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Revised version, two remarks added. To appear in Lett. Math. Phys.
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adiabatic theory, gap condition
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\title{A note on the adiabatic theorem without gap condition}
\author{Stefan Teufel \\
 Zentrum Mathematik, Technische Universit\"at M\"unchen,\\ 80290 M\"unchen, Germany}
\maketitle
\begin{abstract}
We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of
Avron and Elgart \cite{AE} by providing an elementary solution of the ``commutator equation''.
In addition, a minor modification of their argument allows for a more direct treatment of eigenvalue
crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence
in the adiabatic limit.
\end{abstract}


In \cite{AE},  Avron and Elgart prove an adiabatic theorem without gap condition. 
We will show in this note 
how to simplify parts of their proof by giving a considerably simpler solution of
the ``commutator equation'' (\ref{commu}), which is elementary even in the case of 
unbounded Hamiltonians. 
Furthermore, a minor change in the argument allows for a simpler treatment of eigenvalue crossings.

One is interested in the solution  of the initial value problem
\begin{equation}\label{SE}
i\frac{d}{d s} U_\tau (s) = \tau\,H(s)\,U_\tau(s)\,, \quad U_\tau(0)={\bf 1}
\end{equation}
for $s\in[0,1]$ in the limit of large $\tau$. Here $H(s)$ is a family of self-adjoint 
operators on some Hilbert space $\mathcal{H}$ with a common dense domain $\mathcal{D}$ such that 
 $R(i,\cdot)\in C^1([0,1],\mathcal{B})$ and such that, uniformly for $s\in[0,1]$, 
$H(s)\geq \Lambda_1>-\infty$ and  $\|H(s)\dot R(i,s)\|\leq \Lambda_2<\infty$. 
$R(z,s):= (H(s)-z)^{-1}$ and $(\mathcal{B},\|\cdot\|)$ 
denotes the space of bounded operators on $\mathcal{H}$ equipped with the usual norm.
The existence of a family of unitaries $U_\tau(s)$ such that (\ref{SE}) holds on $\mathcal{D}$ 
follows from general theory,  cf.\ \cite{AE}. 

For a discussion of the history, the physical interpretation and applications 
of the following theorem we refer to \cite{AE}. The analogous result in the
context of the space-adiabatic theorem with an application to 
the massless Nelson model will be presented in \cite{T}.\\ 


\noindent {\bf Theorem.}\quad {\em
Suppose that $E(s)$ is an eigenvalue of $H(s)$ and that $P(s)$ is a family of finite 
rank projections such that  $H(s)P(s)=E(s)P(s)$ for all $s\in[0,1]$,
$P(\cdot)\in C^2([0,1],\mathcal{B})$  and such that $P(s)$ is the spectral projection of $H(s)$ on $\{E(s)\}$ for almost all $s\in[0,1]$. 
 Then the solution $U_\tau(s)$ of  (\ref{SE}) satisfies
\begin{equation}\label{dist}
\sup_{s\in[0,1]}\|\,(1-P(s))\, U_\tau(s)\,P(0)\,\| \to 0 \quad\mbox{as}\quad\tau\to\infty\,.
\end{equation}
}

Note that if $E(s)$ crosses a different eigenvalue $E_1(s)$ at some time $s=s_0$, 
i.e.\ $E(s)= E_1(s)$ only for $s=s_0$,
then the rank of the spectral projection $\1I_{\{E(s)\}}(H(s))$ jumps and thus it is 
not even continuous at $s=s_0$. If it is possible to  continue  $\1I_{\{E(s)\}}(H(s))$ through the crossing
in a differentiable way, such a crossing (actually a countable number of them) 
is included in the above formulation of the theorem.

The following proof, as well as the original one in \cite{AE}, actually yields a stronger statement than (\ref{dist}):
Let $U_{\rm A}(s)$ be the adiabatic time evolution given as the solution of (\ref{SE}) with
$H(s)$ replaced by $H_{\rm A}(s):= H(s) + i\tau^{-1}[\dot P(s),P(s)]$,\footnote{Using 
$i\tau^{-1}[\dot P(\cdot),P(\cdot)]\in C^1([0,1],\mathcal{B})$ it is easy to see that for $\tau$
large enough also the family  $H_{\rm A}(s)$ satisfies the general assumptions on 
$H(s)$. Hence the existence of a unitary propagator $U_{\rm A}(s)$ satisfying (\ref{SE}) on $\mathcal{D}$ with 
$H(s)$ replaced by $H_{\rm A}(s)$ follows.}
 then
\begin{equation} \label{dist2}
\sup_{s\in[0,1]} \|\, U_\tau(s) - U_{\rm A}(s)\, \| \to 0 \quad\mbox{as}\quad\tau\to\infty\,.
\end{equation}
The adiabatic evolution exactly intertwines $P(0)$ and $P(s)$, i.e.\
\begin{equation}\label{aint}
U^*_{\rm A}(s)\,P(s)\,U_{\rm A}(s) = P(0)\,,
\end{equation}
and thus (\ref{dist2}) implies (\ref{dist}). To see (\ref{aint}) for $s\in[0,1]$, note that it is clearly true for $s=0$ and that
the time derivative of the left hand side vanishes because, with (\ref{Pdot}),
$\dot P(s) = - i\, \tau\, [ H_{\rm A}(s), P(s)]$.

The first general adiabatic theorem without gap condition based on a completely 
different strategy of proof is due to Bornemann \cite{B}.
His result is considerably weaker since only (\ref{dist}) is obtained and only in the
strong operator topology. However, his result was the motivation for this author to
relax the ``spectrality'' condition from
$P(s)=\1I_{\{E(s)\}}(H(s))$ for all $s\in[0,1]$, as in \cite{AE}, to 
$P(s)=\1I_{\{E(s)\}}(H(s))$ for almost all $s\in[0,1]$, as in \cite{B}. 

\noindent {\em Proof.}\quad
Using the standard Cook's type argument one expresses the difference of the unitaries in terms of the 
difference of the generators. One finds first for $\psi\in \mathcal{D}$ that
\begin{eqnarray}
 \left( U_\tau(s) - U_{\rm A}(s) \right) \psi &=& - \,U_\tau(s) \int_0^sdt\frac{d}{dt}\Big( U_\tau^*(t)U_{\rm A}(t)\Big)\psi
\nonumber\\
&=& -\, U_\tau(s) \int_0^s dt\, U_\tau^*(t)[\dot P(t),P(t)]U_{\rm A}(t)\psi\nonumber
\end{eqnarray}
and thus employing density of $\mathcal{D}$ that
\begin{equation}
 \|\, U_\tau(s) - U_{\rm A}(s) \|= \|\int_0^s dt\, U_\tau^*(t)[\dot P(t),P(t)]U_{\rm A}(t)\|\,. \label{diff0}
\end{equation}
If one can find bounded operators $X(s)$, $Y(s)$ satisfying
\begin{equation}\label{commu}
[\dot P(s),P(s)] = [H(s),X(s)] + Y(s)\,,
\end{equation}
then the integrand in (\ref{diff0}) can be written as a time derivative plus a remainder,
\begin{eqnarray}
U_\tau^*[\dot P,P]\,U_{\rm A} &=& -\frac{i}{\tau} \frac{d}{ds}\Big( U_\tau^* \,X\,U_\tau U^*_\tau U_{\rm A}\Big) 
\nonumber\\ &&\, +\frac{i}{\tau} \Big(
U^*_\tau\,X[\dot P,P]U_{\rm A} + U^*_\tau\dot X U_{\rm A}\Big)+ U^*_\tau\,Y\,U_{\rm A}\,, \label{pi}
\end{eqnarray}
and (\ref{pi}) in (\ref{diff0}) yields the bound
\begin{eqnarray}
\lefteqn{ \sup_{s\in[0,1]} \|\, U_\tau(s) - U_{\rm A}(s) \|\leq}\nonumber\\&& \sup_{s\in[0,1]}
 \frac{2(1+\|\dot P(s)\|)\,\|X(s)\|}{\tau} 
+\int_0^1\,ds\,\left( \frac{\| \dot X(s) \|}{\tau} + \|Y(s)\|\right)\,.\label{bound}
\end{eqnarray}
If there was a gap $\delta$ separating $E(s)$ from the rest of the spectrum of $H(s)$, then 
$X_{\rm gap}(s) := R(E(s),s)\dot P(s)P(s) + P(s)\dot P(s) R(E(s),s)$, $Y_{\rm gap}(s)\equiv 0$ would be a
  solution of (\ref{commu}) with  $X_{\rm gap}(\cdot)\in C^1([0,1], \mathcal{B})$. This is because 
one obtains from differentiating $P(s)^2=P(s)$ that
\begin{equation}\label{Pdot}
\dot P(s) = (1-P(s))\,\dot P(s)\,P(s) + P(s)\,\dot P(s)\,(1-P(s))\,,
\end{equation}
and thus
$\| R(E(s),s)\dot P(s)P(s)\| =\| R(E(s),s) (1-P(s))\,\dot P(s)\,P(s)\|\leq \|\dot P(s)\|/\delta$. 
Hence, in the presence of a gap (\ref{bound})
gives the standard result $ \sup_{s\in[0,1]} \|\, U_{\rm A}(s) - U_\tau(s) \|=\mathcal{O}(\tau^{-1})$.

Instead of using a cutoff function $\widetilde g$ to define
$R(E(s),s)\,\widetilde g((H(s)-E(s))/\Delta)$ in absence of a gap as in \cite{AE}, 
we shift the resolvent into the complex plane and  define
\begin{eqnarray*}
X_\Delta (s) &:=& R(E(s)-i\Delta,s)\, \dot P(s)\,P(s)\, +\, \mbox{adjoint}\\& =&  
X_\Delta (s)\,P(s)  \, +\, \mbox{adjoint}\,, \\
Y_\Delta (s) &:=& i\,\Delta  R(E(s)-i\Delta,s)  \,  \dot P(s)\,P(s)\,  -\,  \mbox{adjoint}\\& = &
Y_\Delta (s)\,P(s)  \, -\, \mbox{adjoint}  \,.
\end{eqnarray*}
 Here ``\,$\pm$\,adjoint'' means that the adjoint of 
the first term in the sum is added resp.\ subtracted. 
With our definition the fact that $X_\Delta(s)$ and $Y_\Delta(s)$ solve (\ref{commu})
is obvious and the following bounds have elementary proofs even in the case of
unbounded $H(s)$.

We will show that $X_\Delta(\cdot)$, $Y_\Delta(\cdot)\in C^1([0,1],\mathcal{B})$ 
 for  $\Delta\in (0,1]$ and that for some constant $C<\infty$
\begin{equation} \label{boundX}
 \sup_{s\in[0,1]}\|X_\Delta(s) \|\leq \,\frac{C}{\Delta}\,,\quad 
 \sup_{s\in[0,1]}\left\|\dot X_\Delta(s)\right\| \leq \,\frac{C}{\Delta^2}\,,
\end{equation}
\begin{equation}
 \sup_{s\in[0,1], \Delta\in(0,1]}\|Y_\Delta(s)\|<\infty \label{boundY1}
\end{equation}
and
\begin{equation}\label{boundY}
\|Y_\Delta(s)\|\to 0 \quad\mbox{as}\quad\Delta\to 0
\end{equation}
for almost all $s\in[0,1]$.
 Then the theorem follows by inserting (\ref{boundX})--(\ref{boundY}) into (\ref{bound}), using dominated convergence
 and chosing $\Delta=\Delta(\tau)$ with $\lim_{\tau\to\infty}\Delta(\tau)=0$ and  
$\lim_{\tau\to\infty}\tau\Delta^2(\tau)=\infty$.

It suffices to establish  (\ref{boundX})--(\ref{boundY}) for $X_\Delta(s)P(s)$ resp.\ $Y_\Delta(s)P(s)$,
 since the adjoints have the same norms.
 (\ref{boundY1}) and the first bound in 
(\ref{boundX}) follow from $\|R(E(s)-i\Delta,s)\|\leq1/\Delta$ and from $P(\cdot)\in C^2([0,1],\mathcal{B})$.
For the second bound in (\ref{boundX}) observe that
\[
\frac{d}{ds} R(E(s)- i\Delta,s) = -  R(E(s)- i\Delta,s) (\dot H(s)-\dot E(s)) R(E(s)- i\Delta,s)
\]
and thus, by inserting ${\bf 1} = R(i,s)\,(H(s)-(E(s) -i\Delta) - i +( E(s) -i\Delta))$, 
\[
 \|\dot  R(E(s)- i\Delta,s)\| \leq \left( 
\frac{\|\dot H(s) R(i,s)\|(2\Delta + 1 + |E(s)|)}{\Delta^2}+ \frac{|\dot E(s)|}{\Delta^2}\right)\,.
\]
Since $E(s)=\mbox{tr}(H(s)P(s))/\mbox{tr}P(s)$, 
it follows from the differentiability of $P(s)$ and $H(s)$ that also $E(s)$ is continuously differentiable
and therefore $ \sup_{s\in[0,1]}(|E(s)|+|\dot E(s)|)<\infty$. According to the assumptions on $H(s)$ we have
$\sup_{s\in[0,1]}$ $\|\dot H(s) R(i,s)\|= \sup_{s\in[0,1]}\| (H(s)+i) \dot R(i,s)\|<\infty$.


The final step in proving the adiabatic theorem without gap is to establish 
(\ref{boundY}) and we repeat  the argument from \cite{AE}. 
Since $P(s)$ has finite rank, $\lim_{\Delta\to 0}$ $\|Y_\Delta(s)P(s)\|=0$ follows
if one can show  $\lim_{\Delta\to 0}\|Y_\Delta(s)P(s)\psi\|=0$ for all $\psi$.
Let $\varphi := \dot P(s)P(s)\psi$, then
\begin{eqnarray*}
\lim_{\Delta\to 0}\|Y_\Delta(s)P(s)\psi\|^2 &=& \lim_{\Delta\to 0}\| i \Delta R(E(s)-i\Delta,s)\varphi\|^2\\&
= & \lim_{\Delta\to 0} \int_\mathbb{R} \mu_\varphi(d\lambda)\frac{\Delta^2}{(\lambda-E(s))^2+\Delta^2}
= \mu_\varphi(E(s))\,,
\end{eqnarray*}
where $\mu_\varphi$ denotes the spectral measure of $H(s)$ for $\varphi$. If $P(s)$ is the spectral projection
on $\{E(s)\}$, then $ \mu_\varphi(E(s))=0$ because, according to (\ref{Pdot}), $\varphi\in\mbox{Ran}(1-P(s))$.
Hence, $\lim_{\Delta\to 0}\|Y_\Delta(s)P(s)\psi\|=0$ for almost all $s\in[0,1]$.
\hfill$\qed$\\


\noindent {\em Remark 1.}\quad
The preceding proof shows, in particular, that
information about the rate of convergence in (\ref{dist}) can be derived from  information
about the limiting  behavior of $R(E(s)-i\Delta,s)\, \dot P(s)\,P(s)$ in the limit $\Delta\to 0$.
Assume that there is a function $\eta:[0,1]\to[0,1]$
with $\eta(\Delta)\geq \Delta$ such that there is a constant $C_1<\infty$ with
\begin{equation}\label{rate1}
\sup_{s\in[0,1]} \|\, R(E(s)-i\Delta,s)\, \dot P(s)\,P(s)\, \|\,\leq \,C_1\, \frac{\eta(\Delta)}{\Delta}
\,.
\end{equation}
Then (\ref{boundX}) and  (\ref{boundY}) can be sharpened to
\[
 \sup_{s\in[0,1]}\|X_\Delta(s) \|\leq \,C\, \frac{\eta(\Delta)}{\Delta} \,,\quad 
 \sup_{s\in[0,1]}\left\|\dot X_\Delta(s)\right\| \leq \,C\,\frac{\eta(\Delta)}{\Delta^2}\,,
\]
\[
 \sup_{s\in[0,1]}\|Y_\Delta(s) \|\leq \,C\,\eta(\Delta)\,,
\]
which inserted into (\ref{bound}) yields for $\Delta(\tau) := \tau^{-\frac{1}{2}}$ and some 
$C<\infty$ that
\begin{equation}\label{dist4}
\sup_{s\in[0,1]}\|\, U_\tau(s) - U_{\rm A}(s)\,\| \,\leq\, C\,\eta(\tau^{-\frac{1}{2}})\,.
\end{equation}
Analogously one concludes that if in addition to (\ref{rate1}) also 
\begin{equation}\label{rate2}
\sup_{s\in[0,1]} \|\, \frac{d}{ds}(R(E(s)-i\Delta,s)\, \dot P(s)\,P(s))\, 
\|\,\leq \,C_2\, \frac{\eta(\Delta)}{\Delta}
\end{equation}
holds for some $C_2<\infty$, then one obtains 
\begin{equation}\label{dist3}
\sup_{s\in[0,1]}\|\, U_\tau(s) - U_{\rm A}(s)\,\| \,\leq\, C\,\eta(\tau^{-1})\,.
\end{equation}
 It was shown in \cite{AE} that
(\ref{rate1}) follows with $\eta(\Delta) = \Delta^\frac{\alpha}{2}$, 
if the spectral measures $\mu_{\varphi(s)}$ are $\alpha$-H\"older continuous uniformly
for $\varphi(s)\in \mbox{Ran}P(s)$ and $s\in[0,1]$. Then, according to (\ref{dist4}),
the adiabatic limit is approached at least at the rate $\tau^{-\frac{\alpha}{4}}$.
This slightly improves the rate $\tau^{-\frac{\alpha}{4+\alpha}}$, which was
obtained in Corollary 1 in \cite{AE} under the same assumptions.\footnote{Note that 
the rate  $\tau^{-\frac{\alpha}{2+\alpha}}$ which is stated  
 in Corollary 1 in \cite{AE} results from a typo in the proof: In Eq.\ (34) in \cite{AE}
$\alpha$ should read $\alpha/2$, which results in the rate  $\tau^{-\frac{\alpha}{4+\alpha}}$.}

For a nontrivial example where the conditions (\ref{rate1}) and (\ref{rate2}) can be checked 
explicitly, we refer to \cite{T}.\\


\noindent {\em Remark 2.}\quad
With one reservation, namely the existence of $U_{\rm A}(s)$,
it is possible to replace the condition $P(\cdot)\in C^2([0,1],\mathcal{B})$ in the theorem by 
$P(\cdot)\in C^1([0,1],\mathcal{B})$. I owe the following argument to Alexander Elgart.
One can approximate $\dot P(s)$ by a family $A_\epsi(s)$, defined, e.g., as
\[
A_\epsi(s) = \frac{1}{\epsi}\int_0^1 dt\,e^{-\frac{(t-s)^2}{\epsi^2}} \,\dot P(t)\,, 
\] 
so that $\|\dot P(s) - A_\epsi(s)\| \to 0$ as $\epsi\to 0$ uniformly
for $s\in [0,1]$. Since $A_\epsi(\cdot)\in C^1([0,1],\mathcal{B})$ with
$\|\dot A_\epsi (s) \|\leq C/\epsi^2$, it can be used to define $\epsi$-dependent
solutions of the commutator equation (\ref{commu}) as
\begin{eqnarray*}
X_{\Delta,\epsi} (s) &:=& R(E(s)-i\Delta,s)\, A_\epsi(s)\,P(s)\, +\, \mbox{adjoint}\,, \\
Y_{\Delta,\epsi} (s) &:=& i\,\Delta  R(E(s)-i\Delta,s)  \,  A_\epsi(s)\,P(s)\,-\,  \mbox{adjoint} \\
&& +\,[(\dot P(s)-A_\epsi(s) ),P(s) ]\,,
\end{eqnarray*}
where in (\ref{boundX}) and (\ref{boundY1}) only $\sup_{s\in[0,1]}\left\|\dot X_{\Delta,\epsi}\right\| 
\leq \frac{C}{\Delta^2\epsi^2}$ changes. Instead of (\ref{boundY}) one obtains
$\lim_{\Delta,\epsi\to 0} \|Y_{\Delta,\epsi}(s) \|= 0$ irrespective of the order in which 
the limits are taken. By inserting the new bounds into (\ref{bound}) and
chosing, for example, $\Delta=\epsi=\tau^{-\frac{1}{5}}$, one obtains
 (\ref{dist2}) without the need for differentiability of
$\dot P(s)$.

A warning is however in order. If we assume only $P(\cdot)\in C^1([0,1],\mathcal{B})$
then $H_{\rm A}(s)$ does not satisfy the general assumptions guaranteeing existence of 
$U_{\rm A}(s)$ anymore, because $(H_{\rm A}(s)+i)^{-1}$ need not be differentiable. 
Hence, if the condition  $P(\cdot)\in C^2([0,1],\mathcal{B})$ in the theorem
is replaced by $P(\cdot)\in C^1([0,1],\mathcal{B})$, one has to assume that  the unitary propagator 
$U_{\rm A}(s)$ with generator $H_A(s)$ exists.
In the special case when $H(s)$ is a family of {\em bounded} operators one has
$H_{\rm A}(\cdot) \in C([0,1],\mathcal{B})$ and then Dyson expansion yields the existence
of the unitary propagator  $U_{\rm A}(s)$ even for $P(\cdot)\in C^1([0,1],\mathcal{B})$.\\



\noindent I thank Alexander Elgart for fruitful discussions and Stephan De Bi\`evre for helpful comments.



\begin{thebibliography}{99}

\bibitem{AE} Avron, J.E.\ and A.\ Elgart.: {\em Adiabatic theorem without a gap condition},
Commun.\ Math.\ Phys.\ {\bf 203}, 445--463 (1999).

\bibitem{B} Bornemann, F.: {\em Homogenization in time of singularly perturbed mechanical systems},
Lecture Notes in Mathematics {\bf 1687}, Springer, Heidelberg, 1998.

\bibitem{T} Teufel, S.: {\em A space-adiabatic theorem without gap condition and the massless Nelson model},
in preparation.

\end{thebibliography}


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