Content-Type: multipart/mixed; boundary="-------------0109281622566"

This is a multi-part message in MIME format.

---------------0109281622566
Content-Type: text/plain; name="01-343.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-343.comments"

10 pages, no figures
---------------0109281622566
Content-Type: text/plain; name="01-343.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-343.keywords"

adiabatic theorem
---------------0109281622566
Content-Type: application/x-tex; name="sot_adiabatic.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="sot_adiabatic.tex"

\documentclass[reqno]{amsart}
\usepackage{amsmath,amsfonts,amsthm}
\newcommand{\kernel}[3]{\left\langle #1\left\vert #2\right\vert#3 \right\rangle}
\newtheorem{thm}{Theorem}
%\renewcommand{\thethm}{}
\newtheorem{lemma}{Lemma}
\newtheorem{lemma1}[lemma]{Lemma}
\newtheorem{lemma2}[lemma]{Lemma}
\newtheorem{lemma3}[lemma]{Lemma}
\newtheorem{conj}{Conjecture}
\newcommand{\remark}{{\it Remark\,\,\,\,}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Hi}{\mathcal H}
\newcommand{\e}{\mathrm e}
\newcommand{\im}{\mathrm i}
\newcommand{\I}{\mathrm I}
\newcommand{\1}{{\mathbf 1}}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\di}{\mathrm d}
\newcommand{\BV}{\mathrm{BV}}
\renewcommand{\thefootnote}{\alph{footnote}}
\DeclareMathOperator*{\sotlim}{SOT-lim}
\def\ip#1{\left < #1 \right >}
\def\norm#1{\left \| #1 \right \|}
\newcounter{list}
\begin{document}
\flushbottom
\title{A strong operator topology adiabatic theorem}
\author{Alexander Elgart and Jeffrey H. Schenker}
\date{September 27, 2001}
\begin{abstract}
We prove an adiabatic theorem for the evolution of spectral
data under a weak additive perturbation.  For continuous functions
of the unperturbed Hamiltonian the convergence is in norm while
for a larger class functions, including the spectral projections
associated to embedded eigenvalues, the convergence is in the
strong operator topology.
\end{abstract} \maketitle
\section{Introduction}
In this note, we discuss an adiabatic theorem for a certain class of 
quantum evolutions.
Adiabatic theorems describe the limit $\tau \rightarrow \infty$ of a 
solution to an
initial value problem of the form
\begin{equation}\label{eq:IVP}
\begin{cases}
\im \dot U_\tau(t) \ =& \ H  ( t / \tau) U_\tau(t)  
\; , \quad t \in [0, \tau]  \\
  U_\tau(0) \ =& \ \1
\end{cases}  \; .
\end{equation}
See, for example, \cite{ae1,b} for background and references therein.  
Here we examine
a class of systems in which the time dependence is produced by a
weak perturbation.

In the context of quantum mechanics the initial value problem is a 
Schr\"odinger equation where $U_{\tau}(\cdot)$ is a unitary evolution in 
Hilbert space and $H(\cdot)$ is a self adjoint operator 
with slow time dependence. 
One is often interested in the evolution of a spectral subspace 
associated to the instantaneous Hamiltonian $H(t)$.  A
prerequisite for an adiabatic theorem is the requirement that the 
spectral subspace  be differentiable in time.  In practice it is 
difficult to verify differentiability -- unless the so called gap 
condition is satisfied.  However for a family of Hamiltonians related 
by a unitary evolution -- {\it i.e.}, $H(s) = V(s) H_{o} V^{\dag}(s)$ -- 
such smoothness is guaranteed by smoothness of $V$.  Equivalently
we may consider a family of the form
\begin{equation}\label{eq:SE} 
H_{\tau}(t/\tau) \ = \ H_{o}\ + \ \frac{1}{\tau}
  \Lambda(t/\tau) 
\end{equation}
where $H_o$ and $\Lambda(s)$, for $s \in [0,1]$, are self adjoint
operators.  These two possibilities are connected via the interaction 
picture when $\Lambda$ is the generator of $V(s)$: 
\begin{equation}
\im \dot V(s) \ = \ \Lambda(s) V(s) \; , \quad V(0) \ = \ \1 \; .
\end{equation}

We discuss here the limit $\tau \rightarrow \infty$ of a
solution, $A_\tau(t) = U_\tau(t) A(0) U^\dag_\tau(t)$, to the
associated Heisenberg equation
\begin{equation}
\im \dot A_\tau(t) \ = \ \left [H_{\tau}(t/\tau) \, 
, \, A_\tau(t) \right ] \;
\end{equation}
when the initial observable is a function of $H_o$,{\it i.e.},
$A(0) = f(H_o)$. Our main result, Theorem~\ref{at}, states that
\begin{equation}\label{eq:limit}
U_\tau(\tau) f(H_o) U^\dag_\tau(\tau) \ \longrightarrow \ f(H_o)
\end{equation}
for a wide class of functions $f$.

The topology in which eq.~\ref{eq:limit} holds depends on the
continuity of $f$ relative to the spectral properties of $H_o$:
for continuous functions we obtain norm convergence while for a class
of discontinuous functions we obtain strong operator convergence. Let
us recall that a family $\tau \mapsto A_\tau$ of operators
converges to $A$ in the {\it strong operator topology} (SOT) if
\begin{equation}
\lim_{\tau \rightarrow \infty} A_\tau \, \psi \  = \ A \, \psi
\end{equation}
for every $\psi \in \Hi$ and converges in norm if
\begin{equation}
\lim_{\tau \rightarrow \infty} \norm{A_\tau - A} \ = \ 0 \; .
\end{equation}
We denote SOT convergence by ``$\sotlim A_\tau = A $''.

Sch\"odinger equations with a Hamiltonian of the form 
\eqref{eq:SE} find direct physical
application in the description of the motion of a quantum particle
in a time dependent potential energy.   In this case, $H_o$
describes the motion of the particle in the absence of time
dependent terms and is generally the Laplacian or some
perturbation thereof, possibly discretized, the underlying Hilbert
space being $\ell^2(\Z^d)$ or $L^2(\R^d)$. The time dependent term
$\Lambda(t)$ is the operator of multiplication by a bounded
function $\Lambda(x,t)$. Theorem~\ref{at} is relevant
to the adiabatic evolution of an ensemble of non-interacting
particles with Fermi statistics. The observables, in this case,
are the Fermi-Dirac distributions $F_{\mu,\beta}(H_o) =
\frac{1}{1 + e^{\beta(H_o - \mu)}}$ at positive temperatures and
the spectral projections $\chi(H_{o} \le \mu)$ and/or 
$\chi(H_{o} < \mu)$ at zero temperature.  We obtain an adiabatic 
evolution even if there is an eigenvalue at the chemical potential $\mu$!

The remainder of this note is organized as follows.  We state
our main result in section~\ref{sec:thm} and there prove those
parts of it which follow from norm resolvent convergence. This
section of the proof is very simple but is unrelated to 
the arguments in the subsequent sections.
Lemma~\ref{lem:ibp}, presented at the start of
section~\ref{sec:proof}, states that the portion of
Theorem~\ref{at} which relates to functions of bounded variation
($BV$) may be reduced to a statement about spectral projections. A
proof of this statement, based on ideas that go back to
Kato~\cite{k}, is presented in section~\ref{sec:proof}. In
section~\ref{sec:BV} we prove Lemma~\ref{lem:ibp}. In
section~\ref{sec:example}, we describe an example, due to Michael
Aizenman, which shows that the norm topology is inadequate when we
consider discontinuous functions of $H_o$. Finally, in
section~\ref{sec:conjecture} we describe a stronger result which
holds when $H_o$ has purely discrete spectrum and motivate a
related conjecture regarding the Schr\"odinger evolution.

\section{The theorem and all we can show with the
resolvent}\label{sec:thm}

Before we state Theorem~\ref{at}, let us recall the definition of
certain classes of functions $f:\R \rightarrow \C$:
\begin{list}{(\arabic{list})} {\usecounter{list}
\setlength{\leftmargin=24pt} \setlength{\labelsep=12pt}
\setlength{\itemindent=12pt}} \item Let $C_b$ denote the bounded
continuous functions.
\item Let $C_o$ denote those functions in $C_b$ which vanish at
$\pm \infty$.
\item Let $BV$ denote the functions of {\em bounded variation},
{\it i.e.}, functions $f$ for which
\begin{equation} {\rm
Var}(f) \ := \ \sup_{n \ge 1} \ \sup_{x_o < \cdots < x_n \in \R} \
\sum_{j=1}^n |f(x_j) - f(x_{j-1})|   \ < \ \infty \; .
\end{equation}
A function in $BV$ can have only countably many points of
discontinuity. See \cite[Ch. 3]{folland} for further discussion.
\end{list}

\begin{thm}\label{at} Let $H_o$ be a self adjoint operator
and suppose that the time evolution $U_\tau$ satisfies the initial
value problem \eqref{eq:IVP} with $H_{\tau}(t/\tau) = H_{0} + (1/\tau) 
\,
\Lambda(t/\tau)$ where $\Lambda(\cdot)$ is a uniformly bounded
continuously differentiable self adjoint family.  Given a
measurable function $f$, consider the statement
\begin{equation}\label{eq:thmlimit}
\lim_{\tau \rightarrow \infty} W_\tau(s)f(H_{0})W_\tau^\dag(s) \ =
\ f(H_o) \; , \mbox{ uniformly for }  s \in [0,1] \; ,
\end{equation}
where $W_\tau$ is the evolution at scaled time, $W_\tau(s) =
U_\tau(\tau \cdot s)$ .
\begin{list}{(\arabic{list})} {\usecounter{list}
\setlength{\leftmargin=24pt} \setlength{\labelsep=6pt}
\setlength{\itemindent=-6pt}}
\item If $f \in C_o$ then eq.~\eqref{eq:thmlimit} is true
in the operator norm topology .
\item If $f = g + h$ with $g \in C_b$ and $h \in BV$ then
eq.~\eqref{eq:thmlimit} is true in the strong operator topology.
\end{list}
\end{thm}
\noindent{\it Remarks}:
\begin{list}{(\arabic{list})}
{\usecounter{list}  \setlength{\leftmargin=24pt}
\setlength{\labelsep=12pt} \setlength{\itemindent=12pt}}

\item Operators $A_\tau(s)$ are said to converge uniformly to
$A$ in the strong operator topology if 
\begin{equation} \lim_{\tau \rightarrow \infty}
\sup_{s} \norm{A_\tau(s) \psi - A \psi} \ = \ 0
\end{equation}
for every $\psi \in \Hi$.  Uniform norm convergence is defined
similarly.

\item The strong operator topology is the strongest topology in
which we can expect an adiabatic limit for discontinuous functions
of $H_o$. In section~\ref{sec:example} we describe an elementary
example of a system for which $W_\tau(s) f(H_o) W_\tau^\dag( s)$
fails to converge in the norm topology.

\item With some extra work one can prove the conclusions of the 
theorem assuming about $\Lambda(\cdot)$ only that it is self adjoint
and essentially bounded.  However, we restrict our attention to 
differentiable perturbations to streamline the presentation.

\item If the operator $H_o$ is unbounded, the distinction between
$C_o$ and $C_b$ is meaningful.  Functions in
$C_b$ may be ``discontinuous at infinity'' which explains the loss
of norm convergence.

\item Among the functions of bounded variation are the {\em
Kr\"onecker} delta functions: $\delta_E(x) = 1$ if $x = E$ and $0$
otherwise. Thus we obtain an adiabatic evolution for the spectral
projection associated to any eigenvalue -- even if it has infinite
degeneracy and is embedded in the essential spectrum!\footnote{The 
general adiabatic theorem for an embedded eigenvalue with {\em finite}
degeneracy was proved previously \cite{ae1}. That work was motivated 
by consideration of the behavior of an atom in a radiation field.}

\item The standard adiabatic theorems describe the limiting behavior
of the Sch\"odinger evolution for a system having a gap in its
spectrum with initial data being a spectral projection onto an
energy band.\footnote{We are aware of one example of an adiabatic
theorem without a gap condition in this context. This is a result
for finite rank perturbations of dense point spectrum \cite{ahs}.}
A projection onto a spectral band is a {\em continuous} function
of $H_o$, thus the convergence occurs in the norm topology. In
such a setting it is possible to find an explicit bound on the
rate of convergence in eq.~\ref{eq:thmlimit} (see, for example,
eq.~\eqref{eq:explicit} and Lemma~\ref{O}).
\end{list}

For a great many functions, eq.~\eqref{eq:thmlimit} follows from
well known convergence theorems and a simple formula --
eq.~\eqref{eq:resolvent} -- which shows that
\begin{equation}\label{eq:nrc}
\sup_{s \in [0,1]} \norm{W_\tau(s) (H_o - z)^{-1} W_\tau^\dag( s)
- (H_o - z)^{-1} } \ \longrightarrow \ 0
\end{equation}
for every $z \not \in \R$, which is to say that $W_\tau(s) H_o
W_\tau^\dag(s) \rightarrow H_o$ uniformly in $s$ in the ``norm
resolvent sense''. The implications of norm resolvent convergence
for Theorem~\ref{at} are that
\begin{list}{(\arabic{list})}
{\usecounter{list}  \setlength{\leftmargin=24pt}
\setlength{\labelsep=12pt} \setlength{\itemindent=12pt}}
\item Eq.~\ref{eq:thmlimit} holds in the norm topology for $f \in
C_o$ \cite[Thm. VIII.20] {reed&simon}.
\item  Eq.~\ref{eq:thmlimit} holds in the strong operator topology for
$f \in C_b$ or when $f$ is the characteristic function of an open
interval $(a,b)$ provided that $a$ and $b$ are not eigenvalues of
$H_o$.  This follows from \cite[Thm.
VIII.20 and VIII.24] {reed&simon} since uniform ``strong resolvent
convergence'' is implied by eq.~\ref{eq:nrc}.
\end{list}
What is remarkable is that with some additional work we can prove
that eq.~\ref{eq:thmlimit} holds, for example, when we take $f$ to
be the characteristic function of an open interval $(a,b)$ and
{\em one or both of $a,b$ is an eigenvalue with arbitrary
degeneracy}.

To verify eq.~\eqref{eq:nrc}, we use the identity
\begin{multline}\label{eq:resolvent}
\left ( H_\tau(s)  - z \right )^{-1} \  - \ W_\tau(s) \left (
H_\tau(0) - z \right )^{-1} W_\tau(s)
\\ =  \ W_\tau(s)  \int_0^{s}  W_\tau(t)^\dag  \left ( \frac{\di}{\di t}
\left ( H_\tau(t) - z \right )^{-1} \right ) W_\tau(t) \, \di t \,
W_\tau(s)^\dag \; ,
\end{multline}
where $H_\tau(s) = H_o + \frac{1}{\tau} \Lambda(s)$.
Eq.~\eqref{eq:resolvent} follows from the fundamental theorem of
calculus and the observation that
\begin{equation}
\frac{\di}{\di t} \Big ( W_\tau(t)^\dag  \left ( H_\tau(t) - z
\right )^{-1} W_\tau(t) \Big ) \ = \ W_\tau(t)^\dag  \left (
\frac{\di}{\di t} \left ( H_\tau(t) - z \right )^{-1} \right )
W_\tau(t) \; .
\end{equation}
Now,  eq.~\eqref{eq:nrc} follows from eq.~\eqref{eq:resolvent}
because the latter implies that
\begin{equation}\label{eq:explicit}
\norm{W_\tau(s) (H_o - z)^{-1} W_\tau(s) - (H_o - z)^{-1} } \ \le
\ \frac{C}{(\mathrm{Im} z)^2 } \frac{1}{\tau}  \; ,
\end{equation}
since
\begin{equation}
\frac{\di}{\di t} \left ( H_\tau(t) - z \right )^{-1}  \ = \
\frac{1}{\tau} \left ( H_\tau(t) - z \right )^{-1} \,  \dot
\Lambda(t) \, \left ( H_\tau(t) - z \right )^{-1} \; ,
\end{equation}
and
\begin{equation}
\left ( H_\tau(s) - z \right )^{-1} \ = \ \left ( H_o - z \right
)^{-1} - \frac{1}{\tau} \left ( H_\tau(s) - z \right )^{-1}
\Lambda(s) \left ( H_o
 - z \right )^{-1} \; .
\end{equation}

Before we proceed, let us describe an example which demonstrates
that we cannot really go any further using only norm resolvent
convergence. Consider the self adjoint operator on $\ell^2(\Z)$
given in Dirac notation by $H_o = \sum_{m \neq 0} \frac{1}{m} |m
\left > \right < m|$.  For each $n$ let $V_n$ be the unitary on
$\ell^2(\Z)$ which ``swaps $0$ and $n$'', {\it i.e.},
\begin{equation}
(V_n \psi)(m) \ = \ \begin{cases} \psi(m) & \text{if $m \neq 0,
n$} \\ \psi(0) & \text{if $m = n$} \\ \psi(n) & \text{if $m = 0$}
\end{cases} \; .
\end{equation}
Then $ V_n H_o V_n^\dag = H_o + \frac{1}{n} \left ( |0 \left
> \right < 0| - |n \left > \right < n| \right )$.  Thus $V_nH_o
V_n^\dag \rightarrow H_o$ in norm, and therefore in norm resolvent
sense, as $n \rightarrow \infty$. Yet, if $P_{0} = |0 \left>
\right < 0| $ -- the spectral projection of $H_o$ associated to
eigenvalue $0$ -- then
\begin{equation}
V_n P_{0} V_n \ = \ |n \left
> \right < n| \ \stackrel{ \text{SOT}}{\longrightarrow} \ 0
\quad n \rightarrow \infty \; .
\end{equation}

\section{SOT convergence for spectral
projections}\label{sec:proof} The claim that
eq.~\eqref{eq:thmlimit} holds whenever $f \in BV$ is, at heart, a
statement about spectral projections as is indicated by the
following lemma:
\begin{lemma}\label{lem:ibp}
Let $H_o$ and $\Lambda(t)$ be as in Theorem~\ref{at}. Then
eq.~\eqref{eq:thmlimit} holds in the SOT for every $f \in BV$ if
and only if it holds whenever $f(x) = \chi(x \le E)$ or $f(x) =
\chi(x \ge E)$.
\end{lemma}
\noindent We postpone the proof of Lemma~\ref{lem:ibp} to
section~\ref{sec:BV} and focus here on proving
eq.~\ref{eq:thmlimit} with $f(x) = \chi(x \ge E)$ and $f(x) =
\chi(x \le E)$ for every $E$ in $\R$.

In what follows we fix $E$ and take $P = \chi(H_o \le E)$. The
other case -- $\chi(H_o \ge E)$ -- is handled in exactly the same
way by changing $\le$ to $\ge$ in the appropriate places.  We must
show that for any $\psi \in \Hi$
\begin{equation}\label{eq:adfermproj}
\lim_{\tau \rightarrow \infty} \, \sup_{s \in [0,1]} \norm{ \left
( W_\tau(s) \, P \, W_\tau(s)^\dag \ - \ P  \right ) \psi} \ = \ 0
\; .
\end{equation}

Our argument is stated most readily with the semigroup
$W_\tau(t,s) = W_\tau(t) W_\tau^\dag(s)$ -- note that $W_\tau(s) P
W_\tau^\dag(s) = W_\tau(s, 0) P W_\tau(0,s)$ and $P = W_\tau(s,s)
P W_\tau(s,s) $. We would like to compare $W_\tau(s,t)$ with the
semigroup associated to $H_o$, so we define
\begin{equation}
\Omega_\tau(t,s) \ := \ \e^{\im \tau (t-s) H_o} W_{\tau}(t)
W_\tau(s)^\dag \; .
\end{equation}
Since the exponential of $H_o$ commutes with $P$ and
$\Omega_\tau(t,s)$ is unitary
\begin{equation}
\begin{split}
\norm{\left ( W_\tau(s) P W_\tau(s)^\dag - P \right ) \psi} \ =& \
\norm{\left ( \Omega_\tau(0,s)^\dag P \Omega_\tau(0,s) \ - \ P
\right ) \psi } \\  =& \ \| [P, \Omega_\tau(0,s)]\psi \| \; .
\end{split}
\end{equation}
Finally, because $P$ is a projection
\begin{equation}\label{eq:ppbar}
[P, \Omega_\tau(t,s)] \ = \ P \, \Omega_\tau(t,s) \bar P - \bar P
\, \Omega_\tau(t,s) P \; ,
\end{equation}
where $\bar P = \1 - P$. Therefore, eq.~\ref{eq:adfermproj} will
follow if we can verify that both terms on the right side of
eq.~\eqref{eq:ppbar} uniformly converge to zero in the SOT.

Consider the first term.  Let $P_\Delta: =\chi(E < H_{0} < E +
\Delta)$, then
\begin{equation}
P \, \Omega_\tau(t,s) \bar P \ = \ P \,  \Omega_\tau(t,s) ( \bar P
- P_{\Delta}) \ + \ P \,\Omega_\tau(t,s) P_{\Delta}.
\end{equation}
We will see below (Lemma~\ref{O}) that the operator norm of $P
\Omega (P - P_\Delta)$ is uniformly bounded by $1/\tau \Delta$.
Thus given $\psi \in \Hi$
\begin{equation}
\norm{P \, \Omega_\tau(t,s) \bar P \psi} \ \le \ \frac{C}{\tau
\Delta} \norm{\psi} \ + \ \norm{P_\Delta \psi} \; .
\end{equation}
If, for instance, $\Delta=1/\sqrt\tau$ then both terms converges
to zero since $\sotlim P_\Delta = 0$ -- whether or not there is an
eigenvalue at $E$.

The second term of \eqref{eq:ppbar} requires a little more care.
Because $E$ may be an eigenvalue, we need to isolate the
contribution from the associated projection $P_E = \chi(H_o = E)$.
Let $P_\Delta' = \chi(E -\Delta < H_o < E)$ and consider
\begin{multline}
\bar P \, \Omega_\tau(t,s) P \ = \ \bar P \, \Omega_\tau(t,s) ( P
- P_{\Delta}' - P_E)  \ + \ \bar P \, \Omega_\tau(t,s) P_{\Delta}'
\\ + \ \bar P \, \Omega_\tau(t,s) P_E \; .
\end{multline}
As above, if we take $\Delta = 1/\sqrt \tau$ then the first and
second terms tend uniformly to zero. That the third term also
converges to zero is the content of the following lemma:
\begin{lemma}\label{lem:PE}
Let $P_E := \chi(H_o = E)$.  Then $ (\1 - P_E ) \,
\Omega_\tau(t,s) P_E $ uniformly tends  to zero in the strong 
operator topology.
\end{lemma}
\begin{proof}
The operator $\Omega_\tau(t,s)$ satisfies a Volterra equation
\begin{equation}\label{eq:volterra}
\Omega_\tau(t,s)\ = \1 + \int_s^t \di r \ K_\tau(r,s)
\Omega_\tau(r,s) \; ,
\end{equation}
with
\begin{equation}
  K_\tau(r,s) \ = \ - \im \e^{\im\tau(r-s)H_o} \Lambda(r) \e^{\im
  \tau(s-r)H_o}  \; .
\end{equation}
By iterating eq.~\eqref{eq:volterra} we obtain a norm convergent
series
\begin{equation}\label{eq:series}
\Omega_\tau(t,s) \ = \ \sum_{n = 0}^\infty A_\tau^{n}(t,s) \;
\end{equation}
where
\begin{equation}\label{eq:intforA}
A_\tau^n(t,s) \ = \ \idotsint\displaylimits_{s \le r_n \le \ldots
\le r_1 \le t} \di r_1 \ldots \di r_n K_\tau(r_1,s) \ldots
K_\tau(r_n,s)
 \; .
\end{equation}
Since $A_\tau^n(t,s)$ is obtained by integrating a product of $n$
factors of $K$ over a simplex of volume $(t-s)^n / n!$ we have the
elementary norm bound
\begin{equation}\label{eq:normbound}
\norm{A_\tau^n(t,s)} \ \le \ \frac{1}{n!} \kappa^n (t-s)^n \; ,
\end{equation}
where $\kappa = \sup_r \norm{\Lambda(r)}$.

We see from \eqref{eq:series}, \eqref{eq:normbound}, and dominated
convergence that it suffices to show for each $n$ that $\bar P_E
A_\tau^n(t,s) P_E \rightarrow 0$ uniformly in the SOT. This may be
proved as follows. First note that
\begin{align}
P_E K_\tau(r,s) P_E \ =& \ - \im P_E \Lambda(r) P_E \; , \\ \bar
P_E K_\tau(r,s) P_E \ =& \ - \im \bar P_E \e^{\im \tau (r-s)(H_o-
E)} \Lambda(r) P_E \; .
\end{align}
Next observe that
\begin{equation}
\int_s^r \di r' \bar P_E \e^{\im \tau (r'-s)(H_o- E)} \
\longrightarrow \ 0
\end{equation}
uniformly in the strong operator topology 
from which it follows via integration by parts that
\begin{equation}\label{eq:barPEB}
\int_s^r \di r' \bar P_E \e^{\im \tau (r'-s)(H_o- E)} B(r')\
\longrightarrow \ 0
\end{equation}
for any differentiable family of operators $B(r)$ which
does not depend on $\tau$.

Now consider the expression for $A^n_\tau$ obtained by inserting
$\1 = \bar P_E + P_E$ between the two right most factors of
$K_\tau$ in the integral which appears in eq.~\eqref{eq:intforA}.
Proceed with the term obtained from $P_E$ by inserting $\bar P_E +
P_E$ between the next two factors of $K$. Continue from right to
left in this way, expanding only the terms obtained from $P_E$. We
obtain an expression for $\bar P_E A^n_\tau(t,s) P_E $ as of sum
of $n$ terms, the $j$th term being
\begin{multline}
(-\im)^j \idotsint\displaylimits _{s \le r_n \le \ldots \le r_1
\le t} \di r_1 \ldots \di r_n \bar P_E K_\tau(r_1,s) \ldots
K_\tau(r_{n-j},s) \\ \times \bar P_E \e^{\im \tau
(r_{n-j+1}-s)(H_o- E)} \Lambda(r_{n-j+1}) P_E  \ldots \Lambda(r_n)
P_E \; ,
\end{multline}
which uniformly converges to zero by virtue of
eq.~\eqref{eq:barPEB}.  Since $A^n_\tau$ is a finite linear
combination of terms which uniformly tend to zero it does so as
well.
\end{proof}

It remains to show that $\norm{P \Omega_\tau(t,s) (\bar P -
P_\Delta)}$ is bounded by $1/\tau \Delta$.
\begin{lemma}\label{O}
Let $ P_1:= \chi(H_o \le E_1)$ and $P_2 := \chi(H_o \ge E_2)$ with
$E_2 > E_1$. Then
\begin{equation}\label{eq:Omegabound}
\Vert P_1 \Omega_\tau(t,s) P_2 \Vert \ \le \ \frac{C}{\Delta\tau}
\; ,
\end{equation}
where  $\Delta = E_2 - E_1$ and $C$ is a constant which does not
depend on $E_1$ or $E_2$.   The same inequality holds with $P_1, \
P_2$ interchanged.
\end{lemma}
\begin{proof}
As in the proof of Lemma~\ref{lem:PE} the idea is to prove a bound
on each term $A_\tau^n$ in the expansion for $\Omega_\tau$.  In
this case, we will show that
\begin{equation}\label{eq:Abound}
\norm{P_1 A_\tau^n(s,t) P_2} \ \le \ \frac{n}{\tau \Delta}
\frac{\alpha^n}{ (n-1)!}
\end{equation}
where $\alpha$ is a constant independent of $s,t$.  Summing these
bounds clearly implies eq.~\eqref{eq:Omegabound} -- see
eq.~\eqref{eq:series}.

The main step is to show that
\begin{equation}\label{eq:Kbound}
\Vert P_1 \int_t^sdrK_\tau(r,s)\ P_2 \Vert \ \le \
\frac{C}{\Delta\tau} \; ,
\end{equation}
and the same with $P_1$ and $P_2$ interchanged.   The idea is
that, since $K_\tau(r,s) = \e^{\im \tau(r-s)H_o} \Lambda(r)
\e^{\im \tau(s-t) H_o}$ and the spectral supports of $P_1$ and
$P_2$ are distance $\Delta$ apart, the integral over $r$ has a
highly oscillating phase of order $\tau \Delta$. For a rigorous
argument, however, it is convenient to use a commutator equation
and integration by parts to extract eq~\eqref{eq:Kbound}. This
method goes back to Kato \cite{k}.

The commutator $[H_o, X]$ might be ill defined if $H_o$ is
unbounded.  Thus we introduce a cutoff and work instead with
$[H_o, P_M X P_M]$ where $P_M = \chi(-M < H_o < M)$ and $M \in (0
, \infty)$.  At the end of the argument we take $M \rightarrow
\infty$.  The $X$ we have in mind is
\begin{equation}
X(r)\ := \ \frac{1}{2 \pi \im } \int_{\Gamma} \di z \, P_1 \, R(z)
\Lambda(r) R(z ) \, P_2 \; .
\end{equation}
where $R(z):= (H_0-z)^{-1}$ and the contour $\Gamma$ is the line
$\{E' + \im \eta \, : \, \eta \in \R\}$ with $E' = (E_2 + E_1)/2$.
A simple calculation yields
\begin{equation}
[H_o, P_M X(r) P_M]\ = \  P_M P_1 \Lambda(r) P_2 P_M\; .
\end{equation}
Therefore
\begin{equation}\label{eq:PKPX}
\begin{split}
P_M P_1 K_\tau(r,s) P_2 P_M \ =& \ [H_o, \e^{\im\tau(r-s)H_o} P_M
X(r) P_M \e^{\im
  \tau(s-r)H_o} ] \\ =& \ \frac{1}{\im \tau} P_M  \left ( \frac{\di}{\di r}
  ( \e^{\im\tau(r-s)H_o} X(r) \e^{\im \tau(s-r)H_o} ) \right . \\ & \qquad \left .
  \phantom{\frac{\di}{\di r}}- \
   \e^{\im\tau(r-s)H_o} \dot X(r) \e^{\im \tau(s-r)H_o} \right
   ) P_M
   \; .
\end{split}
\end{equation}
However $X(r)$ and $\dot X(r)$ are uniformly bounded, $
\norm{X(r)} , \| \dot X(r)\|  \le  C /\Delta $,  so integrating
\eqref{eq:PKPX} yields
\begin{equation}
\Vert P_{M} \, P_1 \int_t^sdrK_\tau(r,s)\ P_2 \, P_{M} \Vert \ \le \
\frac{C}{\Delta\tau} \; .
\end{equation}
In the limit $M \rightarrow \infty$ this implies
eq.~\eqref{eq:Kbound} by lower semi-continuity of the norm.  
The second case with $P_1$ and $P_2$
interchanged follows with an obvious modification of $X$.

The rest of the argument is similar to the proof of
Lemma~\ref{lem:PE}.  We insert a decomposition of the identity $\1
= Q + \bar Q$ between the factors of $K$ in the integral
expression for $A_\tau^n$, eq.~\eqref{eq:intforA}. To apply
\eqref{eq:Kbound}, we should maintain a spectral gap between the
projections which sits to the left and right of $K$. Therefore we
define $Q_{j}:=\chi(H_{0}\le Q_{1}+j/n \Delta)$ for $j=0, ... , n$
and insert $1=Q_{j}+ \bar Q_{j}$ between the $j$th and $(j+1)$th
factors of $K$.   With these insertions, $P_1A_\tau^nP_2$ breaks
into $2^{n}$ terms and each term includes at least one factor of
the type $Q_{j} K_\tau(r_{j+1},s) \bar Q_{j+1}$ or $\bar Q_{j}
K_\tau(r_{j+1},s) Q_{j+1}$ where there is a gap of size $\Delta / n$
between the spectral supports of the two projections.  We apply
integration by parts to the integral over $r_{j+1}$ to obtain a
factor which may be bounded by eq.~\eqref{eq:Kbound}:
\begin{multline}
\int_0^{r_j} \di r_{j+1} \, Q_{j} K_\tau(r_{j+1},s) \bar Q_{j+1}
B(r_{j+1}) \\ = \ \int_0^{r_j} \di r_{j + 1} \, Q_j
\int_{r'}^{r_j} \di r' \, K_\tau(r',s) \bar Q_{j+1} \dot
B(r_{j+1}) \; .
\end{multline}
Elementary norm estimates and eq~\eqref{eq:Kbound} now show that
each of the $2^n$ terms is bounded by $n \beta^{n}/(\Delta\tau
(n-1)!)$ for some $\beta$ which implies
eq.~\eqref{eq:Abound} with $\alpha = 2 \beta$.
\end{proof}

\section{Integration by parts and the proof of lemma~\ref{lem:ibp}}\label{sec:BV}

Turning to the proof of Lemma~\ref{lem:ibp}, we note that the
spectral theorem provides the representation
\begin{equation}\label{eq:spectralthm}
f(H_o) \ = \ \int f(E) \di P_o(E)
\end{equation}
valid for bounded measurable $f$.  The goal is to integrate this
expression by parts thereby obtaining an expression involving $\di
f$ and $P_o(E) = \chi(H_o \le E)$.  This argument works precisely
when $f \in BV$ as we shall now explain.

The projection valued measure $\di P_o(E)$ is the differential of
$P_o(E) = \chi(H_o \le E)$ which is of bounded variation {\em in
the strong operator topology}. That is, for any $\psi \in \Hi$,
\begin{equation}\sup_{n \ge 1} \ \sup_{E_0 < \cdots < E_n
\in \R} \ \sum_{j=1}^n \|P_o(E_j) \psi - P_o(E_{j-1}) \psi \| \ <
\ \infty \; .
\end{equation}
We could equally well work with $P_o(E) = \chi(H_o < E)$ or a
number of other choices -- the distinction being meaningful only
if $H_o$ has point spectrum.

Since the function $P_o$ is SOT-continuous from the left at every
$E$, {\it i.e.}  $P_o(E-0) = P_o(E)$,  we may integrate
\eqref{eq:spectralthm} by parts whenever $f \in BV$ and everywhere
continuous from the right:\footnote{The extension of integration
by parts to functions in $BV$ is a standard part of real analysis
-- we direct the reader to \cite[Ch. 3]{folland} for details.}
\begin{multline}
f(H_o) \ = \ f(\infty) \1 - \int \di f(E) P_o(E) \; , \\ f \in BV
\mbox{ and continuous from the right.}
\end{multline}
For general $f \in BV$ this formula is replaced by
\begin{equation}\label{eq:integratedbyparts}
\begin{split}
f(H_o) \ = \ f(\infty) \1 \ -& \ \int \di f(E) \, \chi(H_o \le E)
\\ +& \ \sum_{E \in \R}
(f(E) - f(E+0)) \, \chi(H_o = E) \; .
\end{split}
\end{equation}
Note that $ \sum_{E \in \R} |f(E) - f(E+0)| \le  {\rm Var}(f)  < \infty$.
In particular, there can be only countably many $E \in \R$ for
which $f(E) \neq f(E+0)$.

Now suppose that
\begin{equation}\label{eq:Alim}
\sotlim \,  W_\tau( s) \, A \, W_\tau(s)^\dag \ = \ A
\end{equation}
uniformly in $s$ whenever $A = \chi(H_o \le E)$  or $A = \chi(H_o
\ge E)$ with $E \in \R$. Since
\begin{equation}
\chi(H_o = E) \ = \ \chi(H_o \le E) + \chi(H_o \ge E) - \1 \; ,
\end{equation}
eq.~\eqref{eq:Alim} also holds with $A = \chi(H_o = E)$. Now given
$f \in BV$, use \eqref{eq:integratedbyparts} to express $f(H_o)$
and find that
\begin{equation}
\sotlim \, W_\tau(s) f(H_o) W_\tau(s)^\dag \ = \ f(H_o)
\end{equation}
uniformly in $s$ by dominated convergence.

\section{Why the norm topology is inadequate -- an example}
\label{sec:example} The following example is due to Michael
Aizenman and is motivated by the consideration of systems with
dense point spectrum. 

Consider a countable collection of non interacting two-level systems:
\begin{equation}
H_{\tau}(t/\tau) \ = \ \sum_{k=0}^\infty\oplus H_k(t/\tau)\; ,\quad H_k(t/\tau)
 \ = \ m_k\sigma_z+
\epsilon \frac{t}{\tau}\sigma_x\,,
\end{equation}
with $m_k:=1/k$ and $\sigma_z, \sigma_x$ the Pauli spin matrices.
Theorem~\ref{at} certainly applies to the unitary evolution $U_{\tau}$
associated with Hamiltonian $H_{\tau}(t/\tau)$.
The unitaries, of course, decompose into a direct sum of
two by two matrices $U^k_\tau$ generated by $H_k$.

Let us choose $f(H(0)) = P_o$ to be the spectral projection onto
negative energies: $P_o =\chi(H(0) < 0)$. We will show that,
although in accordance with Theorem~\ref{at}
\begin{equation}
\sotlim U_\tau(\tau s)P_o U^\dagger_\tau(\tau s) \ = \ P_o \; ,
\end{equation}
we nonetheless have
\begin{equation}
\limsup_{\tau \rightarrow \infty} \norm{U_\tau(\tau s)P_o
U^\dagger_\tau( \tau s)  \ - \ P_o} \ \ge \ \alpha_\epsilon(s)
\end{equation}
where, for sufficiently small $\epsilon$, $\alpha_\epsilon(s) > 0$
for every $s$ in $[0,1]$.

Indeed, choose the particular sequence $\tau_n := n$ of integer
values for $\tau$. For each $n$, we consider the two-level system
with $m_n:=1/\tau_n$.  For this system
\begin{equation} \im \dot U^n_\tau(t) \ = \ H_n U^n_\tau(t)
  \ = \ \frac{1}{\tau_n} (\sigma_z + t \sigma_x)U^n_\tau(t)\, ,
\end{equation}
so that $U^n_{\tau_n}(\tau_n s) =: V(s)$ is independent of $n$.
The matrix $V(s)$ is obtained by integrating
\begin{equation}
\im \dot V(s) \ = \ (\sigma_z + \epsilon s \sigma_x) V(s)
\end{equation}
with initial condition $V(0) \ = \begin{pmatrix} 1 & 0 \\ 0 & 1
\end{pmatrix}$. Clearly, with $\epsilon$ small enough
\begin{equation}
\Vert V(s) \sigma_z V(s)^\dag - \sigma_z\Vert  \ >\ 0\,,
\end{equation}
for all $s\in[0,1]$.
Since, as mentioned above, $U_\tau$ is the direct
sum of the two by two matrices $U^k_\tau$ we have
\begin{equation}
\norm{U_{\tau_n}(\tau_n s) P_o U^\dag_{\tau_n}(\tau_n s)) - P_o}
\ \ge \ \norm{V(s) \theta(-\sigma_z) V(s)^\dag -
\theta(-\sigma_z)} \ > \ 0 \; .
\end{equation}

\section{The Schr\"odinger picture -- a
conjecture}\label{sec:conjecture} Theorem~\ref{at} describes the
adiabatic limit of the {\em Heisenberg picture} of quantum
dynamics.  As for the {\em Schr\"odinger picture}, in general it
is not true that $U_\tau(\tau s) \rightarrow \1$;  nor is there
reason to expect $U_\tau(\tau s)$ to converge to anything at all.
In fact, in case the spectrum of $H_o$ consists of countably many
isolated eigenvalues, a simple modification of our proof shows
that
\begin{equation}
\sotlim_{\tau \rightarrow \infty} \, \e^{\im \tau s H_o}
U_\tau(\tau s) \ = \ \Phi(s) \;
\end{equation}
where $\Phi(s)$ is a unitary operator which commutes with $H_o$.
If there is a uniform lower bound on the spacing between
neighboring eigenvalues then the convergence is in {\em norm}. The
unitary $\Phi(s)$ is related to $\Lambda(\cdot)$ and the spectral
projections $P_E = \chi(H_o = E)$ via the evolution equation:
\begin{equation}
\im \dot \Phi(s) \ = \ \left ( \sum_{E \in \sigma(H_o) } P_E
\Lambda(s) P_E \right ) \, \Phi(s) \; .
\end{equation}
For general $H_o$ it is not clear from our proof that $\e^{\im
\tau s H_o} U_\tau(\tau s)$ even converges.  However if it does
converge, it is clear from Theorem~\ref{at} that the limiting
operator commutes with $H_o$ which motivates the following
conjecture:
\begin{conj}[Sch\"odinger Adiabatic Theorem]Under the assumptions of
Theorem~\ref{at}, $\e^{\im \tau s H_o} U_\tau( \tau s)
\stackrel{SOT}{\longrightarrow} \Phi(s)$ where $\Phi(s)$ is a
unitrary operator which commutes with $H_o$.
\end{conj}

\subsection*{Acknowledgments} We are grateful to M. Aizenman for useful
discussions and the example presented in
Section~\ref{sec:example}. This work was partially supported by
the NSF Grant PHY-9971149 (AE).


\begin{thebibliography}{10}
\bibitem{ae1} J. E. Avron and A. Elgart, {\it Adiabatic theorem without a Gap
Condition}, {Comm.\ Math.\ Phys.\ } \textbf{ 203 }(1999), 445-463.
\bibitem{ahs} J. E. Avron, J. S. Howland and B. Simon, {\it Adiabatic
theorems for dense point spectra},
{Comm.\ Math.\ Phys.\ } \textbf{ 128 }(1990), 497-507.
\bibitem{b} F. Bornemann, {\it Homogenization in time
of singularly perturbed mechanical systems},  Lecture  Notes in
Math. 1687, Springer (1998).
\bibitem{folland}  G. B. Folland, {\it Real analysis : modern techniques and
their applications},  New York,  Wiley (1984).
\bibitem{k} T. Kato, {\it On the adiabatic theorem of quantum
mechanics}, {Phys. Soc. Jap.} \textbf{ 5 }(1958), 435-439.
\bibitem{reed&simon} M. Reed and  B. Simon, {\it Methods of Modern Mathematical Physics I:
Functional Analysis}, San Diego, Academic Press (1980).
\end{thebibliography}
\end{document}
% ----------------------------------------------------------------






--------------------------------------------------------------------------------

---------------0109281622566--
