Content-Type: multipart/mixed; boundary="-------------0008281825562"
This is a multi-part message in MIME format.
---------------0008281825562
Content-Type: text/plain; name="00-326.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-326.keywords"
Absolutely Continuous Spectrum, Schrodinger Operator
---------------0008281825562
Content-Type: text/plain; name="cit_thesis.cls"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="cit_thesis.cls"
%-----------------------------------------------------------------------------
% CIT_THESIS.STY Primary style for Caltech Ph.D. Thesis
%
% USAGE: (LaTeX 2.09) \documentstyle[draftfoot,preprint]{cit_thesis}
% (LaTeX2e) \documentclass[draftfoot,preprint]{cit_thesis}
%
% Modifies the {report} style, so that the title page and chapters
% are handled according to the Caltech Regulations for Format and
% Presentation of Theses (Office of Graduate Studies), and K. Turabian,
% _A Manual for Writers_ (U. Chicago Press, Chicago, 1987).
%-----------------------------------------------------------------------------
% The secondary style of [preprint] sets the conditional ``\ifpreprintsty''
% TRUE. The command ``\preprint'', at the beginning of the document, has
% the same effect. This causes a local publication number (see ``\pubnum''
% below) to be typeset in the upper right corner of the title page. The author
% may also write conditional macros using the ``\ifpreprintsty''
% construction.
%
% For preliminary or working copies of the thesis, the secondary style of
% [draftfoot] may be used, or the command ``\draftfoot'' may be specified at
% the beginning of the document. This produces a line at the bottom of every
% page, with ``DRAFT'' in the lower left corner, and the date and time when
% LaTeX was executed in the lower right.
%
% A number of useful macros are defined for use in this style. The
% title page of a thesis contains several standard phrases with possibly
% ``variable'' information, such as the degree to be awarded, the name
% and address of the institution, and copyright information. The macros
% below are used to specify these data; the macro arguments indicate
% the default values as defined in this style file.
%
% \degreeaward{Doctor of Philosophy} Degree to be awarded
% \university{California Institute of Technology} Institution name
% \address{Pasadena, California} Institution address
% \copyyear{\the\year} Year of diploma award
%
% \pubnum{} Publication/preprint number
%
% The usual LaTeX macros ``\author,'' ``\title,'' and ``\date'' work as
% defined in the LaTeX manual. The argument to \date should be the date
% on which the candidate's final examination (defense) is scheduled.
%
% The publication/preprint number will be typeset in the upper right corner
% of the title page if and only if the [preprint] style is selected, or if
% the ``\preprint'' command was executed before \maketitle.
%
% The command ``\maketitle'' will generate a title page formatted according
% to the Caltech instructions, and including all of the information defined
% by the macros above. Unlike the standard version, ``\maketitle'' may be
% repeated several times in a document: the terms defined for it are _not_
% reset after the page is formatted.
%
% The command ``\extrachapter'' will generate an unnumbered chapter heading,
% similar to the LaTeX ``\chapter*'' command. In addition, ``\extrachapter''
% will produce an entry in the Table of Contents. This command is used in
% the {abstract} and {acknowledgements} environments, listed below.
%
% The initial portion of the thesis document, typically the copyright page,
% abstract, and tables of contents, figures, and tables, are numbered in
% roman numerals, while the body of the thesis is numbered in arabic. The
% {frontmatter} environment handles these page numbering conventions. The
% author should place
%
% \begin{frontmatter}
% ...(abstract, contents, etc.)
% \end{frontmatter}
%
% immediately following the \maketitle command. The first entry in the
% {frontmatter} environment should generally be ``\makecopyright.'' This
% command generates a spearate copyright page, with the text
%
% (C) \copyyear
% \author
% All Rights Reserved
%
% centered at the bottom of the page. Obviously, the items indicated with
% macro commands above are filled in with the actual arguments to those
% commands.
%
% The abstract is produced with the {abstract} environment, which generates
% an unnumbered chapter with an entry in the table of contents. The author
% should use
%
% \begin{abstract}
% ...(body of abstract goes here)
% \end{abstract}
%
% within the {frontmatter} environment.
%
% An acknowledgements page is produced in the same way as the abstract, with
% the {acknowledgements} environment.
%
%
% 950201 Michael H. Kelsey Adapted from original of unknown source.
% 950208 -- Merged with "citthesis.cls" from Blair Zajac.
% Change [draft] and \draft to ``draftfoot''.
% 950525 -- Per request from B. McKenney, chapter headings are done in
% \LARGE font, not \Large.
% 960319 -- Force [letterpaper] option to {report} class
% 960517 -- Allow double-sided generation (remove \@twosidedfalse)
% -- Change \ifpreprintstyle to \ifpreprintsty for REVTeX
% compatibility
% 960521 -- Fix \evensidemargin setting for double-ssided support
%-----------------------------------------------------------------------------
%%%
%%% Pre-define commands required for option-processing
%%%
%
% \draftfoot will put "DRAFT" and the current date and time in the footer
%
\newif\ifthesisdraft
\def\draftfoot{\thesisdrafttrue\thesisfooter{Draft Copy\hss\thesistime}}
%
% \preprint will set the conditional \ifpreprintstyle to be TRUE
%
\newif\ifpreprintsty
\def\preprint{\preprintstytrue}
%%%
%%% Select between LaTeX 2.09 and LaTeX2e, based on \documentclass command
%%%
%
\ifx\undefined\documentclass %%% LaTeX 2.09 configuration %%%
%
\typeout{Document style `cit_thesis'. 8 February 1995, M. Kelsey}
%
% Define actions for style options
%
\@namedef{ds@draftfoot}{\thesisdrafttrue}
\@namedef{ds@preprint}{\preprintstytrue}
%
\input report.sty\relax %% Standard REPORT format
\input setspace.sty\relax %% Load package for double-spacing
%
\else %%% LaTeX2e configuration %%%
\NeedsTeXFormat{LaTeX2e}[1994/06/01]
\ProvidesClass{cit_thesis}[1995/02/08 Caltech Thesis class]
%
% Define actions for style options
%
\DeclareOption{draftfoot}{\thesisdrafttrue}
\DeclareOption{preprint}{\preprintstytrue}
%
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{report}}
\ProcessOptions* %% Process remaining options locally
\LoadClass[letterpaper]{report} %% Standard REPORT format
\RequirePackage{setspace} %% Load package for double-spacing
\fi
%
%%%
%%% Override standard REPORT format for thesis layout requirements
%%%
%
% Define \thispagestyle to null so that \chapter doesn't change page style.
%
\let\thesis@document\document
\def\document{\thesis@document\thispagestyle{empty}\def\thispagestyle##1{}}
%
% Define commands to generate draft-mode footers
%
\def\thesisfooter#1{\def\@oddfoot{#1} \def\@evenfoot{#1}}
%
% Construct timestamp information for draft-mode footer
%
\newcount\timehh\newcount\timemm
\def\thesistime{%
\timehh=\time\divide\timehh by 60%
\timemm=\timehh\multiply\timemm by -60%
\advance\timemm by \time%
\ifnum\timehh<10 0\fi\number\timehh\,:\,%
\ifnum\timemm<10 0\fi\number\timemm%
\ \ \today%
} %% End of \thesistime
%
% Set the headers and footers. Centered page number at top, empty footer.
% Top, bottom and right margins 1 inch, Left marging 1-1/2 inches.
%
\def\@oddhead{\hss\thepage\hss} \def\@evenhead{\hss\thepage\hss}
\let\@oddfoot\@empty \let\@evenfoot\@empty
\ifthesisdraft\draftfoot\fi %% Redefine footers for draft thesis
%
\setstretch{1.5} %% ``Double-spaced'' text
%
\setlength{\oddsidemargin}{.5in} %% Left margin = 1-1/2 inches
\setlength{\evensidemargin}{0pt} %% Left margin = 1 inch (default)
\setlength{\textwidth}{6in} %% Makes right margin = 1 inch
\setlength{\topmargin}{-.3in} %% Top margin = 3/4'' inch (incl pg#)
\setlength{\headsep}{.15in} %% Space between page # and text
\setlength{\textheight}{8.95in} %% Makes bottom margin = 1 inch
\setlength{\footskip}{.4in} %% Space from bottom of text to foot
%
% Define spacing for use on titlepage
%
\def\titleskip{\vskip 4\bigskipamount} %% Whitespace after title
\def\authorskip{\vskip 2\bigskipamount} %% Whitespace after author
%
% Create a command for unnumbered chapters, with a Table of Contents
% entry.
%
\def\extrachapter{\@ifnextchar[{\@lextrachapter}{\@extrachapter}}
\def\@lextrachapter[#1]#2{% User specified optional contents entry
\chapter*{#2}\addcontentsline{toc}{chapter}{#1}%
}
\def\@extrachapter#1{% Chapter heading used as contents entry
\chapter*{#1}\addcontentsline{toc}{chapter}{#1}%
}
%
% Chapter headings should be at top of page, single line format
% with single font size of \Large.
%
\def\@makechapterhead#1{\vspace*{10pt}
{\parindent 0pt \raggedright \LARGE\bf%
\ifnum \c@secnumdepth>\m@ne%
\@chapapp{} \thechapter \hskip 2.3ex plus .2ex
\fi
#1\par \nobreak \vskip 20pt}
} %% End of \@makechapterhead
\def\@makeschapterhead#1{\vspace*{10pt}
{\parindent 0pt \raggedright \LARGE\bf%
#1\par \nobreak \vskip 20pt}
} %% End of \@makeschapterhead
%
% We want roman page numbers in the frontmatter, and arabic afterward.
% use \begin{frontmatter} ... \end{frontmatter} to do this.
%
\newenvironment{frontmatter}%
{\pagenumbering{roman}}{\clearpage\pagenumbering{arabic}}
%
% Additional titlepage definitions
%
\def\degreeaward{\gdef\@degreeaward}
\degreeaward{Doctor of Philosophy} % Default is Ph.D.
\def\university{\gdef\@university}
\university{California Institute of Technology} % Default is Caltech
\def\address{\gdef\@address}
\address{Pasadena, California} % Default is Caltech
\def\copyyear{\gdef\@copyyear}
\copyyear{\the\year} % Default is today
\def\pubnum{\gdef\@pubnum}
\pubnum{} % No publication number
%
% The titlepage must conform to CIT requirements
%
\def\maketitle{%
{\parindent=0pt
\begin{titlepage}
\let\footnotesize\small \let\footnoterule\relax \setcounter{page}{1}
\vskip-\headsep
\ifpreprintsty\ifx\empty\@pubnum\else
{\singlespace\hfill\hbox{\begin{tabular}{l}\@pubnum\end{tabular}}}
\titleskip
\fi\fi
\begin{center}
{\Large\bf \@title\par}
\titleskip
Thesis by\\
{\large \@author\par}
\authorskip
In Partial Fulfillment of the Requirements\\
for the Degree of\\
\@degreeaward\par
\vfill
\@university\\
\@address\par
\authorskip
\@copyyear\par
{\normalsize (Submitted \@date)\par}
\end{center}
\vspace*{.5in}
\end{titlepage}}
} %% End of \maketitle
%
% The copyright page is new, must conform to CIT requirements
%
\def\makecopyright{%
\setcounter{page}{2}
{\parindent=0pt
\null % Forces rest to bottom of page
\vfill
\begin{center}
\copyright\quad \@copyyear\\
\@author\\
All Rights Reserved
\end{center}}
\newpage
} %% End of \makecopyright
%
% The abstract and acknowledgements are treated as unnumbered chapters
% with Table of Contents entries.
%
\def\abstract{\extrachapter{Abstract}} % \begin{abstract}
\def\endabstract{\clearpage} % \end{abstract}
\def\acknowledgements{\extrachapter{Acknowledgements}}
\def\endacknowledgements{\clearpage}
%
% The {thebibliography} environment should put an entry into the Table
% of Contents, just as the {abstract} and {acknowledgements} environments
% do.
%
\let\thesis@thebibliography=\thebibliography
\def\thebibliography#1{\thesis@thebibliography{#1}%
\addcontentsline{toc}{chapter}{Bibliography}}
%
\endinput
---------------0008281825562
Content-Type: text/plain; name="setspace.sty"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="setspace.sty"
%%% ======================================================================
%%% @LaTeX-style-file{
%%% filename = "setspace.sty",
%%% version = "3.1",
%%% date = "25 May 1994",
%%% time = "10:21:09.67 CDT",
%%% author = "George D. Greenwade",
%%% address = "Department of Economics and Business Analysis
%%% College of Business Administration
%%% P. O. Box 2118
%%% Sam Houston State University
%%% Huntsville, Texas, USA 77341-2118",
%%% email = "bed_gdg@SHSU.edu (Internet)
%%% BED_GDG@SHSU (BITNET)
%%% SHSU::BED_GDG (THENET)",
%%% telephone = "(409) 294-1266",
%%% FAX = "(409) 294-3712",
%%% supported = "yes",
%%% archived = "SHSU*",
%%% keywords = "LaTeX style, spacing, doublespace, half spacing,
%%% modifications to original",
%%% codetable = "ISO/ASCII",
%%% checksum = "14962 309 1381 13018",
%%% docstring = "This is a very quick hack to get setspace.sty
%%% to work with LaTeX2e as well as 2.09 with and
%%% without NFSS -- but it does work on the small
%%% test I've run. Comments welcome!
%%%
%%% The primitive logic is that we can test for
%%% the existence the definition of
%%% \selectfont -- not in LaTeX 2.09 without NFSS
%%% (prior toggle -- version 2.x)
%%% \@newbaseline -- defined in NFSS but not in
%%% LaTeX 2e (current toggle -- version 3.0)
%%%
%%% This is a modification to Erica Harris'
%%% setspace.sty. The main aspects of this
%%% modification deal with the definitions of
%%% \doublespacing, \onehalfspacing, and
%%% \singlespacing (these are near the beginning of
%%% the file). Primarily, these deal with adding
%%% fontsize changes to guarantee that the new
%%% baseline is properly defined and placed into
%%% action. The extra \vskip in the definition of
%%% \singlespacing seems to make for a cleaner
%%% transition from multiple spacing back to single
%%% spacing. These did not appear warrranted for
%%% other size changes.
%%%
%%% Modified by GDG on November 1, 1992, to allow
%%% for use of New Font Selection Scheme.
%%%
%%% Modified by GDG on June 4, 1993, to correct
%%% for spacing tokens in defintion of \@setsize
%%% Thanks to Kaja P. Christiansen
%%% for the fix!!
%%%
%%% Modified by GDG on May 24, 1994, to change toggle
%%% definition from \selectfont to \@newbaseline.
%%%
%%% Modified by GDG on May 25, 1994, to add
%%% definition of \everydisplay -- this part of
%%% the code was apparently written by Geoffrey
%%% Tobin on Thu 23 Jan 1992 and was provided by
%%% stanton@haas.berkeley.edu (Richard Stanton).
%%% This should help with some of the awkward math
%%% placements in changing spacings.
%%%
%%% The original headers of the main file cover the
%%% major aspects of this file and its use.
%%%
%%% The checksum field above contains a CRC-16
%%% checksum as the first value, followed by the
%%% equivalent of the standard UNIX wc (word
%%% count) utility output of lines, words, and
%%% characters. This is produced by Robert
%%% Solovay's checksum utility."
%%% }
%%% ======================================================================
%% FILE: setspace.sty in SYS2.TEX.PUB.ISULATEX.STYLES
%% AUTHOR: Erica M. S. Harris
%% DATE: April 1990
%% MOD: March 1991
%% DESCR: LaTeX Document style option "setspace"
%% Usage: \documentstyle[...,setspace,...]{...}
%% Based on the doublespace option created by Stephen Page.
%%
%% This style option provides commands and environments for doing
%% double and one-and-a-half spacing based on pt size.
%%
%% Single spacing is the default.
%%
%% Three commands, \doublespacing, \onehalfspacing, and
%% \singlespacing, are for use in the preamble to set the overall
%% spacing for the document. If a different spacing is required then
%% the \setstretch{baselinestretch} command can be used in the
%% preamble to set the baselinestretch appropriately. The default
%% spacing with this style option is single spacing.
%%
%% Three environments, singlespace, onehalfspace, and doublespace,
%% allow the spacing to be changed within the document. Both the
%% onehalfspace and doublespace environments are intended to increase
%% the spacing, so the onehalfspace environment should not be used in
%% a double spaced document. If an increased spacing different from
%% one-and-a-half or double spacing is required then the spacing
%% environment can be used. The spacing environment takes one
%% argument which is the larger baselinestretch to use,
%% e.g., \begin{spacing}{2.5}.
%%
%% \footins is adjusted the same as \parskip - appears to work. Lose
%% stretch parts but don't consider that to be crucial
%%
%% Removed code for altering spacing before and after displayed
%% equations - just looked too much.
%%
%% MODS:
%% Redefinition of \spacing and \endspacing for consistency with
%% TeX 3.x inserted by George Greenwade. Modification provided by
%% Philip Ross (ROSS@UK.AC.ABDN.BIOMED) and John Byrne via INFO-TeX.
%%
%% PLEASE REPORT ANY BUGS
%%
%% Old Documentation follows:
%% 1. A new environment "singlespace" is provided, within which single
%% spacing will apply.
%% JFL - changed so that it works in regular text and so that
%% vertical space before and after is correctly computed
%% 2. Double spacing is turned off within footnotes and floats (figures
%% and tables).
%% 3. Proper double spacing happens below tabular environments and in
%% other places where LaTeX uses a strut.
%% 4. Slightly more space is inserted before footnotes.
%% 5. JFL - fixes spacing before and after displayed math
%%
%%
%% mods: Jean-Francois Lamy
%% lamy@ai.toronto.edu
%% lamy@utai.uucp
%%
%% POSSIBLE BUGS:
%% . Increasing struts may possibly cause some other obscure part of
%% formatting to fall over.
%% . \begin{singlespace}\begin{quote} produces the wrong spacing before
%% the quote (extra glue is inserted).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\doublespacing{\ifcase \@ptsize \relax % 10pt
\def\baselinestretch{1.667}
\or % 11pt
\def\baselinestretch{1.618}
\or % 12pt
\def\baselinestretch{1.655}
\fi
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
\else% NFSS (2.09) present
\@newbaseline%
\fi%
}
\def\onehalfspacing{\ifcase \@ptsize \relax % 10pt
\def\baselinestretch{1.25}
\or % 11pt
\def\baselinestretch{1.213}
\or % 12pt
\def\baselinestretch{1.241}
\fi
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
\else% NFSS (2.09) present
\@newbaseline%
\fi%
}
\def\singlespacing{\def\baselinestretch{1}
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
\else% NFSS (2.09) present
\@newbaseline%
\fi%
\vskip\baselineskip% Correction for coming into singlespace
}
\def\setstretch#1{\renewcommand{\baselinestretch}{#1}}
%---Stretch the baseline BEFORE calculating the strut size. This improves
% spacing below tabular environments etc., probably...
% Comments are welcomed.
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\def\@setsize#1#2#3#4{\@nomath#1%--modified 1993.04.07--GDG per KPC
\let\@currsize#1\baselineskip
#2\baselineskip\baselinestretch\baselineskip
\parskip\baselinestretch\parskip
\setbox\strutbox\hbox{\vrule height.7\baselineskip
depth.3\baselineskip width\z@}%
\skip\footins \baselinestretch\skip\footins
\normalbaselineskip\baselineskip#3#4}
\else% NFSS (2.09) present
\def\@newbaseline{\setnew@baselineskip
\baselineskip\baselinestretch\baselineskip
\setbox\strutbox\hbox{\vrule\@height.7\baselineskip
\@depth.3\baselineskip \@width\z@}%
\normalbaselineskip\baselineskip}%
\fi%
%---Increase the space between last line of text and footnote rule.
%\skip\footins 20pt plus4pt minus4pt
%---Reset baselinestretch within footnotes and floats.
\def\@xfloat#1[#2]{\ifhmode \@bsphack\@floatpenalty -\@Mii\else
\@floatpenalty-\@Miii\fi\def\@captype{#1}\ifinner
\@parmoderr\@floatpenalty\z@
\else\@next\@currbox\@freelist{\@tempcnta\csname ftype@#1\endcsname
\multiply\@tempcnta\@xxxii\advance\@tempcnta\sixt@@n
\@tfor \@tempa :=#2\do
{\if\@tempa h\advance\@tempcnta \@ne\fi
\if\@tempa t\advance\@tempcnta \tw@\fi
\if\@tempa b\advance\@tempcnta 4\relax\fi
\if\@tempa p\advance\@tempcnta 8\relax\fi
}\global\count\@currbox\@tempcnta}\@fltovf\fi
\global\setbox\@currbox\vbox\bgroup
\def\baselinestretch{1}\@normalsize
\boxmaxdepth\z@
\hsize\columnwidth \@parboxrestore}
\long\def\@footnotetext#1{\insert\footins{\def\baselinestretch{1}\footnotesize
\interlinepenalty\interfootnotelinepenalty
\splittopskip\footnotesep
\splitmaxdepth \dp\strutbox \floatingpenalty \@MM
\hsize\columnwidth \@parboxrestore
\edef\@currentlabel{\csname p@footnote\endcsname\@thefnmark}\@makefntext
{\rule{\z@}{\footnotesep}\ignorespaces
#1\strut}}}
% A single spaced quote (say) is done by surrounding singlespace with quote.
%
\def\singlespace{%
\vskip\baselineskip%
\def\baselinestretch{1}%
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
\else% NFSS (2.09) present
\@newbaseline%
\fi%
\vskip-\baselineskip}
\def\endsinglespace{\par}
% spacing, doublespace and onehalfspace all are meant to INCREASE the
% spacing (i.e. calling onehalfspace from within doublespace will not
% produce a graceful transition between spacings)
%
% Next two definitions fixed for consistency with TeX 3.x
\def\spacing#1{\par%
\begingroup % moved from \endspacing by PGBR 29-1-91
\def\baselinestretch{#1}%
\ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e
\ifx\@currsize\normalsize\@normalsize\else\@currsize\fi%
\else% NFSS (2.09) present
\@newbaseline%
\fi%
}
\def\endspacing{\par%
\vskip \parskip%
\vskip \baselineskip%
\endgroup%
\vskip -\parskip%
\vskip -\baselineskip}
% one and a half spacing is 1.5 x pt size
\def\onehalfspace{\ifcase \@ptsize \relax % 10pt
\spacing{1.25}
\or % 11pt
\spacing{1.213}
\or % 12pt
\spacing{1.241}
\fi}
\let\endonehalfspace=\endspacing
% double spacing is 2 x pt size
\def\doublespace{\ifcase \@ptsize \relax % 10pt
\spacing{1.667}
\or % 11pt
\spacing{1.618}
\or % 12pt
\spacing{1.655}
\fi}
\let\enddoublespace=\endspacing
% gt - EMSH chose to omit display math part that follows.
% She wrote (see above) that the "altered spacing before and after displayed
% equations ... just looked too much".
%
% Fix up spacing before and after displayed math
% (arraystretch seems to do a fine job for inside LaTeX displayed math,
% since array and eqnarray seem to be affected as expected).
% Changing \baselinestretch and doing a font change also works if done here,
% but then you have to change @setsize to remove the call to @nomath)
%
\everydisplay{
\abovedisplayskip \baselinestretch\abovedisplayskip%
\belowdisplayskip \abovedisplayskip%
\abovedisplayshortskip \baselinestretch\abovedisplayshortskip%
\belowdisplayshortskip \baselinestretch\belowdisplayshortskip}
\endinput
---------------0008281825562
Content-Type: application/x-tex; name="It.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="It.tex"
\documentclass[12pt]{cit_thesis}
%
% consise...
\setstretch{1}
%
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
%\usepackage{showkeys}
\hyphenation{di-men-sion-al}
\newcommand{\sigac}{{\sigma_{\mathrm{ac}}}}
\newcommand{\Sigac}{{\Sigma_{\mathrm{ac}}}}
\newcommand{\Reals}{{\bf R}}
\newcommand{\Cmplx}{{\bf C}}
\newcommand{\Ints}{{\bf Z}}
\newcommand{\psipm}{\psi^{ _\pm}}
\newcommand{\psip}{\psi^{ _+}}
\newcommand{\psim}{\psi^{ _-}}
\newcommand{\tri}{\triangle}
\newcommand{\barr}{\overline}
\newcommand{\dde}{\frac{d\phantom{\epsilon}}{d\epsilon}}
\renewcommand{\Im}{{\mathrm{Im}}}
\renewcommand{\Re}{{\mathrm{Re}}}
\renewcommand{\det}{\mathrm{det}}
\newcommand{\tr}{\mathrm{tr}}
%\newcommand{\qed}{\hfill $\square$}
\DeclareMathOperator*{\wlim}{weak-lim}
% Simple list
\newcounter{smalllist}
\newenvironment{SmallList}{%
\begin{list}{{\rm\roman{smalllist})}}%
{\setlength{\topsep}{0mm}\setlength{\parsep}{0mm}\setlength{\itemsep}{0mm}%
\setlength{\labelwidth}{10mm}\usecounter{smalllist}}%
}{\end{list}}
% Theorem environments
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}[chapter]
\newtheorem{prop}[lemma]{Proposition}
\newtheorem{coro}{Corollary}
\newtheorem{definition}{Definition}
\title{Perturbations of One-Dimensional Schr\"odinger Operators Preserving the Absolutely Continuous Spectrum}
\author{Rowan Killip}
\date{the 8$^{\text{th}}$ of August 2000}
\university{California Institute of Technology}
\copyyear{2001}
\begin{document}
\maketitle
\begin{frontmatter}
\makecopyright
\mbox{}
\vfill
\hfill\parbox[c]{3.5in}{\it
To my grandfathers,\\
who taught me how to put things back together\\
---I figured out how to pull them apart myself.
}\hfill\mbox{}
\vfill
\mbox{}
\begin{acknowledgements}
I have recieved financial support from a W.H.~Pickering Fellowship and from the Alfred P. Sloan Foundation for which I am very thankful.
Much of this text was written at the Ecole Polytechnique (France) and I wish to acknowledge the warm hospitality of J.~Magnen and the whole Centre Physique Th\'eorique.
In working on this thesis, I have benefited greatly from the guidence and support of P.~Deift, F.~Gesztesy, A.~Kiselev,
Ch.~Remling and particularly my advisor, B.~Simon.
I would like to thank the Caltech math department: faculty, post-docs and secretaries, especially C.~Galvez.
Thanks also to my friends and fellow students for making my stay at Caltech so pleasant.
Lastly and most of all, my heartfelt thanks to my family for their support, especially in my absence, and to Janet and her family for theirs.
\end{acknowledgements}
\begin{abstract}
We study the stability of the absolutely continuous spectrum of one-dimensional Schr\"odinger operators
$$
[Hu](x) = -u''(x) + q(x)u(x)
$$
with periodic potentials $q(x)$. Specifically, it is proved that any perturbation of the potential,
$V\in L^2$, preserves the essential support (and multiplicity) of the absolutely continuous spectrum.
This is optimal in terms of $L^p$ spaces and, for $q\equiv 0$, it answers in the affirmative a
conjecture of Kiselev, Last and Simon.
By adding constraints on the Fourier transform of $V$, it is possible to relax the decay assumptions
on $V$. It is proved that if $V\in L^3$ and $\hat V$ is uniformly locally square integrable, then
preservation of the a.c.~spectrum still holds. If we assume that $q\equiv0$, still stronger results
follow: if $V\in L^3$ and $\hat V(k)$ is square integrable on an interval $[k_0,k_1]$, then the interval
$[k_0^2/4,k_1^2/4]$ is contained in the essential support of the absolutely continuous spectrum of the
perturbed operator.
\end{abstract}
\tableofcontents
% \listoffigures
% \listoftables
\end{frontmatter}
\chapter{Introduction}
We wish to consider perturbations of one-di\-men\-sion\-al Schr\"odinger operators with bounded periodic
potentials. So given a bounded periodic function $q(x)$ on $\Reals$ we define
\begin{equation*}
[H_0u](x) = -u''(x) +q(x)u(x)
\end{equation*}
as an operator acting in $L^2(\Reals)$. (More strictly, the domain of operator $H_0$ is the Sobolev space $\mathsf H^2$, which
is dense in $L^2$.)
The assumption that $q$ is bounded is made purely to avoid distracting complications. More generally, one may consider
$q$ which are locally integrable. We shall also have cause to discuss the discrete version of this operator, namely
$$
[h_0u](n) = u(n+1) + u(n-1) + q(n)u(n)
$$
which acts in $\ell^2(\Ints)$. For the most part though, we shall discuss the traditional
(non-discretized) Schr\"odinger operator.
The spectral theory of one-dimensional Schr\"odinger operators with periodic potentials is very well understood.
Ultimately this stems from the following fact: if $L$ denotes the period of $q$ and $\psi$ is a solution of
$$
-\psi''+q\psi = z\psi
$$
for some $z\in \Cmplx$, then $\phi(x)=\psi(x+L)$ is also a solution of this differential equation. By iterating this fact,
we see that solutions are either steadily growing (as $x\to \infty$ or $-\infty$) or are uniformly bounded. The set
of $z$ for which there are bounded solutions is a closed subset of $\Reals$. Indeed, it is the spectrum of $H_0$, $\sigma(H_0)$.
It consists of closed intervals (known as the spectral bands) and is bounded from below but unbounded above.
For any $z$ in the interior of the spectrum, all solutions are bounded. This does not hold at the band edges \cite{MAGNUS}.
One of the core results of subordinacy theory is that bounded solutions correspond to absolutely continuous spectrum
\cite{SIMON-BNDD}.
Employing this theorem in the current instance tells us that the spectrum $\sigma(H_0)$ of $H_0$ is purely absolutely
continuous with multiplicity two. The nature of the spectrum has been known for a very long time, in particular from
before the advent of subordinacy theory. However, we choose to present it in this way as it foreshadows some of the
discussion to come.
The question at the centre of this thesis is the following: what perturbations $V$ may be added to the potential without
destroying the absolutely continuous spectrum? In particular, what type of decay at infinity is permissible.
At the moment this question is rather imprecise; in particular, we have yet to explain in what sense we wish the absolutely
continuous spectrum to be preserved.
For the moment though, we introduce the following notation for the perturbed operator
\begin{equation*}
[Hu](x) = -u''(x) +q(x)u(x) + V(x)u(x).
\end{equation*}
Undoubtedly the oldest result on spectral stability is due to Weyl \cite[Theorem X\kern -1pt I\kern -1pt I\kern -1pt
I.14]{REED-SIMON4}: any relatively compact perturbation preserves the essential spectrum. For the operators we
consider, this means that if $V$ decays in some reasonable fashion, for example,
\begin{equation*}
\int_{n-1}^{n} V^2 dx \to 0 \text{ as $n\to\pm\infty$,}
\end{equation*}
then $\sigma_{\mathrm{ess}}(H)=\sigma_{\mathrm{ess}}(H_0)=\sigma(H_0)$. This tells us that we need only focus on what
happens to the spectral type on $\sigma(H_0)$. Any spectrum of $H$, lying outside $\sigma(H_0)$, must consist of discrete
eigenvalues.
In the discrete case, one merely needs $V(n)\to 0$ as $n\to\pm\infty$ to apply the Weyl Theorem.
The next general result we wish to consider is the Kato-Rosenblum Theorem (more accurately, a version of it due to Birman
and Kuroda). As well as being interesting in its own right,
it suggests what type of spectral stability result we should try to prove.
If $V\in L^1$ then $V$ is a relatively trace class perturbation of $H_0$. That is to say, for each
$z\in\Cmplx\setminus\Reals$, $R-R_0=(H-z)^{-1}-(H_0-z)^{-1}$ is trace class. The Kato-Rosenblum Theorem
then tells us that wave operators exist and are complete and hence that the absolutely continuous parts of
$H$ and $H_0$ are unitarily equivalent.
Here we have a very clear statement of what it means for the absolutely continuous spectrum to be preserved---namely,
if the absolutely continuous parts of the perturbed and unperturbed operators are unitarily equivalent. This is much
stronger than the statement $\sigac(H)=\sigac(H_0)$. To better explain this, we introduce the following:
\begin{definition}
Given a self-adjoint operator $A$ on a Hilbert space $\mathsf{H}$, an essential support of the a.c.~spectrum is any set $\Sigac$ such that
\begin{SmallList}
\item for every $\phi\in\mathsf{H}$ the corresponding spectral measure $\mu$ obeys
$\mu_{\mathrm{ac}}(S\cap\Sigac)=\mu_{\mathrm{ac}}(S)$ for any Borel set $S$, and
\item $\exists\phi\in\mathsf{H}$ such that $d\mu \gg \chi_\Sigac(E)dE$.
\end{SmallList}
This determines $\Sigac$ uniquely up to sets of zero Lebesgue measure.
\end{definition}
\noindent
{\it Remark} In order to simplify the presentation, equations involving $\Sigac$ should be interpreted as
holding for {\em some} choice of essential support.
\bigskip
The statement `the absolutely continuous parts of $H$ and $H_0$ are unitarily equivalent' is logically
equivalent to `$\Sigac(H)=\Sigac(H_0)$ and the spectral multiplicities are equal on this set'. It should be clear
that $\Sigac\subseteq\sigac$, indeed $\sigac$ is the smallest closed set which contains (a realization of) $\Sigac$.
However, in general $\Sigac\neq\sigac$; they may differ by any set with empty interior and positive Lebesgue measure.
In the case of $H_0$ or $h_0$, $\Sigac=\sigac$.
As with most recent work on this subject, we shall prefer to talk about equality of the essential support of the
absolutely continuous spectrum rather than unitary equivalence of absolutely continuous parts.
For potentials with Coulombic decay, $V(x) \sim 1/x$, the Kato-Rosenblum Theorem is inapplicable. The wave operators
simply do not exist. If we wish to consider potentials with slower than $L^1$ decay, we must
introduce modified wave operators. The best known conditions for existence and completeness of
wave operators \cite{HORM} require decay restrictions on $V,V'$ and $V''$. Alternatively, this monograph contains a proof
(Theorem 30.2.10) that the absolutely continuous spectrum is preserved under weaker assumptions on just $V,V'$
by directly obtaining estimates on the resolvent.
In the one-dimensional case we are considering, however, these are all subsumed by the following result of Weidmann \cite{WEIDMANN}.
If $V$ is the sum of two terms, one integrable and the other tending to zero
at infinity and of bounded variation, then $\Sigac(H)=\Sigac(H_0)$ with equal multiplicity.
Actually this theorem is due to Weidmann only in the case that $q\equiv 0$. For periodic background potentials,
it is due to Stolz \cite{STOLZ}.
Perhaps the earliest results in the opposite direction arose from Pearson studies of sparse potentials
\cite{PEARSON}. Suppose $W(x)$ is continuous and of compact support and that $g_n\not\in\ell^2$. Pearson
showed that if $x_n\to\infty$ sufficiently quickly, then the operator $H$ with $q\equiv 0$ and
$$
V(x)=\sum g_n W(|x|-x_n)
$$
has purely singular continuous spectrum. This tells us, for example, that if $p>2$ then there exists a
$V\in L^p$ so that $\Sigac(H)=\emptyset$.
More recently, Kiselev, Last and Simon \cite{KLS} treated sparse potentials for $x_n$ obeying
$x_n/x_{n+1} \to 0$ by showing that if $g_n\in\ell^2$ then $H$ has purely absolutely continuous spectrum
on $(0,\infty)$, whilst if $g_n\not\in\ell^2$ then $H$ has purely singular continuous spectrum on this
interval. Remling \cite{REMLING-1} also gave related results.
Whilst these results give a very strong indication of what decay properties are insufficient to prove
the preservation of $\Sigac$, at least in an $L^p$ sense, there is a still more instructive result due to Kotani
and Ushiroya \cite{KOTANI-USHIROYA}. Once again we consider $q\equiv 0$. Given any bounded positive decreasing
function $a(x)\not\in L^2(\Reals^+)$, they construct a probability space of choices for $V$, each obeying
$|V(x)|\leq a(|x|)$ such that $H=H_0+V$ has purely singular spectrum with probability one. It is also shown that if
$|V(x)|(1+|x|)^\epsilon \in L^2$ then the spectrum on $(0,\infty)$ is purely absolutely continuous with probability
one. Later we shall see that $L^2$ perturbations preserve the a.c.~spectrum. The Kotani-Ushiroya result implies that
$L^2$ is the border-line in a very strong sense. For example, if $\Phi$ is a non-negative convex function with $x^{-2}\Phi(x)\to0$ as $x\to0$ then there is a $V$ with $\int \Phi\big(|V(x)|\big) dx<\infty$ that gives rise to purely singular spectrum.
In the discrete case, random decaying potentials were studied by Delyon, Simon and Souillard \cite{DELYON,DELYON-S-S,SIMON-SJM}.
Specifically, they studied potentials of the form $V(n)=a_n X_n$ where $a_n$ decays as $|n|\to\infty$
and $X_n$ are independent identically distributed random variables for each $n\in\Ints$. Suppose, for simplicity, that the
distribution of $X_n$ is compactly supported. If $a_n=(1+|n|)^{-\alpha}$, then the following are known to hold with
probability one:
\begin{SmallList}
\item If $\alpha<1/2$, the spectrum is pure point with rapidly decaying eigenfunctions.
\item If $\alpha=1/2$, then the spectrum is purely singular (continuous at low energies, pure point at higher energies) and
\item if $\alpha>1/2$, then Kotani has proved that the spectrum is purely absolutely continuous.
\end{SmallList}
Using this result as an input to a general method, Simon \cite{SIMON-SCSI} has shown that
if $p>2$ then for a dense $G_\delta$ of $V\in \ell^p$ the operator $u(n+1)+u(n-1)+V(n)u(n)$ has purely
singular continuous spectrum on $[-2,2]$. This set is the essential spectum of the operator. For traditional Schr\"odinger
operators it is shown that for a dense $G_\delta$ of $V\in L^p(\Reals),p>2$, the operator $-u''+Vu$ has purely singular
continuous spectrum on $(0,\infty)$.
The dichotomy between $L^2$ perturbations preserving a.c.~spectrum and non-$L^2$ perturbations destroying
it, at least for sparse and random perturbations, led to the conjecture by Kiselev, Last and Simon \cite{KLS}
that if $V$ is square integrable then $\Sigac(H)=\Sigac(H_0)$, at least when $q\equiv 0$.
Recent positive results began with Kiselev's proof \cite{KISELEV-1} that if $|V(x)|\leq
C(1+|x|)^{-\epsilon-3/4}$ then $\Sigac(H)=\Sigac(H_0)$. This paper actually proves a much stronger result:
for almost every $E\in\Sigac(H_0)$ all solutions of $-\psi''+q\psi+V\psi=E\psi$ are bounded and, if $q\equiv0$, have
WKB-type asymptotics. That this implies $\Sigac(H)=\Sigac(H_0)$ follows from the subordinacy theory
result mentioned earlier. By stating that the solutions have WKB asymptotics we mean that there are solutions
$$
\psi_\pm = \exp\Big\{\pm ikx \mp \tfrac{i}{2k} \int_0^x V(s) ds \Big\} + o(1)
$$
with $k^2=E$. Of course, any other solution of $-\psi''+V\psi=E\psi$ can be written as a linear combination
of $\psi_+$ and $\psi_-$.
Later Kiselev \cite{KISELEV-2} and Molchanov
improved this to $|V(x)|\leq C(1+|x|)^{-\epsilon-2/3}$. Finally, Christ-Kiselev \cite{CHRIST-KIS-JAMS}
and Remling \cite{REMLING-2} obtained the optimal result in this direction, namely if $|V(x)|\leq
C(1+|x|)^{-\epsilon-1/2}$ then for almost every energy $E\in(0,\infty)$ all solutions are bounded and, if $q\equiv0$,
obey WKB-type asymptotics. Consequently, $\Sigac(H)=\Sigac(H_0)$.
(That this is optimal in terms of power-law decay follows from the work of Kotani-Ushiroya mentioned earlier.)
By quoting these power-law theorems we have not quite presented their results in full generality. For example
they are able to admit singularities: $(1+|x|)^\epsilon V(x) \in L^p$ for any $1\leq p <2$.
However, they were not able to find a purely
$L^p$-type condition and offered this as a direction for further study \cite{CHRIST-KISELEV-REMLING}.
All these works also treat power decaying potentials for the discrete Schr\"odinger operator.
Recently Christ and Kiselev extended their method to potentials $V\in L^p$ or $V'\in L^p$ provided
$p<2$. See \cite{CHRIST-KIS-PRE}.
For vanishing background potentials, $q\equiv0$, Deift and Killip \cite{DEIFT-K} were able to prove
if $V\in L^2$ then $\Sigac(H)=\Sigac(H_0)$ and so confirm the conjecture of Kiselev, Last and Simon.
Later we shall present a proof of this result.
The main ingredient of the Deift-Killip proof is one of the Buslaev-Faddeev-Zakharov
trace formulae which are perhaps best known for their role in the study of the KdV equation
\cite{NOVIKOV}. Suppose $V$ is of compact support; then we may consider solutions to
$-u''(x)+V(x)u(x)=k^2u(x)$ which agree with $e^{ikx}$ to the right of the support of the potential.
To the left of the support of the potential we can write $u(x)=a_p(k)e^{ikx} + b_p(k)e^{-ikx}$ where $a_p,b_p$
are analytic functions of the momentum $k\neq 0$. The subscript $p$ is to indicate that $a,b$ are parameterized as
functions of momentum $k$; later we shall define a function $a$ which is parameterized by energy $z=k^2$.
In the time-dependent picture, $e^{\pm ikx}$ represent waves travelling to the right/left.
Thus it is natural to term $t(k)=1/a_p(k)$
the transmission coefficient and $r(k)=b_p(k)/a_p(k)$ the reflection coefficient. The second of the BFK trace formulae is
\begin{equation}\label{BFZ2}
\int_\Reals \log|a_p(k)|k^2dk = \tfrac{\pi}{8}\int V(x)^2dx - \tfrac{2\pi}{3}\sum E_n^{3/2},
\end{equation}
where $-E_n$ are the negative eigenvalues of the Schr\"odinger operator with potential $V(x)$.
We will derive this formula in Chapter \ref{DK}.
All the main results of this thesis rest on the fact that preservation of absolutely continuous spectrum follows from
appropriate a priori bounds on $|a_p|$, the reciprocal of the transmission coefficient. This is made precise
in Proposition \ref{mainprop}. Because the eigenvalue term, $-\tfrac{2\pi}{3}\sum E_n^{3/2}$, is negative, (\ref{BFZ2})
gives an a priori bound on $|a_p(k)|$. This is then sufficient to prove that square summable perturbations
preserve the absolutely continuous spectrum when $q\equiv0$.
Using higher-order trace formulae it is possible to prove similar results with
estimates on derivatives. For example: if $V\in L^4$ and $V''\in L^2$ then
$\Sigac(H)=\Sigac(H_0)=(0,\infty)$. We will also prove this in Chapter \ref{DK}.
Though known to the authors at the time, this theorem does not appear in \cite{DEIFT-K}. The spectral
consequences of the full family of trace formulae have recently been studied systematically by Molchanov, Novitskii and Vainberg \cite{MNV}.
The trace formula method is effective only in the case $q\equiv0$. The natural analogue of (\ref{BFZ2}) in the presence of
a periodic background potential suffers from the following problem: the contribution from the eigenvalues lying in the spectral gaps
is no longer assuredly negative. The greater part of this thesis is devoted
to the presentation of a different, though intimately related, method which is able to treat general periodic $q$.
Specifically we shall interpret $a$ as a Fredholm determinant and then use regularized determinants to study the
integral of $\log|a|$ over bounded intervals in energy $E=k^2$; the trace formula is for the integral over all energies.
Although our main goal is to prove results for the whole-line operator $H$, this will be effected by first proving
results for half-line operators. Define
$$
[H_+u](x) = -u''(x) + q(x)u(x) + V(x)u(x) \text{ in $L^2(\Reals^+)$ with $u(0)=0$}
$$
that is, a Schr\"odinger operator on the positive half-axis with a Dirichlet boundary condition at the origin.
The half-line version of the Deift-Killip results described above is
\begin{theorem}
If $q\equiv 0$ and either
\begin{SmallList}
\item $V\in L^2$ or
\item $V\in L^4$ and $V''\in L^2$,
\end{SmallList}
then $\Sigac(H_+)=(0,\infty)$.
\end{theorem}
In the next section, under the heading {\it Proof that Theorems $\Rightarrow$ Corollaries}, we will explain how one obtains
whole-line results from theorems about the half-line operators. For example
\begin{coro}
Suppose $q\equiv 0$ and either
\begin{SmallList}
\item $V\in L^2(\Reals)$, or
\item $V\in L^4(\Reals)$ and $V''\in L^2(\Reals)$.
\end{SmallList}
Then $\Sigac(H)=\Sigac(H_0)$, that is, $(0,\infty)$ and with equal spectral multiplicity two.
\end{coro}
In the presence of a periodic background potential $q$ we prove
\begin{theorem}
If $q$ is periodic and $V\in L^2(\Reals^+)$ then $\Sigac(H_+)=\Sigac(H_0)$.
\end{theorem}
\begin{coro} Suppose $q$ is periodic and $V\in L^2(\Reals)$. Then $\Sigac(H)=\Sigac(H_0)$ with spectral
multiplicity two.
\end{coro}
Note that $\Sigac(H_0)$ is also the essential support of absolutely continuous spectrum for the
unperturbed ($V\equiv0$) half-line operator. Hence its appearance in the theorem. That this is the
optimal $L^p$ space follows from the work of Kotani-Ushiroya mentioned earlier. By adding a mild
constraint, it is possible to obtain a result for $V\in L^3$.
\begin{theorem}
Suppose that $q$ is periodic, $V\in L^3(\Reals^+)$ and its Fourier transform $\hat V\in
\ell^\infty(L^2)$ {\rm (}that is the distribution $\hat V$ is uniformly locally $L^2${\rm )}. Then
$\Sigac(H_+)=\Sigac(H_0)$.
\end{theorem}
\begin{coro}
Suppose that $q$ is periodic, $V\in L^3(\Reals)$ and $\hat V\in \ell^\infty(L^2)$,
then $\Sigac(H)=\Sigac(H_0)$ with spectral multiplicity two.
\end{coro}
\noindent
For means of later convenience we adopt the following norm on $\ell^\infty(L^2)$
\begin{equation}
\label{ellL}
\big\|f\big\|_{\ell^\infty(L^2)}^2 = \sup_n \int^{2\pi/L}_{-2\pi/L}
\big|f\big(x+\tfrac{2\pi n}{L}\big)\big|^2 dx,
\end{equation}
where $L$ denotes the period of $q$. In the case of zero background, $q\equiv0$, we are able to obtain
information which is local energy:
\begin{theorem} Let $I$ be an interval in $(0,\infty)$ and define $\tilde I=\{k\in\Reals:k^2/4\in I\}$.
Suppose $q\equiv0$, $V\in L^3$ and the distribution $\hat V$ restricts to an $L^2$ function on the set
$\tilde I$. Then $I\subseteq\Sigac(H_+)$.
\end{theorem}
\begin{coro}
Under the assumptions of the previous theorem, $I\subseteq\Sigac(H)$ with multiplicity two.
\end{coro}
\noindent
{\it Remarks} 1. In all the above theorems/corollaries, the theory of relatively trace class perturbations states that one may add a
further $L^1$ perturbation to the potential without changing $\Sigac(H)$.
\noindent 2. Our methods do permit us to obtain a version of Theorem 3 which is local in energy; however,
the criterion on $V$ seems difficult to check in general. Specifically, one needs to bound (\ref{star})
in $L^1(I,dE)$ to show that $I\subset\Sigac(H)$ as this would give a replacement for (\ref{gq0}).
\noindent 3. Let $\dot{\mathsf H}^s$ denote the Sobolev space of functions with square integrable derivative of order
$s$. For non-integer $s$, $\dot{\mathsf H}^s = \{f:|k|^s\hat{f}(k)\in L^2(dk)\}$. Then from Corollary 4: if $V\in L^3\cap \dot{\mathsf H}^s$ for some $s\in\Reals$ then $\Sigac(H)=(0,\infty)$.
\noindent 4. The method presented here extends readily to perturbations of the discrete Schr\"odinger operator $h_0$. One
obtains the following analogues of the corollaries given above.
If $q(n)$ is periodic and $V\in\ell^2(\Ints)$ then $\Sigac(h)=\Sigac(h_0)$ with
equal multiplicity (namely two). Suppose $q\equiv 0$, $V\in\ell^3$. If $\hat V(\theta)=\sum V(n)e^{in\theta}$
is square integrable on an interval $I$ then the interval $\tilde I=\{2\cos(\theta/2):\theta\in I\}$ is
contained in $\Sigac(h)$. Moreover, the spectral multiplicity is equal to two on $\tilde I$.
Notice that because $\hat V$ is uniformly locally $L^2$ iff $V$ is $\ell^2$, the analogues of Corollaries 2 and
3 coincide.
\bigskip
Let us briefly discuss the occurrence of $k^2/4$ in Theorem 4. By the Weyl Theorem, we know that if
$V\in L^3$ then $\sigma_{\mathrm{ess}}(H)=\sigma_{\mathrm{ess}}(H_0)$. Consequently, the only way to
obtain absence of absolutely continuous spectrum is to generate enough singular spectrum to take its
place. Wigner and von Neumann discovered a potential with asymptotics $V(x)\sim\sin(kx)/x$ which has a
positive eigenvalue at $E=k^2/4$. The physical explanation for this is that the separation of the
bumps in the potential is half the wavelength of a particle with energy $k^2/4$. In this way, they set
up a coherent reflection which is sufficient to generate an eigenstate. This gives rise to the heuristic that
singularities in $\hat V$ at frequency $k$ give rise to singular spectum at energy $k^2/4$. If $\hat V$ is square integrable on an interval $I$ then, in some sense, it is regular at almost every frequency in that interval.
Naboko \cite{NABOKO} and Simon \cite{SIMON-DPS} have extended the Wigner-\kern -1pt von Neumann idea to
construct potentials with $xV(x)\to\infty$ arbitrarily slowly which produce eigenvalues at a dense set
of $E\in(0,\infty)$. By the theorems presented earlier, these eigenvalues are embedded in
$\Sigac(H)=(0,\infty)$. In this context we should also mention that Remling \cite{REMLING-3} has given
bounds on the Hausdorff dimension of any possible embedded singular continuous spectrum.
We would like to present some sample applications of these theorems to sparse potentials. Given a $C^\infty$ function
of compact support $f$, we wish to consider potentials of the form
\begin{equation}\label{sparseV}
V(x) = \sum_n a_n f\big(\lambda_n[x-n!]\big).
\end{equation}
From the work of Kiselev-Last and Simon mentioned above, we know that if $\lambda_n\equiv1$ then the Schr\"odinger
operator with potential $V$ has purely absolutely continuous or purely singular continuous spectrum on $(0,\infty)$ depending on whether $a_n\in \ell^2$ or $a_n\not\in \ell^2$ respectively. As mentioned earlier, this shows that Theorem 2 is optimal. Notice also that it also has implications for part ii) of Theorem 1: the condition that $V''\in L^2$ cannot
be replaced by an $L^p,p>2$ condition. Indeed for any $s\in\Reals$, $V\in\dot{\mathsf H}^s$ iff $a_n\in \ell^2$.
Let us now consider $a_n=n^{-\alpha}$, $\lambda_n=n^\beta$. By direct calculation, it is easy to see that $V(x)$ will be $L^3$ if $\sum a_n^3 \lambda^{-1}_n < \infty$, that is, if $3\alpha+\beta>1$. Because the potential is so sparse, it is easy to estimate the $\dot {\mathsf H}^s$ norm of $V$. Specifically, given distinct summands $S_n,S_m$ from (\ref{sparseV}),
the $\dot {\mathsf H}^s$ inner product of $S_n$ and $S_m$ is an extremely rapidly decaying function of $|n-m|$. In fact, for $s\in\Ints$, $S_n$ and $S_m$ are orthogonal.
As a result, $V\in\dot{\mathsf H}^s$ iff $2\alpha + (1-2s)\beta > 1$.
If $1/2>\alpha>(1-\beta)/3$ and $\beta>0$ then by choosing $s$ sufficiently negative, $V\in L^3\cap\dot{\mathsf H}^s$ and
so, by Theorem 4, the corresponding Schr\"odinger operator has absolutely continuous spectrum on $(0,\infty)$. While if $1/2>\alpha>(1-\beta)/3$ and $\beta<0$ then by choosing $s>0$ sufficiently large, $V\in L^3\cap\dot{\mathsf H}^s$ and the preservation of a.c. spectrum again follows from Theorem 4.
To produce an example of the applicability of Theorem 3, we would like to consider a slightly different potential
$$
V(x) = \sum_n a_n f(x-n)\cos(nx).
$$
This is in $L^3$ iff $a_n\in \ell^3$. Taking the Fourier transform gives
$$
\hat V(\xi) = \tfrac{1}{2} \sum_n a_n [\hat f(\xi-n) + e^{2i\xi n}\hat f(\xi+n)],
$$
which is then uniformly locally square integrable. This means that one may apply Theorem 3 to see that for any
periodic background potential $q(x)$, the addition of this perturbation preserves the absolutely continuous spectrum.
The text is arranged as follows: In the next chapter we present some background material and show how the
corollaries follow from their respective theorems. Chapter \ref{PD} introduces the main object for study,
$a(z)$, the reciprocal of the transmission coefficient, as the perturbation determinant and derives some of its basic properties.
In particular, Proposition \ref{mainprop} shows how one may obtain spectral information from estimates on $a(z)$.
In Chapter \ref{DK} we discuss the trace formulae and use them to prove Theorem 1.
To prove the other theorems, we use regularized determinants. These are introduced in Chapter
\ref{RegDet} and some estimates are derived. The final chapter combines this material with
some facts about harmonic functions to prove Theorems 2--4.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Background} \label{bckgrnd}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this chapter we introduce notation and some of the background material that we shall be using.
Of course, proofs and more extensive discussions are left to the references. The reader may find
\cite{COD-LEV} and \cite{EASTHAM} of most use. As a natural part of these discussions we prove that
each of the corollaries follows from its corresponding theorem.
Let us begin with the unperturbed operator
\begin{equation*}
[H_0u](x) = -u''(x) + q(x)u(x) \text{ acting in $L^2(\Reals)$,}
\end{equation*}
where $q$ is periodic and bounded. The spectral theory of such operators is well understood and is
usually studied through the Bloch (or Floquet) solutions. For each $z\in\barr{\Cmplx^+}=\{z:\Im\,z\geq
0\}$, these are the non-zero solutions of
\begin{equation}
\label{ODE}
-\psi''(x)+q(x)\psi(x)=z\psi(x)
\end{equation}
for which there exists $\beta\in\Cmplx\setminus\{0\}$ such that
\begin{equation*}
\psi(x+L) = \beta \psi(x).
\end{equation*}
Here $L$ denotes the period of $q$. When $q$ is identically constant the period is ill-defined.
This case is excluded and will be
treated in the next paragraph. For each $z\in\Cmplx^+ = \{z:\Im\,z > 0\}$ there are only two
possibilities for $\beta$: $\beta^+(z)$ with $|\beta^+|<1$ and $\beta^-(z) = 1/\beta^+(z)$. It is
possible to extend $\beta^\pm(z)$ continuously to $\Reals$; in fact,
$\sigma(H_0)=\{E\in\Reals:|\beta^+(E)|=1\}$. Of course, $\beta^+ = 1/\beta^-$ still holds on $\Reals$.
However, at the discrete set of points where $\beta^+=\pm1$, they are no longer distinct. For this
reason, and others related to it, we wish to discard these points from consideration. So we define
\begin{equation}\label{Sig0def}
\Sigma_0 = \{ E\in\sigma(H_0) : \beta^+(E)\neq \pm1\}.
\end{equation}
Because the set of points excluded is discrete, this set forms an essential support for the absolutely
continuous spectrum of $H_0$. Moreover, by fixing the normalisation $\psipm_0(0;z)=1$, there are
solutions $\psipm_0(x;z)$ of (\ref{ODE}) which obey
$$
\psipm_0(x+L;z) = \beta^\pm(z) \psipm_0(x;z)
$$
respectively and which depend analytically on $z\in\Cmplx^+\cup\Sigma_0$. By the definitions of
$\beta^\pm$, if $z\in\Cmplx^+$ then $\psipm_0$ decay exponentially as $x\to\pm\infty$ respectively. In
particular, they are square integrable near $\pm\infty$ which will be important for defining the Weyl
$m$-functions. Notice however that for $E\in\Sigma_0$, the solutions do not decay at all; in fact, they
are almost periodic.
The case where $q$ is identically a constant reduces easily to $q\equiv0$. In this case,
$\Sigma_0=(0,\infty)$ and $\psipm_0(x;z)=\exp(\pm ix\sqrt{z}\,)$ where the branch of the square root is
chosen to make $\psipm_0$ decay in the appropriate directions.
Before moving on, we wish to make three further remarks about $\psipm_0$. Firstly,
\begin{equation}
\label{conj}
\psip_0(x,E)=\barr{\psim_0(x,E)}\quad\text{for all $E\in\Sigma_0$.}
\end{equation}
Secondly, it is possible to give an explicit formula for the Green function in terms of $\psipm_0$.
That is, the resolvent $R_0(z)=(H_0-z)^{-1}$ is an integral operator with kernel
\begin{equation}
\label{Green}
G(x,x';z)=\frac{\psip_0(x^>;z)\psim_0(x^<;z)}{W[\psip_0,\psim_0]},
\end{equation}
where $x^>=\max\{x,x'\}$ and $x^<$ is the minimum. In the denominator we have the Wronskian of the two
solutions, $W[f,g](x)=f(x)g'(x)-f'(x)g(x)$, which does not depend on $x$. Lastly, in what follows the
$z$-dependence will often be left implicit: $\psip_0(x)=\psip_0(x;z)$ and $\psim_0(x)=\psim_0(x;z)$.
As described in the introduction, our study of the perturbed operator
$$
[Hu](x) = -u''(x) + q(x)u(x) + V(x)u(x) \text{ acting in $L^2(\Reals)$}
$$
will employ its relation to half-line operators, in particular, to
$$
[H_\pm u](x) = -u''(x) + q(x)u(x) + V(x)u(x) \text{ in $L^2(\Reals^\pm)$ with $u(0)=0$.}
$$
These are the half-line Schr\"odinger operators with Dirichlet boundary conditions at the origin which
have the same potential as $H$.
\bigskip
\noindent {\it Proof that Theorems $\Rightarrow$ Corollaries}. This is a well-known application of the
Kato-Rosenblum theorem on relatively trace class perturbations \cite{REED-SIMON3}: The introduction of
a Dirichlet boundary condition at the origin reduces $H$ to $H_+ \oplus H_-$. As it is a
relatively trace class perturbation, the absolutely continuous parts of $H$ and $H_+ \oplus H_-$ are unitarily equivalent.
In our preferred nomenclature: $\Sigac(H)=\Sigac(H_+) \cup \Sigac(H_-)$ with multiplicity two on
the set $\Sigac(H_+) \cap \Sigac(H_-)$. Whilst the theorems do not mention $H_-$, it is naturally equivalent
to $H_+$ with the potential $V(-x)$. In this way each of the corollaries follows directly from its theorem. \qed
\bigskip
The method of Dirichlet decoupling used in this proof is very well known; for example, it is used in the standard
proof of Weyl's law on the eigenvalue asymptotics for vibrating membranes. For a discussion of its use in scattering
theory (in arbitrarily many dimensions) see \cite{DEIFT-SIMON}.
The reason for performing this reduction from the whole line to the half-line is because the spectral
properties of $H_\pm$ are directly accessible from the Weyl $m$-functions. Although we might now
dispense with discussing the whole-line operator $H$ and concentrate just on $H_+$, some of the special
properties of the whole-line operator will make our life easier.
The Weyl $m$-functions are the analytic functions $m^+$ and $m^-$ defined by
\begin{equation} \label{mdef}
m^\pm(z) = \pm \frac{ {\psipm}'(0) }{\psipm(0)}\text{ for each $z\in\Cmplx^+$.}
\end{equation}
Here $\psipm$ are the unique (up to scalar multiples) solutions of
\begin{equation*}
-\psi''(x)+q(x)\psi(x)+V(x)\psi(x)=z\psi(x)
\end{equation*}
which are square integrable at $\pm\infty$ respectively. The functions $\psi^\pm$ are typically called
the Weyl solutions.
Any such $m$-function admits a unique representation as
\begin{equation}
\label{Herg}
m(z) = A + \tfrac{1}{\pi} \int_\Reals \frac{1+Ez}{E-z} d\mu(E)
\end{equation}
with $A\in\Reals$ and $d\mu$ a finite positive measure. In many references this representation is written
with the measure $d\nu=(1+E^2)d\mu$ and the corresponding change to the kernel. The significance of this representation is as
follows: if $d\mu^\pm$ are the measures used to represent $m^\pm$ in (\ref{Herg}) then $H_\pm$ are
unitarily equivalent to the operator $f(E)\mapsto Ef(E)$ acting in $L^2(\Reals,d\mu^\pm)$ respectively.
Moreover, from the representation (\ref{Herg}), one may infer that
\begin{equation}\label{wlim}
(1+E^2)d\mu(E) =\wlim_{\epsilon\to0} \Im[m(E+i\epsilon)]dE.
\end{equation}
For the unperturbed problem, $V=0$, the Weyl solutions are the Bloch solutions and so extend
analytically from $\Cmplx^+$ to $\Sigma_0$. Since we chose the normalisation $\psipm_0(0)=1$, the
$m$-functions for the unperturbed problem, $m^+_0,m^-_0$ also extend analytically to $\Sigma_0$. By
(\ref{conj}) we have the additional property that $m^-_0(E)=-\smash{\barr{m^+_0(E)}}$ for all
$E\in\Sigma_0$.
For a general $V\in L^2$ (or $L^3$) the Weyl solutions need not extend analytically, or even
continuously, to $\Sigma_0$. As in Deift-Killip, we shall circumvent this problem by treating the true
perturbation $V$ as the limit of a sequence of potentials $V_n$ which are $C^\infty$ and of compact
support. A second simplification is that we may assume that $V$ is supported on $[0,\infty)$. This is
possible because the theorems refer to $H_+$ which only depends on $V(x)$ for $x\geq0$. For such
potentials the Weyl solutions behave very nicely:
\begin{lemma}\label{psiExist}
Suppose $V\in C^\infty$ is supported on a compact subset of $[0,\infty)$. For each
$z\in\Cmplx\cup\Sigma_0$ there exist solutions $\psipm$ of
\begin{equation}
\label{pODE}
-\psi''(x)+q(x)\psi(x)+V(x)\psi(x)=z\psi(x)
\end{equation}
such that $\psipm(\pm x)=\psipm_0(\pm x)$ for sufficiently large $x>0$. In fact, $\psim(x)=\psim_0(x)$
for all $x\leq0$. Moreover, $\psipm$ are analytic functions of $z\in\Cmplx^+\cup\Sigma_0$.
\end{lemma}
\noindent {\it Proof.}
This lemma amounts to little more than the existence theorem for linear ordinary differential equations,
though it is quite instructive to first apply variation of parameters with respect to
$\psip_0,\psim_0$.\qed
\bigskip
This lemma implies that $m^-(z)=m^-_0(z)$ and so is analytic on $\Cmplx^+\cup\Sigma_0$. It also shows
that $m^+(z)$ is meromorphic on $\Cmplx^+\cup\Sigma_0$. In fact, if $m^+(z)$ were to have a pole inside
$\Sigma_0$ this would force $H^+$ to have an eigenvalue. This is impossible because as $x\to\infty$ any
non-trivial solution of the perturbed ODE is a non-trivial linear combination of the Bloch solutions and
so not $L^2$. Therefore, $m^\pm(z)$ are analytic on $\Sigma_0$ and, by (\ref{Herg}), on $\Cmplx^+$
also. As a result we may take the weak limit (\ref{wlim}) explicitly to find
\begin{equation}\label{Ewlim}
(1+E^2) d\mu^+(E) = \Im\big[m^+(E)\big] dE \quad \text{ on $\Sigma_0$.}
\end{equation}
This suggests that we need to bound $\Im\,m^+(z)$ away from zero almost everywhere on the spectrum of
the unperturbed operator. Actually, since this equation is only for $V\in C^\infty$ of compact
support, it suggests that we should bound $\Im\;m(z)$ uniformly for a sequence of potentials converging to
the true perturbation. This intuition will be made precise in Proposition \ref{mainprop}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The Perturbation Determinant}\label{PD}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The last chapter closed by suggesting that we shall be able to prove the theorems by bounding
$\Im\,m(z)$ from below. The main result of this chapter, Proposition \ref{mainprop}, shows how
this may be achieved by first obtaining a priori bounds on the function $a(z)$. We introduce the function
$a(z)$ in terms of a Fredholm determinant:
\begin{prop}
Suppose $V\in C^\infty$ is supported on a compact subset of $[0,\infty)$. For each $z\in\Cmplx^+$, the
operator $R_0(z)V$ is of trace class, so we may define
\begin{equation}
\label{det}
a(z)=\det\big(1+R_0(z)V\big),
\end{equation}
which may be written directly in terms of the Bloch and Jost solutions
\begin{equation}
\label{Wrons}
a(z)=\frac{W[\psip,\psim]}{W[\psip_0,\psim_0]}.
\end{equation}
This shows that $a(z)$ may be extended analytically to $\Sigma_0$.
\end{prop}
\noindent
{\it Proof.} Recall that $R_0$ is the resolvent of $H_0$, that is, $R_0(z)=(H_0-z)^{-1}$. For
$q\equiv0$, that $R_0(z)V$ is trace class follows immediately from a theorem of Birman-Solom\kern
-0.04em jak \cite[Theorem 4.5]{SIMON-TI}. This may be extended to $q\not\equiv0$ by employing the
resolvent formula. Section 3 of the above reference proves that the determinant exists and Proposition
5.7 proves (\ref{Wrons}) in the case that $q\equiv0$. This proof can easily be adapted to the present
cause.
Naively (\ref{Wrons}) only shows that $a(z)$ is meromorphic on $\Sigma_0$. However, the theory of
periodic Schr\"odinger operators shows that $W[\psip_0,\psim_0]$ is non-zero throughout
$\Cmplx^+\cup\Sigma_0$. \qed
\bigskip
When $q\equiv0$ we described another function $a_p$ in the introduction as the reciprocal of the transmission coefficient.
An explanation of their equivalence, up to parameterization, will be posponed until the next chapter where the trace formula
\ref{BFZ2} is proved. In order to relate $a$ and the $m$-functions we will need a further
\begin{lemma} \label{Imm}
Suppose $V$ is of compact support. Then for each point $E\in\Sigma_0$,
\begin{equation}
\label{ImmQuot}
\frac{\Im\, m^+(E)}{\Im\, m^+_0(E)} =
\left|\frac{\psip_0(0)}{\psip(0)}\right|^2
\end{equation}
and similarly for $m^-,m^-_0$. Moreover for such $E$, all four $m$-functions have positive imaginary part.
\end{lemma}
\noindent
{\it Proof}. Because $E$ is real, both $\psip(x)$ and $\barr{\psip}(x)$ (its complex conjugate) are
solutions to the perturbed ODE (\ref{pODE}). {From} the invariance of Wronskians, it then follows that
\begin{align*}
-2i\Im(m^+)|\psip(0)|^2 &={\psip}'(0)\barr{\psip(0)} - \psip(0)\barr{{\psip}'(0)} \\
&=W\big[\psip,\barr{\psip}\,\big](0) \\
&=W\big[\psip,\barr{\psip}\,\big](x) \quad \forall x.
\end{align*}
Similarly, for all $x\in\Reals$, $-2i\Im(m_0)|\psip_0(0)|^2=W\big[\psip_0,\barr{\psip_0}\,\big](x)$
which is non-zero for $E\in\Sigma_0$ because $\barr{\psip_0}=\psim_0$ for such $E$ and the Bloch
solutions are linearly independent. Now for $x$ sufficiently large $\psip(x)$ and $\psip_0(x)$ agree.
This means that the Wronskians must be equal, which proves (\ref{ImmQuot}). It also shows that
$\Im\,m^\pm(E)$ and $\Im\,m^\pm_0(E)$ are non-zero. That they are not negative is a consequence of the
representation (\ref{Herg}). \qed
\bigskip
Evaluating the Wronskians of (\ref{Wrons}) at the point $x=0$, we have
\begin{align*}
a(z) &= \frac{\psip(0){\psim}'(0)-\psip(0){\psim}'(0)}{\psip_0(0){\psim_0}'(0)-\psip_0(0){\psim_0}'(0)} \\
&= \frac{\psip(0)\psim(0)}{\psip_0(0)\psim_0(0)} \frac{m^+(z) + m^-(z)}{m^+_0(z) + m^-_0(z)}.
\end{align*}
However, because we only wish to prove theorems about $H_+$ we continue to assume $V$ is supported in $[0,\infty)$.
This means $\psim(0)=\psip_0(0)$ and $m^-(z)=m^-_0(z)$ (refer to Lemma \ref{psiExist}). Employing these identities,
\begin{equation*}
a(z) = \frac{\psip(0)}{\psip_0(0)} \frac{m^+(z) + m^-_0(z)}{m^+_0(z) + m^-_0(z)}
\end{equation*}
for each $z\in\Cmplx^+\cup\Sigma_0$. If we consider just $E\in\Sigma_0$, Lemma \ref{Imm} permits us to write
\begin{align}
\notag
\big|a(E)\big|^2 &= \frac{\Im\, m^+_0(E)}{\Im\, m^+(E)}
\left|\frac{m^+(E) + m^-_0(E)}{m^+_0(E) + m^-_0(E)}\right|^2\\
\label{aandm}
&= \frac{|m^+(E) + m^-_0(E)|^2}{4\,\Im\, m^+(E)\;\Im\, m^-_0(E)}.
\end{align}
To obtain the second equality, we have used the fact that (\ref{conj}) implies $\smash{\barr{m^+_0(E)}}=-m^-_0(E)$
for each $E\in\Sigma_0$.
From the representation (\ref{Herg}) it follows that all $m$-functions have non-negative imaginary
parts. From this and some simple manipulations of (\ref{aandm}) it follows that $|a(E)|^2\geq 1$ for
all $E\in\Sigma_0$. In fact, $a(z)$ does not have any zeros in $\Cmplx^+$ either. If it did,
(\ref{Wrons}) would imply that $\psip,\psim$ were linearly dependent and hence that the perturbed ODE
(\ref{pODE}) has an $L^2$ solution for this $z$. This is impossible because $H$ is self-adjoint and so
cannot have a non-real eigenvalue. Let us repeat our recent deductions for future reference.
\begin{lemma} \label{abelow}
Suppose $V\in C^\infty$ is supported on a compact subset of $[0,\infty)$. The function $a(z)$ is
analytic and non-zero on $\Cmplx^+\cup\Sigma_0$. Consequently, $\log\big|a(z)\big|$ is harmonic there.
Moreover,
\begin{equation}\label{abelowEQ}
\smash{
\big|a(E)\big|^2 = \frac{|m^+(E) + m^-_0(E)|^2}{4\,\Im\, m^+(E)\;\Im\, m^-_0(E)} \geq 1
}
\end{equation}
for all $E\in\Sigma_0$.
\end{lemma}
We have bounded $\log|a(E)|$ from below. But really, as the following proposition suggests, we should
bound it from above. Since $a(z)$ is non-zero, $\log|a(z)|$ is harmonic. This will prove instrumental
in obtaining the desired inequality.
\begin{prop} \label{mainprop}
Let $V\in L^p(\Reals^+)$ {\rm (}for any $1\leq p<\infty${\rm)} and extend it to a function on $\Reals$ by $V(x)=0$
for $x<0$. Further, let $V_n$ be a sequence of $C^\infty$ functions on $\Reals$, each supported on a
compact subset of $[0,\infty)$, which converge to the true perturbation $V$ in $L^p$ sense. To each
such approximate perturbation, associate its own function $a_n(z)=\det[1+R_0(z)V_n]$. Suppose
$I\subset\Sigma_0$ is a compact interval and that there is a bounded continuous function
$w:I\to[0,\infty)$ with support $I$ such that
\begin{equation} \label{mainest}
\int_I \log\big|a_n(z)\big| w(E) dE \leq C
\end{equation}
holds uniformly in $n$. Then $I$ is contained in the essential support of the absolutely continuous
spectrum of $H_+$, in symbols $I\subset\Sigac(H_+)$.
\end{prop}
\noindent {\it Proof.} Let $m^\pm_n$ denote the Weyl $m$-functions for $H_n=H_0+V_n$ and $m^\pm$ those
for the full perturbed problem $H=H_0+V$. For consistency of notation we fix $V_0\equiv0$. Because
$V_n\to V$ in $L^p$, $m^\pm_n(z)\to m^\pm(z)$ uniformly on compact subsets of $\Cmplx^+$. From the
representation (\ref{Herg}) it follows that $d\mu^\pm_n$, the measures occurring in the representation of
$m^\pm_n$, converge weakly to $d\mu^\pm$, the measures for $m^\pm$. From the fact that
$\Im\,m^\pm_n(i)\to\Im\,m^\pm(i)$ one may also infer that the total mass of each $d\mu^\pm_n$ is
uniformly bounded.
From Lemma \ref{Imm} we know that $\Im\,m^\pm_n(E)$ are positive for all $n\geq 0$. Consequently,
$|m^+_n(E)+m^-_0(E)|>\Im\,m^-_0(E)$ and it follows, from (\ref{aandm}), that
$$
\big|a(E)\big|^2 \geq \frac{\Im\,m^-_0(E)}{4\,\Im\,m^+_n(E)} \text{ for all $E\in\Sigma_0$.}
$$
As we just remarked, $\Im\,m^-_0(E)$ is positive on $\Sigma_0$ and hence it is
bounded below on the compact set $I$. As a result, there is a constant $c\geq0$ so that
$$
2\log\big|a(E)\big| \geq -\log\big[\Im\,m^+_n(E)\big] - c.
$$
Integrating and applying (\ref{mainest}) it follows that there exists a constant $C'$ such that
\begin{equation}\label{halfway}
- \int_I \log\big[\Im\,m^+_n(E)\big]w(E)\,dE \leq C'.
\end{equation}
This suggests that $\Im\,m^+_n(E)$ cannot be too small. But first we must exclude the possibility that
cancellations are occurring. Let $\log^+(x)=\max\{0,\log(x)\}$ which obeys $\log^+(x)\leq x$. Then
\begin{align*}
\int_I \log^+\big[\Im\,m^+_n(E)\big] w(E)\,dE &\leq \int_I \Im\,m^+_n(E) w(E)\,dE = \\
=\int_I w(E) d\mu^+_n(E) &\to \int w(E) d\mu^+
\end{align*}
by using (\ref{Ewlim}) and the weak convergence of $d\mu^\pm_n$ discussed at the beginning of this proof.
This means that the integral of $\log^+[\Im\,m^+_n(E)]$ is uniformly bounded and (\ref{halfway}) reduces to
$$
\int_I \log^-\big[\Im\,m^+_n(E)\big] w(E)\,dE \leq C''
$$
uniformly in $n$. Here we have used the notation $\log^-(x)=\max\{0,-\log(x)\}$.
Since $\log^-$ is convex, we may apply Jensen's inequality and then (\ref{Ewlim}) to infer
\begin{gather*}
C'' \geq w(K)\,\log^-\!\left\{\tfrac{1}{w(K)}\int_K \Im\,m^+_n(E) w(E)\,dE\right\} \\
\Rightarrow\qquad\qquad \int_K w(E) \,d\mu^+_n(E) \geq
w(K)\,\exp\!\big\{{-C''}/w(K)\big\}\qquad\qquad\phantom{\Rightarrow}\mbox{}
\end{gather*}
for any compact $K\subset I$. Here $w(K)=\int_K w(E)\,dE$.
Observe the following fact: if the measure $d\nu$ (on $\Reals$) is the weak limit of $d\nu_n$ and $K$ is
a compact set then $\nu(K)\geq\varlimsup \nu_n(K)$. This is a simple consequence of the fact that Borel
measures on $\Reals$ are regular (c.f. Lemma 2 of \cite{DEIFT-K}). Applying this fact with
$d\nu_n=w(E)\,d\mu^+_n(E)$,
$$
\int_K w(E) \,d\mu^+(E) \geq w(K)\exp\!\big\{{-C'}/w(K)\big\}.
$$
Since $w$ is positive Lebesgue almost everywhere on $I$, and so on $K$, this proves that the Lebesgue
measure on $I$ is absolutely continuous with respect to $d\mu^+$. That is, $I$ is a subset of the
essential support of the absolutely continuous part of $d\mu^+$. This completes the proof for, as we
mentioned after introducing the representation (\ref{Herg}), $H_+$ is unitarily equivalent to
multiplication by $E$ in $L^2\big(d\mu^+(E)\big)$. \qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Trace Formulae}\label{DK}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This chapter is dedicated to the case $q\equiv 0$ and presents joint work with P. Deift \cite{DEIFT-K}. This permits some considerable simplifications, especially
if we replace the former spectral parameter $z$ by $k^2$ with $k\in\Cmplx$. For example, for each $z\in\Cmplx^+$,
the Weyl solutions are given by $\psipm_0(x;z)=\exp(ikx)$ where $k$ is the square root of $z$ which lies in the
upper half-plane. With this parameterization, we obtain Weyl solutions which are entire functions of $k$; as functions
of $z$ they have a branch cut. In analogy with Lemma \ref{psiExist} we have
\begin{lemma} If $V$ is supported on a compact subset of $[0,\infty)$ then $\psipm(x)$ are entire functions of $k$ and,
for $k\neq 0$, we may write
\begin{equation}
\label{apbp}
\psip(x) = a_p(k) e^{ikx} + b_p(k) e^{-ikx} \quad \forall x<0
\end{equation}
with $a_p$ and $b_p$ analytic functions of $k\in \Cmplx\setminus\{0\}$.
\end{lemma}
Now we are ready to demonstrate the relation between $a_p(k)$ and the function $a(z)$ of the previous chapter. Evaluating the Wronskians of (\ref{Wrons}) at any point $x<0$ we have
\begin{align*}
a(z) &= \frac{W[a_pe^{ikx} + b_p(k) e^{-ikx},e^{-ikx}]}{W[e^{ikx},e^{-ikx}]} \\
&= a_p(k)
\end{align*}
and so $a(z)=a_p(k)$ provided $k$ is chosen to be the square root of $z$ which lies in the upper half-plane.
In order to apply Proposition \ref{mainprop} we wish to obtain a bound on a weighted integral of $\log|a(E)|$ over $[0,\infty)$ for potentials $V\in C^\infty_c$. The second of the BFZ trace formulae (\ref{BFZ2}) arises from the
evaluation of
$$
\int_0^\infty \log\big|a(E)\big| \sqrt{E}\,dE
$$
and is valid for potentials $V$ with $(1+|x|)V(x)\in L^1$. As we do not need it in this generality, we shall derive it
for $V\in C^\infty_c$.
By taking complex conjugates in (\ref{apbp}), it follows that $a_p(-k)=\overline{a_p(k)}$. As a result we can
rewrite the above integral as
$$
\int_{\Reals} \log\big[a_p(k)\big] k^2 dk.
$$
We would like to evaluate this integral by closing the contour in the upper half-plane. While $a_p(k)$ is analytic,
$\log\big[a_p(k)\big]$ fails to be at the zeros of $a_p(k)$. By (\ref{abelowEQ}) we know that $a_p$ does not have
zeros for $k\in\Reals$. Moreover, as discussed just before Lemma \ref{abelow}, complex zeros can only occur if $z=k^2$ is an eigenvalue of (\ref{pODE}). Therefore, the zeros $a_p(k)$ which lie in the upper half-plane occur at finitely many purely imaginary points $i\beta_1,...,i\beta_m$ corresponding to the negative eigenvalues $-E_1=-\beta_1^2,...,-E_m=-\beta_m^2$
of (\ref{pODE}). To remove the problem of its zeros, we divide $a_p(k)$ by the Blaschke product
$$
B(k) = \prod_{j=1}^m \frac{k-i\beta_j}{k+i\beta_j}.
$$
Then the Cauchy Theorem says
$$
\int_{-R}^R \log\big[a_p(k)/B(k)\big] k^2dk =
- \int_0^\pi \log\big[a(Re^{i\theta})/B(Re^{i\theta})\big] iR^3e^{3i\theta}d\theta.
$$
Because for $k\in\Reals$ both $B(-k)=\overline{B(k)}$ and $|B(k)|=1$, the left-hand integral is unchanged by the
introduction of $B(k)$. So taking the limit $R\to\infty$
\begin{align}
\notag
\int_0^\infty \log\big|a(E)\big| \sqrt{E}\,dE &= -\lim_{R\to\infty}
\int_0^\pi \log\big[a_p(Re^{i\theta})/B(Re^{i\theta})\big] iR^3e^{3i\theta}d\theta \\
\label{contint}
&= - \tfrac{2\pi}{3} \sum_{j=1}^m E_j^{3/2}
- \lim_{R\to\infty} \int_0^\pi \log\big[a_p(Re^{i\theta})\big] iR^3e^{3i\theta}d\theta.
\end{align}
To calculate the remaining integral, we need to
determine the asymptotics of $\log[a(k)]$ as $k$ goes to infinity in the upper half-plane.
If we write $\psip(x)=p(x)e^{ikx}$, then $p(x)$ obeys the integral equation
\begin{align}\label{intEq}
p(x) & = 1 + \tfrac{i}{2k}\int_x^\infty \left[1-e^{2ik(y-x)}\right]V(y)p(y)dy.
\end{align}
This equation is of Volterra type and so may be solved by iteration. If $\Im\,k\geq0$ the kernel is $O(1/|k|)$, so
for $x<0$ and $\Im\,k\geq 0$ we have
\begin{align*}
p(x) & = 1 + \tfrac{i}{2k}\!\int_0^\infty \big[1-e^{2ik(y-x)}\big] V(y)dy \\
& - \tfrac{1}{4k^2}\!\int_0^\infty \!\!\! \int_y^\infty \!\!
\big[1-e^{2ik(y-x)}\big] \big[1-e^{2ik(z-y)}\big] V(y)V(z)dzdy \\
& - \tfrac{i}{8k^3}\!\int_0^\infty \!\!\! \int_y^\infty \!\!\! \int_z^\infty \!\!
\big[1-e^{2ik(y-x)}\big] \big[1-e^{2ik(z-y)}\big] \big[1-e^{2ik(u-z)}\big]V(y)V(z)V(u)dudzdy \\
& + O\big(|k|^{-4}\big).
\end{align*}
Because $x<0$, $\psip(x)=a_p(k) e^{ikx} + b_p(k) e^{-ikx}$ and so $p(x)=a_p(k)+ b_p(k) e^{-2ikx}$. Therefore, to find $a_p$, we just need to collect those terms in the above expansion without a leading factor of $e^{-2ikx}$:
\begin{align*}
a_p(k) =& 1+\tfrac{i}{2k}\!\int\!V(y)dy-\tfrac{1}{8k^2}\!\iint\left[1-e^{2ik|y-z|}\right]V(y)V(z)dydz \\
& \phantom{1} + \int_0^\infty \!\!\! \int_y^\infty \!\!\! \int_{z}^\infty \!\!
\big[1-e^{2ik(z-y)}\big] \big[1-e^{2ik(u-z)}\big]V(y)V(z)V(u)dudzdy \\
& \phantom{1} + O(|k|^{-4}).
\end{align*}
The change in the term with two integrals arises from symmetrizing the integrand then taking half of the
integral over all space.
Now we proceed to expand $\log[a_p(k)]$ using $\log(1+\epsilon)=\epsilon-\epsilon^2/2+\epsilon^3/3+O(\epsilon^4)$;
again this is for $\Im k\geq0$:
\begin{align*}
\log[a_p(k)] =& \tfrac{i}{2k}\!\int\!V(y)dy+\tfrac{1}{8k^2}\!\iint e^{2ik|y-z|} V(y)V(z)dydz \\
&-\tfrac{i}{16k^3}\!\int\!V(x)dx \iint e^{2ik|y-z|} V(y)V(z)dydz \\
&-\tfrac{i}{8k^3}\!\int_0^\infty \!\!\! \int_y^\infty \!\!\! \int_z^\infty \!\!
\big[ e^{2ik(u-y)} - e^{2ik(z-y)} - e^{2ik(u-z)} \big]V(y)V(z)V(z) dydzdu\\
&+O(|k|^{-4}).
\end{align*}
Notice that something remarkable has happened; the three terms of the form $\frac{C}{k^3}\big[\int\! V\big]^3$ which
occur, cancel. To see this we have used the fact that
$$
\int_0^\infty \!\!\! \int_y^\infty \!\!\! \int_z^\infty \!\! V(y)V(z)V(u)\,dydzdu
= \tfrac{1}{6}\!\iiint V(y)V(z)V(u)\;dydzdu.
$$
We are now ready to perform the contour integral
$$
\lim_{R\to\infty} \int_0^\pi \log\big[a_p(Re^{i\theta})\big] iR^3e^{3i\theta}d\theta.
$$
Naive estimation shows that terms $O(|k|^{-4})$ make no contribution as $R\to\infty$. The term of order $|k|^{-1}$,
$\frac{i}{2k}\int V$, gives zero identically for all $R$. The two terms $O(|k|^{-3})$ also give zero contribution
in the limit $R\to\infty$ because both integrands are $O\big(\frac{1}{1+R\sin(\theta)}\big)$.
(Notice that this would not be the case
had the terms of the form $\frac{C}{k^3}\big[\int\! V\big]^3$ not cancelled.) With these observations, (\ref{contint})
reduces to
$$
\int_0^\infty \!\!\log\big|a(E)\big| \sqrt{E}\,dE = - \tfrac{2\pi}{3} \sum_{j=1}^m E_j^{3/2}
- \lim_{R\to\infty} \tfrac{i}{8}\!\!\int_0^\pi\!\!\!\iint e^{-2iR|y-z|e^{i\theta}} V(y)V(z) Re^{i\theta}dydzd\theta.
$$
To perform this last integral, it is better to use the Cauchy Theorem once more---this time to replace the semi-circular
contour $Re^{i\theta}$, $\theta\in[0,\pi]$ with a straight line:
$$
\int_0^\infty \!\!\log\big|a(E)\big| \sqrt{E}\,dE = - \tfrac{2\pi}{3} \sum_{j=1}^m E_j^{3/2}
+ \lim_{R\to\infty} \tfrac{1}{8}\!\!\iint\!\!\int_{-R}^R e^{2ik|y-z|} V(y)V(z) \,dkdydz.
$$
Performing the $k$ integral first gives a factor $\sin[2R(y-z)]/(y-z)$. As $R\to\infty$ this factor converges distributionally to $\pi\delta(y-z)$. Since we have assumed that $V$ is $C^\infty$ and compactly supported, this
leads to
\begin{equation}\label{2ndBFZ}
\int_0^\infty \!\!\log\big|a(E)\big| \sqrt{E}\,dE = - \tfrac{2\pi}{3} \sum_{j=1}^m E_j^{3/2}
+ \tfrac{\pi}{8}\!\!\int V^2
\end{equation}
which is the trace formula we set out to prove.
Proceeding more systematically it is possible to derive the full family of trace formulae. See for example
\cite{NOVIKOV} for a proof of
\begin{prop}
If $V$ is $C^\infty$ and of compact support then
\begin{equation}\label{NthBFZ}
\int^\infty_0 \!\!\log|a(E)| E^{n-1/2} dE = \frac{(-1)^n\pi}{2^{2n+1}} \int \xi_{2n+1}(x) dx
+ \frac{(-1)^n 2\pi}{2n+1} \sum E_m ^{n+1/2},
\end{equation}
where $\xi_0(x)=0,\xi_1(x)=V(x)$ and $\xi_{n+1} + \xi'_{n} + \smash{\displaystyle \sum_{\ell=1}^{n}} \xi_{\ell}\xi_{n-\ell}=0$.
\end{prop}
\noindent
{\it Proof of Theorem 1}. We begin by proving part i). Let $V_n$ be a sequence of $C^\infty$ functions of compact support which converge to the true perturbation $V$ in $L^2$. Then it follows from (\ref{2ndBFZ}) that
\begin{equation*}
\int_0^\infty \!\!\log\big|a_n(E)\big| \sqrt{E}\,dE \leq \tfrac{\pi}{8}\!\!\int V_n^2 \leq C
\end{equation*}
because the eigenvalue term is always negative, or zero. But this provides the conditions of Proposition \ref{mainprop}
and so proves that $\Sigac=[0,\infty)$.
The proof of part ii) is almost identical, but employs (\ref{NthBFZ}) with $n=3$; (\ref{2ndBFZ}) corresponds to $n=1$.
More explicitly we have
\begin{align*}
\int^\infty_0 \log|a(E)| E^{5/2} dE =& -\tfrac{2\pi}{7} \sum E_m ^{7/2} +
\tfrac{\pi}{2^7}\! \int (V'')^2-5V^2V''+5 V^4\, dx \\
\leq& \,\tfrac{\pi}{2^7}\! \int (V'')^2-5V^2V''+5 V^4\, dx \\
\leq& \,\tfrac{7\pi}{2^8} \big\| V'' \big\|_{L^2}^2 + \tfrac{15\pi}{2^8} \big\| V \big\|_{L^4}^4,
\end{align*}
where the last inequality follows from H\"older and $2XY\leq X^2+Y^2$. In this way, if we choose a sequence of
$C^\infty$ functions $V_n$ each of compact support so that $V_n\to V$ in both $L^4$ and $\dot H^2$, then we
have a uniform bound on
$$
\int^\infty_0 \!\!\log|a_n(E)| E^{5/2} dE.
$$
This means that we may once again employ Proposition \ref{mainprop} to complete the proof.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Regularized Determinants} \label{RegDet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This chapter introduces regularized determinants and provides some estimates. As in the previous
chapter, we are interested in a priori bounds and shall assume that $V$ is $C^\infty$ and of compact
support.
If one regards determinants as the product of the eigenvalues, regularized determinants are the analogue
of the convergence factors used in the Weierstrass Factorisation Theorem. A thorough treatment of these
functions may be found in \cite{SIMON-TI}. We shall need only the first two regularized determinants:
\begin{align}
\det_2(1+A) &= \det(1+A)\exp\{-\tr(A)\},\\
\det_3(1+A) &= \det(1+A)\exp\{-\tr(A)+\tfrac{1}{2}\tr(A^2)\}.
\end{align}
Whilst at first these are only defined for elements of trace class, $A\in\mathfrak{I}_1$, they may be
extended to the Hilbert-Schmidt class, $\mathfrak{I}_2$, and $\mathfrak{I}_3=\{A:\tr(|A|^3)<\infty\}$
respectively. The possibility of this extension is underlined by the estimates
\begin{align}
\label{det2est}
\big|\det_2(1+A)\big| &\leq \exp\{C_2\|A\|_2^2\},\\
\label{det3est}
\big|\det_3(1+A)\big| &\leq \exp\{C_3\|A\|_3^3\},
\end{align}
where $C_2,C_3$ are universal constants and $\|A\|_n^n=\tr(|A|^n)$ denotes the $n$th power of the norm
on $\mathfrak{I}_n$. These estimates are proved in Section 9 of \cite{SIMON-TI}. Our application of
these estimates is
\begin{lemma}\label{TIest}
There is a constant $C$ depending only on $q$, so that the estimates
\begin{align*}
\big\| R_0(E+i\epsilon)V\big\|_2 &\leq C \epsilon^{-1/2} (1+\epsilon^{-1/2}) \|V\|_{L^2},\\
\big\| R_0(E+i\epsilon)V\big\|_3 &\leq C \epsilon^{-1/2} (1+\epsilon^{-1/2}) \|V\|_{L^3}
\end{align*}
hold for all $E\in\Reals$ and all $\epsilon>0$. And consequently, for a new constant $C$,
\begin{align*}
\log\big| \det_2\big(1+R_0(E+i\epsilon)V\big) \big|
&\leq C \epsilon^{-1}(1+\epsilon^{-1}) \|V\|_{L^2}^2,\\
\log\big| \det_3\big(1+R_0(E+i\epsilon)V\big) \big|
&\leq C \epsilon^{-3/2}(1+\epsilon^{-3/2}) \|V\|_{L^3}^3
\end{align*}
hold for this same set of $E,\epsilon$.
\end{lemma}
\noindent {\it Proof.} The first two estimates follow directly from Theorem B.9.3 of \cite{SEMIGROUPS}.
More explicitly, since $|E+i\epsilon|^{-1}$ is bounded by a constant multiple of $\epsilon^{-1/2}
(1+\epsilon^{-1/2}) (1+|E|)^{-1/2}$, this theorem shows that there exists $C_p$ with
$$
\big\|R_0(E+i\epsilon)V\big\|_p \leq C_p \epsilon^{-1/2} (1+\epsilon^{-1/2}) \big\|V\big\|_{L^p}
$$
for any $p\geq 2$.
Employing these bounds in (\ref{det2est}) and (\ref{det3est}) gives the second pair of estimates.\qed
\bigskip
Given the specificity of the situation we are treating, these estimates are quite crude. However, they
will prove ample for our purposes.
For $E+i\epsilon\in\Cmplx^+$ we can also obtain lower bounds on $\log|\det(1+R_0V)|$.
\begin{prop}\label{detbelow}
Suppose $V$ is $C^\infty$ and of compact support. Then there is a constant $C$ so that
\begin{align}
\label{det2b}
\log\big| \det_2\big(1+R_0(E+i\epsilon)V\big) \big| &\geq - C \epsilon^{-1}(1+\epsilon^{-1}) \|V\|_{L^2}^2,\\
\label{det3b}
\log\big| \det_3\big(1+R_0(E+i\epsilon)V\big) \big| &\geq - C \epsilon^{-3/2}(1+\epsilon^{-3/2}) \|V\|_{L^3}^3
\end{align}
hold for all $E\in\Reals$ and $\epsilon>0$.
\end{prop}
\noindent {\it Proof.}
Let us begin by calculating some derivatives. Note that all resolvents appearing in this proof are evaluated
at the point $E+i\epsilon\in\Cmplx^+$.
\begin{align}
\notag
\dde \log\big| \det(1+R_0V)\big| &= \Re\;\tr\big( i(1+R_0V)^{-1}R_0R_0V \big) \\
\label{dlog}
&= \Re\;\tr\big( iRR_0V \big) \\
\intertext{by the resolvent formula. Also}
\label{dtr1}
\dde \Re\;\tr(R_0V) &= \Re\;\tr\big( iR_0R_0V \big) \\
\label{dtr2}
\dde \tfrac{1}{2}\Re\;\tr(R_0V\!R_0V) &= \Re\;\tr\big( iR_0V\!R_0R_0V \big)
\end{align}
By combining (\ref{dlog}) and (\ref{dtr1}) we have
\begin{align}
\notag
\dde \log\big| \det_2(1+R_0V)\big| &= \Re\;\tr\big( i(R-R_0)R_0V \big) \\
\label{blah}
&= -\Re\;\tr\big( iR_0V\!R R_0V \big).\\
\intertext{By employing the trace ideal version of H\"older's inequality, this implies}
\left| \dde \log\big| \det_2(1+R_0V)\big| \right| &\leq \|R_0V\|_2^2\, \|R\| \notag
\end{align}
Here $\|\cdot\|$ denotes the operator norm. Of course, $\|R\|\leq\epsilon^{-1}$.
This and Lemma~\ref{TIest} permit us to conclude
$$
\left| \dde \log\big| \det_2(1+R_0V)\big| \right| \leq C' \epsilon^{-2}(1+\epsilon^{-1}) \|V\|^2_{L^2}.
$$
Continuing from (\ref{blah}), one can show
\begin{align*}
\dde \log\big| \det_3(1+R_0V)\big| &= - \Re\;\tr\big( iR_0V\!R R_0V \big) + \Re\;\tr\big( iR_0V\!R_0R_0V \big)
\\
&= \Re\;\tr\big( iR_0V (R_0-R) R_0V \big)
\\
&= \Re\;\tr\big( iR_0V\!R_0V\!R R_0V \big)
\\
\left| \dde \log\big| \det_3(1+R_0V)\big| \right| & \leq \|R_0V\|_3^3 \|R\| \leq
C''\epsilon^{-5/2}(1+\epsilon^{-3/2})\|V\|_{L^3}^3
\end{align*}
by similar means.
Since $V$ is of compact support, it is easy to show that
$$
\lim_{\epsilon\to\infty} \det_n\big[1+R_0(E+i\epsilon)V\big] = 1 \quad \forall E\in\Reals.
$$
Hence the proposition follows from our derivative bounds and the Fundamental Theorem of Calculus.
\qed
\bigskip
Our goal is to obtain (\ref{mainest}) so that we may apply Proposition \ref{mainprop}. This requires us
to understand the behaviour of $a(E)$ for $E\in\Sigma_0$ whilst thus far we have been estimating the
regularized determinants for $z\in\Cmplx^+$. The significance of these estimates at non-real $z$ will
become clearer in the next chapter. For the moment, however, we wish to relate $a(E)$ and the
regularized determinants for $E\in\Sigma_0$. We begin by considering $\det_2$.
\begin{prop}\label{fanda} Suppose $V\!$ is $C^\infty$ and of compact support. Then the harmonic function
$$
f(z) = \log\big|\det_2\big(1+R_0(z)V\big)\big|
$$
extends continuously from $\Cmplx^+$ to $\Sigma_0$. Moreover, $f(E)=\log\big|a(E)\big|$ for $E\in\Sigma_0$.
\end{prop}
\noindent {\it Proof.}
To see why $f(z)$ is harmonic, just recall Lemma \ref{abelow} and
$$
f(z) = \log|a(z)| - \Re\;\tr \big(R_0(z)V\big)
$$
from the definition of $\det_2$. Because $V$ is integrable, we can write
\begin{equation}
\label{Retr}
\Re\;\tr \big(R_0(z)V\big) = \Re \, \int G(x,x;z) V(x) dx
\end{equation}
and so extend the left-hand side continuously to $\Sigma_0$. By Lemma \ref{abelow} this means that
$f(z)$ extends continuously to $\Sigma_0$. To complete the proof we need to show that (\ref{Retr})
vanishes for $z=E\in\Sigma_0$. For such $E$, we can use (\ref{conj}) and (\ref{Green}) to write
$$
\Re \, \int G(x,x;E) V(x) dx = \Re \int \frac{|\psip(x)|^2}{W\big[ \psip,\barr{\psip}\,\big]} V(x) dx
$$
which is zero because $W\big[ \psip,\barr{\psip}\,\big]$ is purely imaginary.\qed
\bigskip
The fact that the real part of the Green function vanishes inside the spectrum is what permitted us to
conclude that $f(E)=\log|a(E)|$. This holds not only for periodic $q$ but on the absolutely continuous spectrum
of any ergodic Schr\"odinger operator \cite{KOTANI,SIMON-KT}. It is possible to extend our results to prove that any interval
contained within the a.c.~spectrum of an operator with ergodic potential remains in the a.c.~spectrum
under an $L^2$ perturbation. For these potentials, the results that we obtained quite easily
from the properties of the Bloch solutions must be obtained more circuitously. For example, the fact that the Weyl
solutions extend analytically from $\Cmplx^+$ to $\Sigma_0$ must be obtained by first observing that the Green
function may be extended analytically by an application of the Schwarz reflection principle.
The equivalent of Proposition \ref{fanda} for $\det_3$ is more involved, especially if $q\not\equiv0$.
\begin{prop} \label{ganda}
Suppose $V\in C^\infty$ and is of compact support. Then the harmonic function
$$
g(z) = \log\big|\det_3\big(1+R_0(z)V\big)\big|
$$
extends continuously from $\Cmplx^+$ to $\Sigma_0$. Suppose $q\equiv 0$ and $I\subset\Sigma_0=(0,\infty)$ is a compact
interval. If we define $\tilde I = \{k\in\Reals:k^2/4\in I\}$ then
\begin{equation}
\label{gq0}
\int_I \Big|g(E)-\log|a(E)|\Big| dE \leq C_I \big\|\hat V\big\|^2_{L^2(\tilde I;dk)}.
\end{equation}
The version for $q\not\equiv 0$ is slightly weaker: for any compact interval $I\subset\Sigma_0$
\begin{equation}
\label{gq}
\int_I \Big|g(E)-\log|a(E)|\Big| dE \leq C_I \big\|\hat V\big\|^2_{\ell^\infty(L^2)}.
\end{equation}
The definition of $\ell^\infty(L^2)$ was given in {\rm (\ref{ellL})} of Chapter~\ref{bckgrnd}.
\end{prop}
\noindent {\it Proof.}
The result on continuous extension is essentially the same as the previous proposition because
\begin{align*}
g(z) &= f(z) + \tfrac{1}{2} \Re\;\tr\big[R_0(z)V\!R_0(z)V\big] \\
&= f(z) + \tfrac{1}{2} \Re \iint G(x,y;z)^2V(x)V(y) dx\,dy.
\end{align*}
As $f(E)=\log|a(E)|$ for all $E\in\Sigma_0$, we need only attend to the additional term.
Given $E\in\Sigma_0$, we may employ (\ref{conj}) and (\ref{Green}) to write
$$
g(E)-f(E) = 2\Re\;\underset{\!\!\!\!x>y}{\iint} \frac{\psip_0(x)^2\barr{\psip_0}(y)^2}{W\big[\psip_0,\barr{\psip_0}\,\big]^2}
V(x)V(y) dx\,dy.
$$
However, the Wronskian is purely imaginary and taking the complex conjugate of $\psip_0(x)^2\barr{\psip_0}(y)^2$
is equivalent to interchanging $x,y$. So
\begin{align} \notag
g(E)-f(E)&=\frac{1}{W\big[\psip_0,\barr{\psip_0}\,\big]^2} \iint \psip_0(x)^2\barr{\psip_0}(y)^2 V(x)V(y) dx\,dy \\
\label{star}
&=\frac{1}{W\big[\psip_0,\barr{\psip_0}\,\big]^2} \left| \int \psip_0(x)^2 V(x) dx \right|^2.
\end{align}
Controlling this when $q\equiv0$ is the easiest because $\psip_0(x;E)=\exp(ikx)$ where $k^2=E$ and
$k>0$. In this case, therefore,
$$
g(E)-f(E) = \frac{-1}{4k^2} \big|\hat V(2k)\big|^2
$$
and for each compact $I\subset(0,\infty)$ we have
$$
\int \big| g(E)-f(E) \big| dE \leq C_I \big\|\hat V \big\|^2_{L^2(\tilde I)},
$$
which proves (\ref{gq0}) because $f(E)=\log|a(E)|$.
The proof for $q\not\equiv0$ is more involved. What we present now is actually just a minor alteration
of an argument from the ``Proof of Theorem 1.8'' in \cite{KISELEV-PRE}. Recall from
Chapter~\ref{bckgrnd} that there is a unimodular complex number $\beta^+(E)$ associated to each
$E\in\Sigma_0$ so that
\begin{equation}
\label{betaagain}
\psip_0(x+L;E) = \beta^+(E) \psip_0(x;E),
\end{equation}
where $L$ is the period of $q$. Moreover, $\beta^+(E)\neq\pm 1$ so we may define
$\gamma(E)=\arg(\beta^+(E))\in(-\pi,\pi)$. It is well known that $\gamma(E)$ is strictly monotonic on
each connected subset of $\Sigma_0$. Indeed, there exists $\epsilon_I>0$ so that
$|\gamma'(E)|>\epsilon_I$ for all $E$ in the compact set $I$. From (\ref{betaagain}) we see that for
each $E\in\Sigma_0$, $\sigma(x;E)=\exp[-i\gamma(E)x/L]\psip_0(x;E)$ is $L$-periodic (in $x$) and hence
so is $\sigma(x;E)^2$. Because $I$ is compact the family $\{\sigma(x;E)^2:E\in I\}$ is uniformly
continuously differentiable in $x$. That is to say,
$$
\sup_{E\in I} \|\sigma(\cdot,E)\|_{C^1} \leq C_I < \infty.
$$
As a result, when we develop $\sigma^2$ as a Fourier series $$ \sigma(x;E)^2 = \sum_n c_n(E)
e^{2\pi i nx/L} $$ the sequence $\tilde c_n = \sup_E |c_n(E)|$ is bounded in $\ell^1$.
Let us now apply what we have learned to the estimation of (\ref{star}):
\begin{align*}
\int \psip_0(x)^2 V(x) dx &= \int \sigma(x;E)^2 e^{2i\gamma(E)x/L} V(x) dx \\
&= \int \sum_n c_n(E) \exp\big\{i\big[2\pi n + 2\gamma(E)\big]x/L\big\} V(x) dx\\
&= \sum_n c_n(E) \hat V\big(\tfrac{2\pi n + 2 \gamma(E)}{L}\big).
\end{align*}
By using the triangle inequality, then H\"older's inequality, this implies
$$
\left\| \int \psip_0(x)^2 V(x) dx \right\|_{L^2(I;dE)} \leq \sum_n \tilde c_n
\big\|\hat V\big(\tfrac{2\pi n + 2 \gamma(E)}{L}\big) \big\|_{L^2(I;dE)}.
$$
Remember that $|\gamma'(E)|>\epsilon_I$ for $E\in I$. So, if $\gamma(I)=[\gamma_0,\gamma_1]$, then by a
change of variables,
$$
\int_I \big| V\big(\tfrac{2\pi n + 2 \gamma(E)}{L}\big) \big|^2 dE \leq
\int_{2\gamma_0/L}^{2\gamma_1/L} \big| V\big(\tfrac{2\pi n}{L} + y\big) \big|^2 \epsilon_I^{-1}\,dy.
$$
But $[\gamma_0,\gamma_1]\subset(-\pi,\pi)$ and so, by our choice of $\ell^\infty(L^2)$ norm, it is immediate that
$$
\left\| \int \psip_0(x)^2 V(x) dx \right\|_{L^2(I;dE)} \leq
\epsilon_I^{-1} \|\tilde c_n\|_{\ell^1} \|V\|_{\ell^\infty(L^2)}.
$$
Because $W\neq 0$ this provides sufficient control on (\ref{star}) to prove (\ref{gq}). \qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Proofs of Theorems 2--4}
In Chapter \ref{bckgrnd} we showed that if $V$ is of compact support, then the function
$$
\log\big|a(z)\big|= \Re\,\log\big\{ \det\big(1+R_0(z)V\big) \big\}
$$
is harmonic on the set $\Cmplx^+\cup\Sigma_0$. Recall that $\Sigma_0$ is an essential support for the
absolutely continuous spectrum of $H_0$. Indeed, $\Sigma_0$ is just $\sigma(H_0)$ with a discrete set of
points removed.
In Chapter \ref{RegDet} we introduced regularized determinants and defined
\begin{align}
f(z) &= \log\big|\det_2\big(1+R_0(z)V\big)\big|, \\
g(z) &= \log\big|\det_3\big(1+R_0(z)V\big)\big|.
\end{align}
We further showed that these functions are also harmonic on $\Cmplx^+\cup\Sigma_0$, obtained estimates
for $z\in\Cmplx^+$ and bounded $f(E)-a(E)$ and $g(E)-a(E)$ for $E\in\Sigma_0$.
The current chapter is devoted to obtaining the estimate (\ref{mainest}) and so proving Theorems 2--4.
We will use the fact that $f,g$ are harmonic to convert our bounds for non-real $z$ to bounds for real
$z\in\Sigma_0$. For this reason we begin with some results about harmonic functions.
Given a closed interval $I\subset\Sigma_0$, we define a triangle over $I$ as follows: If $I=[a,b]$,
define $c=(a+b)/2+i(b-a)/2$. Then the points $a,b,c$ form a right-angled isosceles triangle with
hypotenuse $I$. The remaining two sides we denote by $\Lambda$. Let $\tri=\{x+iy:a+y\leq x\leq
b-y,y\geq 0\}$, that is, the solid triangle with boundary $\Lambda\cup I$, and let $z_0\in\tri$ be the
centroid (centre of mass) of $\tri$, $z_0=(a+b)/2+i(b-a)/6$. The choice of the centroid for $z_0$ is
essentially arbitrary.
This construction is depicted more clearly in Figure 1.
\begin{figure}[ht]
\begin{center}
%\fbox{
\setlength{\unitlength}{1mm}
\begin{picture}(80,40)(-40,-5)
\put(-22,24){$\Lambda$}
\put(-19,25){\vector(2,-1){8}}
\put(-19,25){\vector(4,-1){30}}
\put(-30, 0){\circle*{2}}\put(-31,-3.7){$a$}
\put( 30, 0){\circle*{2}}\put( 29,-3.7){$b$}
\put( 0,30){\circle*{2}}\put( 1,31){$c$}
\put( 0,10){\circle*{2}}\put( 2,9.5){$z_0$}
\put(-1,-3.7){$I$}
\put(-40, 0){\line(1, 0){80}}
\put(-30, 0){\line(1, 1){30}}
\put( 0,30){\line(1,-1){30}}
\linethickness{2pt}
\put(-30, 0){\line(1,0){60}}
\end{picture}\\
%}
Figure 1: The definition of $\Lambda,z_0$.
\end{center}
\end{figure}
Let $\mathcal{P}_{z_0}(z)$ denote the Poisson kernel for our triangle evaluated at the point $z_0$. That
is to say, for any function $F$, continuous and harmonic throughout the solid triangle $\tri$,
$$
F(z_0) = \int_{\Lambda\cup I} \mathcal{P}_{z_0}(z) F(z) |dz|.
$$
Moreover, we have the following estimate on $\mathcal{P}_{z_0}(z)$.
\begin{lemma} \label{PoisLemma} There is a constant $C_{z_0}$ such that
\begin{equation}
\label{PoisEst}
\mathcal{P}_{z_0}(z) \leq C_{z_0} \Im(z)^3 \quad \forall z\in\Lambda.
\end{equation}
\end{lemma}
\noindent {\it Proof.} By scaling and translation, it suffices to consider just one triangle and its
centroid, say one with $a=0$. By conformal mapping it is easy to calculate the Poisson kernel for the
infinite sector $\{z:0\leq\arg(z)\leq\pi/4\}$ with the chosen $z_0$. Doing so, one obtains a bound of
the form (\ref{PoisEst}) for this kernel. The chosen triangle is a subset of this sector and harmonic
measure is monotone with respect to the domain. This means that (\ref{PoisEst}) holds for all $z$ from
the line $\barr{ac}$. For $z$ on the other side, $\barr{bc}$, the same estimate holds by symmetry.
The fact that we obtain a third power in (\ref{PoisEst}) is because the angles $\angle cab$ and $\angle
cba$ are $\pi/4$. With angles $\alpha$ one would obtain the power $(\pi/\alpha)-1$. \qed
\bigskip
We have now accumulated enough lemmas and propositions to prove Theorems 2--4. We begin with the
\bigskip
\noindent {\it Proof of Theorem 2}. Let $V\in L^2(\Reals^+)$ and extend it to the whole of $\Reals$
by $V(x)=0$ for $x<0$. Next we choose a sequence of $C^\infty$ functions, $V_n$, each supported on a
compact subset of $[0,\infty)$ which converge to $V$ in $L^2$ sense. To each $V_n$ we associate
functions
\begin{align*}
a_n(z) &= \det\big(1+R_0(z)V_n\big)\\
\intertext{as in Proposition \ref{mainprop}, and}
f_n(z) &= \log\big|\det_2\big(1+R_0(z)V_n\big)\big|
\end{align*}
as in Proposition \ref{fanda}.
We shall prove the theorem by showing that any compact interval $I\subset\Sigma_0$ is also contained in
$\Sigac(H)$. For in this way $\Sigma_0\subset\Sigac(H_+)$. By Weyl's Relative Compactness Theorem, this
implies that $\Sigma_0=\Sigac(H_+)$, as discussed in the introduction. So let us fix $I$ and define
$\Lambda,z_0$ as was done earlier (c.f. Fig. 1).
{From} the fact that the functions $f_n(z)$ are harmonic (Proposition \ref{fanda}) and the definition of
the Poisson kernel, we know that
\begin{equation}
f_n(z_0) = \int_{\Lambda\cup I} \mathcal{P}_{z_0}(z) f_n(z) \,|dz|.
\end{equation}
As the sequence is $V_n$ uniformly bounded in $L^2$, Lemma \ref{TIest} tells us that $f_n(z_0)$ is
bounded uniformly from above. For the same reason, Propositions \ref{detbelow} and \ref{PoisLemma}
combine to show that the integral over $\Lambda$ is bounded uniformly from below. Combining these two
deductions, it follows that
$$
\int_I \mathcal{P}_{z_0}(E) f_n(E) \,dE \leq C
$$
uniformly in $n$. But $f_n(E)=\log|a(E)|$ by Proposition \ref{fanda}. Therefore, we may apply Proposition
\ref{mainprop} with $w(E)=\mathcal{P}_{z_0}(E)$ and so conclude that $I\subset\Sigac(H_+)$. As $I$ was
chosen arbitrarily, this completes the proof. \qed
\bigskip
The proofs of Theorems 3 and 4 follow along the same lines. However, for completeness we include a brief
sketch of each.
\bigskip
\noindent {\it Proof of Theorem 3}. In this case we choose $V_n$ so that $V_n\to V$ in $L^3$ sense and
such that $\hat V_n$ are uniformly bounded in $\ell^\infty(L^2)$. Then, repeating the deductions of the
previous proof, but with Proposition~\ref{ganda} replacing Proposition~\ref{fanda}, we find
\begin{equation}\label{guni}
\int_I \mathcal{P}_{z_0}(E) g_n(E) \,dE \leq C
\end{equation}
uniformly in $n$. But now we may employ (\ref{gq}) to infer that Proposition \ref{mainprop} holds for any compact
$I\subset\Sigma_0$. This completes the proof just as it did for Theorem 1. \qed
\bigskip
\noindent {\it Proof of Theorem 4}. It makes no harm to assume that the interval $I$ in the theorem is
compact, for any interval is a countable union of compact intervals. Next we choose $V_n\to V$ in $L^3$
such that $\int_{\tilde I} |\hat V|^2 dk$ is bounded. Lemma \ref{TIest} may again be used to estimate
$g_n(z_0)$ from above while Proposition \ref{PoisLemma} and (\ref{det3b}) of Proposition \ref{detbelow}
combine to bound the integral over $\Lambda$ from below. This gives (\ref{guni}) and then (\ref{mainest})
follows from (\ref{gq0}) of Proposition \ref{ganda}. With this input, we can apply Proposition
\ref{mainprop} and so complete the proof. \qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{10}
\bibitem{CHRIST-KIS-JAMS} Christ, M., Kiselev, A., {\it Absolutely continuous spectrum for one-di\-men\-sion\-al Schr\"o\-dinger
operators with slowly decaying potentials: some optimal results}, J. Amer. Math. Soc. {\bf 11} (1998) 771--797.
\bibitem{CHRIST-KIS-PRE} Christ, M., Kiselev, A., {\it WKB asymptotic behavior of almost all generalized eigenfunctions for
one-dimensional Schr\"o\-dinger operators with slowly decaying potentials}, Preprint Feburary 2000.
\bibitem{CHRIST-KISELEV-REMLING} Christ, M., Kiselev, A., Remling, C., {\it The absolutely continuous spectrum of
one-dimensional Schr\"o\-dinger operators with decaying potentials}, Math. Res. Lett. {\bf 4} (1997), no.~5, 719--723.
\bibitem{COD-LEV} Coddington, E. A., Levinson, N., {\it Theory of ordinary differential equations}, McGraw-Hill Book Company, Inc.,
New York-Toronto-London, 1955.
\bibitem{DEIFT-K} Deift, P., Killip, R., {\it On the absolutely continuous spectrum of one-di\-men\-sion\-al Schr\"o\-dinger
operators with square summable potentials}, Comm. Math. Phys. {\bf 203} (1999) 341--347.
\bibitem{DEIFT-SIMON} Deift, P., Simon, B., {\it On the decoupling of finite singularities from the question of asymptotic
completeness in two body quantum systems}, J. Funct. Anal. {\bf 23} (1976) 218--238.
\bibitem{DELYON} Delyon, F., {\it Appearance of purely singular continuous spectrum in a class of random
Schr\"odinger operators}, J. Statist. Phys. {\bf 40} (1985) 621--630.
\bibitem{DELYON-S-S} Delyon, F., Simon, B., Souillard, B., {\it From power pure point to continuous spectrum in
disordered systems}, Ann. Inst. H. Poincar\'e {\bf 42} (1985) 283--309.
\bibitem{EASTHAM} Eastham, M. S. P., {\it The Spectral Theory of Periodic Differential Equations,}
Scottish Academic Press, Edinburgh and London, 1973.
\bibitem{HORM} H\"ormander, L. {\it The analysis of linear partial differential operators IV.}, Springer Verlag, New York,
1995.
\bibitem{KISELEV-1} Kiselev, A., {\it Absolutely continuous spectrum of one-dimensional Schr\"o\-dinger operators and
Jacobi matrices with slowly decreasing potentials}, Comm. Math. Phys. {\bf 179} (1996), 377--400.
\bibitem{KISELEV-PRE} Kiselev, A., {\it Preservation of the absolutely continuous spectrum of Schr\"o\-dinger equation under
perturbations by slowly decreasing potentials and a.e. convergence of integral operators}, Preprint number 96-470 at
{\tt http://rene.ma.utexas.edu/\\mp\verb!_!arc/}
\bibitem{KISELEV-2} Kiselev, A., {\it Stability of the absolutely continuous spectrum of the Schr\"o\-dinger equation
under slowly decaying perturbations and a.e. convergence of integral operators}, Duke Math. J. {\bf 94} (1998), 619--646.
\bibitem{KLS} Kiselev, A., Last, Y., Simon, B., {\it Modified Pr\"ufer and EFGP transforms and the spectral
analysis of one-dimensional Schr\"o\-dinger operators,} Comm. Math. Phys. {\bf 194} (1998), 1--45.
\bibitem{KOTANI} S. Kotani, {\it Ljapunov indices determine absolutely continuous spectra of
stationary random one-dimensional Schr\"odinger operators} in {\it Stochastic analysis
(Katata/Kyoto, 1982)}, 225--247, North-Holland, Amsterdam, 1984
\bibitem{KOTANI-USHIROYA} Kotani, S., Ushiroya, N., {\it One-dimensional Schr\"o\-dinger operators with random
decaying potentials}, Comm. Math. Phys. {\bf 115} (1988), 247--266.
\bibitem{MAGNUS} Magnus, W., Winkler, S., {\it Hill's Equation} Dover, New York, 1979.
\bibitem{MNV} Molchanov, S., Novitskii, M., Vainberg, B., {\it First KdV integrals and absolutely continuous spectrum
for 1-D Scr\"odinger operator}, Preprint number 00-216 at {\tt http://rene.ma.utexas.edu/mp\verb!_!arc/}
\bibitem{NABOKO} Naboko, S. N., {\it On the dense point spectrum of Schr\"o\-dinger and Dirac operators}, Teoret. Mat. Fiz.
{\bf 68} (1986) 18--28.
\bibitem{NOVIKOV} Novikov, S. et al. {\it Theory of Solitons: the Inverse Scattering Methods}, a translation of
{\it Teoriia solitonov}, Consultants Bureau, New York, c1984.
\bibitem{PEARSON} Pearson, D. B., {\it Singular Continuous measures in scattering theory}, Comm. Math. Phys. {\bf 60} (1978), 13--36.
\bibitem{REED-SIMON4} Reed, M., Simon, B., {\it Methods of Modern Mathematical Physics. IV. Analysis of Operators},
Academic Press, New York-London, 1978.
\bibitem{REED-SIMON3} Reed, M., Simon, B., {\it Methods of Modern Mathematical Physics. III. Scattering Theory},
Academic Press, New York-London, 1979.
\bibitem{REMLING-1} Remling, C., {\it A probabilistic approach to one-dimensional Schr\"o\-dinger operators with sparse
potentials}, Comm. Math. Phys. {\bf 185} (1997) 313--323.
\bibitem{REMLING-2} Remling, C., {\it The absolutely continuous spectrum of one-dimensional Schr\"o\-dinger operators with
decaying potentials}, Comm. Math. Phys. {\bf 193} (1998) 151--170.
\bibitem{REMLING-3} Remling, C., {\it Bounds on embedded singular spectrum for one-dimensional Schr\"o\-dinger operators},
Proc. Amer. Math. Soc. {\bf 128} (2000) 161--171.
\bibitem{SIMON-TI} Simon, B., {\it Trace Ideals and Their Applications}, London Mathematical Society Lecture Note Series
35, Cambridge University Press, 1979.
\bibitem{SEMIGROUPS} Simon, B., {\it Schr\"o\-dinger semigroups}, Bull. Amer. Math. Soc. {\bf 7} (1982) 442--526.
\bibitem{SIMON-SJM} Simon, B., {\it Some Jacobi matrices with decaying potential and dense point spectrum},
Comm. Math. Phys. {\bf 87} (1982) 253--258.
\bibitem{SIMON-KT} Simon, B., {\it Kotani theory for one dimensional stochastic Jacobi matrices},
Comm. Math. Phys. {\bf 89} (1983), 227-234
\bibitem{SIMON-SCSI} Simon, B., {\it Operators with singular continuous spectrum: I. General operators},
Ann. of Math. {\bf 141} (1995) 131--145.
\bibitem{SIMON-BNDD} Simon, B., {\it Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schr\"odinger
operators}, Proc. Amer. Math. Soc. {\bf 124} (1996) 3361--3369.
\bibitem{SIMON-DPS} Simon, B., {\it Some Schr\"o\-dinger operators with dense point spectrum},
Proc. Amer. Math. Soc. {\bf 125} (1997) 203--208.
\bibitem{STOLZ} Stolz, G., {\it On the absolutely continuous spectrum of perturbed Sturm-Liouville
operators,} J. Reine Angew. Math. {\bf 416} (1991) 1--23.
\bibitem{WEIDMANN} Weidmann, J., {\it Spectral Theory of Ordinary Differential Operators}, Lecture Notes in Mathematics 1258,
Springer-Verlag, Berlin, 1987.
\end{thebibliography}
\end{document}
---------------0008281825562--