Content-Type: multipart/mixed; boundary="-------------0003050318259" This is a multi-part message in MIME format. ---------------0003050318259 Content-Type: text/plain; name="00-100.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-100.comments" AMS-codes: 81V70, 81Q70, 46C05 ---------------0003050318259 Content-Type: text/plain; name="00-100.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-100.keywords" Fredholm, Toeplitz, quantum Hall effect, index ---------------0003050318259 Content-Type: application/x-tex; name="fredholm8.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fredholm8.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} %\if@twoside m \oddsidemargin 14truemm\evensidemargin 0mm \marginparwidth 85pt %\else \oddsidemargin 7truemm\evensidemargin 7truemm \marginparwidth 68pt \fi %\topmargin 5mm \headheight 0mm \headsep 0mm %\textheight 225truemm \textwidth 150truemm %\usepackage{amsmath,amsfonts} \parindent=7mm %\renewcommand{\baselinestretch}{1.8} \setcounter{equation}{0} \newtheorem{Def}{Definition} \newtheorem{theorem}{Theorem} \newtheorem{cor}[theorem]{Corollary} \newtheorem{lem}[theorem]{Lemma} \def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt \hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt} \hrule height.#2pt}}}} \def\square{\mathchoice\sqr54\sqr54\sqr{2.1}5\sqr{1.5}5} \def\qed{\hbox{\hskip 6pt\vrule width6pt height7pt depth1pt \hskip1pt}} \def\R{\real} \def\C{\complex} \def\nd{\noindent} \def\real{{\rm I\kern-.2em R}} \def\complex{\kern.1em{\raise.47ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bearray{\begin{eqnarray}} \def\eearray{\end{eqnarray}} \def\bra{\langle} \def\ket{\rangle} %\pagestyle{empty} \title{Generic Jumps of Fredholm Indices and the Quantum Hall Effect} \author{J.~E.~Avron and L.~Sadun\footnote{On leave from the Department of Mathematics, University of Texas, Austin, TX 78712 USA} \\ Department of Physics, Technion, 32000 Haifa, Israel} \begin{document} \maketitle \begin{abstract} We describe the generic behavior of Fredholm indices in the space of Toeplitz operators. We relate this behavior to certain conjectures and open problems that arise in the context of the Quantum Hall Effect. \end{abstract} \input epsf \section{Introduction and Motivation} Suppose one interpolates between Fredholm operators with different indices. What can one say about the way the indices change? The answer to this question depends on the choice of the embedding space for the Fredholm operators in question. In the space of bounded operators, little can be said. But, in the space of Toeplitz operators, (and then also for Toeplitz modulu compacts), as we shall explain, the indices change by abrupt discontinuous jumps that tend to be small. We relate this behavior to certain conjectures and open problems that arise in the context of the Quantum Hall Effect (QHE) \cite{qhe}. \subsection{Physical background} In the theory of the integer quantum Hall effect (of non-interacting electrons) \cite{bel,ass} one identifies the Hall conductance with the Fredholm index of a rather special operator, namely $PUP$, thought of as an operator on the range of $P$. Here $P=P(E)$ is an (infinite dimensional) projection in the Hilbert space $L^2(C)$, namely the projection on the spectrum of the one electron Hamiltonian below the Fermi energy $E$. $U$ is the multiplication operator $z\over |z|$ associated with a singular gauge transformation that introduces an Aharonov-Bohm flux tube at the origin of the Euclidean plane. $PUP$ is Fredholm provided the integral kernel of the projection, $p(z,z';E)$ has good decay properties as $|z-z'|$ gets large \cite{ass}. Recent progress in the rigorous theory of random Schr\"odinger operators relevant to the QHE \cite{aizenman} guarantees good decay properties for $p(z,z';E)$ provided $E$ lies in certain energy intervals. Percolation arguments \cite{trugman} and scaling theories of localization \cite{khmel} give theoretical evidence that these decay properties persist for all but a discrete set of energies. This implies that the graph of the Hall conductace as a function of $E$ should be a step function. Indeed, experimentally, the Hall conductance in the integer Hall effect, is close to a monotonic step function with $\pm 1$ and $\pm 2$ jumps \cite{laugh}. (Jumps by 2 occur when the Hall conductance is larger than 6 and is attributed to the smallness of the magnetic moment of the electron in these systems.) The smallness of the jumps of the Fredholm indices in the QHE might, of course, be a special property of a special system. Here, instead, we want to explore the opposite point of view, namely the possibility that the existence of steps and the smallness of the jumps reflects a generic property of Fredholm indices and has little to do with the specific properties of the system in question. Some support to this point of view comes from the relation of Chern numbers and Fredholm indices. In non-commutative geometry \cite{connes} Chern numbers and Fredholm indices are intimately related. This is also the case in the index theory of elliptic operators \cite{atiyah}. For Chern numbers that arise from studies of spectral bundles (of Hamiltonians with discrete spectra), a generic deformation of the Hamiltonian leads to a step function with $\pm 1$ jumps in the first Chern number \cite{simon}. This follows from the Wigner von Neumann codimension 3 rule for eigenvalue crossing \cite{wvn} and the fact that a generic crossing is a conic crossing and is not system specific. As far as the QHE goes one might argue that since the Hall conductance can be directly related to a Chern number \cite{qhe,tknn}, the genericity of small jumps follows immediately. The difficulty with this argument has to do with the thermodynamic limit. Normally, the QHE is associated with large systems. The genericity result quoted above for Chern numbers is for operators with discrete spectrum. This is the case for finite sytems, but is in general not the case for extended systems, and in particular does not apply to models of the quantum Hall effect. The main attractive feature of the Fredholm approach to the Hall effect is that it is phrased directly in the thermodynamic limit. Another way of phrasing the main theme of this paper is: What, if any, is the analog for Fredholm operators of the genericity of small jumps in Chern numbers? \subsection{The mathematical problem} We wish to interpolate between two (or more) Fredholm operators. If the indices of these operators are different this cannot be done within the space of Fredholm operators. At some points in the interpolation the Fredholm property will be lost and the index will be ill defined. For ``generic'' interpolations, what is the nature of this bad set? Near such a bad point, how big a range of indices can be found? Working in the space of bounded operators, little can be said. The space is simply too large, and when the Fredholm property is lost we lose all analytic control. However, in the space of sufficiently smooth Toeplitz operators interesting results can be obtained. In systems without symmetry, we find the following behavior: Almost every operator is Fredholm, and sets of codimension $n$ appear as boundaries between regions of Fredholm operators whose indices differ by $n$. We speak simply of the index ``jumping by $n$'' on a set of codimension $n$. In systems with a $Z_2$ symmetry (e.g. time reversal symmetry or complex conjugation symmetry), sets of codimension $n$ appear as common boundaries of regions of Fredholm operators whose indices differ by as much as $2n$. That is, the index can jump by as much as $2n$ on a set of codimension $n$. \section{Basic Definitions and Properties} We review here the basic definitions and properties of Fredholm operators on separable Hilbert spaces. For a more complete treatment see \cite{douglas}. \begin{Def} A bounded operator $A$ on a separable Hilbert space if {\em Fredholm} if there exists another bounded operator $B$ such that $1-AB$ and $1-BA$ are compact. \end{Def} In particular, the kernel and cokernel of $A$ are finite dimensional, and we define \begin{Def} The {\em index} of a Fredholm operator $F$ is \be Index(F)= dim\, Ker(F)- dim\,Ker(F^\dagger).\ee \end{Def} Fredholm operators are stable under compact perturbations and under small bounded perturbations. That is, if $A$ is Fredholm, there exists an $\epsilon >0$ such that, for any bounded operator $B$ with operator norm $\| B \| < \epsilon$ and for any compact operator $K$, the operator $A+B+K$ is Fredholm with the same index as $A$. The simplest example of a Fredholm operator with nonzero index is the shift operator. Let $e_0, e_1, e_2, \ldots$ be an orthonormal basis for a Hilbert space, and let the operator $a$ act by \be a(e_n) = \cases{e_{n-1} & if $n>0$ \cr 0 & if $n=0$}. \ee The adjoint of $a$ acts by \be a^\dagger(e_n) = e_{n+1} \ee Since $a a^\dagger= a^\dagger a + |e_0 \ket \bra e_0 |$ is the identity, $a$ is Fredholm. The kernel of $a$ is 1-dimensional. The cokernel of $a$, which is the same as the kernel of $a^\dagger$, is 0 dimensional. Thus the index of $a$ is 1. Similarly, $a^\dagger$ is Fredholm with index $-1$. The following theorem is standard: \begin{theorem} If $A_1, \ldots A_n$ are Fredholm operators, then the product $A_1 A_2 \cdots A_n$ is also Fredholm, and $Index(A_1\cdots A_n) = \sum_{i=1}^n Index(A_i)$. \end{theorem} Finally we consider connectedness in the space of Fredholm operators. If $A$ and $A'$ are Fredholm operators on the same Hilbert space, then there is a continuous path of Fredholm operators from $A$ to $A'$ if and only if $Index(A)=Index(A')$. (By continuous, we mean relative to the operator norm). Put another way, the path components of $Fred(H)$, the space of Fredholm operators on $H$, is indexed (pun intended) by the integers. The $n$-th path component is precisely the set of Fredholm operators of index $n$ \cite{douglas}. \section{Fredholm Operators in the Space of Bounded Operators} The most natural setting for our problem is consider arbitrary bounded operators, with the topology defined by the operator norm. We ask how many parameters must be varied in order to reach the common boundary of two regions, whose indices differ by $k$. Unfortunately, the answer is independent of $k$: \begin{theorem} Let $U_n$ be the set of Fredholm operators of index $n$. Every point on the boundary of $U_n$ is also on the boundary of $U_m$, for every integer $m$. \end{theorem} Proof: Let $A$ be a (not Fredholm) operator on the boundary of $U_n$. Given $\epsilon >0$, we must find an operator in $U_m$ within a distance $\epsilon$ of $A$. Suppose that the kernel and cokernel of $A$ are infinite dimensional, and that there is a gap in the spectrum of $A^\dagger A$ at zero. (If this is not the case, we may perturb $A$ by an arbitrarily small amount to make it so). Now let $B$ be a unitary map from the kernel of $A$ to the cokernel. Let $P,\ (P')$ be the orthogonal projection onto $ker(A), \ (coker(A))$, and let $a$ be a shift operator on $ker(A)$. For each $m \ge 0$, $A(\epsilon)=A + \epsilon B a^m P$ has a bounded right inverse \be A^\dagger {1\over P'+ AA^\dagger} P'_\perp + {1\over\epsilon} (a^\dagger)^m B^\dagger P'. \ee It follows that the cokernel of $A(\epsilon)$ is empty. It is easy to see that the kernel of $A(\epsilon)$ is $m$ dimensional hence $Index(A(\epsilon))=m$. Similarly, $A + \epsilon B (a^\dagger)^{m} P$ has index $-m$. \hfill$\qed$ This theorem tells us that, in the space of all bounded operators there is no specific notion of being at a transition point from index $n$ to index $m$. As long as an operator stays Fredholm, its index cannot change, and when it fails to be Fredholm it can change into anything. To achieve useful results, we must work on a smaller space. \section{Linear Combinations of Shifts} In this section and the next we show that ``generic" behavior is indeed achieved in some finite dimensional spaces, and in some infinite-dimensional spaces with sufficiently fine topologies. We see also how control is lost as the space is enlarged and the topology is coarsened. \subsection{Shift by one} We begin by considering linear combinations of the shift operator $a$ and the identity operator 1. That is, we consider the operator $$ A = c_1 a + c_0 $$ where $c_1$ and $c_0$ are constants. \begin{theorem} \label{0-1} If $|c_1| \ne |c_0|$, then $A$ is Fredholm. The index of $A$ is 1 if $|c_1|>|c_0|$ and zero if $|c_1|<|c_0|$. If $|c_1| = |c_0|$, then $A$ is not Fredholm. \end{theorem} \nd Proof: First suppose $|c_0| > |c_1|$. Then $A$ is invertible: $$ A^{-1} = c_0^{-1} (1 + (c_1/c_0) a)^{-1} = \sum_{n=0}^\infty {(-1)^n c_1^n \over c_0^{n+1}} a^n, $$ as the sum converges absolutely. Thus $A$ has neither kernel nor cokernel, and has index zero. If $|c_1| > |c_0|$, then the kernel of $A$ is 1-dimensional, namely all multiples of $|\psi \ket = \sum_{n=0}^\infty z_0^n e_n$, where $z_0 = -c_0/c_1$. Notice how the norm of $|\psi \ket$ goes to infinity as $|z_0| \to 1$. However, $A^\dagger$ has no kernel, since for any unit vector $|\phi \ket$, $\|A^\dagger | \phi \ket\| = \|\bar c_1 a^\dagger |\phi \ket + \bar c_0 |\phi \ket\| \ge \|\bar c_1 a^\dagger |\phi \ket \| - \|\bar c_0 |\phi \ket\| = |c_1| - |c_0|$. Thus the index of $A$ is 1. If $|c_1| = |c_0| $, then $A$ is at the boundary between index 1 and index 0, and so cannot be Fredholm.\hfill$\qed$ \subsection{Finite linear combinations of shifts} Next we consider linear combinations of $1, a, a^2, \ldots$ up to some fixed $a^n$. That is, we consider operators of the form \be A = c_n a^n + c_{n-1} a^{n-1} + \cdots + c_0. \label{A} \ee This is closely related to the polynomial \be p(z) = c_n z^n + \cdots + c_0. \label{p} \ee \begin{theorem} If none of the roots of $p$ lie on the unit circle, then $A$ is Fredholm, and the index of $A$ equals the number of roots of $p$ {\em inside} the unit circle, counted with multiplicity. If any of the roots of $p$ lie {\em on} the unit circle, then $A$ is not Fredholm. \end{theorem} \nd Proof: The polynomial $p(z)$ factorizes as $p(z) = c_k \prod_{i=1}^k (z-\zeta_i)$, where $k$ is the degree of $p$ (typically $k=n$, but it may happen that $c_n=0$). But then $A = c_k \prod_{i=1}^k (a - \zeta_i)$. If none of the roots $\zeta_i$ lie on the unit circle, then each term in the product is Fredholm, so the product is Fredholm, and the index of the product is the sum of the indices of the factors. By Theorem \ref{0-1}, this exactly equals the number of roots $\zeta_i$ inside the unit circle. If any of the roots lie on the unit circle, then a small perturbation can push those roots in or out, yielding Fredholm operators with different indices. This borderline operator therefore cannot be Fredholm.\hfill$\qed$ The last theorem easily generalizes to linear combination of left-shifts and right-shifts. The index of an operator \be A = c_n a^n + \cdots + c_1 a + c_0 + c_{-1} a^\dagger + \cdots + c_{-m} (a^{\dagger})^m \ee equals the number of roots of \be p(z) = \sum_{i=-m}^n c_i z^i \ee inside the unit circle, minus the degree of the pole at $z=0$ (that is $m$, unless $c_{-m}=0$). This follow from the fact that \be A = (\sum_{i=-m}^n c_i a^{i+m}) (a^\dagger)^m. \ee Since there is no qualitative difference between combinations of left-shifts and combinations of both left- and right-shifts, we restrict our attention to left-shifts only, and consider families of operators of the form (\ref{A}). \begin{theorem}\label{finite-combinations} In the space of complex linear combinations of 1, $a$, \ldots, $a^n$, almost every operator is Fredholm. For every $k \le n$, the points where the index can jump by $k$ (by which we mean the common boundaries of regions of Fredholm operators whose indices differ by $k$) is a set of real codimension $k$. In the space of real linear combinations of 1, $a$, \ldots, $a^n$, almost every operator is Fredholm. For every $k \le n$, the points where the index jumps by $k$ is a stratified space, the largest stratum of which has real codimension $\lfloor (k+1)/2 \rfloor $, where $ \lfloor x \rfloor$ denotes the integer part of $x$. \end{theorem} \nd Proof: Our parameter space is the space of coefficients $c_i$, or equivalently the space of polynomials of degree $\le n$. This is either $\R^{n+1}$ or $\C^{n+1}$, depending on whether we allow real or complex coefficients. In either case, the set $U_k$ of Fredholm operators of index $k$ is identical to the set of polynomials with $k$ roots inside the unit circle and the remaining $n-k$ roots outside (if $c_n=0$, we say there is a root at infinity; if $c_n=c_{n-1}=0$, there is a double root at infinity, and so on. Counting these roots at infinity, there are always exactly $n$ roots in all.) The boundary of $U_k$ is the set of polynomials with at most $k$ roots inside the unit circle, at most $n-k$ outside the unit circle, and at least one root on the unit circle. (Strictly speaking, the zero polynomial is also on this boundary. This is of such high codimension that it has no effect on the phase portrait we are developing.). We consider the common boundary of $U_k$ and $U_{k'}$. If $k0$ and a left-shift if $m<0$. All our results about shifts can therefore be understood in the context of Toeplitz operators. Theorem \ref{finite-combinations} refers to operators $T_f$, where $f$ is a polynomial in $z^{-1}$ of limited degree. Theorem \ref{limit-space} considers polynomials or arbitrary degree in $z$ and $z^{-1}$. We will see that the results carry over to analytic functions on an annulus around $S^1$, and to a lesser extent to $C^k$ Toeplitz operators, but with results that weaken as $k$ is decreased. Here are some standard results about Toeplitz operators. For details, see \cite{douglas}. \begin{theorem} A $C^1$ Toeplitz operator $T_f$ is Fredholm if and only if $f$ is everywhere nonzero on the unit circle. In that case the index of $T_f$ is minus the winding number of $f$ around the origin, namely \be Index(T_f) = -Winding(f) = {-1\over 2 \pi i} \int_{S^1} {df \over f}, \label{wind} \ee \end{theorem} Given the first half of the theorem, the equality of index and winding number is easy to understand. We simply deform $f$ to a function of the form $f(z)=z^n$, while keeping $f$ nonzero on all of $S^1$ throughout the deformation (this is always possible, see e.g. \cite{gp}). In the process of deformation, neither the index of $T_f$ nor the winding number of $f$ can change, as they are topological invariants. Since the winding number of $z^n$ is $n$, and since $T_{z^n} = (a^\dagger)^n$ (if $n\ge 0$, $a^{-n}$ otherwise), which has index $-n$, the result follows. We now consider functions $f$ on $S^1$ that can be analytically continued (without singularities) an annulus $r_0 \le |z| \le r_1$, where the radii $r_0 < 1$ and $r_1>1$ are fixed. This is equivalent to requiring that the Fourier coefficients $\hat f_n$ decay exponentially fast, i.e. that the sum \be \sum_{n=-\infty}^\infty |\hat f_n |(r_0^n + r_1^n) \ee converges. For now we do not impose any reality constraints or other symmetries on the coefficients $\hat f_n$. This space of functions is a Banach space, with norm given by the sup norm on the annulus. This norm is stronger than any Sobolev norm on the circle itself. The analysis of the corresponding Toeplitz operators is straightforward and similar to the proof of Theorem \ref{finite-combinations}. Since $f$ has no poles in the annulus, we just have to keep track of the zeroes of $f$. For the index of $T_f$ to change, a zero of $f$ must cross the unit circle. For the index to jump from $k$ to $k'$, $|k-k'|$ zeroes must cross simultaneously. In the absence of symmetry, the locations of the zeroes are independent and can be freely varied, so this is a codimension-$|k-k'|$ event. If we impose a reality condition: $f(\bar z) = \overline{f(z)}$, then zeroes appear only on the real axis or in complex conjugate pairs. In that case, changing the index by 2 is merely a codimension-1 event. Combining these observations we obtain \begin{theorem}\label{analytic} In the space of Toeplitz operators that are analytic in a (fixed) annulus containing $S^1$, almost every operator is Fredholm. For every integer $k \ge 1$, the points where the index can jump by $k$ is a set of real codimension $k$. If we impose a reality condition $f(\bar z)=\overline{f(z)}$ then, for every $k \le n$, the points where the index jumps by $k$ is a stratified space, the largest stratum of which has real codimension $\lfloor (k+1)/2 \rfloor $. \end{theorem} Finally we consider Toeplitz operators that are not necessarily analytic, but are merely $\ell$ times differentiable, and we use the $C^\ell$ norm. Our result is \begin{theorem}\label{C^ell} In the space of Toeplitz $C^\ell$ operators, almost every operator is Fredholm. For every integer $k$ with $1 \le k \le 2\ell+1$, the points where the index can jump by $k$ is a set of real codimension $k$. For every integer $k \ge 2\ell+1$, the points where the index can jump by $k$ is a set of real codimension $2\ell+1$. \end{theorem} In other words, our familiar results hold up to codimension $2\ell+1$, at which point we lose all control of the change in index. \smallskip \nd Proof: As long as $f$ is everywhere nonzero, $T_f$ is Fredholm. To get a change in index, therefore, we need one or more points where $f$, and possibly some derivatives of $f$ with respect to $\theta$, vanish. Suppose then that for some angle $\theta_0$, $f(\theta_0)=f'(\theta_0)=\cdots=f^{(n-1)}(\theta_0)=0$ for some $n \le \ell$, but that the $n$-th derivative $f^{(n)}(\theta_0) \ne 0$. This is a codimension $2n-1$ event, since we are setting the real and imaginary parts of $n$ variables to zero, but have a 1-parameter choice of points where this can occur. Without loss of generality, we suppose that this $n$-th derivative is real and positive. By making a $C^\ell$-small perturbation of $f$, we can make the value of $f$ highly oscillatory near $\theta_0$, thereby wrapping around the origin a number of times. However, since a $C^\ell$-small perturbation does not change the $n$-th derivative by much, the sign of the real part of $f$ can change at most $n$ times near $\theta_0$, so the argument of $f$ can only increase or decrease by $n\pi$ or less. The difference between these two extremes is $2n\pi$, or a change in winding number of $n$. To change the index by an integer $m$, therefore, we must have the function vanish to various orders at several points, with the sum of the orders of vanishing adding to $m$. The generic event is for $f$ (but not $f'$) to vanish at $m$ different points -- this is a codimension $m$ event, analagous to having $m$ zeroes of a polynomial cross the unit circle simultaneously at $m$ different points. All other scenarios have higher codimension and are analogous to having 2 or more zeroes of the $m$ zeroes crossing the unit circle at the same point. The situation is different, however, when the function $f$ and the first $\ell$ derivatives all vanish at a point $\theta_0$. Then the higher-order derivatives are not protected from $C^\ell$-small perturbations and, by making such a perturbation, we can change $f$ into a function that is identically zero on a small neighborhood of $\theta=\theta_0$. By making a further small perturbation, we can make $f$ wrap around the origin as many times as we like near $\theta=\theta_0$. More specifically, if $f$ is zero on an interval of size $\delta$, then, for small $\epsilon$, $\tilde f(\theta) = f(\theta) + \epsilon e^{iN\theta}$ will wrap around the origin approximately $N\delta/2\pi$ times near $\theta_0$. By picking $N$ as large (positive or negative) as we wish, we can obtain arbitrarily positive or negative indices. As long as we take $\epsilon \ll N^{-\ell}$, this perturbation will remain small in the $C^\ell$ norm.\hfill\qed \section{The Quantum Hall Effect} We have seen in the previous section that the Fredholm index of a generic one dimensional family of Toeplitz operators is a step function with small jumps. This is reminiscent of what one observes for the Hall conductance for random Schr\"odinger operators. In this section we want to discuss some of the difficulties, and what one would still need to know, for the strategy in this paper to yield useful results for the QHE. \subsection{Landau levels} The Hall conductance is related to the Index of $PUP$ (on $Range\ P$) with $P$ a spectral projection in $L^2(\complex)$ and $U$ a multiplication by $z\over |z|$. This operator is closely related to a Toeplitz operator in the case of a basic paradigm for the Hall effect: \begin{theorem} Let $P$ be a projection on the lowest Landau level in $\real^2$, then $PUP$ differs from a Toeplitz operator by a compact operator. \end{theorem} \nd Proof: A basis for the lowest Landau level is \be |n\ket={1\over \sqrt{\pi\, n!}}\, z^n\,e^{-|z|^2/2},\quad n\ge 0. \ee As a consequence \be \bra n|U|m\ket=\delta_{n,m+1}\, {(m+1/2)!\over m!\sqrt{m+1}}\approx \delta_{n,m+1}\left(1-{1\over 8m}\right). \qed \ee The same result also holds if $P$ is a projection on a higher Landau level, but the calculation is more involved. If $P$ is a projection onto multiple Landau levels, then $PUP$ is a compact perturbation of a direct sum of Toeplitz operators, one for each Landau level. This suggests that the class of Toeplitz operators is indeed related to the QHE. For (spinless) electrons/holes on the Euclidean and hyperbolic planes, with homogeneous magnetic field, and without disorder, $Index (PUP)(E)$ has been explicitly computed as a function of the ``Fermi energy'' $E$. In the Euclidean plane one finds a monotonic step function with jumps $\pm 1$ \cite{pnueli}. (One needs both signs for electrons and holes.) The same results apply in the hyperbolic plane for all energies below the continuous spectrum \cite{pnueli}. This implies that also for (relatively) compact perturbations of these Hamiltonians the Fredholm index in the QHE behaves as does the Fredholm index of Toeplitz operators. The situation is, however, quite different for Schr\"odinger operators with periodic potentials where $PUP(E)$ failes to be Fredholm on intervals of ``energy bands" and where the Fredholm index in adjacent gaps can jump by large integers \cite{tknn}. \subsection{An open problem} For applications to the Hall effect one considers $PUP$ (on the range of $P$) where the projection $P$ depends on a parameter such as the Fermi energy or the external magnetic field. The family $PUP$ is therefore defined on different spaces, since the range of $P$ is not fixed. Our strategy, so far, has been to study a family of operators on a {\em fixed} Hilbert space. To adapt the QHE to this strategy one must replace $PUP$ by something like \be C=PUP+1-P, \label{C} \ee acting on the full Hilbert space, as $Index(C)$ on the full space coincides with $Index(PUP)$ on $Range(P)$. Now, a deformation of $P$ leads to a deformation of $C$ and gives a family of bounded operators on a fixed space, say, $L^2(\complex)$. However, this modification is not without a price since now, even for the simple case of a full Landau level, $C$ is not strictly a Toeplitz operator. It is a rather silly generalization of a Toeplitz operator to a direct sum of a Toeplitz operator and the identity. A more serious problem has to do with what should one pick as a good family $P$. In particular, when one considers a variation of the Fermi energy $E$ the corresponding projection $P(E)$ is not continuous in the operator norm. Hence, a smooth variation of $E$ is not even a smooth variation of $C$ in the operator norm (much less in the sharper norms considered above). Using the fact that the Fredholm index does not change under small changes in the norm of the operator, there is no harm done if one replaces the spectral projection $P(E)$ by the Fermi function \be P_\beta(E)= {1\over \exp \beta (H-E)+1}, \ee for $\beta$ large. Unlike $P(E)$, $P_\beta(E)$ is a smooth function of $E$, and so the family $C_\beta(E)$ is smooth. The price one pays is that $P_\beta(E)$ is not a projection, which leads to ambiguities as to what one might want to choose for $C_\beta(E)$ . For example, instead of (\ref{C}) one might choose \be C_\beta(E)=P_\beta(E)UP_\beta(E)+(1-P_\beta^2(E)). \ee The trouble is that it is not clear what, if anything, the results about families of Toeplitz operators imply for the family $C_\beta(E)$. We therefore pose the following questions: For random Scr\"odinegr operators on the plane, with $\beta$ sufficiently large, what are the properties of the family of operators $C_\beta(E)$? Is it Fredholm away from a discrete set of energies $E$, or does it fail to be Fredholm on bigger sets? If it fails to be Fredholm at isolated points, are the jumps generically small? \section*{Acknowledgments} This research was supported in part by the Israel Science Foundation, the Fund for Promotion of Research at the Technion, the DFG, the National Science Foundation and the Texas Advanced Research Program. \begin{thebibliography}{article} \bibitem[Aiz]{aizenman}M. Aizenman, {\em Localization at weak disorder: Some elementary bounds}, Rev. Math. Phys. {\bf 6}1163 (1994); M. Aizenman and S. Molchanov, {\em Localization at large disorder and extreme energies} Comm. Math. Phys. {\bf 157 },245, (1993); M. Aizenman and G.M. Graf, {\em Localization Bounds for an Electron Gas}, J. Phys. A: Math. Gen. 31 (1998) 6783-6806, cond-mat/9603116. \bibitem[APn]{pnueli} J.\ E. Avron and A.\ Pnueli, {\em Landau Hamiltonians on symmetric space}, in {\it ``Ideas and Methods in Mathematical analysis, stochastics, and applications Vol II"}, S.\ Albeverio, J.\ E.\ Fenstad, H. Holden and T.\ Lindstr\o m, Editors, Cambridge University Press, (1992). \bibitem[ASS]{ass} J.~E. Avron, {\em Charge deficiency, charge transport and comparison of dimensions} R.~Seiler and B.~Simon, Comm.~Math.~Phys. {\bf 159}, 399 (1994) \bibitem[Ati]{atiyah} M. Atiyah, {\em Algebraic topology and operators in Hilbert space}, Lecture Notes in Mathematics {\bf 103}, 101--122 (1969). \bibitem[BvES]{bel}J. Bellissard, A. van Elst, H. Schultz-Baldes, {\em The noncommutative geometry of the quantum Hall effect}, J. Math. Phys. {\bf 35}, 5373 (1994). \bibitem[Con]{connes}A. Connes, {\em Noncommutative Geometry}, Academic Press ,1994 \bibitem[Dou]{douglas}Douglas, R.G. , {\em Banach algebra techniques in operator theory}, Academic Press (1972) \bibitem[GuP]{gp} V.W.~Guillemin and A. Pollack, {\em Differential Topology}, Prentice Hall (1974). \bibitem[Khm]{khmel} D. Khmelnitsky,{\em Quantisation of Hall conductivity}, JETP lett {\bf 38}, 552-556,(1983), and {\em Quantum Hall effect and Additional Oscillations of Conductivity in Weak Magnetic Fileds}, Phys. Let. A {\bf 106}, 182,(1984). \bibitem[Lau]{laugh} URL http://www.nobel.se/announcement-98/physics98.html \bibitem[Sim]{simon}B.Simon, {\it Holonomy, the quantum adiabatic theorem and Berry's phase}, Phys.\ Rev.\ Lett. {\bf 51} 2167-2170 (1983). \bibitem[Sto]{qhe} M.~Stone, {\em The Quantum Hall Effect} , World Scientific, Singapore, (1992); D.~J. Thouless, J.\ Math.\ Phys.\ {\bf 35}, 1-11, (1994); A. H. MacDonald, Les Houches LXI, 1994, E. Akkermans, G. Montambaux, J.~L. Pichard and J. Zinn Justin Eds., North Holland 1995. \bibitem[TKNN]{tknn} D.~J.~Thouless, M.~Kohmoto,~P.~Nightingale and M. den Nijs, {\it Quantum Hall conductance in a two dimensional periodic potential}, Phys.~Rev.~Lett. {\bf 49}, 40, (1982). \bibitem[Tru]{trugman}S.A. Trugman, {\em Localization, percolation and the quantum Hall effect}, Phys. Rev. B,{\bf 27}, 7539-7545, (1983) \bibitem[vNW]{wvn} J.~von Neumann and E.~P. Wigner, Phys. 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