Content-Type: multipart/mixed; boundary="-------------0112050243280" This is a multi-part message in MIME format. ---------------0112050243280 Content-Type: text/plain; name="01-450.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-450.comments" Revised version, two remarks added. To appear in Lett. Math. Phys. ---------------0112050243280 Content-Type: text/plain; name="01-450.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-450.keywords" adiabatic theory, gap condition ---------------0112050243280 Content-Type: application/x-tex; name="AEFin.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="AEFin.tex" \documentclass{article} \usepackage{amssymb} \setlength{\topmargin}{-2.5cm} \setlength{\oddsidemargin}{1.5cm} \setlength{\evensidemargin}{1.5cm} \setlength{\textwidth}{13cm} \setlength{\textheight}{22cm} \begin{document} \def\epsi{\varepsilon} \def\kasten#1{\mathop{\mkern0.5\thinmuskip \vbox{\hrule \hbox{\vrule \hskip#1 \vrule height#1 width 0pt\vrule}\hrule}\mkern0.5\thinmuskip}} \def\qed{\mathchoice{\kasten{8pt}} {\kasten{6pt}} {\kasten{4pt}} {\kasten{3pt}}} \def\1I{\mathchoice{ \hbox{${\rm 1}\hspace{-2.3pt}{\rm l}$} } { \hbox{${\rm 1}\hspace{-2.3pt}{\rm l}$} } { \hbox{$ \scriptstyle {\rm I}\!{\rm N}$} } { \hbox{$ \scriptscriptstyle {\rm I}\!{\rm N}$}}} \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} \end{document} ---------------0112050243280--