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\def\at{adiabatic theorem}
\def\qm{Quantum Mechanics}
\def\e{\varepsilon}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{ $\scriptscriptstyle
|$}}\kern-.40em{\rm C}}
\newtheorem{main}{Theorem}[section]
\newtheorem{adiabatic}[main]{Lemma}
\newtheorem{XY}[main]{Lemma}
\newtheorem{dos}[main]{Corollary}
\newtheorem{f}[main]{Definition}
\newtheorem{ac}[main]{Lemma}
\newtheorem{gauss}[main]{Lemma}
\newtheorem{pdotp}[main]{Proposition}
\title{The Adiabatic Theorem of Quantum Mechanics}
\author{ J.~E.~Avron and A. Elgart
\\ Department of Physics, Technion, 32000 Haifa, Israel}
\begin{document}
\flushbottom
\maketitle
\begin{abstract}
We prove the adiabatic theorem for quantum evolution without the traditional
gap condition. We show that the theorem holds essentially in all cases
where it can
be formulated. In particular, our result implies that the adiabatic
theorem holds
also for eigenvalues embedded in the continuous spectrum. If there is
information on
the H\"older continuity of the spectral measure, then one can also estimate
the rate
at which the adiabatic limit is approached.
\end{abstract}
%\newpage
The adiabatic theorem of Quantum Mechanics describes the long time behavior of
the solutions of an initial value problem where the Hamiltonian generating the
evolution depends slowly in time. Traditionally, the theorem is stated
for Hamiltonians that have an eigenvalue which is separated by a gap from the
rest of the spectrum. Folk wisdom is that a gap condition is a {\it
sine qua non} for the adiabatic theorem to hold. In particular, according to
this folk wisdom, one does not expect a general adiabatic theorem for
Hamiltonians
that have an eigenvalue embedded in, say, the continuous spectrum. Our purpose
is to show that this folk wisdom is wrong, and there is a general adiabatic
theorem without a gap condition. All one really needs for the adiabatic theorem
is a spectral projection for the Hamiltonian that depends smoothly on time.
To formulate the problem more precisely it is convenient, and traditional, to
replace the physical time $t$ by the scaled time $s=t/\tau$ where
$\tau \to\infty$ is the adiabatic time scale. In this notation, the
adiabatic theorem is concerned with the solution of the initial value problem
\begin{equation} i\, \dot \psi_\tau (s) = \tau H(s)\, \psi_\tau (s),\quad s\in
[0,1]\label{scrod}
\end{equation}
in the limit of large $\tau$. $H(s)$ is a
self-adjoint Hamiltonian which depends smoothly\footnote{For the applications
to quantum mechanics, $H(s)$ is a Schr\"odinger operator, which is unbounded, so
the notion of smoothness requires some discussion, see e.g.
\cite{asy}. To avoid getting into technical issues that may obscure the basic
mechanism we want to expose, we assume that $H(s)$ is a smooth family
of bounded operators. From a physical point of view the
adiabatic theorem is an infrared (i.e.\ low energy) problem. As such, the
unboundedness of the Hamiltonian, which is an ultraviolet property, should not
play a role.}
on $s\in
[0,1]$ and
$\psi$ is a vector (in Hilbert space) valued function. Hence $H(t/\tau)$
evolves slowly in time for a long interval of time with finite overall change in
the Hamiltonian $H(s)$.
The quantum adiabatic theorem says that the solution to the initial value
problem is characterized, in the adiabatic limit $\tau \to\infty$, by spectral
information.
The first satisfactory formulation and rigorous proof of an adiabatic theorem
in the then new quantum mechanics was given in 1928 by Born and Fock
\cite{bf}. They were motivated by a point of view advocated by Ehrenfest, which
identified classical adiabatic invariants with quantum numbers. The theorem
they proved was geared to show that quantum numbers are invariant under
adiabatic deformations. The class of Hamiltonian operators they
considered was so that $H(s)$ has simple discrete spectrum. Their proof was a
variant of the method of variation of constants. A formal version of this
result, usually without the careful analysis of Born and Fock, is what one
finds in most textbooks on quantum mechanics.
In 1958 Kato \cite{kato} initiated a new strategy for proving adiabatic
theorems. He introduced a notion of adiabatic evolution which is purely
geometric. It is associated with a natural connection in the bundle of spectral
subspaces. Kato's method was to compare
the geometric evolution with the evolution generated by $H(s)$ and to show
that in the adiabatic limit the two coincide. Using this idea, Kato extended
the results of Born and Fock to the case where
$H(s)$ had non-simple spectrum and, more
significantly, to operators that had more general types of spectra. This is of
importance for applications to quantum mechanics of atoms and molecules where
absolutely continuous spectrum is always present, see e.g. \cite{hvz}. Kato
proved an adiabatic theorem when the initial data lie in a subspace
corresponding to an isolated eigenvalue of
$H(0)$, provided that the corresponding subspace of $H(s)$ has constant
multiplicity and was separated by a gap from the rest of the spectrum for all
$s$. No assumption on the spectral type of
$H(s)$ in the rest of the spectrum, that is, beyond the gap, need be made.
Kato's results were further extended in \cite{asy} who showed that
an adiabatic theorem can be formulated and proven without any assumption
on the nature of the spectral type also for the initial data, thereby dropping
the condition that the initial data lie in a subspace corresponding to an
eigenvalue with fixed multiplicity. For example, the initial data could lie
in a subspace corresponding to an energy band with, say, absolutely continuous
spectrum. The one crucial condition that appears to make the theorem in
\cite{asy} go is the
gap condition. For the example above, the gap condition is that this
energy band be separated by a gap from the rest of the spectrum. This result was
applied to the study of quantization of transport in the Hall effect
\cite{thouless}.
There are examples of adiabatic theorems without a gap condition. But, these
examples all appear to be special, in that some special property of
the Hamiltonian intervenes and
appears to play the role of a gap. One such example is the adiabatic
theorem for crossing eigenvalues which was studied by Born and Fock.
We return to this example below. Another example is an adiabatic
theorem for rank one perturbations of dense point spectra studied in \cite{ahs}.
The current status of the general adiabatic theorem is that, modulo
technicalities, provided the initial data lie in a spectral subspace of
$H(0)$ and the corresponding spectral subspace of $H(s)$ is separated by a gap
from the rest of the spectrum for all $s$, the time evolution respects the
spectral splitting in the adiabatic limit.
If one examines the existing adiabatic theorems one sees that while the
gap condition plays a central and crucial role in the proofs, it appears to
play no such role in the formulation. To formulate an adiabatic
conjecture all one needs is a smooth family of spectral projections. If that
is the case, the initial data have a distinguished spectral subspace to
cling to. We shall show that this condition is essentially all one needs to
prove the adiabatic theorem.
More precisely, we prove an adiabatic theorem provided
the Hamiltonian has a smooth and finite dimensional spectral projection.
The gap condition is dispensed with.
The adiabatic theorem is sometimes understood to be the statement that the
adiabatic limit is approached exponentially fast for all times that lie
outside the
support of $\dot H(s)$. A general result of this kind, using the gap
condition, is
described in \cite{ks,n}.
We do not prove such a strong result here. Rather, we stick to the
traditional usage
of Born, Fock and Kato, where by the adiabatic theorem we refer to the
remarkable
fact, quite unlike perturbation theory, that there is a precise control on the
evolution for Hamiltonians that undergo a {\em finite} variation.
The spectral gap in the adiabatic theorem controls the rate at which the
adiabatic limit is approached. A finite gap
guarantees that the rate is at least $O(1/\tau)$.
Giving up the gap condition does not go without price. To see what this price
should be let us recall a result of Born and
Fock who studied also crossing eigenvalues where the spectral
projections have a smooth continuations through the crossing point
\cite{bf}.
Born and Fock
established an adiabatic theorem in this case, where the physical
evolution clings to the spectral projection, and picks the smooth
continuation at the crossing. The rate at which the
adiabatic limit is approached is only
$O(1/\sqrt\tau )$ for linear crossing. This suggests that the price for
giving up the gap condition
is the sacrificing of knowledge about the rate at which the adiabatic
limit is approached.
Our approach to an adiabatic theorem without a gap condition has some of the
flavor of an operator analog of the Riemann-Lebesgue lemma. If a function and
also its derivative are in $L^1(\real)$ then it
is an elementary exercise that its Fourier transform decays at infinity at
least as fast as an inverse power of the argument. Riemann-Lebesgue lemma says
that, in fact, the Fourier transform of any $L^1(\real)$ function vanishes at
infinity. The loss of {\it a-priori} information about the derivative
translates to loss of information about the
rate at which the function vanishes at infinity. In this analogy
differentiability is the analog of the gap condition,
and the $L^1(\real)$ condition is the analog of the smoothness condition on the
spectral projection.
To simplify the
presentation, we shall stay away from making optimal assertions.
Let
$H(s)$ be a family of bounded\footnote{See footnote 1.}
self-adjoint Hamiltonians that
generates unitary evolution as the solution of the initial value problem:
\begin{equation} i\,\dot U_\tau (s) = \tau H(s) U_\tau(s),\quad U_\tau(0)=1,
\quad s\in [0,1].\label{schrodinger}
\end{equation} We assume that $H(s)$ has eigenvalue $\lambda(s)$ and this
eigenvalue has finite multiplicity. For this eigenvalue we formulate and prove
our main result:
\begin{main}\label{main} Suppose that $P(s)$ is finite rank spectral projection,
which is at least twice differentiable (as a bounded operator), for the
Hamiltonian
$H(s)$, which is bounded and differentiable for all $s\in[0,1]$. Then, the
evolution of the initial state
$\psi(0)\in Range P(0)$, according to Eq.~(\ref{scrod}), is such that in
the adiabatic limit
$\psi_\tau (s)\in Range P(s)$ for all $s$.
\end{main}
Remark: In the case that there is eigenvalue crossing $P(s)$ is not smooth
and $Tr\,
P(s)$ is discontinuous. We shall not treat this case. The method we
describe can, in
fact, handle crossings provided these occur at finite number of points in
time. But,
keeping with our policy of avoiding making optimal results in order to
keep the
basic ideas in the forefront, we shall not treat crossings here.
We recall the notion of adiabatic evolution
\cite{kato,asy}. Let
$U_A(s)$ be the solution of the initial value problem:
\begin{equation} i\,\dot U_A(s) =\tau\,\left( H(s)+\frac{i}{\tau}\, [\dot
P(s),P(s)]\right)\, U_A(s),\quad U_A(0)=1,
\quad s\in [0,1].\label{kato}
\end{equation} It is known that this unitary evolution has the
intertwining property \cite{asy}:
\begin{equation} U_A(s)\, P(0) =P(s)\, U_A(s).
\end{equation} That is $U_A(s)$ maps $Range\ P(0)$ onto $Range\ P(s)$.
In particular, the solution of the initial value problem
\begin{equation}
i\,\dot \psi(s) = \tau\,\left( H(s)+\frac{i}{\tau}\, [\dot
P(s),P(s)]\right)\, \psi(s), \ \psi(0)\in Range P(0),
\end{equation}
has the property that $\psi(s)\in Range P(s)$. We shall show that the
Hamiltonian evolution,
$U_\tau(s)$, is close to the adiabatic evolution $U_{A}(s)$.
We first formulate the basic lemma:
\begin{adiabatic} Let $P(s),\ s\in[0,1],$ be a
differentiable family of spectral projections for the self-adjoint Hamiltonian
$H(s)$ with (operator) norm
$\Vert
\dot P(s)\Vert<\infty$. Suppose that the commutator equation
\begin{equation} [\dot P(s), P(s)] = [H(s),X (s)]
+Y(s),\label{commutators}
\end{equation} has operator valued solutions, $X (s)$ and $Y (s)$ with
$X (s)$, $\dot X (s)$ and $Y (s)$ bounded.
Then
\begin{equation}
\Vert (U_\tau(s)-U_A(s))\,P(0)\Vert\le
\begin{array}{c}
\max\\
s\in [0,1]
\end{array}\left(
\frac{2\,\Vert X (s)\,P(s)\Vert+\Vert \dot{ \Big(X(s)
\,P(s)\Big)}P(s)\Vert }{\tau}+\Vert \,Y (s) \,P(s)\Vert\right).\label{estimate}
\end{equation}
\end{adiabatic}
Remark: The commutator equation, Eq.~(\ref{commutators}), can be
viewed as a definition of $Y(s)$. The issue is not to find a solution to
this equation, but rather to find solutions that make $Y$ as
small as possible.
In the case that there is a gap $\Delta$ separating
the eigenvalue from the rest of the spectrum, a solution of the
commutator equation is
\begin{equation}
X(s)=\frac{1}{2\pi i}\,\int_\Gamma \,R(z,s)\, \dot P(s) R(z,s) \,
dz,\quad Y(s)=0.\label{gap}
\end{equation}
Here $\Gamma$ is a circle in the complex plane, centered at the eigenvalue,
and of
radius $\Delta/2$. $R(z,s)$ is the resolvent at scaled time $s$, see
\cite{asy}. In this case the rate at which the adiabatic limit is
obtained, is seen from Eq.~(\ref{estimate}) to be $1/\tau$.
The strategy for
proving the adiabatic theorem without a gap condition is to show that one
can pick
$Y$ so that its norm is arbitrarily small, possibly at the expense of large
norm for
$X$ and
$\dot X$. So long as the norm of $X$ and $\dot X$ is finite, it can be
compensated by taking $\tau$ large. This means that one can make the
right hand side of Eq.~(\ref{estimate}) arbitrarily small. The price
paid is that there is, generally speaking, no information about the rate at
which the
adiabatic limit is obtained.\footnote{With additional information about
structure of the spectra, one can sometimes get estimates on the rate
at which the adiabatic limit is approached. See corollary (\ref{dos}).}
Proof:
Let $W(s) = U_\tau^\dagger(s) U_A(s)$ be the wave operator comparing the
adiabatic and Hamiltonian evolution. Since
\begin{equation}
\Vert
U_\tau(s)-U_A(s)\Vert=\left\Vert U_\tau(s)\Big(1-W(s)\Big)\right\Vert=\Vert
1-W(s)\Vert,
\end{equation} we need to bound $W(s)-1.$ From the definition of the
adiabatic evolution, the commutator equation, and the equation of motion
\begin{eqnarray}
\dot W(s)&=&U^\dagger_\tau(s)\left( [\dot P(s),P(s)]
\right)\,U_A(s)\nonumber \\
&=& U^\dagger_\tau(s)\left( [\dot P(s),P(s)]
\right)\,U_\tau(s)\,W(s)
\nonumber \\
&=& U^\dagger_\tau(s)\Big( [ H(s),X(s)] +Y(s)
\Big)\,U_\tau(s)\,W(s)
\\
&=& -\frac{i}{\tau}\Big(\dot U^\dagger_\tau(s)X(s)U_\tau(s) +
U^\dagger_\tau(s)X(s)\dot U_\tau(s)\Big)\,W(s)+U^\dagger_\tau(s)
Y(s)U_A(s)\nonumber \\
&=& -\frac{i}{\tau}\Big(\dot{( U^\dagger_\tau(s)X(s)U_\tau(s))} -
U^\dagger_\tau(s)\dot X(s)U_\tau(s)\Big)\,W(s)+U^\dagger_\tau(s)
Y(s)U_A(s)
\nonumber \\
&=& -\frac{i}{\tau}\left\{\dot{\Big(
U^\dagger_\tau(s)X(s)U_\tau(s)\,W(s)\Big)} -
U^\dagger_\tau(s) X(s)U_\tau(s)\,\dot W(s)\right.\nonumber \\
&&\left .\phantom{\frac{i}{\tau}}-
U^\dagger_\tau(s)\dot X(s)U_\tau(s)\,W(s)\right\}+\,U^\dagger_\tau
Y(s)U_A(s)\nonumber
\\
&=& -\frac{i}{\tau}\left\{\dot{\Big( U^\dagger_\tau(s)X(s)U_A(s)\Big)} -
U^\dagger_\tau(s) X(s)\,[\dot P(s),P(s)]\,U_A(s)\right .\nonumber
\\&&\left.\phantom{\frac{i}{\tau}}- U^\dagger_\tau(s)\dot
X(s)U_A(s)\right\}+\,U^\dagger_\tau(s)
Y(s)U_A(s)\nonumber.
\end{eqnarray}
The lemma
then follows by integration since
$W(s)$ is unitary with
$W(0)=1$.\hfill$\Box$
Let us describe a solution of the commutator equation which is motivated
by the solution Eq.~(\ref{gap}) in the case of a gap. In order to
have explicit error estimates and also in order to make the
presentation simple and as elementary as possible, we choose a
Gaussian regularizer.
\begin{f}
Let $g$ and $e$ denote the Gaussian and Error functions\footnote{The
error function we use differs by a scaling of the argument, an overall factor
and shift from the
canonical error function.}, and $\Phi$ be
the special function defined below:
\begin{equation}
g(\omega)= e^{-\pi\omega^{2}},\quad e(t)=\int_{-\infty}^{t}ds\,
g(s),\quad \Phi(t)=\theta(t)- e(t),
\end{equation}
$\theta$ is the usual step function.
\end{f}
An elementary lemma is:
\begin{gauss}
$\Phi$ has finite $L^{1}$ norm and finite moments. In particular:
\begin{equation}
\Vert \Phi(t) \Vert_{1} = \frac{1}{\pi},\quad \Vert t\Phi(t) \Vert_{1}=
\frac{1}{4\pi}.
\end{equation}
Under scaling, $\Delta>0$:
\begin{equation}
\left\Vert \Phi\left({t}{\Delta}\right) \right\Vert_{1} = \frac{1}{\pi\Delta},
\quad \left\Vert
t\Phi\left({t}{\Delta}\right) \right\Vert_{1}=\frac{1}{4\pi\Delta^{2}}.
\end{equation}
\end{gauss}
We assume, without loss, that the spectral projection $P(s)$ is
associated with the eigenvalue zero.
\begin{XY}\label{estimates}
Let $P(s)$ be a smooth spectral projection for $H(s)$ associated with the
eigenvalue zero. Let
$\Gamma$ be an infinitesimal contour around the origin in the complex
plane.\footnote{At this point the choice of Gaussian is not optimal. It
would be more
convenient to choose a regularizer which is a better approximant to a
characteristic function and the reader may want to think
of a Gaussian which is flattened at the top.} Let
\begin{equation} F_\Delta(s)=g\left(\frac{H(s)}{\Delta}\right)-P(s).
\end{equation}
Then the commutator
equation has the solution
\begin{eqnarray}
X_\Delta(s)&=& A+A^\dagger,\quad A=P(s)\, \dot P(s)
R(0,s)\,\left(1-g\left(\frac{H(s)}{\Delta}\right)\right);\nonumber \\
Y_\Delta(s)&=& -F_\Delta(s)\dot P(s) P(s)+P(s)\dot P(s)
F_\Delta(s),\label{XY}
\end{eqnarray}
with \begin{eqnarray}
\Vert X_\Delta(s)P(s)\Vert &\le & \frac{2\Vert \dot P(s)P(s)
\Vert}{\Delta},\\
\Vert \dot {\Big(X_\Delta(s)P(s)\Big)} \Vert&\le& \frac{2\Vert \ddot P(s)
\Vert}{\Delta}+
\frac{\pi\,\Vert \dot P(s)
\Vert\, \Vert \dot H\Vert }{\Delta^{2}}.\nonumber
\end{eqnarray}
\end{XY}
Remark: In the case that the family of Hamiltonians is related by unitaries
\begin{equation} H(s)=V(s)\, H_0\,V^\dagger(s),\quad V(s)=\exp (i\,s\,\sigma)
\end{equation} such that $\sigma$ is bounded operator,
one can improve\footnote{ see proposition (\ref{unitary})} the estimate to
$\Vert \dot {\Big(X_\Delta(s)P(s)\Big)} \Vert \le \frac{\Vert[X,\sigma]\Vert}
{\Delta}$.
Proof:
We start with a formal calculation. Let
\begin{equation}
X_\Delta(s)=\frac{1}{2\pi i}\,\int_\Gamma dz\, (1-F_\Delta(s))\,R(z,s)\,
\dot P(s)
R(z,s)\,(1-F_\Delta(s)).
\end{equation}
Since $\dot P(s) = P(s)\dot P (s) +\dot P (s)P(s)$,
$X_\Delta (s)$ can be written as a sum of two adjoint terms, one of them is
\begin{eqnarray}
\frac{1}{2\pi i}\,\int_\Gamma dz\, (1-F_\Delta(s))\,R(z,s)\,P(s) \dot P(s)
R(z,s)\,(1-F_\Delta(s))&
=&\nonumber\\
\frac{1}{2\pi i}\, (1-F_\Delta(s))\,P \dot P(s)
\left(\int_\Gamma dz\,\frac{R(z,s)}{z}\right)\,(1-F_\Delta(s))&=&\nonumber \\
\,P(s) \dot P(s)
{R(0,s)}(1-P(s))\,(1-F_\Delta(s))=
P(s) \dot P(s)
{R(0,s)}(1-P(s)-F_\Delta(s))&=&\nonumber \\
P(s) \dot P(s)
{R(0,s)}\left(1-g\left(\frac{H(s)}{\Delta}\right)\right)
&=&A.
\end{eqnarray}
We have used
\begin{equation}
P(s)\,F_\Delta(s)=F_\Delta(s)\,P(s)=\Big(g(0)-1 \Big)\,P(s)=0.
\end{equation}
Using this integral representation of $X_\Delta (s)$ we now find $Y_\Delta
(s)$. By our choice of
$F_\Delta(s)$ we have $[F_\Delta(s),H(s)]=0$.
Hence,
\begin{eqnarray}
[X_\Delta(s),H(s)]&=&\frac{1}{2\pi i}\,\int_\Gamma\,dz\,\Big[ (1-F_\Delta(s) )\,
R(z,s)\, \dot P(s) R(z,s) \,(1-F_\Delta(s)) ,H(s)-z\Big]\, \nonumber \\
&=&\frac{1}{2\pi i}\,\int_\Gamma\,dz\,(1-F_\Delta(s))\Big[ \,R(z,s)\, , \dot
P(s)\Big](1-F_\Delta(s))
\nonumber \\ &=&(1-F_\Delta(s))\,\Big[ P(s) ,\dot P(s)
\Big]\, (1-F_\Delta(s))\nonumber \\
&=&[ P(s) ,\dot P(s)] -\Big\{F_\Delta(s),[ P(s), \dot P(s)]\Big\}+
F_\Delta(s)[P(s),\dot P(s)]F_\Delta(s)\nonumber \\ &=&[ P(s) ,\dot P(s)] +
F_\Delta(s)\dot P(s)
P(s)-P(s)\dot P(s) F_\Delta(s).
\end{eqnarray}
So a solution of the commutator equation is
\begin{equation}
Y_\Delta(s)= -F_\Delta(s)\dot P(s) P(s)+P(s)\dot P(s) F_\Delta(s).\label{Y}
\end{equation}
It remains to estimate the norms of $X$ and $\dot X$.
Using the fact the a Gaussian is its own Fourier transform,\begin{equation}
g\left(\frac{H(s)}{\Delta}\right)=\Delta \int_\real g(\Delta\,t)\, \exp
[{2\pi i t
H(s)}] dt,
\end{equation}
one checks that with our choice of $\Phi$
\begin{equation}
R(0, s) \left(1-g\left(\frac{H(s)}{\Delta}\right)\right)=
2\pi i\,\int_\real \Phi\left({t}{\Delta}\right)\, \exp
[{2\pi i t
H(s)}] \,dt.
\end{equation}
Hence
\begin{equation}
\left\Vert R(0, s) \left(1-g\left(\frac{H(s)}{\Delta}\right)\right)
\right\Vert\le
2\pi\,\left\Vert \Phi\left({t}{\Delta}\right)\,\right\Vert_1=
\frac{2}{\Delta}.
\end{equation}
Using the equation for $X(s)$ this estimate proves the bound on $X(s)$.
To get a bound on $\dot {\Big(X_\Delta(s)P(s)\Big)}$, use Duhammel formula
\begin{equation}
\dot{\Big(\exp (2\pi i t H(s))\Big)}= 2\pi i \, t\, \int_{0}^{1} dz\,e^{
2\pi iz t
H(s)}\, \dot H(s) \,e^{ 2\pi i(1-z) t
H(s)}.
\end{equation}
Collecting the various terms give the claimed estimate.
\hfill$\Box$
\begin{pdotp}\label{unitary} Let $H(s)$ be the family
\begin{equation} H(s)=V(s)\, H\,V^\dagger(s),\quad V(s)=\exp (i\,s\,\sigma),
\end{equation}
with $P$ a finite-dimensional projection onto the $Ker\,H$.
It is enough to solve for the commutator equation
\begin{equation}
[H, X] +Y=i\,\{\sigma,P\} -2\,i\, P\sigma P,\label{commute}
\end{equation} for fixed $X$ and $Y$.
$X(s)$ and $Y(s)$ are then determined by the obvious unitary conjugation,
and $\Vert \dot X(s)\Vert=\Vert[X,\sigma]\Vert$.
\end{pdotp} Proof:
Since
$P(s)=V(s)\, P\,V^\dagger(s)$, we have
\begin{equation}
\dot P(s)=i\,V(s)\, [\sigma,P]\,V^\dagger(s),
\end{equation} and
\begin{eqnarray} [\dot P(s),P(s)]&=&i\,V(s)\,
\Big[[\sigma,P],P\Big]\,V^\dagger(s)\nonumber \\ &=&i\,V(s)\,
\Big(\{\sigma,P\}-2P\,\sigma\,P\Big)\,V^\dagger(s).
\end{eqnarray}
\hfill$\Box$
As lemma (\ref{estimates}) shows, as
$\Delta$ shrinks, the norms of $X(s)$ and $\dot X(s)$ may, and in general, will,
grow. This, however is of no concern, as long as the norms remain finite, for
one can always compensate for this growth by choosing $\tau$ large enough. The
good thing about shrinking $\Delta$ is that this can be used to make the
norm of $Y_\Delta$
arbitrarily small. Hence, we can always make the right hand side of
Eq.~(\ref{estimate}) arbitrarily small.
%The price we do pay is the loss of
%information on the rate at which the adiabatic limit is approached.
\begin{ac}
Suppose that $H(s)$ is smooth with a zero eigenvalue with spectral
projection
$P(s)$ smooth and of finite rank. Let $F_\Delta(s)$ be as above. Then
$\Vert Y_\Delta(s)\,P(s)\Vert=\Vert F_\Delta(s)\dot P(s)P(s)\Vert
\to 0$ uniformly as $\Delta$ shrinks to zero.
\end{ac}
Proof: For the sake of simplicity suppose that $P(s)$ has rank one with
$P(s)\psi(s)=\psi(s)$, $\langle \psi|\psi\rangle=1$. Let $\varphi=\dot
P(s)\psi
$. Then, using property $P(s)\varphi(s)=P(s)\dot P(s)\psi(s)=P(s)\dot
P(s)P(s)\psi(s)=0$, we obtain
\begin{eqnarray}
\Vert F_\Delta\dot P(s) P(s)\Vert^2=\Vert F_\Delta \dot P(s)
\psi(s)\Vert^2&=&
\Vert F_\Delta
\varphi(s)\Vert^2=\nonumber \\
\left\Vert\left( g\left(\frac{H(s)}{\Delta}\right) -P(s)\right)
\varphi(s)\right\Vert^2=
\left\Vert g\left(\frac{H(s)}{\Delta}\right)
\varphi(s)\right\Vert^2&=&
\left\langle \,\varphi \,|g^2\left(\frac{H(s)}{\Delta}\right)
| \,\varphi \, \right\rangle=\nonumber\\
\int_{\sigma(H(s))}g^2(x/\Delta)d\mu_\varphi(x),
\end{eqnarray}
where $\mu_\varphi$ denotes the spectral measure.
Now, $g\left(\frac{x}{\Delta}\right) $ is bounded by one, and goes
monotonically
to zero for all $x\neq 0$, and $g(0)=1$. Hence
\begin{equation}
\lim_{\Delta\to 0}\int_{\sigma(H(s))}g^2(x/\Delta)d\mu_\varphi(x)=
\mu_\varphi(0)=0.\label{measure}
%\lim_{\Delta\to 0}g\left(\frac{H(s)}{\Delta}\right)=g(0) \, P(s)=P(s).
\end{equation}
It follows that
there is a sequence of $\Delta$ that make $Y_\Delta (s)$ arbitrarily small.
\hfill$\Box$
By taking $\tau$ large enough one can therefore make the
right hand side of Eq.~(\ref{estimate}) as small as one pleases. This completes
the proof of the theorem. \hfill$\Box$
The physical interpretation of the general adiabatic theorem is that although
the adiabatic theorem ``always'' holds, it does so for different physical
mechanisms. In the case that there is a gap in the spectrum the
adiabatic theorem holds because the eigenstate is protected by a gap from
tunneling out of the spectral subspace. In the case that there is no gap and the
spectrum near the relevant eigenvalue is essential, the adiabatic theorem holds
for a different reason: The essential spectrum is associated with states
that are supported near spatial infinity, and there is little tunneling
to these states because of small overlap with the wave function
corresponding to an eigenvalue which is supported away from infinity.
If one has additional information about the nature of the spectrum
embedding the eigenvalue, one can sometimes get estimates on the rate
at which the adiabatic limit is approached. An illustration of this is
given below.
Recall \cite{last} that a (Borel) measure $\mu$ is called (uniformly)
$\alpha$-H\"older continuous, $\alpha\in[0,1]$, if there is a constant $C$ such
that for every interval $\Delta$ with $|\Delta|<1 $ \footnote{$|\cdot|$ denote
Lebesgue measure.}
\begin{equation}
\mu(\Delta)