\alpha})/x^{\beta}.$ Interest in this class of potentials was in part fueled by the Wigner-von Neumann example \cite{WvN}, where $V(x) \sim c\sin (2x)/x$ at infinity, which leads to a positive eigenvalue embedded in the absolutely continuous spectrum. %A reasonable question %is whether such long-range potential can have strong effect on the % absolutely continuous spectrum too. For a wide class of potentials of this type (including the original example of \cite{WvN}), existence and asymptotic completeness of (sometimes modified) wave operators has been shown. In the opposite direction, Pearson \cite{Pe} constructed examples of potentials in $\cap_{p>2} L^p(\reals)$ for which the spectrum is purely singular, and hence wave operators do not exist. Kotani and Ushiroya \cite{KU} also provided power decaying examples where the spectrum is purely singular (pure point) for the rate of decay $x^{-\alpha},$ $\alpha<1/2.$ In the last few years, there has been a series of works %\cite{Ki1,CK1,Re1,DK,CK2,K,CK3,CK4} studying existence of absolutely continuous spectrum for slowly decaying potentials in full generality, with no additional conditions on derivatives or specific representation. In \cite{christkiselevpowerdecay,Re1} it was shown that the absolutely continuous spectrum is preserved if $|V(x)| \leq C(1+x)^{-\alpha},$ $\alpha > 1/2,$ and moreover the generalized eigenfunctions have WKB-type plane wave asymptotics. Recently, we improved the result of \cite{christkiselevpowerdecay} to treat potentials in $L^p,$ $p<2$ \cite{christkiselevdecaying}. The sharpest result on the stability of the absolutely continuous spectrum is due to Deift and Killip \cite{DK}, who showed it for $V \in L^2$. This result is optimal in the $L^p$ scale, as Pearson's examples show. The method of \cite{DK} is quite different from \cite{christkiselevpowerdecay,Re1} and yields much less information concerning the nature of the generalized eigenfunctions. Although the absolute continuity of the spectrum is an important characterization of an operator with direct consequences for the dynamical behavior of the particle, wave operators provide a much finer description of long-time dynamics. The purpose of this paper is to use the additional information contained in the asymptotic behavior of generalized eigenfunctions \cite{christkiselevdecaying} to prove the existence of (modified) wave operators for potentials $V \in L^p,$ $p<2.$ Let \begin{equation}\label{phasec} W(\lambda,t) = -(2\lambda)\rp\int_0^{2\lambda t} V(s)\,ds. \end{equation} Define $e^{-iH_0t\pm iW(H_0,\mp t)}$ to be the Fourier multiplier operator on $L^2(\reals^+)$ that maps $\int_0^\infty F(\lambda)\sin(\lambda x)\,d\lambda$ to $\int_0^\infty e^{-i\lambda^2t \pm iW(\lambda^2,\mp t)} F(\lambda)\sin(\lambda x)\,d\lambda$, for all $F\in L^2(\reals^+,d\lambda)$. Define \begin{equation} \label{modifiedwaveoperatorsdefn} \Omega^m_\pm f = \lim_{t\to\mp\infty} e^{itH_V}e^{-it H_0 \pm iW(H_0,\mp t)}f \end{equation} for all $f\in L^2(\reals^+)$. Among our main conclusions are the following two theorems. \begin{theorem} \label{thm:waveoperators} Let $V$ be a potential in the class $L^1+L^p(\reals^+)$ for some $1x}f(y)dy|$. Such a thing cannot be bounded by %a square function $g$ with a favorable weight $2^{-\delta m}$, %and there must be a problem at the level of our applications %to operators, since almost everywhere convergence doesn't hold %for general orthogonal series, and so forth. \begin{proposition} \label{prop:numericalbound} There exists $C<\infty$ such that for any martingale structure $\{E^m_j\}$, any $\delta\ge 0$, any $f_1,\dots,f_n\in L^1(\reals)$, and any $n\ge 2$, \begin{equation} |M_n(f_1,\dots,f_n)| \le \frac{C^{n+1}}{\sqrt{n!}} g_{-\delta}(f_1)\cdot g_\delta(\{f_k:k\ge 2\})^{n-1}. \end{equation} Moreover for any $\delta'>\delta \ge 0$, there exists $C<\infty$ such that for all $\{f_i\}$ and all $n\ge 2$, \begin{equation} |M_n^*(f_1,\dots,f_n)| \le \frac{C^{n+1}}{\sqrt{n!}} g_{-\delta}(f_1)\cdot g_{\delta'}(\{f_k:k\ge 2\})^{n-1}. \end{equation} \end{proposition} In previous work \cite{christkiselevfiltrations} we proved the simpler analogue with $\delta=0$ and with the bound (for $M_n$) $C^{n+1}(n!)^{-1/2}g(\{f_k: k\ge 1\})^n$, and applied it to the analysis of generalized eigenfunctions, which can be expanded as sums over $n$ of such iterated multiple integrals, where $f_k$ is essentially $\exp(\pm 2i\lambda x)V(x)$, $V$ is the potential, and $\lambda^2$ is a spectral parameter; $g(f)$ is then a function of $\lambda$. In the present work, we need a refinement in which $f_1$ is essentially $\exp(\pm i\lambda x)h(x)$, and $h$ is an arbitrary $L^2$ function, unrelated to $V$. The quantity $g(h)$ is not appropriately bounded for $h\in L^2$, forcing the introduction of the mollifying factors $2^{-\delta m}$ in its definition. This in turn forces compensating factors of $2^{+\delta m}$, leading to the above formulation. \begin{proof} It is proved in \cite{christkiselevfiltrations} that there exist positive constants $b_n$ satisfying $b_n\le C^{n+1}/\sqrt{n!}$ and $n^{1/2}b_{n+1}/b_n\to C$ as $n\to\infty$, such that for all nonnegative real numbers $x,y$, \begin{equation} \label{eq:modifiedbinomial} b_n y^n + \sum_{i=2}^{n-2} b_i b_{n-i}x^iy^{n-i} + b_n x^n \le b_n (x^2+y^2)^{n/2}. \end{equation} It is also shown in \cite{christkiselevfiltrations} that \begin{equation} \label{eq:onefunctionversion} \begin{split} |M_n(f,\dots,f)| &\le b_n g_0(f)^n \\ M_n^*(f,\dots,f) &\le C_\e^n b_n g_\e(f)^n \end{split} \end{equation} for every $\e>0$. Moreover, for distinct functions $f_i$, \begin{equation} \label{eq:modifiedonefunctionversion} \begin{split} |M_n(f_1,\dots,f_n)| &\le b_n g_0(\{f_i\})^n \\ M_n^*(f_1,\dots,f_n)|&\le C_\e^n b_n g_\e(\{f_i\})^n. \end{split} \end{equation} Although this bound is not explicitly formulated in \cite{christkiselevfiltrations}, it follows directly from exactly the argument given there. For $n\ge 2$ define $\tilde b_n = R^{n} b_{n-2}$, where $R$ is a sufficiently large positive constant, to be determined later in the proof. In order to simplify notation, we will prove the result in the special case $f_2=f_3=\cdots =f_n=f$, and will write $f_1=\tilde f$. The proof will be by induction on $n$. First we will treat only the case where $n\ge 6$, assuming the result for all $n\le 5$, and at the end will discuss the modification for small $n$. We write $f^m_j = \chi_{E^m_j}\cdot f$ and $\tilde f^m_j = \chi_{E^m_j}\cdot \tilde f$, where $\chi_{E}$ denotes always the characteristic function of a set $E$. \begin{lemma} If $R$ is chosen to be sufficiently large then for all $n\ge 2$ and any $\delta\ge 0$, \begin{equation} |M_n(\tilde f,f,\dots,f)|\le \tilde b_n g_{-\delta}(\tilde f) \cdot g_\delta(f)^{n-1}. \end{equation} \end{lemma} \begin{proof} By inequality (4.6) of \cite{christkiselevfiltrations}, \begin{equation}\begin{split} \label{eq:inductivestep} |M_n(\tilde f,f,\dots,f)| \le &|M_n(\tilde f^1_1,f^1_1,\dots,f^1_1)| \\ & + |M_{n-1}(\tilde f^1_1,f^1_1,\dots,f^1_1)|\cdot |{\textstyle\int} f^1_2| \\ & + \sum_{i=2}^{n-2} |M_{n-i}(\tilde f^1_1,f^1_1,\dots,f^1_1)| \cdot |M_i(f^1_2,\dots,f^1_2)| \\ & + |{\textstyle\int}\tilde f^1_1|\cdot M_{n-1}(f^1_2,\dots,f^1_2)| + |M_n(\tilde f^1_2,f^1_2,\dots,f^1_2)| \\ \le & |M_n(2^{-\delta}\tilde f^1_1,2^\delta f^1_1,\dots,2^\delta f^1_1)| \\ & + |M_{n-1}(2^{-\delta} \tilde f^1_1, 2^\delta f^1_1,\dots,2^\delta f^1_1)| \cdot |{\textstyle\int} 2^\delta f^1_2| \\ & + \sum_{i=2}^{n-2} |M_{n-i}( 2^{-\delta}\tilde f^1_1,2^\delta f^1_1,\dots, 2^\delta f^1_1)| \cdot |M_i(2^\delta f^1_2,\dots,2^\delta f^1_2)| \\ & + |{\textstyle\int} 2^{-\delta}\tilde f^1_1| \cdot |M_{n-1}(2^\delta f^1_2,\dots,2^\delta f^1_2)| \\ & + |M_n(2^{-\delta}\tilde f^1_2,2^\delta f^1_2,\dots,2^\delta f^1_2)| . \end{split}\end{equation} We have assumed that $n\ge 2$ and $\delta\ge 0$ to ensure that at least one factor of $2^{\delta}$ offsets the factor of $2^{-\delta}$. The first and last terms in the preceding bound involve $M_n$ itself, but the former involves only the restrictions of $\tilde f,f$ to $E^1_1$, while the latter involves only their restrictions to $E^1_2$; thus these expressions are in a sense simpler than the original expression $M_n$. We will therefore use as part of our induction hypothesis the desired inequalities for $M_n(\tilde f^1_1,f^1_1,\cdots,f^1_1)$ and for $M_n(\tilde f^1_2,f^1_2,\cdots,f^1_2)$. For the justification of this method of reasoning see the two paragraphs immediately following inequality (4.12) of \cite{christkiselevfiltrations}. The collection of all those sets $E^m_j$ with $m\ge 1$ and $j\le 2^{m-1}$ forms a martingale structure on $E^1_1$; however, when it is considered as such, the index $m$ should be replaced by $m-1$. Thus the induction hypothesis, for the first term on the right-hand side of the preceding bound, may be stated as \begin{multline} |M_n(2^{-\delta} \tilde f^1_1,2^{\delta} f^1_1,\cdots, 2^\delta f^1_1)| \\ \le \tilde b_n \sum_{m=2}^\infty 2^{-m\delta} (\ \sum_{j=1}^{2^{m-1}} |{\textstyle\int} \tilde f^m_j|^2 )^{1/2} \cdot \Big[ \sum_{m=2}^\infty 2^{m\delta} (\ \sum_{j=1}^{2^{m-1}} |{\textstyle\int} f^m_j|^2 )^{1/2}\Big]^{n-1}\ . \end{multline} There is a corresponding bound for $|M_n(2^{-\delta}\tilde f^1_2,2^\delta f^1_2,\cdots,2^\delta f^1_2)|$, with the inner sum running instead over $2^{m-1}From the induction hypothesis and \eqref{eq:modifiedonefunctionversion} we obtain \begin{equation} \begin{split} |M_n(\tilde f,f,\dots,f)| \le & \tilde b_n \tilde g^1_1(g^1_1)^{n-1} + \tilde b_{n-1}\tilde g^1_1(g^1_1)^{n-2} |{\textstyle\int} 2^\delta f^1_2| \\ & + \sum_{i=2}^{n-2} \tilde b_{n-i}b_i \tilde g^1_1(g^1_1)^{n-1-i} (g^1_2)^i + |{\textstyle\int} 2^{-\delta}\tilde f^1_1| \cdot b_{n-1}(g^1_2)^{n-1} + \tilde b_n \tilde g^1_2(g^1_2)^{n-1}. \end{split}\end{equation} Consider now the sum of the first term on the right, together with all terms of the summation for which the index $i$ is either in $[2,n-4]$, or equals $n-2$: \begin{align*} \tilde b_n \tilde g^1_1(g^1_1)^{n-1} & + \sum_{i=2}^{n-4} \tilde b_{n-i}b_i \tilde g^1_1(g^1_1)^{n-1-i} (g^1_2)^i + \tilde b_2 b_{n-2}\tilde g^1_1 g^1_1(g^1_2)^{n-2} \\ & = R^n\tilde g^1_1 g^1_1 \Big( b_{n-2}(g^1_1)^{n-2} + \sum_{i=2}^{n-4}R^{-i} b_{n-i-2}b_i (g^1_1)^{n-2-i}(g^1_2)^i + R^{-n+2}b_0 b_{n-2}(g^1_2)^{n-2} \Big) \\ & \le R^n\tilde g^1_1 g^1_1 \Big( b_{n-2}(g^1_1)^{n-2} + \sum_{i=2}^{n-4}b_{n-i-2}b_i (g^1_1)^{n-2-i}(g^1_2)^i + b_0 b_{n-2}(g^1_2)^{n-2} \Big) \\ & \le R^n\tilde g^1_1 g^1_1 b_{n-2} \big( (g^1_1)^{2(n-2)}+(g^1_2)^{2(n-2)} \big)^{1/2} \\ & = \tilde b_n \tilde g^1_1 g^1_1 (g^1)^{n-2} . \end{align*} To pass from the third line to the fourth we have invoked \eqref{eq:modifiedbinomial}. We now have \begin{equation} \label{intermediateboundforMn} \begin{split} |M_n(\tilde f,f,\dots,f)| & \le \tilde b_n \tilde g^1_1 g^1_1 (g^1)^{n-2} + \tilde b_3 b_{n-3} \tilde g^1_1(g^1_1)^2(g^1_2)^{n-3} + \tilde b_n\tilde g^1_2(g^1_2)^{n-1} \\ &+ \tilde b_{n-1}\tilde g^1_1(g^1_1)^{n-2}|{\textstyle\int} 2^\delta f^1_2| + |{\textstyle\int} 2^{-\delta}\tilde f^1_1| b_{n-1}(g^1_2)^{n-1} . \end{split}\end{equation} Set $\beta = \tilde b_3 b_{n-3}/\tilde b_n = R^{3-n}b_1b_{n-3}/b_{n-2}$ and note that $\beta\le CR^{3-n}n^{1/2}\le 1$, if $R$ is chosen to be sufficiently large, since we are assuming $n\ge 6$. Applying Cauchy-Schwarz to the sum of the first three terms on the right-hand side of the preceding inequality, we obtain \begin{multline} \tilde b_n \tilde g^1_1 g^1_1 (g^1)^{n-2} + \tilde b_3 b_{n-3} \tilde g^1_1(g^1_1)^2(g^1_2)^{n-3} + \tilde b_n\tilde g^1_2(g^1_2)^{n-1} \\ \le\tilde b_n\tilde g^1\cdot \Big( (g^1_1)^2(g^1)^{2n-4} + (g^1_2)^{2n-2} + 2\beta(g^1_1)^3(g^1_2)^{n-3}(g^1)^{n-2} + \beta^2(g^1_1)^4(g^1_2)^{2n-6} \Big)^{1/2}. \end{multline} We claim that \begin{equation} (g^1_1)^2(g^1)^{2n-4} + (g^1_2)^{2n-2} + 2\beta(g^1_1)^3(g^1_2)^{n-3}(g^1)^{n-2} + \beta^2 (g^1_1)^4(g^1_2)^{2n-6} \le (g^1)^{2n-2}, \end{equation} provided that $n\ge 6$ and $\beta$ is sufficiently small. Writing $x=g^1_1,y=g^1_2$, this is simply \begin{equation*} x^2(x^2+y^2)^{n-2}+2\beta x^3y^{n-3}(x^2+y^2)^{(n-2)/2} +\beta^2 x^4 y^{2n-6}+y^{2n-2} \le (x^2+y^2)^{n-1}, \end{equation*} which is equivalent to \begin{equation*} 2\beta x^3 y^{n-3}(x^2+y^2)^{(n-2)/2} +\beta^2 x^4 y^{2n-6}+y^{2n-2} \le y^2(x^2+y^2)^{n-2}. \end{equation*} By homogeneity we may assume that $x^2+y^2=1$, so we wish to have \begin{equation*} 2\beta x^3 y^{n-3} +\beta^2 x^4 y^{2n-6}+y^{2n-2} \le y^2. \end{equation*} This holds whenever $x^2+y^2=1$ and $x,y\ge 0$, provided $\beta$ is sufficiently small, provided that each of the exponents on $y$ on the left-hand side is strictly larger than $2$; this holds provided $n\ge 6$. We have thus established that for $n\ge 6$, \begin{multline} \label{eq:lastMnstep1} |M_n(\tilde f,f,\dots,f)| \le\tilde b_n\tilde g^1(g^1)^{n-1} \\ + \tilde b_{n-1}\tilde g^1_1(g^1_1)^{n-2}|{\textstyle\int} 2^\delta f^1_2| + |{\textstyle\int} 2^{-\delta}\tilde f^1_1| b_{n-1}(g^1_2)^{n-1} . \end{multline} Representing the right-hand side as $\tilde b_n\tilde x x^{n-1} + \tilde b_{n-1}\tilde x x^{n-2}y + b_{n-1}\tilde y x^{n-1}$ where $x=g^1,y=|\int 2^\delta f^1_2|$ and $\tilde x = \tilde g^1,\tilde y = |{\textstyle\int} 2^{-\delta}\tilde f^1_1|$, we seek to bound this right-hand side by \begin{equation} \label{eq:lastMnstep2} \tilde b_n (\tilde x+\tilde y)(x+y)^{n-1} \ge \tilde b_n \tilde x x^{n-1} + \tilde b_n\tilde x(n-1)x^{n-2}y + \tilde b_n\tilde y x^{n-1}. \end{equation} Now\footnote{It is in order to be able to assert this last inequality that we incorporate an $\ell^1$ with respect to $m$, rather than an $\ell^2$ norm, in the definitions of the functionals $g_\delta$.} $\tilde b_{n-1} \lesssim R n^{1/2}\tilde b_n\le(n-1)\tilde b_n$ for all sufficiently large $n$. Likewise, $b_{n-1} \le Cn^{1/2}R\rp\tilde b_n \le (n-1)\tilde b_n$. Hence a term-by-term comparison completes the proof of the bound for $M_n$, for $n\ge 6$. \end{proof} Observe that the restriction $n\ge 6$ was only used above to control the first line of the right-hand side of \eqref{intermediateboundforMn}; once that term is majorized by $\tilde b_n\tilde g^1(g^1)^{n-1}$, the reasoning of the final paragraph allows us to absorb the two remaining special terms. The cases $n=2,3$ are quite simple; beginning again with \eqref{eq:inductivestep}, we obtain $n+1$ terms, all of which are rather simple for $n=2,3$. The details are left to the reader. Consider $n=5$, the most complicated case remaining. From \eqref{eq:inductivestep} we obtain an upper bound of \begin{equation} \label{eq:casefivemainpart} C_5 b_5 \tilde g^1_1 (g^1_1)^4 + C_3 b_2\tilde g^1_1(g^1_1)^2(g^1_2)^2 + C_2 b_3\tilde g^1_1 g^1_1(g^1_2)^3 + b_5 C_5\tilde g^1_2(g^1_2)^4 \end{equation} plus the two special terms involving $\int_{E^1_1} \tilde f$, $\int_{E^1_2}f$. By Cauchy-Schwarz we may bound the square of \eqref{eq:casefivemainpart} by \begin{equation*} b_5(C_5\tilde g^1)^2 \big[(g^1_1)^4 + \beta(g^1_1)^2(g^1_2)^2 + \beta g^1_1(g^1_2)^3\big]^2 + (g^1_2)^4, \end{equation*} where $\beta$ may be made as small as desired by choosing $C_5$ to be sufficiently large relative to $C_2,C_3$. Thus the desired inequality reduces to \begin{equation*} (x^4+\beta x^2y^2+\beta xy^3)^2+y^8\le (x^2+y^2)^4, \end{equation*} which holds for small $\beta$. The case $n=4$ is similar but simpler since one fewer term appears in the analogue of \eqref{eq:casefivemainpart}. We next discuss $M_n^*$. Let $\chi_y$ denote the characteristic function of $(-\infty,y]$ and apply the result just proved to $M_n(\tilde f,f\cdot\chi_y, \cdots,f\cdot\chi_y)$ to obtain \begin{equation} M_n^*(\tilde f,f,\dots,f) \le \frac{C^{n+1}}{\sqrt{n!}} g_{-\delta}(\tilde f) \cdot[\sup_y g_\delta(f\cdot\chi_y)]^{n-1}. \end{equation} It was shown in the proof of Proposition~4.2 of \cite{christkiselevfiltrations} that for any function $F$, \begin{equation} \sup_y \sum_{m=1}^\infty (\sum_j |{\textstyle\int}_{E^m_j} F\chi_y|^2)^{1/2} \le C \sum_{m=1}^\infty m (\sum_j |{\textstyle\int}_{E^m_j} F|^2)^{1/2}, \end{equation} and we may dominate the multiplicative coefficient $m$ by $C_\epsilon 2^{m\epsilon}$ on the right-hand side. Exactly the same proof allows the insertion of a factor of $2^{\delta m}$ after the first sum on both sides of the inequality, provided that $\delta\ge 0$, yielding the bound asserted. We have so far discussed only the special case where $f_2=f_3=\cdots = f_n =f$, but the general case is treated by exactly the same argument, simply bounding $|\int_{E^m_j}f_i|$ by $\max_k |\int_{E^m_j}f_k|$ wherever the former arises in the proof. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Multilinear operators} \label{section:multilinear} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The bounds derived in the preceding section for multiple integrals will be used to obtain Lebesgue space norm bounds for certain multilinear operators. Consider a family of integral operators \begin{equation} T_if(\lambda) = \int_\reals K_i(\lambda,x)f(x)\,dx. \end{equation} Denote by $\|T\|_{p,q}$ the norm of $T$ as an operator from $L^p(\reals)$ to $L^q(\reals)$. Consider associated multilinear operators \begin{equation} \scriptt_n(f_1,f_2,\dots,f_n)(\lambda) = \iint_{x_1\le x_2\le\cdots\le x_n} \prod_{i=1}^n K_i(\lambda,x_i)f_i(x_i)dx_i \end{equation} and their maximal variants \begin{equation} \scriptt_n^*(f_1,f_2,\dots,f_n)(\lambda) = \sup_y\big| \iint_{x_1\le x_2\le\cdots\le x_n\le y} \prod_{i=1}^n K_i(\lambda,x_i)f_i(x_i)dx_i \big|\ . \end{equation} \begin{theorem} \label{thm:multilinearoperators} For any $N<\infty$, $p0$ be a constant to be specified. By Proposition~\ref{prop:numericalbound}, \begin{equation} \label{eq:boundfromtheorem1} \scriptt_n^*(f_1,\dots,f_n)(\lambda) \le \frac{C^{n+1}}{\sqrt{n!}} \tilde G(\lambda) G(\lambda)^{n-1} \end{equation} where \begin{align} \tilde G(\lambda) & = \sum_{m=1}^\infty 2^{-\delta m} (\sum_j |T_1(f_1\cdot \chi^m_j)(\lambda)|^2)^{1/2} \\ G(\lambda) & = \sum_{m=1}^\infty 2^{\delta m} (\sum_j \max_i|T_i(f_i\cdot \chi^m_j)(\lambda)|^2)^{1/2}. \end{align} Now \begin{multline} \|\tilde G\|_{L^2} \le \sum_m 2^{-\delta m} (\sum_j\|T_1(f_1\cdot\chi^m_j)\|_{L^2}^2)^{1/2} \\ \le \sum_m 2^{-\delta m} \|T_1\|_{2,2}(\sum_j\|f_1\cdot\chi^m_j)\|_{L^2}^2)^{1/2} \le C_\delta \|T_1\|_{2,2}\|f_1\|_{L^2} \end{multline} provided that $\delta>0$. As for $G$, we may majorize \begin{equation} G(\lambda) \le \sum^*_i \sum_{m=1}^\infty 2^{\delta m} (\sum_j |T_i(f_i\cdot \chi^m_j)(\lambda)|^2)^{1/2}. \end{equation} It is shown on page 413 of \cite{christkiselevfiltrations} (see also the proof of Corollary~\ref{cor:nontangentialmaxbound} below) that since $p<2\le q$, there exists $\e>0$ such that \begin{equation} \|(\sum_j |T_i(f_i\cdot \chi^m_j)|^2)^{1/2}\|_{L^q} \le C2^{-\e m}, \end{equation} under the condition that $\|f_i\chi^m_j\|_{L^p}^p\le C' 2^{-m}$ for all $j,m$. Choosing $\delta<\e$ and summing over $m$ gives \begin{equation} \|G\|_{L^q} \le C<\infty, \end{equation} of course under the hypothesis that $\|f_i\|_{L^p}=1$ for all $i\ge 2$. An application of H\"older's inequality concludes the proof. \end{proof} Our main theorems are based on estimates for such multilinear operators; however, we require not the conclusion of Theorem~\ref{thm:multilinearoperators}, but rather the more detailed information contained in \eqref{eq:boundfromtheorem1} together with the norm bounds for $\tilde G,G$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Nontangential convergence and the maximal function}\label{nontan} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Throughout the paper we write $\complex^+=\{z\in\complex: \Im(z)>0\}$. Consider the cones \begin{align*} \Gamma_{\alpha}(w) &=\{z\in\complex^+: |\Re(z)-w|<\alpha\Im(z)\}, \\ \Gamma_{\alpha,\delta}(w) &=\{z\in\complex^+\cap\Gamma_\alpha(w): \Im(z)<\delta\}. \end{align*} A function $f(z)$ defined on $\complex^+$ is said to converge to a limit $a$ as $z\to w$ nontangentially if for every $\alpha<\infty$, $f(z)\to a$ as $z\to w$ within the cone $\Gamma_\alpha(w)$. The nontangential maximal function of $f$ is defined by \begin{equation} Nf(w) =N_{\alpha,\delta}f(w)= \sup_{z\in \Gamma_{\alpha,\delta}(w)}|f(z)|. \end{equation} We will need the following local variant of a standard property of functions holomorphic in the whole half plane $\complex^+$. \begin{lemma}\label{analyticf} Let $\delta>0$. Let $\Lambda$ be an open subinterval of $\reals$, and let $\Lambda'\Subset\Lambda$ be a relatively compact subinterval. Let $0\le q\le\infty$. Let $B$ be a Banach space. Suppose that $F$ is a holomorphic $B$-valued function in $\{\lambda+i\e: 0<\e<\delta,\ \lambda\in\Lambda\}$. Then for any $\alpha<\infty$, there exist $C<\infty$ and $\delta'>0$ depending on $\alpha,\Lambda,\Lambda'$, but not on $F$, such that \begin{equation} \|N_{\alpha,\delta'}F(\cdot)\|_{L^q(\Lambda')} \le C \sup_{0<\e<\delta} \|F(\cdot+i\e)\|_{L^q(\Lambda)} . \end{equation} \end{lemma} By a $B$-valued holomorphic function we mean a continuous function $f$ from an open subset of $\complex$ to $B$, such that $z\mapsto \ell(f)(z)$ is holomorphic for every bounded linear functional $\ell$ on $B$. \begin{proof} Suppose first that $f$ is continuous on the closed rectangle $\Lambda+i[0,\delta]$, and that $10$. Fixing any $0s$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Generalized eigenfunctions associated to complex spectral parameters} \label{section:complex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %{\bf Remark for us.} %Main results of paper should generalize to slowly varying case where %some derivative is in $L^p$ \dots %\smallskip Consider the generalized eigenfunction equation \begin{equation} \label{eigenfneqn} -u'' + V(x)u = zu, \end{equation} where $z$ is permitted to be complex. In earlier work we have analyzed the solutions to this equation for $z$ real and positive, and have shown that for almost every such $z$ there exists a solution with certain asymptotic behavior as $x\to+\infty$. Our present purpose is to analyze solutions for complex $z$ and to show that they tend almost everywhere to the solutions for real $z$, as $z$ approaches the positive real axis. We restrict attention to parameters $z\in\complex^+\cup \reals^+$, and choose a branch of $\sqrt{z}$ which has nonnegative imaginary part for such $z$. Define the phases \begin{equation} \xi(x,z) = \sqrt{z}x-(2\sqrt{z})\rp\int_0^x V. \end{equation} \begin{theorem} \label{thm:complexeigenfunctions} Let $1\le p<2$ and assume that $V\in L^1+L^p$. For each $z\in\complex^+$ there exists a (unique) solution $u(x,z)$ of the generalized eigenfunction equation \eqref{eigenfneqn} satisfying \begin{equation} \label{wkbasymptotics} u(x,z) - e^{i\xi(x,z)} \to 0 \text{ and } \partial u(x,z)/\partial x - i\sqrt{z}e^{i\xi(x,z)} \to 0 \qquad\text{as } x\to +\infty. \end{equation} $u(x,z)$ is continuous as a function on $\complex^+\times\reals$, and is holomorphic with respect to $z$ for each fixed $x$. Likewise, there exists such a (unique) solution for almost every $z\in\reals^+$. For almost every $E\in\reals$, $u(x,z)$ converges to $u(x,E)$ uniformly for all $x$ in any interval bounded below, as $z\to E$ nontangentially. \end{theorem} Here ``almost every'' means with respect to Lebesgue measure. The existence of such solutions for almost every $z\in\reals^+$ is proved in \cite{christkiselevdecaying}. For $z\in\complex^+$ it is well known under weaker hypotheses on $V$. The new point here is the convergence as $z\to\reals^+$, and in particular, the fact that it is globally uniform in $x$. It suffices to prove the theorem for $x\in [0,\infty)$, since the conclusion for $x\in [\rho,\infty)$, for any $\rho>-\infty$, then follows via the eigenfunction equation \eqref{eigenfneqn}. By rewriting \eqref{eigenfneqn} as a first-order system, performing a couple of algebraic transformations, reducing to an integral equation, and solving it by iteration, one arrives \cite{christkiselevdecaying} at a formal series representation for solutions of \eqref{eigenfneqn}: \begin{equation} \label{formalsolutionseries} \begin{pmatrix} u(x,z) \\ u'(x,z) \end{pmatrix} = \begin{pmatrix} e^{i\xi(x,z)} & e^{-i\xi(x,z)} \\ i\sqrt{z} e^{i\xi(x,z)} & -i\sqrt{z} e^{-i\xi(x,z)} \end{pmatrix} \cdot \begin{pmatrix} \sum_{n=0}^\infty T_{2n} (V, \dots, V)(x,z) \\ -\sum_{n=0}^\infty T_{2n+1} (V, \dots, V)(x,z) \end{pmatrix} \end{equation} where \begin{equation} T_n (V_1, \dots, V_n)(x,z)= (2\sqrt{z})^{-n} \int_{x \leq t_1 \leq \dots \leq t_n} \prod_{j=1}^n e^{2i (-1)^{n-j} \xi(t_j,z)} V_j(t_j)\,dt_j, \end{equation} with the convention $T_0(\cdot) \equiv 1$. To prove Theorem~\ref{thm:complexeigenfunctions} we will show that each multilinear expression $T_n$ is well-defined for all $z\in\complex^+$, that $T_n(\cdot)(x,z)\to 0$ as $x\to+\infty$ for all $n\ge 1$, that they have the natural limits as $z\to\reals^+$ nontangentially, and that these expressions satisfy bounds sufficiently strong to enable us to sum the infinite series to obtain the desired conclusions. Substitute $\zeta=\sqrt{z}$ and write $\zeta = \lambda+i\e$, noting that $\e>0$. Also write $\phi(x,\zeta) = \xi(x,z)$, $S_n(V_1,\dots)(x,\zeta) = T_n(V_1,\dots)(x,z)$. Let $\{E^m_j\}$ be a martingale structure on $\reals^+$. Denote by $t_{m,j}^\pm$ respectively the right ($+$) and left ($-$) endpoints of the interval $E^m_j$. The real part of the exponent $2i\sum_{j=1}^n (-1)^{n-j} \xi(t_j,z)$ is to leading order \begin{equation*} -2\Im(\sqrt{z}[(t_n-t_{n-1})+(t_{n-2}-t_{n-3})+\cdots]) = -2\e\cdot [(t_n-t_{n-1})+(t_{n-2}-t_{n-3})+\cdots], \end{equation*} which is nonnegative (for $x\ge 0$) for all $z\in\complex^+$ since $t_1\le t_2\cdots$; the exponential factor decays rapidly as $[(t_n-t_{n-1})+(t_{n-2}-t_{n-3})+\cdots]\to\infty$. This accounts for the difference between $z\in\complex^+$ and $z\in\reals^+$. Define \begin{equation}\begin{split} G_m(V)(\zeta) &= \Big( \sum_{j=1}^{2^m} |s^{m,-}_j(V,\zeta)|^2 + | s^{m,+}_j(V,\zeta)|^2\Big)^{1/2} \\ G(V) &= \sum_{m=1}^\infty mG_m(V) \end{split}\end{equation} where \begin{equation} \label{emjintegrals} \begin{split} s^{m,-}_j(V,\zeta) &= \int_{E^m_j} e^{2i[\phi(t,\zeta)-\phi(t^-_{m,j},\zeta)]}V(t)\,dt \\ s^{m,+}_j(V,\zeta) &= \int_{E^m_j} e^{2i[\phi(t^+_{m,j},\zeta)-\phi(t,\zeta)]}V(t)\,dt. \end{split}\end{equation} When $j=2^m$, and only then, the right endpoint of $E^m_j$ is infinite. To simplify notation we make the convention that for $j=2^m$, the second term $|s^{m,+}_j(V,\zeta)|^2$ is always to be omitted, in the definition of $G$ and anywhere else that the quantities $\phi(t^+_{m,j},\zeta)$ arise. The following definitions will allow us to regard $G$ as a linear operator, and hence to exploit properties of holomorphic functions. \begin{definition} $\scriptb$ denotes the Banach space consisting of all sequences $\{\complex^2\owns s^m_j: m\ge 0,\ 1\le j\le 2^m\}$, with the norm $\|s\|_\scriptb = \sum_m m\big(\sum_{j=1}^{2^m} |s^m_j|^2\big)^{1/2}$. $\frakG: L^p\mapsto\scriptb$ denotes the operator \begin{equation} \frakG(V)(\zeta) = \{(s^{m,+}_j(V,\zeta),s^{m,-}_j)(V,\zeta): 1\le m<\infty, 1\le j\le 2^m\}. \end{equation} \end{definition} Thus \begin{equation*} G(V)(\zeta) = \|\frakG(V)(\zeta)\|_{\scriptb}. \end{equation*} Likewise we may write $G_M(V) = \|\frakG_M(V)\|_{\scriptb}$ with the analogous definition of $\frakG_M$. The real parts of $i\phi(t,\zeta)-i\phi(t^-_{m,j},\zeta)$ and $i\phi(t^+_{m,j},\zeta)-i\phi(t,\zeta)$ are bounded above uniformly for $0<\Im(\zeta)\le 1$, and are $\le -c\Im(\zeta)(t-t^-_{m,j})$ and $\le -c\Im(\zeta)(t^+_{m,j}-t)$, respectively; see \eqref{realpartnonpositive}. Therefore for $V\in L^1+L^\infty$ and $\zeta\in\complex^+$, each of these integrals converges absolutely, and each defines a holomorphic scalar-valued function of $\zeta\in\complex^+$. Thus $\frakG(V)$ may be regarded as a $\scriptb$--valued holomorphic function\footnote{ By this we mean simply that it is a continuous mapping into $\scriptb$ with respect to the norm topology, and that each $s^{m,\pm}_j$ is a holomorphic scalar-valued function.} in any open set where it can be established that the series defining $\|\frakG(V)\|_\scriptb$ converges uniformly. We will also use the following variant. Given a collection of functions $(V_1,\dots,V_n)$, we define \begin{equation} G(\{V_i\})(\zeta) = \sum_{m=1}^\infty m\cdot \Big( \sum_{j=1}^{2^m} \sum^*_i |s^{m,+}_j(V_i,\zeta)|^2+|s^{m,-}_j(V_i,\zeta)|^2 \Big)^{1/2}, \end{equation} where $\sum\limits^*$ indicates that the sum is taken over a maximal set of indices $i$ for which the functions $V_i$ are all distinct. % ??? do we need this paragraph at all? \begin{lemma} \label{lemma:Gmajorizationwithmodifiedphases} For all $V,n$ and all $\zeta\in \complex^+\cup\reals^+$, \begin{equation} \sup_{x\in\reals^+}|S_n(V,V,\dots,V)(x,\zeta)| \le \frac{C^{n+1}}{\sqrt{n!}} G(V)(\zeta)^n. \end{equation} More generally, for all $n$ and all $\{V_1,\dots,V_n\}$, \begin{equation} \sup_{x\in\reals^+}|S_n(V_1,V_2,\dots,V_n)(x,\zeta)| \le \frac{C_k^{n+1}}{\sqrt{n!}} G(\{V_i\})(\zeta)^n, \end{equation} provided that $\{V_i\}_{i=1}^n$ has cardinality $\le k$. \end{lemma} Write $p'=p/(p-1)$. \begin{lemma} \label{lemma:parseval} For any compact interval $\Lambda\Subset(0,\infty)$, there exists $C<\infty$ such that for any $1\le p\le 2$, for all $t'\in\reals$ and $f\in L^p(\reals)$, for every $\e\ge 0$, \begin{align} &\|\int_{t\ge t'} e^{2i[\phi(t, \lambda+i\e) - \phi(t', \lambda+i\e)]} f(t)\,dt\|_{L^{p'}(\Lambda,d\lambda)} \le C\|f\|_{L^p} \\ &\|\int_{t\le t'} e^{2i[\phi(t', \lambda+i\e) - \phi(t, \lambda+i\e)]} f(t)\,dt\|_{L^{p'}(\Lambda,d\lambda)} \le C\|f\|_{L^p}. \end{align} \end{lemma} For $\e=0$ these integrals need not converge absolutely, hence require interpretation. They are initially well-defined for compactly supported $f$, and the lemma asserts an {\em a priori} bound for such functions. Then they are defined for general $f\in L^p$ by approximating in $L^p$ norm by compactly supported functions, and passing to the limit in $L^{p'}$ norm. Let an exponent $p<\infty$ be specified. Recall that a martingale structure is said to be adapted to $f$ in $L^p$ if $\int_{E^m_j}|f|^p = 2^{-m}\int|f|^p$ for all $m,j$. Recall from the preceding section the definitions of the cones $\Gamma_{\alpha,\delta}$ and associated nontangential maximal functions $N_{\alpha,\delta}$. \begin{corollary} \label{cor:nontangentialmaxbound} Let $\alpha<\infty$, let $1\le p \le 2$, and let $\Lambda\Subset(0,\infty)$ be any compact subinterval. Then there exist $C<\infty,\delta>0$ such that for any $f\in L^p(\reals)$ and for any martingale structure $\{E^m_j\}$ on $\reals^+$ % adapted to $f$, \begin{equation} \label{nonadaptedbound} \|N_{\alpha,\delta} G_m(f)(\lambda)\|_{L^{p'}(\Lambda,d\lambda)} \le C\|f\|_{L^p}. \end{equation} Moreover for each $1\le p<2$ there exists $\rho>0$ such that for any $f\in L^p$ and for any martingale structure adapted to $f$ in $L^p$, \begin{equation} \label{GMdecaybound} \|N_{\alpha,\delta} G_m(f)(\lambda)\|_{L^{p'}(\Lambda,d\lambda)} \le C2^{-\rho m}\|f\|_{L^p}. \end{equation} Consequently under these additional hypotheses, \begin{equation} \|N_{\alpha,\delta} G(f)(\lambda)\|_{L^{p'}(\Lambda,d\lambda)} \le C\|f\|_{L^p}. \end{equation} Moreover, for almost every $\lambda\in\Lambda$, \begin{equation} \|\frakG(f)(\zeta)-\frakG(f)(\lambda)\|_{\scriptb} \to 0 \end{equation} as $\zeta\to\lambda$ nontangentially. \end{corollary} We postpone the proofs of Lemmas~\ref{lemma:Gmajorizationwithmodifiedphases}, \ref{lemma:parseval} and Corollary~\ref{cor:nontangentialmaxbound} until the end of the section. For any $t\ge t'$ and any $\zeta=\lambda+i\e$ with $\e\ge 0$, \begin{equation} \label{realpartnonpositive} \Re(i\phi(t,\lambda+i\e)-i\phi(t',\lambda+i\e)) = -\e(t-t') -\e(\lambda^2+\e^2)\int_{t'}^t V, \end{equation} and $|\int_{t'}^t V|\le |t-t'|^{1/p'}\|V\|_{L^p}$. Therefore % ?? phi, xi depend on V; need to explain/deal with this ... ?? \begin{equation} |e^{i[\phi(t,\lambda+i\e)-\phi(t',\lambda+i\e)]}| \le Ce^{-c\e|t-t'|} \end{equation} where $C,c\in\reals^+$ are constants which depend only on the $L^p$ norm of $V$. Hence for all $n\ge 1$, \begin{align} |S_{2n}(V,V,\dots,V)(x,\lambda+i\e)| & \le \iint_{x\le t_1\le\cdots\le t_{2n}} e^{-c\e(t_{2n}-t_{2n-1}+t_{2n-2}-\cdots)} \prod_j V(t_j)\,dt_j \notag \\ & \le \frac{C^n}{n!} \Big(\iint_{x\le t'\le t} e^{-c\e(t-t')}|V(t')V(t)|\,dt'\,dt\Big)^n \notag \\ & \le \frac{C^n}{n!} \e^{-2n(1-p\rp)} \|V\|_{L^p([x,\infty)}^{2n}. \end{align} In the same way, for $x\ge 0$, one obtains for $n\ge 0$ \begin{equation} |S_{2n+1}(V,V,\dots,V)(x,\lambda+i\e)| \le \frac{C^n}{n!} \e^{-(2n+1)(1-p\rp)} \|V\|_{L^p([x,\infty)}^{2n+1}. \end{equation} Let $z$ belong to any compact subset of $\complex^+$. Then $\e=\Im(\sqrt{z})$ has a strictly positive lower bound. Therefore the individual terms of the series \eqref{formalsolutionseries} defining a formal solution of \eqref{eigenfneqn} do define uniformly bounded functions of $(x,\zeta)$ for $x\ge 0$ and $\zeta$ in any compact subset of $\complex^+$, and moreover, the series are uniformly absolutely convergent. As in Lemma~4.2 of \cite{christkiselevdecaying}, it follows that the sums of these series define solutions of the generalized eigenfunction equation \eqref{eigenfneqn} for all such $z$. Because the $L^p$ norm of $V$ over $[x,\infty)$ tends to zero as $x\to+\infty$, only $T_0(x)$ contributes in that limit, so these solutions do have the desired WKB asymptotics $\exp(i\xi(x,z))$. Clearly each summand depends holomorphically on $\zeta$, hence so do the sums. Existence of a (unique) solution for almost every $z\in\reals^+$ is proved in \cite{christkiselevdecaying}. We come now to the main step, where we relate $z\in\complex^+$ to $z\in\reals^+$. For compactly supported $f\in L^1$, the quantities $s^{m,\pm}_j(f,\zeta)$ are clearly holomorphic functions of $\zeta$ where $\Im(\zeta)>0$, and are continuous at $\e=0$ for each $0\ne \lambda\in\reals$. The same holds for $S_n(f,f,\dots,f)(x,\zeta)$, for each $x\in\reals$. $\frakG(f)$ is likewise a $\scriptb$--valued holomorphic function of $\zeta$, continuous on $\complex^+\cup\reals^+$, for compactly supported $f\in L^1$. This follows by combining the holomorphy of the individual terms \eqref{emjintegrals} with the rapid convergence bound \eqref{GMdecaybound}. \begin{lemma} \label{lemma:Snconvergence} Let $1\le p<2$ and $V\in L^p(\reals)$. For almost every $E\in\reals$, for every $n\ge 1$, $T_n(V,V,\dots,V)(x,z)\to T_n(V,V,\dots,V)(x,E)$ uniformly for all $x\ge 0$ as $\complex^+\owns z\to E$ nontangentially. \end{lemma} \begin{proof} It is equivalent to show that \begin{equation*} \sup_{x\ge 0}\big| S_n(V,V,\dots,V)(x,\zeta)-S_n(V,V,\dots,V)(x,\lambda)\big| \to 0 \end{equation*} as $\zeta\to\lambda$ nontangentially, for almost all $\lambda\in\Lambda$, for any fixed compact interval $\Lambda\subset\reals^+$. For $V=W\in L^1$ with compact support, we have already established convergence uniformly in $x,\lambda$, as $\complex^+\owns\zeta\to \lambda$ unrestrictedly (rather than merely nontangentially), since the phases $\phi(t,\zeta)$ converge uniformly to $\phi(t,\lambda)$ for $t,\zeta,\lambda$ in any compact set. Let $V$ be given, and remain fixed for the remainder of this proof. Set $\phi_V(t,\zeta)=\zeta t - (2\zeta)\rp\int_0^t V$. Whenever we write $S_n(f_1,f_2,\dots,f_n)$, it is defined in terms of the phases $\phi(t_i,\zeta) = \phi_V(t_i,\zeta)$, independent of $f_1,f_2,\dots$. Thus the $S_n$ are here genuine multilinear operators. Let $\e>0$ be arbitrary, and fix a martingale structure $\{E^m_j\}$ adapted to $V$ in $L^p$ on $\reals^+$. Decompose $V=W+(V-W)$ where $W(x) =V(x)\chi_{(0,R]}(x)$, with $R$ chosen so that $\|V-W\|_{L^p}<\e$ and moreover so that $\|NG(V-W)\|_{L^{p'}(\Lambda)}<\e$. Such a choice is possible, since \begin{equation*} \|NG_M(V\chi_{[R,\infty)})\|_{L^{p'}(\Lambda)} \le C\min(2^{-M\delta}\|V\|_{L^p},\,\| V\chi_R\|_{L^p}). \end{equation*} Then \begin{multline} |S_n(V,V,\dots,V)(x,\zeta)-S_n(V,V,\dots,V)(x,\lambda)| \\ \le |S_n(W,W,\dots,W)(x,\zeta)-S_n(W,W,\dots,W)(x,\lambda)| \\ + |S_n(V,V,\dots,V)(x,\zeta)-S_n(W,W,\dots,W)(x,\zeta)| \\ + |S_n(V,V,\dots,V)(x,\lambda)-S_n(W,W,\dots,W)(x,\lambda)| . \end{multline} The first term on the right tends to zero, in the sense desired. Majorize the second by \begin{multline} |S_n(V,V,\dots,V)(x,\zeta)-S_n(W,W,\dots,W)(x,\zeta)| \\ \le \sum_{i=1}^n |S_n(V,\dots,V,V-W,W,\dots,W)(z,\zeta)| \end{multline} where in the $i$-th summand, the argument of $S_n$ has $i-1$ copies of $V$ and $n-i$ copies of $W$. Fix any aperture $\alpha\in\reals^+$. Thus as established in the proof of Proposition~4.1 of \cite{christkiselevdecaying}, \begin{multline} \sup_{x\ge 0} \sup_{\zeta\in \Gamma_{\alpha,\delta} (\lambda)} |S_n(V,V,\dots,V)(x,\zeta)-S_n(W,W,\dots,W)(x,\zeta)| \\ \le C_n \sum_{i=1}^n \sup_{\zeta\in \Gamma_{\alpha,\delta}(\lambda)} G(V)^{i-1}(\zeta)G(W)^{n-i}(\zeta)G(V-W)(\zeta) \\ \le C_n \sum_{i=1}^n NG(V)^{i-1}(\lambda)NG(W)^{n-i}(\lambda)NG(V-W)(\lambda). \end{multline} Let $q=p'$. By Chebyshev's inequality, for any $\beta>0$, \begin{equation}\begin{split} &|\{\lambda\in\Lambda: \sup_{\zeta\in \Gamma_{\alpha,\delta}(\lambda)}\sup_{x\ge 0} |S_n(V,V,\dots,V)(x,\zeta)-S_n(W,W,\dots,W)(x,\zeta)|>\beta\}| \\ &\qquad\qquad \le C_n\beta^{-q/n} \sum_{i=1}^n \| NG(V)^{i-1}NG(W)^{n-i}NG(V-W)\|_{L^{q/n}(\Lambda)}^{q/n} \\ &\qquad\qquad \le C_n\beta^{-q/n} \sum_{i=1}^n (\|NG(V)\|_{L^q(\Lambda)}^{i-1} \|NG(W)\|_{L^q(\Lambda)}^{n-i} \|NG(V-W)\|_{L^q(\Lambda)})^{q/n} \\ &\qquad\qquad \le C_n\beta^{-q/n} \sum_{i=1}^n (\|V\|_{L^p}^{n-1}\|V-W\|_{L^p})^{q/n} \\ &\qquad\qquad \le C_n\beta^{-q/n} \|V\|_{L^p}^{q(n-1)/n}\e^{q/n}. \end{split}\end{equation} The same reasoning gives \begin{multline} |\{\lambda\in\Lambda: \sup_{x\ge 0} \big|S_n(V,V,\dots,V)(x,\lambda)-S_n(W,W,\dots,W)(x,\lambda)|>\beta\}\big| \\ \le C_n\beta^{-q/n} \|V\|_{L^p}^{q(n-1)/n}\e^{q/n}. \end{multline} Consequently \begin{multline} |\{\lambda\in\Lambda: \limsup_{\Gamma_\alpha(\lambda)\owns\zeta\to\lambda}\,\, \sup_{x\ge 0}\, \big|S_n(V,V,\dots,V)(x,\zeta)-S_n(V,V,\dots,V)(x,\lambda)\big| >\beta\} \\ \le C_n\beta^{-q/n} \|V\|_{L^p}^{q(n-1)/n}\e^{q/n}, \end{multline} for all $\beta,\e\in\reals^+$. Letting $\e\to 0$, we conclude that the $\limsup$ vanishes for almost every $\lambda$. \end{proof} It is now straightforward to sum the series to obtain the same conclusion regarding convergence of $u(x,z)=u(x,\zeta^2)$ to $u(x,E)=u(x,\lambda^2)$: \begin{equation} |u(x,\zeta^2)-u(x,\lambda^2)| \le \sum_{n=0}^\infty |S_n(V,V,\dots,V)(x,\zeta)-S_n(V,V,\dots,V)(x,\lambda)| \end{equation} and \begin{equation}\begin{split} &\sup_{\zeta\in\Gamma_{\alpha,\delta}(\lambda)}\sup_{x\ge 0} \sum_{n=M}^\infty |S_n(V,V,\dots,V)(x,\zeta)-S_n(V,V,\dots,V)(x,\lambda)| \\ &\qquad\qquad\le \sup_{\zeta\in\Gamma_{\alpha,\delta}(\lambda)}\sup_{x\ge 0} \sum_{n=M}^\infty (|S_n(V,V,\dots,V)(x,\zeta)|+|S_n(V,V,\dots,V)(x,\lambda)|) \\ &\qquad\qquad\le \sum_{n=M}^\infty \frac{C^{n+1}}{\sqrt{n!}} \big(NG(V)(\lambda)+G(V)(\lambda)\big)^n \\ &\qquad\qquad\le \frac{C^{M+1}}{\sqrt{M!}}\big(NG(V)(\lambda)+G(V)(\lambda)\big)^M \sum_{k=0}^\infty \frac{C^{k+1}}{\sqrt{k!}} \big(NG(V)(\lambda)+G(V)(\lambda)\big)^k \\ &\qquad\qquad\le \frac{C^{M+1}}{\sqrt{M!}}\big(NG(V)(\lambda)+G(V)(\lambda)\big)^M \exp(C \big(NG(V)(\lambda)+G(V)(\lambda)\big)^2). \end{split}\end{equation} For almost every $\lambda\in\reals$, $\big(NG(V)(\lambda)+G(V)(\lambda)\big)<\infty$, and hence this expression tends to zero as $M\to\infty$. Coupled with the convergence established for the individual terms $S_n$ in Lemma~\ref{lemma:Snconvergence}, this completes the proof of Theorem~\ref{thm:complexeigenfunctions}, modulo the proofs of Lemmas~\ref{lemma:parseval} and \ref{lemma:Gmajorizationwithmodifiedphases}, and Corollary~\ref{cor:nontangentialmaxbound}. \hfill\qed \begin{proof}[Proof of Lemma~\ref{lemma:parseval}] The proofs of the two inequalities are essentially the same, so we discuss only the first. The exponent here is $i\Phi(t,\lambda+i\e) = i\phi(t,\lambda+i\e)- i\phi(t',\lambda+i\e) = (i\lambda-\e)(t-t') - (i\lambda+\e)(\lambda^2+\e^2)\rp(\int_{t'}^t V)$. Since $t\ge t'$ and $|\int_{t'}^t V| \le C+C|t-t'|^{1/2}$, the real part of $\Phi$ is bounded above, uniformly for all $\lambda$ in any compact subinterval $\Lambda\Subset\reals\backslash\{0\}$ and $\e\ge 0$. Thus $L^1$ is mapped boundedly to $L^\infty(\Lambda)$, uniformly in $\e$; by interpolation it suffices to prove the $L^2$ estimate. Fix any cutoff function $\eta\in C^\infty(\reals\backslash\{0\})$ and consider \begin{equation} \int \Big|\int_{t>t'} e^{i\Phi(t,\lambda+i\e)}f(t)\,dt\Big|^2\eta(\lambda)\,d\lambda = \iint_{s,t\ge t'} f(t)\bar f(s) K(t,s)\,dt\,ds \end{equation} where \begin{equation} K(t,s) =\int e^{\Psi(t,s,\lambda+i\e)} \eta(\lambda) \,d\lambda \end{equation} with \begin{multline} \Psi(t,s,\lambda+i\e) = {2i\Phi(t,\lambda+i\e)-2i\bar\Phi(s,\lambda+i\e)} \\ = 2i\big[\lambda(t-s)-\lambda(\lambda^2+\e^2)\rp{\textstyle\int}_s^t V\big] \\ -2\e\big[(t-t')+(s-t') + (\lambda^2+\e^2)\rp({\textstyle\int}_{t'}^t V) + (\lambda^2+\e^2)\rp({\textstyle\int}_{t'}^s V)\big]. \end{multline} We claim that $|K(t,s)|\le C(1+|s-t|)^{-2}$, uniformly in $\e\ge 0$; this would suffice to imply the $L^2$ bound. The integrand itself is bounded, uniformly in all parameters, so it suffices to restrict attention to the case where $|s-t|\ge C_0$, where $C_0$ is a sufficiently large constant. In that case we integrate by parts, integrating $\exp(2i[\lambda(t-s)-\lambda(\lambda^2+\e^2)\rp\int_s^t V])$, and differentiating $\eta(\lambda)\cdot \exp(-2\e[(t-t')+(s-t') + (\lambda^2+\e^2)\rp(\int_{t'}^t V) + (\lambda^2+\e^2)\rp(\int_{t'}^s V))$, noting that $\partial [\lambda(t-s)-\lambda(\lambda^2+\e^2)\rp\int_s^t V] /\partial\lambda\ge |s-t|/2$ provided $C_0$ is chosen to be sufficiently large. Thus we gain a factor of $(s-t)\rp$. On the other hand, differentiating the other exponential with respect to $\lambda$ brings in an unfavorable term $O(\e \int_{t'}^{\max(s,t)}|V|)$. After two integrations by parts, the integrand is \begin{equation} O(|s-t|^{-2})\cdot e^{-2\e(s-t')-2\e(t-t')} \cdot O( 1+\e^2( \int_{t'}^{\max(s,t)}|V| )^2 ) = O(|s-t|^{-2}), \end{equation} uniformly in $\e$. \end{proof} \begin{proof}[Proof of Lemma~\ref{lemma:Gmajorizationwithmodifiedphases}] The new feature here is the introduction of the modifying factors $\exp( \pm 2i\phi(t_{m,j}^\pm,\zeta) )$; without these, this is proved in \cite{christkiselevfiltrations}. We will merely indicate the modification needed in the argument, referring to \cite{christkiselevfiltrations} for the rest. Consider \begin{equation} \iint_{x\le t_1\le\cdots\le t_n}e^{2i[\phi(t_n,\zeta)-\phi(t_{n-1},\zeta) + \phi(t_{n-2},\zeta)-\cdots ]} f(t_1)f(t_2)\cdots f(t_n)\,dt_1\cdots\,dt_n. \end{equation} Decompose the region of integration $\{t=(t_1,\cdots,t_n): x\le t_1\le\cdots\le t_n\}$ as $\cup_{k=0}^n\Omega_k$ where $\Omega_k=\{t: x\le t_1\le\cdots\le t_k\le t^+_{1,1} = t^-_{1,2}\le t_{k+1}\le\cdots\le t_n\}$. The total integral is \begin{equation}\begin{split} &\sum_k\iint_{\Omega_k} e^{2i[\phi(t_n,\zeta)-\phi(t_{n-1},\zeta)+\cdots\pm\phi(t_{k+1},\zeta) \mp\phi(t^-_{1,2},\zeta)]} \cdot e^{2i[\pm\phi(t^+_{1,1} \mp \phi(t_k,\zeta)\pm\phi(t_{k-1},\zeta)\mp\cdots]} \prod_{j=1}^n f(t_j)\,dt_j \\ & \qquad= \sum_k \Big( \iint_{t^-_{1,2}\le t_{k+1}\le\cdots\le t_n} e^{2i[\phi(t_n,\zeta)-\phi(t_{n-1},\zeta)+\cdots +(-1)^{n-k-1}\phi(t_{k+1},\zeta)]} \prod_{j=k+1}^n f(t_j)\,dt_j \Big) \\ &\qquad\qquad\cdot \Big( \iint_{x\le t_1\le\cdots\le t_k\le t^+_{1,1} } e^{2i(-1)^{n-k}[\phi(t_k,\zeta) - \phi(t_{k-1},\zeta) + \cdots + (-1)^{n-1}\phi(t_1,\zeta)]} \prod_{j=1}^k f(t_j)\,dt_j \Big). \end{split}\end{equation} For each $k$, each of the two factors on the right-hand side has the same form as the multiple integral with which we began, except that when $n-k$ is odd, an extra factor of $-1$ appears in the exponent in the integral with respect to $dt_k\cdots dt_1$; this minus sign destroys the bounds we seek, as is clear from \eqref{realpartnonpositive}. Therefore when $n-k$ is odd, we rewrite the corresponding term as the modified product \begin{multline*} \Big( \iint_{t^-_{1,2}\le t_{k+1}\le\cdots\le t_n} e^{2i[\phi(t_n,\zeta)-\phi(t_{n-1},\zeta)+\cdots +\phi(t_{k+1},\zeta)-\phi(t^-_{1,2},\zeta)]} \prod_{j=k+1}^n f(t_j)\,dt_j \Big) \\ \cdot \Big( \iint_{x\le t_1\le\cdots\le t_k\le t^+_{1,1} } e^{2i[\phi(t^+_{1,1},\zeta) -\phi(t_k,\zeta) + \phi(t_{k-1},\zeta) + \cdots + (-1)^{n-1}\phi(t_1,\zeta)]} \prod_{j=1}^k f(t_j)\,dt_j \Big). \end{multline*} Suppose now that $n$ is even. The proof in \cite{christkiselevfiltrations} is an induction based on a repeated application of this decomposition; each step of that recursion involves a ``cut point'' $t^+_{m,j} = t^-_{m,j+1}$ playing the same role as $t^+_{1,1}=t^-_{1,2}$ in the above formula. At each step, the region of integration is decomposed into subregions as above, and corresponding to each subregion there is a splitting of the terms in the phase into two subsets. At any step which results in an odd number of terms appearing in one (hence both) subsets, we modify the resulting phases by introducing a factor $1 = \exp(\pm 2i[\phi(t^+_{m,j},\zeta)-\phi(t^-_{m,j+1},\zeta)])$, factoring it as a product of one exponentials, and splitting those two exponential factors as above. This, together with the argument in \cite{christkiselevfiltrations}, yields the assertion of the lemma for even $n$. For odd $n$ we introduce a factor of $1 = \exp(2i[\phi(x,\zeta)-\phi(x,\zeta)])$, incorporate $\exp(-2i\phi(x,\zeta))$ into the phase, thus reducing matters again to the case where there an even number of terms. The remaining factor of $\exp(2i\phi(x,\zeta))$ is bounded above, uniformly for $0\le\Im(\zeta)\le 1$ and $x\ge 0$, so is harmless. \end{proof} \begin{proof}[Proof of Corollary~\ref{cor:nontangentialmaxbound}] Let $\Lambda\Subset(0,\infty)$ be a compact interval, and let $1< p<2$, the case $p=1$ being trivial. Set $q = p'=p/(p-1)$. We discuss only the contributions of terms involving $\phi(t^-_{m,j},\zeta)$ to $G$ and $G_M$; those involving $t^+_{m,j}$ are treated in exactly the same way. For any $m\ge 1$ and any $f\in L^p$ we have, since $q/2\ge 1$, \begin{multline} \|\Big( \sum_{j=1}^{2^m} \Big|\int_{E^m_j} e^{2i[\phi(t,\lambda+i\e)-\phi(t^-_{m,j},\lambda+i\e)]} f(t)\,dt\Big|^2 \Big)^{1/2}\|_{L^{q}(\Lambda,d\lambda)}^q \\ \le \Big(\sum_j \Big[\int_{\Lambda} \Big|\int_{E^m_j} e^{2i[\phi(t,\lambda+i\e)-\phi(t^-_{m,j},\lambda+i\e)]} f(t)\,dt\Big|^q \,d\lambda\Big]^{2/q}\Big)^{q/2}, \end{multline} by Minkowski's integral inequality. % marker By Lemma~\ref{lemma:parseval}, the right-hand side is \begin{equation*} \le C \Big(\sum_j \|f\cdot \chi_{E^m_j}\|_{L^p}^2\Big)^{q/2}. \end{equation*} Since $p\le 2$, this is \begin{equation*} \le C\Big(\sum_j \|f\|_{L^p}^{2-p} \|f\cdot \chi_{E^m_j}\|_{L^p}^p )^{q/2} = C\|f\|_{L^p}^q. \end{equation*} %where $\rho>0$ if $p<2$, and $\rho=0$ if $p=2$. If we assume that the martingale structure is adapted to $f$ in $L^p$, then for $p<2$ we have the improved majorization \begin{equation*} \|f\cdot \chi_{E^m_j}\|_{L^p}^2 \le 2^{-m(2-p)/p}\|f\|_{L^p}^{2-p}\|f\cdot \chi_{E^m_j}\|_{L^p}^p, \end{equation*} whence the final bound is $2^{-\rho m}\|f\|_{L^p}^q$ for some $\rho(p)>0$. Therefore \begin{equation} \int_\Lambda \|\frakG_M(f)(\lambda+i\e)\|_{\scriptb}^q\,d\lambda \le C2^{-\rho M}\|f\|_{L^p}^q, \end{equation} uniformly for all $\e>0$. The first two conclusions of the Corollary now follow from Lemma~\ref{analyticf}, since $\Lambda$ is an arbitrary compact interval. That $\frakG(f)(\zeta)$ converges almost everywhere to $\frakG(f)(\lambda)$ in the $\scriptb$ norm as $\zeta\to \lambda$ nontangentially, is an immediate consequence of the bound $\|NG_M(f)\|_{L^q}\le C2^{-\varepsilon M}\|f\|_{L^p}$, since \begin{equation} \int_{t\ge t'} e^{2i[\phi(t, \zeta) - \phi(t', \zeta)]} f(t)\,dt \to \int_{t\ge t'} e^{2i[\phi(t, \lambda) - \phi(t', \lambda)]} f(t)\,dt \end{equation} almost everwhere as $\zeta\to\lambda$ nontangentially, for all $f\in L^p$. This holds for all $f$ in the dense subspace $L^1\cap L^p$, and then follows for general $f$ by Lemma~\ref{analyticf}, Lemma~\ref{lemma:parseval}, and standard reasoning. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Resolvents and spectral projections} \label{section:limiting absorption} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The half-line case} \label{section:resolvents} Consider the operator \begin{equation}\label{so} H_V = -\frac{d^2}{dx^2} +V(x) \end{equation} on $\reals^+$, with Dirichlet boundary condition at the origin. Any other selfadjoint boundary condition can be treated in a similar way. For each $z\in\complex$, let $u_1(x,z), u_2(x,z)$ be the unique solutions of \begin{equation}\label{efeq} -u''+V(x)u = zu \end{equation} satisfying the boundary conditions $u_1(0,z) = (0,1)^{t}, u_2(0,z)=(1,0)^t$; the superscripts $t$ denote transposes. The classical theory of second order differential operators (see e.g. \cite{CL,Tit}) tells us that if \eqref{so} is in the limit point case, then for any $z \in \complex \setminus \reals$ there exists a unique complex number $m(z)$, called the Weyl $m$-function, such that \[ f(x,z) = u_1(x,z)m(z)+u_2(x,z) \in L^2(\reals^+). \] We will always consider potentials which lead to the limit point situation, as is the case if $V \in L^1+L^p$ for some $1\le p<\infty$; see for example \cite{ReSi2}. The Weyl $m$ function is a Herglotz function, that is, it is analytic in the upper half-plane and has positive imaginary part there. %The simplest way to see this positivity %is to consider the Wronskian %$W[f,\ov{f}]$, where in general, $W[f,g]= f'{g}-f{g}'$. %Then $W[f,\ov{f}](0)= 2i \Im m(z)$, $W[f,\ov{f}](\infty)=0$, %and $W'[f,\ov{f}](x) = -2i \Im z |f(x,z)|^2$. Hence %\[ \Im m(z) = \Im z \int_0^\infty |f(x,z)|^2\,dx. \] By direct computation, the resolvent $(H_V-z)^{-1}g$ for $z\in\complex^+$ is given by \begin{equation}\label{resol} (H_V-z)^{-1} g(x) = u_1(x,z) \int_x^\infty f(y,z)g(y)\,dy + f(x,z) \int_0^x u_1(y,z) g(y)\,dy. \end{equation} Denote by $P_{(a,b)}$ the spectral projection associated to $H_V$ and to the interval $(a,b)$. >From the resolvent formula \eqref{resol} we may derive a formula for the projections $P_{(a,b)}$. In doing so, we will use the following three routine facts. \\ 1. The functions $u_1(x,z), u_2(x,z)$ are continuous in $x$ for each $z$, and are entire holomorphic functions of $z$ for each $x$. \\ 2. The $m$ function (in fact, any Herglotz function) %satisfying an appropriate moment condition) has a representation \begin{equation} m(z) = C_1 + C_2z+ \int_\reals \left( \frac{1}{t-z} - \frac{t}{1+t^2} \right) d\mu(t) \end{equation} for some positive Borel measure $\mu$ satisfying $\int (1+|t|^2)^{-1} d\mu(t) < \infty,$ see e.g. \cite{AD}. In the $m$ function context, $\mu$ is often called the spectral measure. For Dirichlet boundary conditions, as considered here, it is the spectral measure corresponding to the generalized vector $\delta_0'$, defined by $\delta_0'(u)=u'(0)$ for any function $u$ in the domain of $H_V.$ The moment condition above corresponds to the fact that the derivative $\delta'_0$ belongs to the Sobolev-like space $H_{-2}(H_V)$ associated to $H_V$ (see e.g.\ \cite{BeSh} for details on families of Sobolev-like spaces associated with any selfadjoint operator $A$). \\ 3. $\Im m(E+i\epsilon)$ converges weakly to $\pi \mu$ as $\epsilon \rightarrow 0^+$. Moreover, $\Im m(E+i\epsilon)$ has limiting boundary values for Lebesgue-almost every $E$, and the density of the absolutely continuous part of $\mu$ satisfies \begin{equation}\label{acpart} d\mu_{\rm{ac}}(E)= \frac{1}{\pi}\Im m(E+i0) \,dE, \end{equation} where $m(E+i0) = \lim_{\e\to 0^+}m(E+i\e)$. Since the imaginary part of $m$ is simply a Poisson integral of $\mu$, this is straightforward. Fix functions $h$ and $g$ with compact support. We integrate the resolvent element $\langle (H_V-z)^{-1}g,h \rangle$ over the contour $\gamma_\epsilon$ in complex plane consisting of two horizontal intervals $(a \pm i \epsilon, b \pm i \epsilon)$, and two vertical intervals at the ends connecting them. In the limit $\epsilon \rightarrow 0^+$, the contributions of the vertical intervals disappear unless $a$ or $b$ is an eigenvalue (point mass of $\mu$); we will assume this is not the case. By the spectral theory (see, e.g. \cite{ReSi1}, Stone formula) we get then the following expression for $\langle P_{(a,b)}g,h \rangle$: \begin{multline}\label{spro} \langle P_{(a,b)}g,h \rangle = \frac{1}{2\pi i}\lim_{\epsilon \rightarrow 0} \int_{\gamma_\epsilon} \langle (H_V-z)^{-1}g,h \rangle \, dz \\ = \int_a^b \int_\reals \int_\reals u_1(x,E)u_1(y,E) g(x)\overline{h}(y)\,dx\,dy \,d\mu(E). \end{multline} In passing to the last line, we have taken into account the resolvent formula \eqref{resol}, the properties of $u_{1},u_2$ (in particular, $u_2$ drops out because of analyticity), and the fact that $\pi^{-1}\Im m(E+i\epsilon)$ converges weakly to $\mu$. The compact supports of $h,g$ ensure that the integral is well-defined. Similar formulas can be found in \cite{CL, Tit}. Since $P_{(a,b)}$ is by its definition an orthogonal projection, an immediate consequence of \eqref{spro} is that the mapping $g\mapsto \int_\reals u_1(x,E)g(x)\,dx$, initially defined for continuous $g$ having compact support, extends to an orthogonal projection from $L^2(\reals^+,dx)$ to $L^2((a,b),\mu)$. Dually, from \eqref{spro} we see that for each $g\in L^2(\reals,d\mu)$, \begin{equation} U_0 g(x) =\lim_{N \rightarrow \infty} \int_{-N}^N u_1(x,E)g(E) d\mu(E) \end{equation} exists in $L^2(\reals^+,dx)$ norm, and that the linear operator $U_0$ thus defined is a unitary bijection from $L^2(\reals,d\mu)$ to $L^2(\reals,dx)$ with inverse \begin{equation} U_0^{-1} g(E) = \lim_{N \rightarrow \infty} \int_{-N}^N u_1(x,E) g(x) dx, \end{equation} where the limit is again taken in $L^2$ norm. With the formula \eqref{spro} for the spectral projections in hand, we can invoke general spectral theory to find expressions for the spectral representation and other functions of $H_V$ (see, e.g.\ \cite{BS}). To $H_V$ and any interval $(a,b)$ are associated a maximal closed subspace of $L^2(\reals^+)$ on which $H_V$ has purely absolutely continuous spectrum, and spectrum contained in $(a,b)$. By \eqref{spro} and \eqref{acpart} as well as by definition of the absolutely continuous part of the spectral projection, the projection $P^{\rm{ac}}_{(a,b)}$ of $L^2(\reals^+)$ onto this subspace can be written as \begin{equation}\label{acpro} P_{(a,b)}^{\rm{ac}}g(x) = \frac{1}{\pi} \int_a^b u_1(x,E) \Big( \int_\reals u_1(y,E) g(y) \,dy \Big) \Im m(E+i0)\,dE. \end{equation} The integral over $\reals$ here is generally understood in the $L^2$-limiting sense; we will omit such explanatory remarks in the future. Consequently the operator $U$ mapping continuous functions with compact support to $L^2(\reals, dx)$, defined by \begin{equation} Uh(x) = \frac{1}{\pi}\int_{\reals} u_1(x,E) h(E) \Im m(E+i0) \,dE \end{equation} extends to an isometry of $L^2(\reals,\Im m(E+i0) \,dE)$ onto the absolutely continuous subspace associated to $H_V$. The unitary evolution operator on the absolutely continuous subspace is given by \begin{equation}\label{evol} e^{-iH_V t} g(x) = \frac{1}{\pi}\int_\reals e^{-iEt}u_1(x,E) \tilde{g}(E) \Im m(E+i0)\,dE, \end{equation} where \[ \tilde{g}(E) = \frac{1}{\pi}\int_\reals u_1(y,E) g(y)\,dy. \] Finally, in the case $V=0$ we can compute explicitly $u_1(x,E) = \sqrt{E}^{-1} \sin \sqrt{E}x,$ $m(z) = \sqrt{z},$ and so the evolution operator can be written as \begin{equation}\label{freeevol} e^{-iH_0 t} g(x) = \frac{1}{\pi}\int_\reals e^{-iEt} \sin(\sqrt{E}x) \hat{g}(E) \,\frac{dE}{\sqrt{E}}, \end{equation} where \[ \hat{g}(E) = \int \sin (\sqrt{E}x) g(x)\,dx. \] For almost every $E>0$, define the scattering coefficient $\gamma(E)\in\complex$ by \begin{equation} \label{reflectioncoefficientdefn} \gamma(E) = 1/ u(0,E) \end{equation} where $u(x,E)$ is the unique generalized eigenfunction asymptotic to $\exp(i\xi(x,E))$ as $x\to+\infty$, whose existence was established in Theorem~\ref{thm:complexeigenfunctions}. The following proposition connects the formulae of this section with the generalized eigenfunctions analyzed in \S\ref{section:complex}. \begin{proposition}[Limiting absorption principle]\label{relim} Assume that $V\in L^1+L^p(\reals)$ for some $10$ \[ u_1(x,E)m(E+i0)+u_2(x,E)=\frac{u(x,E)}{u(0,E)}= \gamma(E) e^{i \xi(x,E)}(1+o(1)) \] as $x \rightarrow \infty$ (there is no absolutely continuous spectrum for $E<0$). The relation between $\gamma$ and $m(E+i0)$ follows by comparing the Wronskians of the left and right hand sides (taken with their complex conjugates). \end{proof} %\noindent \it Remark. {\bf to be deleted later} %\rm Notice that the whole proof used very few specific %properties of $\xi(x,z),$ namely \eqref{lbph}, \eqref{ph1}, \eqref{ph2} % and \eqref{ph3}. %In particular, it can be readily adapted %to the case of slowly varying potentials or Stark operators. Now we are going to rewrite the spectral representation in a manner convenient for the scattering theory. Notice that \[ u_1(x,E) = \frac{1}{2i \Im m(E+i0)} \left( \gamma(E)u(x,E) - \ov{\gamma}(E)\ov{u}(x,E) \right). \] Introduce \begin{equation} \psi(x,\lambda) = \lambda \ov{\gamma}(\lambda^2) u_1(x,\lambda^2) = \frac{1}{2i}\Big( u(x,\lambda^2) - \frac{{\ov{\gamma}(\lambda^2)}}{\gamma(\lambda^2)} \cdot \ov{u}(x,\lambda^2) \Big). \end{equation} where $u(x,E)$ continue to denote the generalized eigenfunctions whose existence was established in Theorem~\ref{thm:complexeigenfunctions}, now for $E\in\reals^+$. Then the results of this subsection may be summarized as follows. \begin{proposition} Suppose that $V\in L^1+L^p(\reals^+)$ for some $1From the preceding section recall the generalized eigenfunctions $\psi(x,\lambda) = (2i)\rp(u - e^{i\omega(\lambda)}\ov{u}))$, where $\exp(i\omega(\lambda)) = \ov\gamma(\lambda^2)/\gamma(\lambda^2)$. Recall the unitary bijection $U_V:L^2(\reals^+,d\lambda)\mapsto \scripth_{\text{ac}}$ defined by $U_Vf(x) = \sqrt{2/\pi}\int_0^\infty f(\lambda)\psi(x,\lambda)\,d\lambda$. Finally, recall that $e^{-iH_Vt}(U_V f)(x) = \sqrt{2/\pi}\int_0^\infty e^{-i\lambda^2 t}f(\lambda )\psi(x,\lambda)\, d\lambda$. Fix a martingale structure $\{E^m_j\}$ on $\reals^+$ that is adapted to $V$, in the sense that $\int_{E^m_j}|V|^p = 2^{-m}\int_{\reals^+}|V|^p$ for all $m\ge 0$ and all $j$. For any sufficiently small $\delta>0$, recall the functional \begin{equation*} g_\delta(f)(\lambda) = \sum_{m=1}^\infty 2^{m\delta} \Big( \sum_{j=1}^{2^m} \big|\int_{E^m_j} e^{2i\phi(x,\lambda)}f(x)\,dx\big|^2 + \big|\int_{E^m_j} e^{-2i\phi(x,\lambda)}f(x)\,dx\big|^2 \Big)^{1/2}. \end{equation*} We have shown that for all sufficiently small $\delta_0$, $g_{\delta_0}(V)(\lambda)<\infty$ for almost every $\lambda\in\reals^+$. Fix some $\delta<\delta_0$. \begin{definition} A compact set $\Lambda\Subset(0,\infty)$ is said to be a {\em set of uniformity} if $g_\delta(V)$ is a bounded function of $\lambda\in\Lambda$, and if $u(x,\lambda)-e^{i\phi(x,\lambda)}\to 0$ as $x\to+\infty$, uniformly for all $\lambda\in\Lambda$. \end{definition} \begin{lemma} For any $f\in L^2(\reals^+)$, for any $R<\infty$, \begin{equation*} \|e^{-itH_V}U_V f\|_{L^2([0,R])} \to 0 \ \ \text{as } t\to\infty. \end{equation*} \end{lemma} \begin{proof} Since $\exp(-itH_V)$ is unitary, it suffices to prove this for all $f$ in some dense subspace of $L^2(\reals^+)$; we choose the subspace consisting of all $f$ supported on some set of uniformity $\Lambda$ (which depends on $f$). The functions $\psi(x,\lambda)$ are uniformly bounded on $[0,\infty)\times\Lambda$, as are their derivatives with respect to $x$, from which it follows that $U_V$ is a compact mapping from $L^2(\Lambda)$ to $L^2([0,R])$. Thus it suffices to establish weak convergence to zero. Now for any $h\in L^2([0,R])$, \begin{equation} \langle e^{-itH_V}U_V f,h\rangle = \int_\Lambda e^{-i \lambda^2 t }f(\lambda)U_V^*h(\lambda)\,d\lambda, \end{equation} which converges to zero by the Riemann-Lebesgue lemma, since $U_V^*h\in L^2(\reals^+)$. \end{proof} Define \begin{align} \psi_0(x,\lambda) & = (2i)\rp\big( e^{i\phi(x,\lambda)} -e^{i\omega(\lambda)}e^{-i\phi(x,\lambda)}\big) \\ U_V^\dagger f(x) & = \sqrt{2/\pi} \int_0^\infty f(\lambda) \psi_0(x,\lambda)\,d\lambda; \end{align} these are approximations to $\psi,U_V$, respectively. The next lemma is the key step in showing that only the leading-order approximation $\psi_0$ to $\psi$ contributes to the long-time asymptotics of the wave group. \begin{lemma} \label{truekeylemma} For any set of uniformity $\Lambda$ and any $R>0$, there exists $C(R,\Lambda)<\infty$ such that for all $f\in L^\infty(\Lambda)$, \begin{equation} \|(U_V-U_V^\dagger)f\|_{L^2([R,\infty))} \le C(R,\Lambda)\|f\|_{L^\infty}, \ \text{ where }\ C(R,\Lambda)\to 0 \text{ as } R\to\infty. \end{equation} \end{lemma} \begin{proof} It suffices to prove this with $\psi(x,\lambda)-\psi_0(x,\lambda)$ replaced by \begin{multline} \Psi(x,\lambda) - e^{i\phi(x,\lambda)} \\ = e^{i\phi(x,\lambda)}\sum_{n=1}^\infty S_{2n}(V,V,\dots,V)(x,\lambda) - e^{-i\phi(x,\lambda)}\sum_{n=0}^\infty S_{2n+1}(V,V,\dots,V)(x,\lambda), \end{multline} for the conclusion for the other terms follows from this by complex conjugation. We argue by duality; let $h$ be an arbitrary function in $L^2([R,\infty))$ of norm $1$, and consider \begin{multline} \int_R^\infty e^{i\phi(x,\lambda)}h(x)S_{2n}(V,V,\dots,V)(x,\lambda) \\ = \iint_{R\le t_0\le t_1\le\cdots\le t_{2n}} e^{i\phi(t_0,\lambda)}h(t_0)\,dt_0\, \prod_{k=1}^{2n}e^{\pm_k 2i\phi(t_k,\lambda)} V(t_k)\,dt_k. \end{multline} Here $\pm_k$ denotes a plus or minus sign depending on $k$ in any manner; in fact these signs alternate in our expansion, but that is of no importance here. Introduce \begin{equation} g_{-\delta}(h)(\lambda) = \sum_{m=1}^\infty 2^{-m\delta} \Big( \sum_{j=1}^{2^m} \big|\int_{E^m_j} e^{i\phi(x,\lambda)}h(x)\,dx\big|^2 + \big|\int_{E^m_j} e^{-i\phi(x,\lambda)}h(x)\,dx\big|^2 \Big)^{1/2}. \end{equation} By Proposition~\ref{prop:numericalbound}, \begin{equation} \Big|\iint_{R\le t_0\le t_1\le\cdots\le t_{2n}} e^{i\phi(t_0,\lambda)}h(t_0)\,dt_0\, \prod_{k=1}^{2n} e^{\pm_k 2i\phi(t_k,\lambda)}V(t_k)\,dt_k \Big| \le \frac{C^{n+1}}{\sqrt{2n!}} g_{-\delta}(h)(\lambda)g_\delta(V_R)^{2n}(\lambda) \end{equation} where $V_R(x) = V(x)\chi_{[R,\infty)}(x)$; there is a corresponding bound for the terms arising from the multilinear expressions $S_{2n+1}$. We may dominate $\sup_R g_\delta(V_R)(\lambda)^2$ by $C' g_{\delta'}(V)(\lambda)^2$ for any $\delta'>\delta$; to simplify notation we replace $\delta'$ again by $\delta$. Consequently by summing over $n$ and comparing with the Taylor expansion for the exponential function, \begin{multline} \Big| \langle \int_\Lambda \int_R^\infty f(\lambda)\left( u(x,\lambda) -e^{i \phi(x,\lambda)}\right) h(x)\,dx\,d\lambda \rangle\Big| \\ \le \int_\Lambda Cg_{-\delta}(h)(\lambda) g_\delta(V_R)(\lambda) e^{Cg_\delta(V)(\lambda)^2} |f(\lambda)|\,d\lambda. \end{multline} Recall that for any $\delta>0$, $\|g_{-\delta}(h)\|_{L^2(\Lambda)} \le C_{\Lambda,\delta}\|h\|_{L^2}\le C_{\Lambda,\delta}<\infty$, for any compact $\Lambda\Subset(0,\infty)$, uniformly over all martingale structures, not necessarily adapted to $h$. The factor $\exp(Cg_\delta(V)(\lambda)^2)$ is bounded uniformly for $\lambda\in\Lambda$, by definition of a set of uniformity. Thus it suffices to show that $\|g_\delta(V_R)(\lambda)\|_{L^2(\Lambda,d\lambda)}\to 0$ as $R\to\infty$. Now in the sum \eqref{eq:defnofgdelta} defining $g_\delta(V)$, the $\ell^2$ sum over $j$ for fixed $m$ is $\le C2^{-\e m}\|V_R\|_{L^2}$ in $L^2(\Lambda)$ for some $\e>0$, so it suffices to show that $\int_{E^m_j}e^{2i\phi(x,\lambda)}V_R(\lambda)\,dx \to 0$ in $L^2(\Lambda)$. This holds by Lemma~\ref{lemma:parseval}, since $V_R\to 0$ in $L^1+L^p$ norm. \end{proof} \begin{corollary} \label{cor:dropmultilinearterms} Let $\rho>0$. For any $f\in L^2(\reals^+,d\lambda)$ supported on $[\rho,\infty)$, \begin{equation} \|e^{-itH_V}U_V f-\sqrt{2/\pi} \int_0^\infty e^{-i\lambda^2 t}f(\lambda)\psi(x,\lambda) \,d\lambda\|_{L^2(\reals^+)} \to 0 \text{ as } |t|\to\infty. \end{equation} \end{corollary} The restriction on the support of $f$ comes about because of the factor of $\lambda\rp$ in the definition of $\phi$, which causes difficulties as $\lambda\to 0$. \begin{proof} It is straightforward to show that $\int_0^\infty e^{-i\lambda^2 t}f(\lambda)\psi(x,\lambda) \,d\lambda \to 0$ in $L^2([0,R])$ for any finite $R$. Let $\e>0$. Choose a set of uniformity $\Lambda\subset[\rho,\infty)$ and a function $F\in L^\infty (\Lambda)$ such that $\|f-F\|_{L^2(\reals^+)}<\e$. Then $\|e^{-itH_V}(U_V f-U_V F)\|_{L^2(\reals^+)}<\e$ for every $t$, as well. By the preceding lemma, there exists $R<\infty$ such that $\|e^{-itH_V}U_V F-\sqrt{2/\pi} \int_0^\infty e^{-i\lambda^2 t}F(\lambda)\psi_0(x,\lambda) \,d\lambda\|_{L^2([R,\infty))}<\e$ for all $t\in\reals$. Fixing such an $R$, there exists $T<\infty$ such that both $e^{-itH_V}U_V F$ and $\sqrt{2/\pi}\int_0^\infty e^{-i\lambda^2 t}F(E)\psi_0(x,\lambda) \,d\lambda$ have $L^2([0,R])$ norms $<\e$, for all $|t|\ge T$. Thus \begin{equation} \|e^{-itH_V}U_V f-\sqrt{2/\pi}\int_0^\infty e^{-i\lambda^2 t}F(\lambda)\psi_0(x,\lambda) \,d\lambda\|_{L^2(\reals^+)}<4\e\text{ for all } |t|\ge T. \end{equation} Finally, note that $\|\int_\rho^\infty g(\lambda) u_0(x,\lambda)\,d\lambda\|_{L^2(\reals^+)} \le C_\rho \|g\|_{L^2([\rho,\infty))}$ for all $g$, where $C_\rho<\infty$ for all $\rho>0$; this follows from the proof of Lemma~\ref{lemma:parseval}. Applying this to $g=f-F$ allows us to replace $F$ again by $f$ in the preceding inequality, at the expense of replacing $4\e$ by $C_\rho\e$. Since $\e>0$ was arbitrary, this completes the proof. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{A time-dependent phase correction} \label{section:phasereduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The goal of this section is to convert the leading-order asymptotics $\exp(\pm i\lambda x \mp i(2\lambda)\rp\int_0^x V)$ to a more standard form, $\exp(\pm i\lambda x \mp i(2\lambda)\rp\int_0^{2\lambda |t|} V)$. \begin{lemma} \label{lemma:easyballisticapproximation} Let $V\in L^1+L^2$. For any $f\in L^2$ supported in a compact subinterval of $(0,\infty)$, \begin{multline} \label{eq:twoeasyterms} \Big\|\int_0^\infty e^{-i\lambda^2 t+i\lambda x -i(2\lambda)\rp\int_0^x V} f(\lambda)\,d\lambda \Big\|_{L^2(\reals^+)} \\ + \Big\|\int_0^\infty e^{-i\lambda^2 t+i\lambda x -i(2\lambda)\rp\int_0^{2\lambda |t|} V} f(\lambda)\,d\lambda \Big\|_{L^2(\reals^+)} \to 0 \text{ as } t\to -\infty. \end{multline} An analogous statement holds as $t\to+\infty$, and follows by taking complex conjugates. \end{lemma} \begin{proof} Since the mapping $f\mapsto \int_0^\infty e^{-i\lambda^2 t+i\lambda x} e^{-i(2\lambda)\rp\int_0^x V} f(\lambda)\,d\lambda$ is bounded from $L^2([\rho,\infty))$ to $L^2(\reals^+)$ for every $\rho>0$, and since the variant defined by replacing $\int_0^x V$ by $\int_0^{2\lambda t}V$ is unitary, it suffices to prove this merely for $f\in C^\infty_0$, which we assume henceforth. Let $\Lambda\Subset\reals^+$ be the support of $f$. By integrating by parts once with respect to $\lambda$, integrating $\exp(i[-\lambda^2 t+\lambda x])$ and differentiating the rest, and by moving absolute value signs inside the integral, we obtain a pointwise in $(x,t)$ bound \begin{multline} |\eqref{eq:twoeasyterms}| \le C\int_\Lambda |x-2\lambda t|\rp \big(1+ |{\textstyle\int}_0^x V| + |{\textstyle\int}_0^{2\lambda |t|}V| + |tV(2\lambda t)| \big) \,d\lambda \\ \le C(|x|+|t|)\rp (\big(1+ |{\textstyle\int}_0^x V| + |{\textstyle\int}_0^{C |t|}V| \big) \end{multline} for the integrands in \eqref{eq:twoeasyterms}. For $t\le -1$, $|x-2\lambda t|\rp\sim (x+|t|)\rp$, which tends to zero in $L^2(\reals^+,dx)$. Thus the first term behaves as desired. The second and third terms do also, if $V\in L^1$. For $V\in L^2$, $x\rp\int_0^x V\in L^2$ as well, so the term $(x+|t|)\rp \int_0^x V$ is an $L^2$ function times $x/(x+|t|)$, hence $\to 0$ in $L^2$ norm. Lastly, $\int_0^{C|t|}V=o(|t|^{1/2})$, while $|t|^{1/2}/(x+|t|)$ is $O(1)$ in $L^2(\reals^+)$. \end{proof} \begin{lemma} \label{lemma:ballisticapproximation} Let $V\in L^1+L^2$. For any $f\in L^2$ supported in a compact subinterval of $(0,\infty)$, \begin{equation} \label{differenceofcorrections} \Big\| \int_0^\infty e^{-i\lambda^2 t+i\lambda x} \Big[ e^{-i(2\lambda)\rp\int_0^x V}- e^{-i(2\lambda)\rp\int_0^{2\lambda |t|} V} \Big] f(\lambda)\,d\lambda \Big\|_{L^2(\reals^+)}\to 0 \text{ as } t\to +\infty. \end{equation} \end{lemma} \begin{proof} As in the preceding lemma, it suffices to prove this for $f\in C^\infty_0(\reals^+)$. Fix a cutoff function $\eta\in C^\infty_0(\reals)$ supported on $(-2,2)$ and $\equiv 1$ on $[-1,1]$, with $0\le\eta\le 1$. Let $\epsilon$ be a strictly positive function, such that $\epsilon(t)\to 0$ as $t\to+\infty$, at a rate to be specified below. Set \begin{equation} \eta(\lambda,t,x) = \eta(\epsilon(t) t^{-1/2} (x-2\lambda t)). \end{equation} Consider the function of $(x,t)$ \begin{equation} \label{eq:functionofxandt} \int_0^\infty e^{-i\lambda^2 t+i\lambda x} \Big[ e^{-i(2\lambda)\rp\int_0^x V}- e^{-i(2\lambda)\rp\int_0^{2\lambda t} V} \Big] f(\lambda)\,\eta(\lambda,t,x)\,d\lambda\ . \end{equation} We have \begin{equation}\begin{split} |\eqref{eq:functionofxandt}| &\le C \int \int_{|y-2\lambda t|\le 2\epsilon(t)\rp t^{1/2}} |V(y)|\,dy \chi_{|x-2\lambda t|\le 2\epsilon(t)\rp t^{1/2}} \,d\lambda \\ & \le C \int \int_{|y-x|\le 4\epsilon(t)\rp t^{1/2}} |V(y)|\,dy \chi_{|x-2\lambda t|\le 2\epsilon(t)\rp t^{1/2}} \,d\lambda \\ &\le C\epsilon(t)\rp t^{-1/2} \int_{|y-x|\le 4\epsilon(t)\rp t^{1/2}} |V(y)|\,dy \\ &\le C\epsilon(t)^{-2}MV(x), \end{split}\end{equation} where $M$ is the maximal function of Hardy and Littlewood. Observe that the restriction $\eta(\lambda,t,x)\ne 0$ for some $\lambda\in\Lambda$ implies that $ct\le x\le Ct$ for some $c,C\in\reals^+$ depending only on $\Lambda$, provided $\epsilon(t)\ll t^{1/2}$, and moreover that $V$ may be replaced by its restriction to such an interval. If $V\in L^2$ then the $L^2$ norm of $M$ applied to the restriction of $V$ to such an interval tends to zero as $t\to\infty$, and by choosing $\epsilon(t)$ to tend to zero sufficiently slowly we find that $\eqref{eq:functionofxandt}\to 0$ in $L^2$. For $V\in L^1$ we have the pointwise bound $C\epsilon(t)\rp t^{-1/2}\int_{ct}^{Ct}|V|$, which is $o(1)\cdot\epsilon(t)\rp$ in $L^2(x\sim t)$; this tends to zero provided $\epsilon(t)$ does so sufficiently slowly. In the general case $V\in L^1+L^2$, we decompose \eqref{eq:functionofxandt} into two parts, and estimate them separately. There remains the contribution of $1-\eta(\lambda,t,x)$: \begin{equation} \int_0^\infty e^{-i\lambda^2 t+ i\lambda x} f(\lambda) \Big(e^{-i(2\lambda)\rp \int_0^{x}V} -e^{-i(2\lambda)\rp \int_0^{2\lambda t}V} \Big) (1-\eta)(\lambda,t,x) \,d\lambda. \end{equation} We would like to apply to it the method of stationary phase. However, the phase function $-t\lambda^2+\lambda x-(2\lambda)\rp\int_0^{2\lambda t}V$ is not well behaved; its partial derivative with respect to $\lambda$ is $x-2\lambda t +(2\lambda)^{-2}(\int_0^{2\lambda t}V) -(2\lambda)\rp 2tV(2\lambda t)$, and the final term is not well under control unless one assumes $V(x) = O(|x|^{-1/2})$. Instead, we integrate by parts, integrating $\exp(i[\lambda x-\lambda^2 t])$ and differentiating the rest, to obtain \begin{equation} \label{eq:abigjob} i\int_0^\infty e^{-i\lambda^2 t+ i\lambda x} f(\lambda) \frac{\p}{\p\lambda} \Big[(x-2\lambda t)\rp \Big(e^{-i(2\lambda)\rp \int_0^{x}V} -e^{-i(2\lambda)\rp \int_0^{2\lambda t}V} \Big) (1-\eta)(\lambda,t,x) \Big] \,d\lambda. \end{equation} When the derivative is expanded according to Leibniz's rule, various terms result. The main terms are those in which $\partial/\partial\lambda$ acts on either of the two exponentials $\exp( i(2\lambda)\rp(\int_0^* V) )$, and we discuss these first. One such term is a constant multiple of \begin{multline} \int_0^\infty e^{-i\lambda^2 t+ i\lambda x -i(2\lambda)\rp \int_0^{2\lambda t}V} f(\lambda) (x-2\lambda t)\rp (1-\eta)(\lambda,t,x) \lambda\rp tV(2\lambda t)\,d\lambda \\ = c \int_0^\infty e^{-i(y^2/4t)+ i(xy/2t) -it y\rp \int_0^{y}V} \tilde f(y/2t) (x-y)\rp (1-\eta)(\lambda,t,x) v(y) \,dy \end{multline} where we have substituted $y = 2\lambda t$ and written $\tilde f(\lambda) = \lambda\rp f(\lambda)\in C^\infty_0$, and where $v=V$ (for the present moment only). The integral operators with kernels $\exp(iAxy)(x-y)\rp \eta(\delta(x-y))$ are bounded on $L^2(\reals)$, uniformly in $A,\delta\in\reals^+$. Thus the $L^2(dx)$ norm of this last expression is $O(\|v\|_{L^2})$. Moreover, since $f$ is supported where $\lambda\ge\rho>0$, only the restriction of $v$ to $[2\rho t,\infty)$ comes into play, so we obtain a bound of $C\|v\|_{L^2([2\rho t,\infty))}$. This holds uniformly in all real-valued functions $V$ appearing in the exponent. There is also an easy alternative bound $O(t^{-1/4}\|v\|_{L^1})$, obtained directly by inserting absolute values inside the integral. Thus the general case $V\in L^1+L^2$ may be treated by decomposing $v$ as a sum, and estimating the two terms separately. Another term arising from \eqref{eq:abigjob} differs only in that $tV(2\lambda t)$ is replaced by $c\lambda^{-2}\int_0^{2\lambda t}V$; in terms of the new variable $y$, $v(y)$ is replaced by $t\rp\int_0^y v$. Since $\lambda$ ranges over a compact interval $\Lambda$, $y$ ranges over an interval $[ct,Ct]$ where $00$, by the relation \eqref{reflectioncoefficientdefn} $\gamma(\lambda^2) = 1/\psi(0,\lambda)$. Writing $\gamma(\lambda^2)= |\gamma(\lambda^2)|e^{i\arg{\gamma(\lambda^2)}}$, we change the definition of $\psi(x,\lambda)$ to \begin{equation} \psi(x,\lambda) = e^{i\arg{\gamma(\lambda^2)}} u(x,\lambda^2) - e^{-i\arg{\gamma(\lambda^2)}}\ov{u(x,\lambda^2)}. \end{equation} In the formal expressions for the spectral projectors and wave group, $\psi$ is now replaced by this new $\psi$. Retracing the above analysis, we find that the wave operators exist, and \begin{equation}\begin{split} \Omega_- &= U_V^{-1}\circ e^{-i\arg{\gamma(\lambda^2)}} \circ U_0, \\ \Omega_+ &= U_V^{-1}\circ e^{i\arg{\gamma(\lambda^2)}} \circ U_0, \label{mwo} \\ S &= U_0\rp\circ e^{2i\arg{\gamma(\lambda^2)}} \circ U_0. \end{split}\end{equation} \end{proof} {\em Remark.\/} The exact expressions for the modified wave operators $\Omega^m_\pm$ depend on a choice of normalization of the solutions $\psi$; see below. These solutions can be modified by factors $e^{i\kappa(\lambda)}$, leading to different $U_V$ and hence different looking expressions for the wave operators, like in \eqref{mwo} and in the proof of Theorem~\ref{thm:waveoperators}. However, the scattering matrix $S(\lambda)$ is invariant under choice of such normalization. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The whole-line case} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In the full-axis Schr\"odinger case, the results are similar. One defines modified wave operators by \begin{equation} \label{eq:wholeaxiswaveoperatorsdef} \Omega_\pm^m f =\lim_{ t \rightarrow \mp \infty} e^{itH_V}e^{-iW_a(-i\partial_x,t)} f, \end{equation} where $\partial_x$ is the operator of one differentiation in $x$, and \begin{equation} W_a(\lambda,t) = \lambda^2 + \frac{1}{2\lambda} \int_0^{2\lambda t} V(s)\,ds. \end{equation} The modified free evolution operator can be written as (compare with \eqref{frevwa}) \begin{equation}\label{modevol} e^{-iW_a(-i\partial_x,t)}f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-i\lambda ^2t +i\lambda x - \frac{i}{2\lambda} \int_0^{2\lambda t} V(s)\,ds} \hat{f}(\lambda)\,d\lambda, \end{equation} where $\hat{f}(\lambda)= \int \exp(-i\lambda x)f(x)\,dx$ is the Fourier transform of $f$. Recall the representation \eqref{perevwa} for the perturbed evolution (we set here $E=\lambda^2$) \begin{equation}\label{impevol} e^{-iH_Vt} f(x) = \frac{1}{2\pi} \int_0^\infty e^{-i\lambda^2t} (\psi_+(x,\lambda)\tilde{f}_+(\lambda) + \psi_-(x,\lambda)\tilde{f}_-(\lambda) )\,d\lambda. \end{equation} Denote by $U_V$ the operator $f\mapsto (f_+,f_-)$, where $f_\pm(\lambda) = \int_\reals \ov{\psi}_\pm (x,\lambda) f(x)\,dx$. Denote by $\scripth_{\text{ac}}$ the maximal closed subspace of $\scripth=L^2(\reals)$ on which $H_V$ has purely absolutely continuous spectrum. \begin{theorem} Let $V\in L^1+L^p(\reals)$ for some $10$, so asymptotic completeness in the relatively weak sense in which we have defined it implies a stronger form. Let us say that the half-axis operator $H_V^\beta$ has boundary condition $\beta$ at the origin if $u(0)+\beta u'(0)=0$ for any function $u$ in the domain of $H_V^\beta.$ \begin{theorem}\label{bcascom} Let $H_V^\beta$ be a Schr\"odinger operator defined on a half-axis with the boundary condition $\beta.$ Assume that $V \in L^1+L^p$ for some $10$ for a.e. $E.$ The variation of the boundary condition can be regarded as rank one perturbation \cite{SVan}, Section I.6. The standard rank one perturbation theory then implies that for any $\beta$, the singular part of the spectral measure $\mu_{\rm{s}}^{\beta}$ can only be supported on a fixed set of energies $S \subset \reals^+$ of zero Lebesgue measure \cite{SVan}, Theorem II.2. But then again by rank one theory for almost every $\beta$ the singular spectrum on $\reals^+$ is empty \cite{SVan}, Theorem I.8. Since the potential belongs to $L^1+L^p,$ the spectrum below zero is discrete (with only $0$ as a possible accumulation point). \end{proof} Next, let us consider the following random model: \begin{equation}\label{rex} V(x) = \sum\limits_{n=1}^\infty a_n(\omega)g(n) f(x-n), \end{equation} where $f(x) \in C_0^\infty (0,1),$ $g \in l^p$ for some $p<2$, and $a_n(\omega)$ are independent identically distributed bounded random variables with zero expectation. We have \begin{theorem}\label{ranascom} Let $H_V$ be a one dimensional Sch\"odinger operator with random potential \eqref{rex}. Then with probability one, the wave operators $\Omega_\pm$ exist and are asymptotically complete. \end{theorem} Indeed, Theorem 9.1 of \cite{KLS} shows that almost surely, the spectrum of the Schr\"odinger operator with potential \eqref{rex} is purely absolutely continuous on $\reals^+.$ Notice that our assumptions on the potential easily imply that the improper integral $\int_0^\infty V$ exists almost surely. Theorem~\ref{thm:waveoperators} and decay of the potential then imply asymptotic completeness. This illustrates another type of situation where asymptotic completeness holds. We remark that the result holds in a variety of more general random models for which \eqref{rex} is just an illustration. For a more general setting, see \cite{KLS}. \smallskip {\it Discussion.\/} The above is only one of various possible definitions of asymptotic completeness. The notion of asymptotic completeness is intended to describe a situation where the Hilbert space is split into two orthogonal subspaces \cite{RS3}: $\scripth_{\text{pp}}(H_V)$ and the range of wave operators, $\scripth_{\text{ac}}(H_V).$ On $\scripth_{\text{ac}}(H_V)$ the perturbed dynamics is close to the modified free evolution at large times, and corresponds to the scattering states. On $\scripth_{\text{pp}}(H_V)$ the dynamics is supposed to be bounded in some sense. However, the intuitive physical assumption that pure point spectrum leads to dynamics which is bounded needs to be clarified, and in recent years there have appeared examples with very non-trivial transport on the pure point component. A %reasonable and widely accepted way to calibrate transport properties is to consider evolution of the averaged moments of coordinate operator: \begin{equation}\label{moments} \langle \langle |X|^m \rangle_\phi \rangle_T = \frac{1}{T} \int\limits_0^T | \langle e^{-iHt}\phi, |X|^m e^{-iHt}\phi \rangle | \, dt, \end{equation} where $\phi$ is the initial state, $\langle \phi_1, \phi_2 \rangle$ is the inner product and $|X|^m$ the operator of multiplication by $(|x|+1)^m$ in coordinate representation. The paper \cite{DJLS} contains an example of a (discrete) Schr\"odinger operator $h$ with pure point spectrum and exponentially decaying eigenfunctions, such that \begin{equation}\label{pointran} \limsup_{t \rightarrow \infty} \langle \langle |X|^2 \rangle_{\delta_0} \rangle_T /T^{\alpha} = \infty \end{equation} for any $\alpha <2.$ Here the initial state is $\delta_0,$ the vector localized at the origin. Given that the rate of growth $T^2$ for the second moment corresponds to ballistic motion, as for the free Laplace operator, this example shows that in some sense the transport associated with point spectrum can be very fast. %Because of such examples, it is clear that the definition given above does not completely %correspond to the %intuitive picture we would like to capture. However, there is still an important difference between transport associated with point and singular continuous spectrum. Namely, let $B_R$ denote the ball of radius $R$ centered at the origin and $B_R^c$ its complement. Let $P_{{\rm pp}}$ and $P_{{\rm c}}$ be the orthogonal projections on $\scripth_{\text{pp}}(H_V)$ and the continuous subspace $\scripth_{\text{c}}(H_V)$ respectively. %Then if the projection of the vector $\phi$ %on $\scripth_{\text{pp}}(H_V)$ is nonzero, Then %we have that for any $\epsilon >0$ there exists $R_\epsilon$ such that \begin{equation}\label{ppbound} \|e^{-iH_Vt} P_{{\rm pp}}\phi \|^2_{L^2(B^c_{R_\epsilon})} < \epsilon \end{equation} for all $t.$ The growth of the moments in \eqref{pointran} is achieved not because of the motion of the whole wavepacket, but because of thin tails escaping to infinity. On the other hand, we have \begin{equation}\label{probfin} \frac{1}{T} \int\limits_0^T \| e^{-iH_Vt} P_{{\rm c}} \phi \|^2_{L^2(B_R)} \,dt \stackrel{T \rightarrow \infty}{\longrightarrow} 0 \end{equation} %converges as $T \rightarrow \infty$ to a positive number for any finite $R.$ %which is sufficiently large. %That it is, the average probability to find a particle in a finite ball is greater than a %fixed positive number. Equation \eqref{probfin} is one of the statements of the RAGE theorem (see, e.g. \cite{CFKS}) and is basically a corollary of Wiener's theorem on Fourier transforms of measures. %with pure point component, %which also implies that conversely, %if the projection of $\phi$ on $\scripth_{\text{pp}}(H_V)$ is zero, %then \eqref{probfin} tends to zero as $T \rightarrow \infty.$ Morever, there exist examples of Schr\"odinger operators \cite{KL} in which the dynamics corresponding to the singular continuous subspace is almost ballistic in a sense that the whole wavepacket is moving to infinity at a fast rate: for any $\rho>0$ there exists $C_\rho$ such that \[ \frac{1}{T} \int\limits_0^T \| e^{-iH_Vt} \phi \|^2_{L^2(B_{C_\rho T^{1-\epsilon}})} \, dt < \rho \] for all $T$ and $\phi$ lying in the singular continuous subspace of $H_V$. \section{Potentials in $\ell^p(L^1)$} Following \cite{christkiselevdecaying}, we sketch here the small modifications needed to extend the analysis of potentials in $L^p$ to those in $L^1+L^p$, and indeed those in the larger class $\ell^p(L^1)$. A locally integrable function $f$ is said to belong to the amalgamated space $\ell^p(L^1)$ if \begin{equation*} \sum_{n\in\integers} \big(\int_n^{n+1}|f(x)|dx \big)^p<\infty. \end{equation*} The norm $\|f\|_{\ell^p(L^1)}$ is the $p$-th root of this expression. For any $1\le p<\infty$, this defines the Banach space $\ell^p(L^1)$, which contains $L^1+L^p$. A martingale structure $\{E^m_j\}$ is said to be adapted to $f$ in $\ell^p(L^1)$ if \begin{equation} \|f\cdot\chi_{E^m_j}\|_{\ell^p(L^1)}^p \le 2^{-m} \|f\|_{\ell^p(L^1)}^p \end{equation} for all $m,j$. For any $f \in\ell^p(L^1)$, there does exist an adapted martingale structure \cite{christkiselevdecaying}. Lemma~\ref{lemma:parseval} extends to $\ell^p(L^1)$: this Banach space is mapped boundedly to $L^{p'}(\Lambda)$ for any compact interval $\Lambda\Subset(0,\infty)$, for any $1\le p\le 2$. The proof is essentially unchanged; see the analogous proof of Proposition~3.5 of \cite{christkiselevdecaying}. Corollary~\ref{cor:nontangentialmaxbound} may now be refined by replacing $\|f\|_{L^p}$ by $\|f\|_{\ell^p(L^1)}$ on the right-hand side of each inequality. With these bounds for the operator $G$ in hand, the remainder of the proof is unchanged. \medskip {\bf Acknowledgement} We thank Barry Simon for useful discussions. \\ \begin{thebibliography}{99} \bibitem{AK} P.K.~Alsholm and T.~Kato, \it Scattering with long range potentials, \rm Partial Diff. Eq., Proc. Symp. Pure Math. Vol. {\bf 23}, Amer. Math. Soc., Providence, Rhode Island, 1973, 393--399 \bibitem{AD} N.~Aronszajn and W.~Donoghue, \it On exponential representations of analytic functions in the upper half-plane with positive imaginary part, \rm J. Analyse Math. {\bf 15} (1957), 321--388 \bibitem{BA} M.~Ben-Artzi, \it On the absolute continuity of Schr\"odinger operators with spherically symmetric, long-range potentials. I, II. \rm J. Differential Equations {\bf 38} (1980), 41--50, 51--60 \bibitem{BAD} M.~Ben-Artzi and A.~Devinatz, \it Spectral and scattering theory for the adiabatic oscillator and related potentials, \rm J. Math. Phys. {\bf 20} (1979), 594--607 \bibitem{BeSh} F.~Berezin and M.~Shubin, \it The Schr\"odinger Equation, \rm Mathematics and its applications (Kluwer Academic Publishers). Soviet series 66, 1991 \bibitem{BS} M.~Birman and M.~Solomyak, \it Spectral Theory of Selfadjoint Operators in Hilbert Space, \rm D.~Reidel Publishing, Dordrecht, 1987 \bibitem{BM} V.S.~Buslaev and V.B.~Matveev, \it Wave operators for Schr\"odinger equation with slowly decreasing potentials, \rm Theor. Math. Phys. {\bf 2} (1970), 266--274 \bibitem{christkiselevpowerdecay} M.~Christ and A.~Kiselev, \it Absolutely continuous spectrum for one-dimensional Schr\"odinger operators with slowly decaying potentials: Some optimal results, \rm J. Amer. Math. Soc. {\bf 11} (1998), 771--797 \bibitem{christkiselevdecaying} \bysame, % = CK1 {\em WKB asymptotics of generalized eigenfunctions of one-dimensional Schr\"odinger oeprators}, J. Funct. Anal. 179 (2001), 426-447. \bibitem{christkiselevfiltrations} % = CK2 \bysame, {\em Maximal functions associated to filtrations}, J. Funct. Anal. 179 (2001), 409-425. \bibitem{christkiselevslowlyvarying} \bysame, {\em WKB and spectral analysis of one-dimensional Schr\"odinger operators with slowly varying potentials}, Comm. Math. Phys. 218 (2001), 245-262. \bibitem{ipamnotes} \bysame, {\em One-dimensional Schr\"odinger operators with slowly decaying potentials: Spectra and Asymptotics}, Institute for Pure and Applied Mathematics lecture notes, March 2001. \bibitem{CL} E.A.~Coddington and N.~Levinson, \it Theory of Ordinary Differential Equations, \rm McGraw-Hill, New York, 1955 \bibitem{CFKS} H.~Cycon, R.~Froese, W.~Kirsch and B.~Simon, \it Schr\"odinger Operators, \rm Springer-Verlag, 1987 \bibitem{DJLS} R.~del Rio, S.~Jitomirskaya, Y.~Last, and B.~Simon, \it Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, \rm J.\ Anal.\ Math.\ {\bf 69} (1996), 153--200 \bibitem{DK} P.~Deift and R.~Killip, \it On the absolutely continuous spectrum of one-dimensional Schr\"odinger operators with square summable potentials, \rm Commun.\ Math.\ Phys. {\bf 203} (1999), 341--347 \bibitem{denisov} S.~A.~Denisov, {\em On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potential}, preprint. \bibitem{D1} A.~Devinatz, \it The existence of wave operators for oscillating potentials, \rm J. Math. Phys. {\bf 21} (1980), 2406--2411 \bibitem{Dol} J.~D.~Dollard, \it Asymptotic convergence and Coulomb interaction, \rm J. Math. Phys. {\bf 5} (1964), 729--738. \bibitem{dollard} \bysame, {\em Quantum mechanical scattering theory for short-range and Coulomb interactions}, Rocky Mountain J. Math. 1 (1971), 5-88. \bibitem{East} M.S.P.~Eastham, \it The Asymptotic Solution of Linear Differential Systems, \rm Clarendon Press, Oxford, 1989 %\bibitem{GK} F.~Germinet and A.~Kiselev, \rm in preparation \bibitem{hormanderwaveoperators} L.~H\"ormander, {\em The existence of wave operators in scattering theory}, Math. Z. 146 (1976), 69-91. \bibitem{KL} A.~Kiselev and Y.~Last, \it Solutions, spectrum and dynamics for Schr\"odinger operators on infinite domains, \rm Duke Math. Journal Vol. 102 (2000), 125--150 \bibitem{KLS} A.~Kiselev, Y.~Last, and B.~Simon, \it Modified Pr\"ufer and EFGP transforms and the spectral analysis of one-dimensional Schr\"odinger operators, \rm Commun.\ Math.\ Phys. {\bf 194} (1998), 1--45 \bibitem{KRS} A.~Kiselev, C.~Remling, and B.~Simon, \it Effective perturbation methods for one-dimensional Schr\"odinger operators, \rm J. Differential Equations 151 (1999), no. 2, 290--312 \bibitem{KU} S.~Kotani and N.~Ushiroya, \it One-dimensional Schr\"odinger operators with random decaying potentials, \rm Commun. Math. Phys. {\bf 115} (1988), 247--266. \bibitem{KR} T.~Kriecherbauer and C.~Remling, \it Finite gap potentials and WKB asymptotics for one-dimensional Schr\"odinger operators, \rm preprint. \bibitem{Nab} S.N.~Naboko, \it Dense point spectra of Schr\"odinger and Dirac operators, \rm Theor.-math. {\bf 68} (1986), 18--28. \bibitem{Nov} S.~Novikov, S.V.~Manakov, L.P.~Pitaevskii and V.E.~Zakharov, \it Theory of Solitons. The Inverse Scattering Method, \rm translated from the Russian. Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York-London, 1984 \bibitem{Pe} D.~Pearson, {\em Singular continuous measures in scattering theory}, Comm. Math. Phys. {\bf 60} (1978), 13--36. \bibitem{ReSi1} M.~Reed and B.~Simon, \it Methods of Modern Mathematical Physics, I. Functional Analysis, \rm Academic Press, New York, 1972 \bibitem{ReSi2} \bysame, \it Methods of Modern Mathematical Physics, II. Fourier Analysis, self-adjointness, \rm Academic Press, New York, 1972 \bibitem{RS3} \bysame, {\it Methods of Modern Mathematical Physics, III. Scattering Theory}, Academic Press, London-San Diego, 1979 \bibitem{Re1} C.~Remling, \it The absolutely continuous spectrum of one-dimensional Schr\"odinger operators with decaying potentials, \rm Comm. Math. Phys. {\bf 193} (1998), 151--170 \bibitem{Re2} \bysame, \it Bounds on embedded singular spectrum for one-dimensional Schr\"odinger operators, \rm Proc. Amer. Math. Soc. 128 (2000), no. 1, 161--171 \bibitem{Re3} \bysame, \it Schr\"odinger operators with decaying potentials: some counterexamples, \rm Duke Math. J. 105 (2000), no. 3, 463--496 \bibitem{rodnianskischlag} I.~Rodnianski and W.~Schlag, {\em Classical and quantum scattering for a class of long range random potentials}, preprint. \bibitem{simondenseembedded} B.~Simon, {\em Some Schr\"odinger operators with dense point spectrum}, Proc. Amer. Math. Soc. 125 (1997), 203-208. \bibitem{SVan} \bysame, \it Spectral analysis of rank one perturbations and applications, \rm Proc. 1993 Vancouver Summer School in Mathematical Physics \bibitem{Tit} E.C.~Titchmarsh, \it Eigenfunction Expansions, \rm 2nd ed., Oxford Univeristy Press, Oxford, 1962 \bibitem{WvN} J.~von Neumann and E.P.~Wigner, \it \"Uber merkw\"urdige diskrete eigenwerte - \rm Z. Phys. {\bf 30}(1929), 465--467 \bibitem{W1} D.A.W.~White, \it Schr\"odinger operators with rapidly oscillating central potentials, \rm Trans. Amer. Math. Soc. {\bf 275} (1983), 641--677 \end{thebibliography} \end{document}\end ---------------0110161419415--