when $\lambda$ is small (at least when (8.22) or (8.25) holds) is that the
divergent piece is rank one. To see this, [634] writes

$K_{\alpha}=L_{\alpha}+M_{\alpha}$ (8.27)
$L_{\alpha}(x,y)=|V|^{1/2}(x)V^{1/2}(y)/2\alpha$ (8.28)
$M_{\alpha}(x,y)=(2\alpha)^{-1}|V|^{1/2}(x)\left[e^{\alpha|x-y|}-1\right]V^{1/2}(y)$
(8.29)

so $\lim_{\alpha\downarrow 0}M_{\alpha}\equiv M_{0}$ exists where

$M_{0}(x,y)=|V|^{1/2}(x)|x-y|V^{1/2}(y)$ (8.30)

Thus, the Birman-Schwinger principle is equivalent to

$\alpha=-\tfrac{\lambda}{2}\langle V^{1/2},(1+\lambda
M_{\alpha})^{-1}|V|^{1/2}\rangle$ (8.31)

The leading term on the right is $-\tfrac{\lambda}{2}\int V(x)\,dx$ and the
next is $\tfrac{\lambda^{2}}{4}\langle V^{1/2},M_{0}|V|^{1/2}\rangle$.

This work lead to several threads of later work. In [643], I asked when a new
eigenvalue, $E(\lambda)$, issuing from $E=0$ at $\lambda=\lambda_{0}$ is
$\text{O}(\lambda-\lambda_{0})$ and proved that

######  Theorem 8.6.

Suppose that $B$ is a relatively compact symmetric perturbation of a self-
adjoint operator, $A$, that $\sigma_{ess}(A)$ includes $[0,\varepsilon]$ for
some $\varepsilon>0$ and that $\lambda_{0}>0$ is such that $A+\lambda B$ has
exactly one more negative eigenvalue (counting multiplicity) for
$\lambda\in(\lambda_{0},\lambda_{0}+\delta)$ than for
$\lambda\in(\lambda_{0}-\delta,\lambda_{0})$. Then there is one eigenvalue,
$E(\lambda)$, near $0$ for $\lambda\in(\lambda_{0},\lambda_{0}+\delta)$ and

$\lim_{\lambda\downarrow\lambda_{0}}\frac{E(\lambda)}{\lambda-\lambda_{0}}=\alpha$
(8.32)

exists. Moreover, $\alpha\neq 0$ if and only if $A+\lambda_{0}B$ has
eigenvalue $0$, and in that case, the eigenvalue is simple and
$\alpha=\langle\varphi,B\varphi\rangle$ where $(A+\lambda_{0}B)\varphi=0$ with
$\lVert\varphi\rVert=1$.

In [655, 409], I made what turned out to be an important definition.

######  Definition 8.7.

Let $\nu\geq 3$ and $V\in L^{p}({\mathbb{R}}^{\nu})\cap
L^{q}({\mathbb{R}}^{\nu})$ for some $p<\tfrac{\nu}{2}<q$. $V$ is called

_supercritical_ $\iff\,\,\inf\text{\rm{spec}}(-\Delta+V)<0$

_subcritical_ $\iff\,\,\inf\text{\rm{spec}}(-\Delta+(1+\varepsilon)V)=0\text{
for some }\varepsilon>0$

_critical_ $\iff\,\,-\Delta+(1+\varepsilon)V\text{ is supercritical for all
}\varepsilon>0\text{ and }\inf\text{\rm{spec}}(-\Delta+V)=0$

Thus critical $V$’s are ones that like $-\Delta$ in $1$ and $2$ dimensions are
about to give birth to bound states. A main result of [655] is that $V$ is
subscritical $\iff\sup_{t}\lVert
e^{-(-\Delta+V)t}\rVert_{\infty,\infty}<\infty$ (the norm from $L^{\infty}$ to
$L^{\infty}$). Both [655] and [409] show that if $V$ and $W$ are both critical
in three dimensions, then $-\Delta+V(x)+W(x-R)$ has a bound state of energy,
$E(R)<0$ for large $R$ with $E(R)\sim-\beta R^{-2}$. [409] even computes the
universal value of $\beta$. This result is connected to the Effimov effect. A
considerable literature has developed on the study of critical operators;
[537] reviewed the literature as of 2005. I note that while [409] focused on
${\mathbb{R}}^{3}$, it has a remark about what happens in
${\mathbb{R}}^{\nu};\nu\geq 4$ which is wrong; for the correct results, see
[537, Theroems 8.5-8.6].

In [410, 411], Klaus and I consider the variety of coupling constant threshold
behavior that can occur for $-\Delta+V+\lambda W$ (with $V$ and $W$ short
range) when as $\lambda\downarrow\lambda_{0}$, some eigenvalue is absorbed in
the continuous spectrum (the first paper deals with the two body problem and
the second with a limited set of $N$ body systems). The results are quite
complicated and supplement/illuminate some work of Jensen-Kato [354].

Our next topic concerns the situation where $V$ goes to infinity at infinity
(or, at least, is bounded away from zero there) but $\min(V(x))=0$ and we are
interested in the lowest eigenvalues of $-\Delta+\lambda^{2}V$ as
$\lambda\to\infty$. In [665] I proved

######  Theorem 8.8 ([665]).

Let $V,W$ be two $C^{\infty}$ functions on ${\mathbb{R}}^{\nu}$ so that

(1) For some $A,R>0$, one has $V(x)\geq A$ if $|x|\geq R$

(2) $V(x)\geq 0$ for all $x$

(3) $V(x)$ vanishes only at $\\{x^{(j)}\\}_{j=1}^{M}$ ($M\geq 1$) and at each
such minimum, the matrix $A^{(j)}_{k\ell}=\frac{\partial V}{\partial
x_{k}\partial x_{\ell}}(x^{(j)})$ is strictly positive.

(4) $W$ is bounded from below

Let $H^{(j)}=-\Delta+\sum_{k,\ell}A^{(j)}_{k\ell}x_{k}x_{\ell}+W(x^{(j)})$.
Let $\\{e_{\alpha}\\}_{\alpha=1}^{\infty}$ be an ordering of the union over
$j$ of the eigenvalues of $H^{(j)}$ (counting multiplicity) so that $e_{1}\leq
e_{2}\leq\dots$. Then

$H(\lambda)\equiv-\Delta+\lambda^{2}V+\lambda W$ (8.33)

has eigenvalues, $\\{E_{\alpha}(\lambda)\\}_{\alpha=1}^{\infty}$, with
$E_{1}(\lambda)\leq E_{2}(\lambda)\leq\dots$ at the bottom of its spectrum and
for any $\alpha=1,2,\dots$ we have that

$\lim_{\lambda\to\infty}\frac{E_{\alpha}(\lambda)}{\lambda}=e_{\alpha}$ (8.34)

Moreover, each $E_{\alpha}(\lambda)$ has an asymptotic series in
$\lambda^{-1}$ to all orders

$E_{\alpha}(\lambda)=e_{\alpha}\lambda+a_{\alpha}^{(0)}+a_{\alpha}^{(1)}\lambda^{-1}+\dots$
(8.35)

I always thought of the paper in which this theorem appeared as Ed Witten’s
homework assignment because one motivation for this work was his wonderful
paper on the supersymmetric proof of Morse inequalities and the Morse index
theorem [771]. In it, he used this theorem (or rather its generalization when
functions on ${\mathbb{R}}^{\nu}$ are replaced by the tangent bundle on a
compact manifold and $-\Delta$ by a Laplace-Beltrami operator (also discussed
in [665])). When using this result, Witten says _Although the rigorous theory
has apparently not been developed for operators acting on vector bundles on
manifolds, the method used in Reed and Simon[555], pp. 34–38, to treat the
double well potential should suffice with some elaboration for this case._ In
fact, the argument in [555] he refers to is one-dimensional and uses some
other properties of the simple double well. The general one-dimensional case
had been done by Combes et al [114], but their arguments also depended on one
dimension and are somewhat involved, so I wrote my paper in part to get a
proof that works in multiple dimensions and also one that is fairly simple.

[665] was the first paper of a series. The other papers dealt with eigenvalue
splitting in the situation where multiple wells have the same eigenvalue. It
is simpler to discuss the case of the lowest eigenvalue assuming a double
degeneracy. A basic role is played by the Agmon metric, mentioned in Section 6
(in the page after (6.37)), which was known to determine the rate of decay of
eigenfunctions. In this situation it is defined as the distance in the Riemann
metric $V(x)(dx)^{2}$, i.e.

$\rho(x,y)=\inf\left\\{\int_{0}^{1}\sqrt{V(\boldsymbol{\gamma}(s))}|\dot{\boldsymbol{\gamma}}(s)|\,ds\,\middle|\,\boldsymbol{\gamma}(0)=x,\boldsymbol{\gamma}(1)=y\right\\}$
(8.36)

over all smooth paths, $\boldsymbol{\gamma}(s),\,0\leq s\leq 1$, between $x$
and $y$. The main result of [666, 669] is

######  Theorem 8.9 ([669]).

Let $V$ be a $C^{\infty}$ function on ${\mathbb{R}}^{\nu}$ (and $W=0$) obeying
conditions (1)-(3) of Theorem 8.8. Suppose there are two points $a\neq b$
where $V$ vanishes and that $e_{1}=e_{2}<e_{3}$ so that $e_{1}$ and $e_{2}$
are eigenvalues associated to the operators, $H^{(j)}$, at the points $a$ and
$b$. Let $j_{a}$ (resp $j_{b}$) be characteristic functions of small balls
about $a$ (resp $b$), balls that are so small that they are disjoint. Let
$\Omega_{\lambda}$ be the normalized ground state of the operator,
$H(\lambda)$, of (8.33). Suppose that

$\liminf\lVert j_{a}\Omega_{\lambda}\rVert\lVert
j_{b}\Omega_{\lambda}\rVert>0$ (8.37)

Then

$\lim_{\lambda\to\infty}-\lambda^{-1}\log\left[E_{2}(\lambda)-E_{1}(\lambda)\right]=\rho(a,b)$
(8.38)

######  Remarks.

1\. (8.37) says that the ground state lives near both minima. One condition
that guarantees this is if there is a Euclidean rotation or reflection of
order $2$ that leaves $V$ invariant with $Ra=b$. In that case the limit is
exactly $1/4$. That holds for the famous $1D$ double well where
$V(x)=x^{2}(x-1)^{2}$. In that case $\rho$ is given by a WKB integral and this
results was proven by various authors in the ten years before my result (see
[669] for references) but my work was the first rigorous result in more than
one dimension.

2\. The proof controls eigenfunction decay by writing the eigenfunctions in
terms of path integrals and using the method of large deviations to single out
a minimum action solution. It is a basic fact of classical mechanics that
minimum action is equivalent to minimum distance in a suitable metric.

3\. The importance of minimum action paths to leading order tunnelling in
multi-dimensions was noted in the theoretic physics literature several years
before my work. These solutions were called _instantons_ ; see [669] for
references to that literature.

4\. Shortly after my work, Helffer-Sjöstrand [302] developed a powerful
microlocal analysis approach to these problems and recovered the results of
Theorems 8.8 and 8.9 that got higher order terms, established some of Witten’s
conjectures and worked in a more general setting as they discussed in a number
of later papers.

5\. I wrote two later papers [670, 672] on some specialized situations related
to Theorem 8.9.

Kirsch and I [401] proved an interesting universal tunnelling bound

######  Theorem 8.10 (Kirsch-Simon [401]).

Let $V$ be a continuous function on ${\mathbb{R}}$ so that
$-\tfrac{d^{2}}{dx^{2}}+V(x)$ has discrete eigenvalues $\\{E_{j}\\}_{j=1}^{N}$
below any essential spectrum. Let $n<N+1$ and suppose for some $\alpha>0$, we
have that $V(x)\geq E_{n}+\alpha^{2}$ on ${\mathbb{R}}\setminus[a,b]$. Let
$\lambda=\max_{E\in[E_{n-1},E_{n}];x\in(a,b)}\sqrt{|E-V(x)|}$. Then

$E_{n}-E_{n-1}\geq\pi\lambda^{2}\alpha(\lambda+\alpha)^{-1}e^{-\lambda(b-a)}$
(8.39)

The last topic that I discuss in this section concerns another non-quasi-
classical situation. Just as Theorem 8.6 was motivated by a question posed to
me by Goldberger and Theorem 8.8 by Witten, this work was motivated by a query
from some theoretical, non-mathematical, physicists. In this case, I was asked
by Jeffrey Goldstone and Roman Jackiw if the two dimensional Schrödinger
operator

$H_{1}=-\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial
x^{2}}+x^{2}y^{2}$ (8.40)

has purely discrete spectrum or not. They noted that one was used to the
condition for purely discrete spectrum of $-\Delta+V$ being that
$V(x)\to\infty$ as $x\to\infty$. While this failed for $V(x,y)=x^{2}y^{2}$
since $V$ vanished on the axes, they suspected the spectrum was discrete since
it went to infinity in all but four directions. In fact, the natural quasi-
classical condition of finite phase space volume, i.e.
$|\\{(x,p)\,|\,p^{2}+V(x)\leq E\\}|<\infty$ for all $E$ also fails in this
case. A closely related question involves the operator

$H_{2}=-\Delta_{D}^{\Omega};\qquad\Omega=\\{(x,y)\in{\mathbb{R}}^{2}\,\mid\,|xy|\leq
1\\}$ (8.41)

Motivated by their question, in [667], I proved that

######  Theorem 8.11 ([667]).

The operators $H_{1}$ of (8.40) and $H_{2}$ of (8.41) both have purely
discrete spectrum.

######  Remarks.

1\. [667] gives six proofs that $H_{2}$ has purely discrete spectrum. The
simplest proves that $H_{1}$ has purely discrete spectrum (and that easily
implies that so does $H_{2}$) as follows: It follows by scaling and the fact
that $-\tfrac{d^{2}}{dq^{2}}+q^{2}$ has smallest eigenvalue $1$ that
$-\tfrac{d^{2}}{dq^{2}}+\omega^{2}q^{2}\geq|\omega|$, which implies that
$-\tfrac{\partial^{2}}{\partial x^{2}}+x^{2}y^{2}\geq|y|$. Interchanging $x$
and $y$, adding the two and multiplying by $1/2$ shows that

$H_{1}\geq\tfrac{1}{2}(-\Delta+|x|+|y|)\equiv H_{3}$ (8.42)

Since $H_{3}$ has purely discrete spectrum, by the min-max principle [711,
Theorem 3.14.5], so does $H_{1}$. The “defect” in this proof is that it turns
out that $H_{1}$ for large energies is much larger than $H_{3}$. The number of
eigenvalues of $H_{3}$ larger than $E$ grows like $E^{3}$ (by a quasi-
classical estimates) while for $H_{1}$ only like $E^{3/2}\ln E$ (as discussed
below) which is much smaller.

2\. The operator $H_{1}$ that Goldstone and Jackiw asked me to look at was a
toy model for a more involved model they were really interested in. Let
${\mathbb{A}}$ be a semi-simple Lie algebra and let $-\Delta_{A}$ be the
Laplacian in the inner product on ${\mathbb{A}}$ given by the negative of the
Killing form. For $\nu\geq 2$, let ${\mathbb{A}}^{\nu}$ be the set of
$\nu$-tuples, $(A_{1},\dots,A_{\nu})$, of elements of ${\mathbb{A}}$. Then
they were interested in the operator on $L^{2}({\mathbb{A}}^{\nu})$

$H_{4}=-\sum_{i}\Delta_{A_{i}}-\sum_{i<j}\text{\rm{Tr}}\left(\left[A_{i},A_{j}\right]^{2}\right)$
(8.43)

This had been proposed as a model of zero momentum Yang-Mills fields. In
[667], I also found one proof that this more involved operator with a
potential that stays zero on unbounded narrow sets has purely discrete
spectrum. Another model that I considered had been proposed by Feynman -
namely take three particles in three dimensions and let the interaction be the
area of the triangle whose vertices are positions of the three particles. This
potential also stays zero on an unbounded narrow set and I proved it has
purely discrete spectrum.

3\. There was much work earlier on Dirichlet Laplacians like $H_{2}$ where
$\Omega$ can have infinite volume but still have purely discrete spectrum. In
1948, Rellich [561] considered a class of Dirichlet operators that includes
$H_{2}$ and proved that they have purely discrete spectrum. In 1953, Molčanov
[500] found necessary and sufficient conditions on $\Omega$ for
$-\Delta_{D}^{\Omega}$ to have purely discrete spectrum in terms of fairly
involved conditions involving capacity, with an optimal version of Molčanov’s
result in Maz’ya-Shubin [492]. Maz’ya [491] has a review of the uses of these
kinds of capacity conditions. For results of this type with magnetic field,
see Helffer-Mohamed [300]

4\. Fefferman and Phong, using ideas described and summarized in [191], have
an illuminating picture of when the naive phase space picture of eigenvalue
counting fails. Their ideas are used in the most general result in [667] and,
in particular, in the proof that $H_{4}$ has purely discrete spectrum.

5\. I revisited the issues connected to Theorem 8.11 many years later [704].

Robert [568], Solomyak [726], Tamura [743] and I [668] also obtained results
on the eigenvalue counting asymptotics of operators like $H_{1}$ and $H_{2}$.
In one sense, one can say this is consistent with the quasi-classical
expectation in that, for example, (8.1) holds for $H_{2}$ since both sides are
infinite. But the growth in this case is not the $\text{O}(E)$ that (8.1)
gives when $|\Omega|<\infty$ but at a different rate which we think of as non-
quasi-classical. In particular, [668] proves that

$\lim_{E\to\infty}E^{-1}(\log E)^{-1}N_{H_{2}}(E)=1/\pi$ (8.44)

and

$\lim_{E\to\infty}E^{-3/2}(\log E)^{-1}N_{H_{1}}(E)=1/\pi$ (8.45)

There is also discussion in the literature of the analog of $H_{2}$ when
Dirichlet boundary conditions are replaced by Neumann boundary conditions. As
I described in Section 6 around equation (6.41), if one looks at the Neumann
Laplacian of

$\Omega=\\{(x,y)\in{\mathbb{R}}^{2}\,\mid\,x>1,|y|<f(x)\\}$ (8.46)

then if the $V$ of (6.40) goes to zero slowly, $-\Delta_{N}^{\Omega}$ has some
a.c. spectrum. Evans and Harris [187] found necessary conditions on when this
operator has purely discrete spectrum. For many such operators, Jakšić,
Molčanov and I [351] found the leading asymptotics for the number of
eigenvalues, $N_{N}^{\Omega}(E)$, of asymptotics as $E\to\infty$. In
particular, we found for the interesting case $f(x)=\exp(-x^{\alpha})$ that

$N_{N}^{\Omega}(E)\sim\left\\{\begin{array}[]{ll}\frac{1}{2}|\Omega|E,&\hbox{
if }\alpha>2\\\ \frac{1}{2}\left(|\Omega|+\frac{1}{2}\right)E,&\hbox{ if
}\alpha=2\\\ C_{\alpha}E^{1/2+1/(2(\alpha-1))},&\hbox{ if
}1<\alpha<2\end{array}\right.$ (8.47)

where

$C_{\alpha}=\frac{1}{4(\alpha-1)\sqrt{\pi}}\left(\frac{\alpha}{2}\right)^{1/(1-\alpha)}\frac{\Gamma(1/(2(\alpha-1)))}{\Gamma(3/2+1/(2(\alpha-1)))}$

For $\alpha>2$, we have a quasi-classical Weyl behavior, but for other alpha,
we have non-quasi-classical behavior.

A final remark on non-quasi-classical eigenvalue behavior. Kirsch and I [403]
(motivated again by a question from a non-mathematical physicist - in this
case, Michael Cross) found such behavior for the growth of the number of
eigenvalues below $E$ as $E\uparrow 0$ for ${-\Delta+c(1+|x|)^{-1}}$.

##  9\. Almost Periodic and Ergodic Schrödinger Operators

In AY 1980-81, I visited Caltech as a Fairchild Distinguished Scholar (I got
an offer during the year and stayed). I was looking forward to a year with no
teaching, only one postdoc (Yosi Avron, who also had a leave from Princeton)
and only one grad student (Peter Perry), a year where I expected to be able to
focus on research with few distractions. I had the impression that many of the
areas I had focused on were winding down, at least as far as my involvement.
CQFT was mainly using involved expansions and estimates, not my forte, and the
leap to four dimensional space-time which required going beyond
superrenormalizable theories seemed daunting (and still hasn’t happened!). The
hottest open question in $N$-body NRQM was asymptotic completeness and, while
I was hopeful the $N$-body Mourre estimates that I’d recently proven with
Perry and Sigal (see Section 6) would be useful, I had no plan for how to
proceed. So I suggested to Yosi that we look at moving into a new area. There
seemed to be two to consider: Schrödinger operators with almost periodic
potentials (where I was aware of some interesting non-rigorous work of Aubry
[24, 20]) and quasi-classical eigenvalue counting (where I was aware of a then
recent preprint of Helffer and Robert [301] \- a kind of multidimensional
version of Bohr-Sommerfeld quantization rules). They both seemed interesting
and promising. After thinking about it, I said to Yosi: “Let’s try to do both.
We’ll do almost periodic first - it doesn’t look very complicated or involved.
We’ll finish it up in six months and then we can turn to quasi-classical”.
Little did I realize! Almost periodic Schrödinger operators was a major focus
of my work for more than 15 years and, now, 40 years later, while there has
been remarkable progress, it is still an active area with its own separate
conferences. While, in the few years after that, I did some quasi-classical
research (see Section 8), I never worked on detailed eigenvalue locations and
related issues although it has become an active area (see, for example,
Zelditch [774] for a recent review of some aspects).

Before turning to the details of this subject, I should point out that it is
intimately connected to the subject of Section 11 (random Schrödinger
operators) and to the subject of Section 12 (singular spectrum) so some papers
may only be mentioned here and discussed in more detail in later sections.
Moreover, our formal discussion below will start with the general framework of
Schrödinger operators and Jacobi matrices with ergodic potentials which
encompasses random and almost periodic potentials as special cases. We will be
very brief in this presentation referring the reader to the relevant sections
of my book with Cycon et al. [125], the lovely review of Jitomirskaya [359]
or, for more comprehensive discussion, the books of Aizenman-Warzel [16],
Carmona-Lacroix [97], Damanik-Filman [128], Pastur-Figotin [528], or Stollmann
[733].

My initial work, much of it with Avron [46, 49, 51, 63, 121, 122, 662, 144]
was a big part of my research during the three year period 1981-1983 (which
was extremely fruitful including also my early work on TKNN integers and
Berry’s phase and their geometric significance [41, 664] (see Section 10), my
work on ultracontractivity [136] (see Section 3), my work on multiwell
problems [665, 669] and on nonclassical eigenvalue asymptotics [667, 668] (see
Section 8), my discovery of localization for slowly decaying random potentials
[660] (see Section 11), the completion and publication of my influential
review article on Schrödinger semigroups [658] (with over 1400 citations on
Google Scholar), several miscellaneous papers on NRQM with Coulomb potentials
[70, 464] as well as the preparation of my 45 hour, Bayreuth lecture course in
the summer of 1982 which turned into [125]).

I gave a review talk on the early work on almost periodic Schrödinger
operators at the 1981 Berlin ICMP [659] and the paper based on the talk became
known as the almost periodic flu paper because I started by remarking on the
fact that there seemed to be a worldwide explosion of work in this new area
that I dubbed the almost periodic flu. Indeed, besides my work in California
with Avron, there was work by Bellissard and collaborators in France (reviewed
with lots of references in [62]; notable was his use of $C^{*}$-algebra
methods), Chulaevsky [112] in Moscow and by Moser [504], Johnson-Moser [367]
and Sarnak [582] in New York (notable was the Johnson-Moser invention of
rotation number and the resulting gap labelling).

The basic framework is a probability measure space $(\Omega,\Sigma,\mu)$ with
expectation, ${\mathbb{E}}$, a distinguished bounded function,
$f:\Omega\mapsto{\mathbb{R}}$, and a distinguished group $g\mapsto T_{g}$ of
ergodic measure preserving maps indexed by the reals or the integers (see
[710, Sections 2.6-2.9] for more on the ergodic and subadditive ergodic
theorems). In the continuous case, one considers a potential
$V_{\omega}(x)=f(T_{x}(\omega))$ and ergodic Schrödinger operator
$H_{\omega}=-\tfrac{d^{2}}{dx^{2}}+V_{\omega}(x)$ acting on
$L^{2}({\mathbb{R}})$ and in the discrete case, one takes diagonal elements
$b_{n}(\omega)=f(T_{n}(\omega))$ and ergodic discrete Schrödinger operator

$(H_{\omega}u)_{n}=u_{n+1}+u_{n-1}+b_{n}(\omega)u_{n}$ (9.1)

acting on $\ell^{2}({\mathbb{Z}})$. While this is the simplest example, one
often generalizes (and we will occasionally below) in three ways: one can
allow suitable unbounded $f^{\prime}s$ (often bounded from below), one can
replace ${\mathbb{R}}$ or ${\mathbb{Z}}$ by ${\mathbb{R}}^{\nu}$ or
${\mathbb{Z}}^{\nu}$ with the multidimensional Laplacians, and, finally, one
can consider ergodic Jacobi matrices rather than only discrete Schrödinger
operators (i.e. allow ergodic $a_{n}$’s).

Two special cases are the random (discussed mainly in Section 11) and almost
periodic cases (the latter is the subject of this section after a general
discussion of some common objects). For the discrete random case,
$\Omega=[a,b]^{\mathbb{Z}}$,
$f(\\{\omega_{j}\\}_{j\in{\mathbb{Z}}})=\omega_{0},T_{1}(\\{\omega_{j}\\}_{j\in{\mathbb{Z}}})=\\{\omega_{j+1}\\}_{j\in{\mathbb{Z}}},$
and

$d\mu(\\{\omega_{j}\\}_{j\in{\mathbb{Z}}})=\otimes_{j\in{\mathbb{Z}}}d\kappa(\omega_{j})$
(9.2)

so $b_{j}(\omega)$ is a sequence of independent identically distributed random
variables (aka, iidrv). The special case where $d\kappa$ is uniform
distribution on an interval is usually called the _Anderson model_ (after [19]
for which Anderson got the Nobel prize for claiming the model had localized
states as we’ll discuss in Section 11). I’ll sometimes call the general iidrv
case the _generalized Anderson model_. Sometimes what I called the generalized
Anderson model is called just the Anderson model but it pays to have separate
names. One sometimes studies unbounded distributions or even non-independence
but demands the $b_{j}$ be “really random”, at least defined by a Markov
process. Since I never worked directly on continuum random operators, I’ll
leave the description of those models to the books mentioned above, especially
Aizenman-Warzel [16] and Stollmann [733]. I do however note that to
accommodate models with say a fixed potential localized about lattice points
in ${\mathbb{R}}^{\nu}$ with iidrv coupling constants, one needs to modify the
set up to only require ergodicity under a discrete group even for
${\mathbb{R}}^{\nu}$ models.

The other case is almost periodic functions. (For more on the general theory
of almost periodic functions, see, e.g. [711, Section 6.6]). In this case,
$\Omega$ is a separable, compact, abelian group, called the _hull_ , there is
a homomorphism $S:G\to\Omega$ (with $G={\mathbb{R}}$ or ${\mathbb{Z}}$) so
that $T_{g}(\omega)=\omega S(g)$ and $f:\Omega\to{\mathbb{R}}$. Two important
special cases are where $S(G)$, the image, is a winding line on a finite
dimensional torus, $\Omega$, viewed as a product of copies of
$\partial{\mathbb{D}}$ with complex product as the group product, in which
case the potential is called _quasiperiodic_ , and the case where the
potential is a uniform limit of periodic functions of longer and longer
commensurate periods (e.g. $V(x)=\sum_{n=1}^{\infty}2^{-n}\cos(x/2^{n})$), in
which case the potential is called _limit periodic_. The most famous example
is

$(H_{\alpha,\lambda,\theta}u)_{n}=u_{n+1}+u_{n-1}+\lambda\cos(\pi\alpha
n+\theta)u_{n}$ (9.3)

the _almost Mathieu operator_ (henceforth AMO). In much recent literature,
what I call $\lambda$ is called $2\lambda$ (so the self dual point is
$\lambda=1$) but I’ll follow the convention of the older literature. I like to
joke that there have been more papers in the _Annals of Mathematics_ about the
AMO than about any other single mathematical object. In the physics
literature, this is called Harper’s equation when $\lambda=2$ and arose as a
tight binding approximation to a 2D electron in magnetic field ($\alpha$ is
then the magnetic flux per unit cell). The name almost Mathieu equation is one
I introduced in [51] and [659]. I took it from the fact that the differential
equation

$-\frac{d^{2}u}{dx^{2}}+\lambda\cos(x)u(x)=Eu(x)$ (9.4)

is called the Mathieu equation (with Avron [47], I had then recently studied
the asymptotics of its gap widths as $E\to\infty$). My name is a joke based on
the fact that (9.3) is almost (9.4) and is also only almost periodic if
$\alpha$ is irrational (while (9.4) is periodic).

Two basic objects associated to one dimensional ergodic operators are the
density of states (DOS) and the Lyapunov exponent. The DOS, unlike the
Lyapunov exponent, makes sense in higher dimensions, but, for simplicity, let
us mainly focus on the one dimensional discrete case. For each $\omega$,
$H_{\omega}$ defines a self-adjoint operator on $\ell^{2}({\mathbb{Z}})$, and
so defines a spectral measure, $d\mu_{\omega}(E)$, defined by

$\int f(E)d\mu_{\omega}(E)=\langle\delta_{0},f(H)\delta_{0}\rangle$ (9.5)

The DOS measure $dk(E)$ is defined by

$dk={\mathbb{E}}(d\mu_{\omega})$ (9.6)

The integrated density of states (_IDS_) is then defined by

$k(E)=dk((-\infty,E))$ (9.7)

If $\chi_{L}$ is the characteristic function of $\\{n\,\mid\,-L\leq n\leq
L\\}$, then translation covariance shows that if $P_{B}(H_{\omega})$ is the
spectral projection for a set, $B$, then

$(2L+1)^{-1}{\mathbb{E}}(\text{\rm{Tr}}(\chi_{L}P_{B}(H_{\omega})\chi_{L}))=\int_{B}dk(E)$
(9.8)

This together with translation covariance and the Birkhoff ergodic theorem
[710, Theorem 2.6.9] imply that for a.e. $\omega$ and all continuous functions
$f$ on ${\mathbb{R}}$ of compact support one has that

$\lim_{L\to\infty}(2L+1)^{-1}\text{\rm{Tr}}(\chi_{L}f(H_{\omega})\chi_{L}))=\int
f(E)dk(E)$ (9.9)

A simple argument (e.g. restricting to moments) then shows that $dk$ is also
the limit of the eigenvalue density of $H_{\omega}$ restricted to large boxes
with either periodic or Dirichlet boundary conditions.

The earliest mathematical work on the DOS was by Benderskiĭ-Pastur [67] who
defined it in the random case as a limit of box eigenvalue counting. See [125,
pg. 175] for additional references on work on the random case prior to Avron-
Simon [51] whose principal theme was the DOS for the almost periodic case
(and, as we’ll discuss below, the Thouless formula, Aubry duality and spectral
properties of the AMO). We introduced the definition via (9.6) and the formula
(9.9) which we proved held for every $\omega$ in the hull, rather than just
almost every $\omega$. We also proved the equality to the definition via (9.6)
to the definition via periodic or Dirichlet boundary conditions and also (up
to a factor of $\pi$) to the then recently defined _rotation number_ of
Johnson-Moser [367]. This equality under boundary conditions was natural given
the statistical mechanical analogy and the proof was not hard. Twenty-five
years later, its analog turned out to be very useful to prove a result [705]
in the theory of orthogonal polynomials that surprised many experts in that
subject.

We also proved that for any $\omega$, the spectrum of $H_{\omega}$ is equal to
the support of the measure $dk$ and, in particular,
$\text{\rm{spec}}(H_{\omega})$ is the same for all $\omega$ in the hull. We
also proved that $k$ was continuous in $E$ (and in $\alpha$ at irrational
values of $\alpha$ \- it is definitely _not_ continuous at rational $\alpha$!)
in 1D and noted the importance of continuity since that implies that for every
$E$ one has that (here $H_{\omega}^{L,D}$ is the restriction of the
Hamiltonian, $H_{\omega}$, to the box $\\{n\,\mid\,-L\leq n\leq L\\}$ with
Dirichlet boundary conditions which one could replace by periodic boundary
conditions).

$k(E)=\lim_{L\to\infty}(2L+1)^{-1}\\#\\{\text{eigenvalues of
}H_{\omega}^{L,D}\leq E\\}$ (9.10)

Our proof of the continuity of $k$ in 1D depended on the fact that in 1D
eigenvalues have multiplicity at most $1$ (all that mattered was the
finiteness) so we suggested, but couldn’t prove, that $k$ was continuous in
general dimension. Moreover, the proof only showed that
$\lim_{\varepsilon\downarrow 0}[k(E+\varepsilon)-k(E-\varepsilon)]=0$ (which
implies continuity for monotone functions) with nothing about how small
differences are. Later, Craig and I [121] proved log Hölder continuity (using
subharmonic function methods introduced in the subject by Herman [313]) and
then extended this to any dimension [122] to get the continuity in any
dimension that [51] had conjectured. Since the proofs of [121, 122] use the
Thouless formula, I’ll discuss it below. I note that shortly after us, Delyon-
Souillard [160] found a distinct, really short, proof of the multidimensional
continuity (but not log Hölder continuity).

Before leaving the DOS, I should mention gap labelling, not because I
contributed to it, but because it will be relevant to the discussion of Cantor
spectrum and the ten martini problem. If $H$ is a discrete Schrödinger
operator of period $L$, the usual Bloch wave analysis shows the spectrum can
have up to $L-1$ gaps and that if there is such a gap, the IDS value in it
(which has to be constant) is an integral multiple of $1/L$. Johnson-Moser
[367] (once one has the equality of their rotation number and the IDS) found
an analog for the 1D almost periodic continuum case (their method was extended
to the discrete case by Delyon-Souillard [159]). Independently, Bellissard
[59], proved the same result using $C^{*}$-algebra methods, eventually using
the same idea for certain higher dimensional operators [61]. Bellisard’s work
caused Johnson [366] to find another approach to gap labelling. For any almost
periodic function, $f$, its average,
${\mathbb{A}}(f)=\lim_{R\to\infty}(2R)^{-1}\int_{-R}^{R}f(x)\,dx$ is easily
shown to exist. Given a real, $\omega$, $f$ is said to have a non-zero Fourier
coefficient at $\omega$ if and only if ${\mathbb{A}}(e^{-2\pi ix\omega}f)\neq
0$. Since in the case that $x\in{\mathbb{Z}}$, these Fourier coefficients only
depend on the fractional part of $\omega$, we view $\omega$ as an element of
${\mathbb{R}}/{\mathbb{Z}}$ which we write as $[0,1)$ and add mod
${\mathbb{Z}}$. One can prove that the set of $\omega$ with non-zero
coefficient is a countable set. The set of reals which are finite sums and
differences of those $\omega$ with non-zero coefficients is called the
_frequency module of $f$_. It is easy to see that $f$ is quasiperiodic if and
only if the frequency module is finitely generated. Moreover, unless the
potentials are periodic, the frequency module is dense in $[0,1)$ or
$[0,\infty)$.

_Gap labelling_ is the assertion that in any gap of the spectrum, the value of
the IDS is a number in the frequency module. What this says in case $V$ or
$a,b$ are periodic, where the frequency module is multiples of $1/p$ (with $p$
the “true” period), is that the constant value of the IDS in a gap is among
$j/p,j=1,\dots$ (where in the discrete case $j$ runs through $p-1$). We speak
of a gap being _open_ for a given $j$ if there is such an interval of
constancy and _closed_ if there is a single energy with $k(E)=j/p$. Earlier in
1976, I had proven in [635] that in the continuum case for a dense
$G_{\delta}$ of $V$’s of period one, all allowed gaps are open. For the
discrete case there is a much more precise analysis [169, 493] that shows the
set of period $p$ Jacobi parameters with at least one gap closed is a finite
union of closed varieties with codimension $2$ so the set with all gaps open
is an open set that is much more than generic. A little thought shows that if
all allowed gaps are open in the almost periodic case, the set of gaps is
dense so that the spectrum of $H$ is a Cantor set (i.e. a closed, perfect,
nowhere dense set). Of course, it can be a Cantor set even if only many,
rather than all, gaps are open.

Avron and I were struck by this Cantor spectrum. Just as we were doing this
work, pictures appeared from the Voyager flyby of Saturn which showed many
more gaps in that planet’s rings than previously known, so many that it almost
appeared that the rings were nowhere dense! We wrote a speculative paper [48]
suggesting the structure might be due to an almost periodic Hill equation,
although we pointed out that naive perturbation estimates of gap size were too
small by several orders of magnitude so there would need to be some then not
understood phenomena increasing this gap size. Alas, there does not seem to be
such a phenomenon and nature chose a different mechanism.

At this time, Avron and I [49], Chulaevskiĭ [112], Moser [504] and Pastur-
Tkachenko [529] independently found classes of limit periodic Schrödinger
operators with Cantor spectrum. We also proved the spectrum remained purely
absolutely continuous so the spectrum was a positive measure Cantor set. (Much
more is now known about limit-periodic Schrödinger operators - a.c. spectrum
is not typical; see Damanik-Gan [129].) We also discovered that such
absolutely continuous spectrum still had all states with slow decay leading us
to develop a refinement of a.c. spectrum [46] .

Mark Kac had moved to USC about the time I was visiting Caltech and at lunch
one day in 1981, he and I discussed Cantor spectrum and the AMO. We agreed
that it was an interesting conjecture to prove that the operator
$H_{\lambda,\alpha,\theta}$ of (9.3) had a Cantor spectrum for all irrational
$\alpha$ and $\lambda\neq 0$ (if $\alpha$ is irrational, it is known (Avron-
Simon [51]) that the spectrum is $\theta$ independent). “That’s a grand
conjecture”, said Mark, “I’ll offer ten martinis for its solution.” He later
repeated this offer at an AMS meeting and I popularized it as the ten martini
problem. Added to the interest was the famous _Hofstadter butterfly_ [319,
320], a picture (see Figure 2) showing the spectrum at the critical value
$\lambda=2$ of the spectrum as a function of $\alpha$ (computed numerically
for $\alpha=p/q$ with $q$ not too large) which looks like a fractal. The ten
martini problem was solved in full in 2004 by Avila-Jitomirskaya [26]
(mentioned in Avila’s Fields Medal citation) after an important partial result
by Puig [544]. This is weaker than the result that all gaps are open,
something known as the dry form of the ten martini problem (still partially
open).

Figure 2. The Hofstadter Butterfly

A year after my lunch with Kac, Bellissard and I [63] used the strategy of my
periodic result [635]. We first proved that if $\alpha=p/q$ is rational and
$q\theta$ is not a multiple of $\pi$, then all gaps were open (i.e. the
spectrum has $q-1$ gaps). This non-trivial analytic fact was proven using
ideas motivated by the classical result of Ince [341] that the continuum
Mathieu operator (9.4) has all gaps open. Once we knew that, with further
analysis and some continuity results on $k(E)$ from Avron-Simon [51], the
magic of the Baire category theorem showed that for a Baire generic set of
$(\alpha,\lambda)$ the spectrum is a Cantor set! It is remarkable that with
one’s Baire hands one can learn something about the irrational case (Cantor
spectrum) by knowing something about the rational case even though, of course,
in the rational case, the spectrum is never Cantor.

When I told Mark about this on the phone admitting it wasn’t the full result,
he remarked “But it is still interesting! I’ll give you three martinis for
it.” So I always think of this as the three Martini result. Alas, before we
met again, Mark was dead of abdominal cancer (the same disease that felled the
other half of the Feynman-Kac formula not too long afterwards).

Returning to my basic series with Avron, I need to define the Lyapunov
exponent. I’ll do it first for the discrete case. Given a pair of potential
Jacobi parameters, $a>0,b\in{\mathbb{R}}$ and $z\in{\mathbb{C}}$, one defines
the single step transfer matrix:

$A(a,b;z)=\frac{1}{a}\left(\begin{array}[]{cc}z-b&-1\\\ a^{2}&0\\\
\end{array}\right)$ (9.11)

so the difference equation

$a_{n}u_{n+1}+b_{n}u_{n}+a_{n-1}u_{n-1}=zu_{n}$ (9.12)

is equivalent to $\left(\begin{array}[]{c}u_{n+1}\\\ a_{n}u_{n}\\\
\end{array}\right)=A(a_{n},b_{n};z)\left(\begin{array}[]{c}u_{n}\\\
a_{n-1}u_{n-1}\\\ \end{array}\right)$. I learned the trick of putting a factor
of $a$ in the lower component which yields an $A$ with
$\operatorname{det}(A)=1$ from Killip in about 2000 and it didn’t appear in
the earlier papers.

One defines the _transfer matrix_

$T_{n}(\\{a_{j},b_{j}\\}_{j=1}^{n};z)=A(a_{n},b_{n};z)A(a_{n-1},b_{n-1};z)\ldots
A(a_{1},b_{1};z)$ (9.13)

We use $T_{n}(z;\omega)$ for the transfer matrix with
$a_{n}(\omega),b_{n}(\omega)$. The Furstenberg-Kesten theorem [710, Theorem
2.9.1 ] then implies that for each fixed $z$, for a.e. $\omega$, one has that
the Lyapunov exponent

$\gamma(z)=\lim_{n\to\infty}n^{-1}\log\left(\lVert
T_{n}(z;\omega)\rVert\right)$ (9.14)

exists and is a.e. $\omega$ independent. More can be proven: The
multiplicative ergodic theorem [710, Theorem 2.9.10] says that for each $z$
and a.e. $\omega$, not only does (9.14) hold but there is a one dimensional
subspace $V_{z;\omega}\subset{\mathbb{C}}^{2}$ so that

$\lim_{n\to\infty}n^{-1}\log\left(\lVert
T_{n}(z;\omega)v\rVert\right)=\left\\{\begin{array}[]{ll}-\gamma(z),&\hbox{ if
}v\in V\setminus\\{0\\}\\\ \gamma(z),&\hbox{ if }v\in{\mathbb{C}}^{2}\setminus
V\end{array}\right.$ (9.15)

Thus for such $z,\omega$, if $\gamma>0$, then all solutions of (9.12) on a
half line either grow or decay exponentially. We emphasize that the need for
a.e. $\omega$ rather than every is not a mere technicality but, as we will see
(in the discussion two paragraphs prior to (9.24)), can have dramatic
consequences.

An important role is played by what is called the _Thouless formula_ :

$\gamma(E)=\int\log|E-E^{\prime}|dk(E)$ (9.16)

which relates the Lyapunov exponent, $\gamma$, to the IDS, $k(E)$ in the
discrete case. This is the form for the discrete Schrödinger case where
$a_{n}\equiv 1$; in general, one has an extra term
${\mathbb{E}}(-\log(a(\omega))$ with the added condition that this expectation
is finite. It has the name because of the 1974 work of Thouless [749] although
it appeared earlier in the physics literature in a paper of Herbert and Jones
[307]. In fact, closely related ideas, although not the exact formula, go back
to Szegő in 1924 [741] who realized an important connection to two dimensional
potential theory (for discussion of the basics of potential theory, see [304,
490, 549] or [710, Sections 3.6-3.7]) since the right side of (9.12) is the
(negative of the) logarithmic potential of $dk$. I was not aware of this
related work from the OP community in the 1980’s but only many years later, at
which point I wrote a summary article [702] that explained the use of
potential theory ideas to spectral theorists and the opposite direction to the
OP community as well as some new insights.

Thouless’ basic idea is that the elements of the transfer matrix when all
$a_{n}\equiv 1$ are monic polynomials, $P_{n}(z)$, whose zeros are the
eigenvalues of the Hamiltonian in a box. Since
$\log(P_{n}(z))=\sum_{j=1}^{n}\log|z-E_{j}|$, where the sum is over the
eigenvalues, (9.16) then follows from the fact that $dk$ is the limit of the
density of eigenvalues in a box. Avron and I realized this argument worked
flawlessly when $z$ lay outside the (convex hull) of the spectrum of $H$, but
because of infinities in the log was problematic for $z$ on the real axis.
Indeed, we noted in the almost periodic case for $z$ off the real axis, it
held for all $\omega$ rather than just a.e. $\omega$. In [51], we were able to
use the fact that the integral on the right side of (9.16) is the Hilbert
transform of $k(E)$ and the $L^{2}$ continuity of Hilbert transform to prove
that (9.16) holds for Lebesgue a.e. $E$ in ${\mathbb{R}}$ and this suffices
for some applications we made.

Slightly later, Craig and I [121] were able to prove (9.16) for all
$E\in{\mathbb{R}}$. The key was the observation of Herman [313] that the limit
in (9.14) was subharmonic. Since the integral on the right side of (9.16) is
also subharmonic and, by Thouless’ argument, the equation holds for $z$ non-
real, it holds for all $z$ by a regularity result on subharmonic functions.
Craig and I realized that in general for fixed $\omega$, the quantity
$\limsup_{n\to\infty}n^{-1}\lVert T_{n}(z;\omega)\rVert$ might not be upper
semicontinuous which implied that this $\limsup$ might only have the right
side of (9.16) as an upper bound. We also realized that since $\gamma(E)\geq
0$, the measure $dk$ in (9.16) can’t give too great weight to small sets which
implied the log Hölder continuity. By looking at averages of positive Lyapunov
exponents on strips, we could even extend the continuity result to higher
dimension.

Avron-Simon [51] also began the study of a fascinating subject, the possible
$\omega$ dependence of spectral components. Recall [711, Theorem 5.1.12] that
one can refine the spectrum into pure point, a.c. and s.c. pieces. It is a
theorem of Kunz-Souillard [426] that these spectral pieces are a.e. constant
in the general ergodic case (one might think this is obvious by the ergodic
theorem but the subtlety is proving the measurability in $\omega$ of the
projections onto various spectral pieces). As mentioned above, Avron and I
proved a.e. could be replaced by all in the almost periodic case for the
spectrum but as we’ll see shortly, that is not true for two of the three
spectral components.

For the AMO, (9.3), there are interesting dependencies of spectral types on
the coupling constant and frequency. A key aspect is what is called Aubry
duality [24, 20]. Formally, the Fourier transform maps $H_{\alpha,\lambda}$ to
$\tfrac{\lambda}{2}H_{\alpha,4/\lambda}$ since it maps the finite difference
operator to multiplication by $\cos$ and turns multiplication by $\cos$ into a
finite difference operator. Of course, this can only be formal since Fourier
transform in $\ell^{2}({\mathbb{Z}})$ maps not to itself but to
$L^{2}(\partial{\mathbb{D}},d\theta/2\pi)$! One version of _Aubry duality_
says that the IDS, $k(\alpha,\lambda;E)$, of $H_{\alpha,\lambda}$ obeys

$k(\alpha,\lambda;E)=k(\alpha,4/\lambda;2E/\lambda)$ (9.17)

One way of understanding the dual relation is to view the direct integral of
$H_{\alpha,\lambda,\theta}$ over $\theta$ as an operator on
$L^{2}({\mathbb{R}})$ and apply the appropriate Fourier transform.
Alternatively, following [20], one looks at $\alpha=p/q$ with $\theta=0$ on a
set with $q$ points on which finite Fourier transform maps
$\ell^{2}({\mathbb{Z}}_{q})$ to itself. One obtains (9.17) by approximating
irrational $\alpha$ by rationals. Aubry-André’s argument [20] for the limit
was formal; Avron and I [51] proved the necessary continuity (which only holds
at $\alpha$ irrational!) to get the first rigorous proof of (9.17). Two
immediate consequences of Aubry duality are (here $\alpha$ is irrational)

$\displaystyle\text{\rm{spec}}(H_{\alpha,\lambda})$
$\displaystyle=\frac{\lambda}{2}\text{\rm{spec}}(H_{\alpha,4/\lambda})$ (9.18)
$\displaystyle\gamma(\lambda,\alpha;E)$
$\displaystyle=\gamma(4/\lambda,\alpha;2E/\lambda)+\log(\lambda/2)$ (9.19)

Aubry-André [20] have a number of conjectures about the almost Mathieu
equation which I (and others) made some progress on in my work. The first
involves the conjecture about the Lebesgue measure,
$|\text{\rm{spec}}(H_{\alpha,\lambda})|$, of the spectrum

$|\text{\rm{spec}}(H_{\alpha,\lambda})|=2|2-|\lambda||$ (9.20)

based on numeric calculations. We note this implies zero Lebesgue measure when
$\lambda=2$ (which has led to a lot of literature on what the Hausdorff
measure is of the set in that case; [361] has a summary of some of that
literature as well as new results), an earlier conjecture of Hofstadter [319,
320].

In [53], Avron, von Mouche and I attempted to prove (9.20) and proved the
equality for rational $\alpha$ if the left side is the Lebesgue measure of the
intersection over $\theta$ of $\text{\rm{spec}}(H_{\alpha,\lambda,\theta})$
and proved the convergence of the Lebesgue measure of the union over $\theta$
as any sequence of rationals approaches an irrational $\alpha$. This implies
that the left side of (9.20) is $\geq$ the right side for $\alpha$ irrational.
The techniques of [53] have been used in many later works studying this
problem. For $\alpha$ whose continued fraction expansions are not bounded,
Last [437, 438] proved the complete (9.20) for all $\lambda$. The set of
$\alpha$ with bounded integers in their continued fraction expansion is easily
seen to have Lebesgue measure zero and to be a nowhere dense $F_{\sigma}$ so
Last’s result covers “most” irrationals but not all and, in particular, it
does not cover the golden mean which has been used in many numeric
calculations. The result is now known for all irrational $\alpha$ through a
series of papers. The history is reviewed in Jitomirskaya-Krasovsky [361]
which has a simple proof of the general result.

The other conjecture in [20] concerns spectral types. One starts with (9.19)
and the fact that always $\gamma\geq 0$ which implies that for $\lambda>2$

$\gamma(E)\geq\log(\lambda/2)$ (9.21)

(proven rigorously by Avron-Simon [51] and Herman [313]) which later was
proven to hold with equality on the spectrum by Bourgain-Jitomirskaya [82]. By
a result of Pastur [527] and Ishii [343], strict positivity of the Lyapunov
exponent implies that the spectrum there has no a.c. component so [20]
suggested that when $\lambda>2$, the spectrum is pure point. The Fourier
transform of a rapidly decaying eigenfunction looks like a plane wave so their
conjecture on spectral type was a.c. spectrum when $0<\lambda<2$ and pure
point spectrum when $\lambda>2$. They realized that $\lambda=2$ would be
subtle and suggested perhaps there would be eigenfunctions with power law
decay.

When Avron and I started thinking about the issue of spectral type for the
AMO, Peter Sarnak, then a graduate student, suggested to me that spectral
properties might depend on the Diophantine properties of the irrational
frequencies (see also [582]), that is how well those irrationals are
approximated by rationals. The _Liouville numbers_ are those irrationals
$\alpha$ for which there exist rationals $p_{k}/q_{k}$ with

$\left|\alpha-\frac{p_{k}}{q_{k}}\right|\leq k^{-q_{k}}$ (9.22)

while we say that $(\alpha_{1},\dots,\alpha_{\ell})$ have _typical Diophantine
properties_ if there exist $C$ and $k$ so that for all integers, not all zero,
we have that

$\min_{m\in{\mathbb{Z}}}\left|m-\sum_{j=1}^{\ell}n_{j}\alpha_{j}\right|\geq
C(n_{1}^{2}+\dots+n_{\ell}^{2})^{-k}$ (9.23)

We say that $\alpha$ is _Diophantine_ if (9.23) holds with $\ell=1$ and
$\alpha_{1}=\alpha$. It is well known that the Diophantine rationals in
$[0,1]$ have full Lebesgue measure while the disjoint set of Liouville numbers
is a dense $G_{\delta}$ (providing an interesting demonstration that the two
notions of generic are distinct). Avron and I decided that the picture of
Aubry-André was likely wrong when $\alpha$ was a Liouville number. I visited
Moscow in the spring of 1981 and explained our expectation and Molchanov came
up to me with a young mathematician, Sasha Gordon, who had shown [261] that if
a potential, $V$, is well enough approximated by periodic potentials, then
$-\tfrac{d^{2}}{dx^{2}}+V$ on $L^{2}({\mathbb{R}},dx)$ has no square
integrable eigenfunctions. Avron and I found the easy extension to the
discrete case and used it and the Pastur-Ishii result to prove [50, 51] that
when $\alpha$ is a Liouville number and $\lambda>2$, then the AMO,
$H_{\alpha,\lambda,\theta}$, has purely singular continuous spectrum for all
$\theta$. This was only one of the times that Gordon had a significant impact
on my work - the other most significant one was his impact on my work on
generic singular continuous spectrum (see Section 12). We also had two joint
papers [264, 265]. I think Gordon, who was a very inventive mathematician,
never got the recognition that he deserved and I felt guilty that I might have
benefitted from his brilliance as much or more than he did!

The result with Avron on examples with purely singular spectrum and $\gamma>0$
on the spectrum shows the subtlety of tracking measure zero sets. By Fubini’s
theorem and the multiplicative ergodic theorem, one concludes that for a.e.
$\theta$, one has that for a.e. $E\in{\mathbb{R}}$, every solution of the
difference equation $H_{\alpha,\lambda,\theta}u=Eu$ either decays or grows
exponentially at both $\pm\infty$. Gordon’s lemma implies that in the
Liouville case, the solutions decaying in one direction can’t in the other so
the spectral measures of $H_{\alpha,\lambda,\theta}$ live on the sets where
Lyapunov behavior fails to hold. The moral is that the a.e. set of $\theta$
where (9.14) might not hold, not only exists but can be where all the
important stuff is happening!

For Diophantine $\alpha$, with $\lambda>2$, the Aubry-André conjecture was
proven by Jitomirskaya [358] who proved for such values of the parameters, one
has dense point spectrum of $H_{\alpha,\lambda,\theta}$ for Lebesgue a.e.
$\theta$. It is not a limitation that the proof is only for a.e. $\theta$
since the spectrum is purely singular continuous for a dense $G_{\delta}$ of
$\theta$ (see the discussion of my work with Jitomirskaya [365] in the next
paragraph). That leaves $\alpha$ which is neither a Liouville number nor
Diophantine, a non-empty, uncountable, set of irrationals that is both of
Lebesgue measure zero and a subset of a nowhere dense $F_{\sigma}$ (so, in a
sense, rare). For such $\alpha$ one looks at the continued fraction
approximations $p_{n}/q_{n}$ which are, the best rational approximations [709,
Section 7.5], and defines

$\beta(\alpha)\equiv\limsup_{n\to\infty}\left(\frac{\log
q_{n+1}}{q_{n}}\right)$ (9.24)

(This measure of approximation in the context of almost periodic Schrödinger
operators goes back to my paper on the Maryland model [671] in an equivalent
form $\beta(\alpha)\equiv\limsup_{n\to\infty}-n^{-1}\log(|\sin(\pi\alpha
n)|)$. Diophantine $\alpha$ have $\beta=0$ and Liouville $\alpha$ have
$\beta=\infty$. Avila-You-Zhou [29] (see also Jitomirskaya-Liu [363]) proved a
conjecture from [357] that if $2<\lambda<2e^{\beta(\alpha)}$ the spectrum of
$H_{\alpha,\lambda,\theta}$ is purely singular continuous for all $\theta$ and
if $\lambda>2e^{\beta(\alpha)}$, then the spectrum of
$H_{\alpha,\lambda,\theta}$ is dense pure point for a.e. $\theta$. See
Jitomirskaya-Liu [362] for more on this case including a review of the
literature and a detailed analysis of the eigenfunctions.

One of the results of the singular continuous revolution that I’ll discuss in
Section 12 is that Jitomirskaya and I [365] proved that if $a_{n}(\omega)=1$
and $b_{n}(\omega)$ is an even almost periodic function, then for a dense
$G_{\delta}$ of $\omega$ in the hull, $h_{\omega}$ has no eigenvalues. If it
is a model where $\gamma>0$ on the spectrum so that the Pastur-Ishii theorem
implies no a.c. spectrum, this yields purely s.c. spectrum for a Baire generic
set of $\omega$. Since there are models (like AMO for $\lambda>2$ and $\alpha$
Diophantine) where it is known there is dense point spectrum for a.e.
$\omega$, we see there are examples where neither the point spectrum nor the
s.c. spectrum are $\omega$ independent even though we know that they are a.e.
$\omega$ independent. However, independently, Kotani [419] and Last and I
[440] showed that a.c. spectrum is always $\omega$ independent.

I later wrote a paper with Hof and Knill [317] in which we proved, using a
relative of the ideas in [365] that certain weakly almost periodic potentials
taking only finitely many values (which are known to have no a.c. spectrum
[418]) have purely s.c. spectrum for a dense $G_{\delta}$ set in their hull.

Next, I turn to AMO when $0<\lambda<2$. The results on point spectrum with
exponentially decaying eigenfunctions for $\alpha$ Diophantine and $\lambda>2$
plus Aubry duality immediately imply lots of a.c. spectrum when $\alpha$ is
Diophantine and $\lambda<2.$ For several years after [51], it was assumed that
the dual of singular continuous spectrum must be purely singular continuous,
so it came as a big surprise when Last [437] proved that for _all_ irrational
$\alpha$ one has that $|\sigma_{ac}(H_{\alpha,\lambda,\theta})|\geq
4-2|\lambda|$. At the time, Last was a graduate student of Avron at Technion
and Avron told me of Last’s result. I assured Avron that I was sure Last was
wrong. Since I would be coming soon to Israel, rather than plow through the
paper and figure out the error, I suggested we meet so I could determine where
the error was and tell him! To my surprise, Last convinced me that his proof
was correct and that he could prove a kind of lower semicontinuity on
$|\sigma_{ac}(H)|$ and then use the fact that Avron, van Mouche and I [53] had
proven the inequality for rational $\alpha$. Once the blinders were removed, I
realized that my work then in progress with Gesztesy [237] provided a new
proof of Last’s result! Indeed, we could slightly improve his result since
where he had $|\sigma_{ac}|$, we could obtain the potentially smaller
$|\Sigma_{ac}|$ ($\Sigma_{ac}$ is the essential support of the ac spectrum,
that is the minimal class of sets mod sets of Lebesgue measure $0$ that
supports all the a.c. spectral measures). We proved that

######  Theorem 9.1 (Gesztesy-Simon [237]).

Let $H^{[n]}$ be a sequence of periodic discrete Schrödinger operators so that
for each fixed $m$, $b^{[n]}_{m}$ has limit $b_{m}$ and let $H$ be the
discrete Schrödinger operator with potential $b$. Then for any open interval
$(\alpha,\beta)\subset{\mathbb{R}}$, we have that

$|(\alpha,\beta)\cap\Sigma_{ac}(H)|\geq\limsup_{n\to\infty}|(\alpha,\beta)\cap\Sigma_{ac}(H^{[n]})|$
(9.25)

For AMO with $|\lambda|<2$, this leaves the question of the point and singular
continuous spectra which were expected to be empty and proving that first for
a.e. $\theta$ and then for all $\theta$ was open for many years; indeed,
proving this for non-Diophantine $\alpha$ (Jitomirskaya [358] had handled the
Diophantine case for a.e. $\theta$) was one of the list of problems I sent to
the 2000 ICMP [690]. The full result was settled by Avila [25] (It is most
unfortunate that this paper has never appeared. It seems the blame is shared
by the top journal that rejected it and by the author who then refused to send
it elsewhere).

Finally, in our discussion of AMO, I mention the self-dual point $|\lambda|=2$
which is often quite subtle. In [265], Gordon, Jitomirskaya, Last and I
claimed that if the spectrum of $H_{\alpha,\lambda,\theta}$ has zero measure
for $\lambda=2$ and some irrational $\alpha$ (the spectrum is $\theta$
independent), then for a.e. $\theta$ the spectrum is purely singular
continuous. At the time the zero measure result was known for most, but not
all, irrational $\alpha$ but, as just mentioned, it is now known for all
irrational $\alpha$. As explained in the discussion under (2) below, [265] was
fooled by a sloppily stated result in [144] so there was a gap in the proof
and the result was only established in Avila et al. [27]. That left the
question of whether there might be an exceptional set with some (or even all)
point spectrum. Very recently, Jitomirskaya [360] proved there are no point
eigenvalues for any $\alpha$ and any $\theta$. This paper includes a
discussion of earlier work between the 1997 paper of Gordon et al [265] and
her 2020 breakthrough.

That concludes our survey of the refined spectral analysis of AMO and I
conclude this section with a summary of some of my other papers on almost
periodic operators.

(1) _Kotani Theory_. In 1982, I received a brilliant paper by S. Kotani [416]
which dealt with ergodic continuum 1D Schrödinger operators. Since it dealt
with a.c. spectrum, it was mainly of interest for the almost periodic case. It
had three main results

(a) A kind of converse of the Pastur-Ishii theorem, namely, if $\gamma(E)=0$
on a Borel subset $A\subset{\mathbb{R}}$ of positive Lebesgue measure, then
for a.e. $\omega$, $H_{\omega}$ has a.c. spectrum on $A$.

(b) If $\gamma(E)=0$ on an open interval $I\subset{\mathbb{R}}$, then the
spectrum is purely a.c. on $I$.

(c) If $\gamma(E)=0$ on a Borel subset $A\subset{\mathbb{R}}$ of positive
Lebesgue measure, then, the process ${\mathbf{x}}\mapsto V_{\omega}(x)$ is
deterministic which means there is no a.c. spectrum in “truly random” cases.

I discovered it was not so straightforward to extend this to the case of
discrete Schrödinger operators but I succeeded in [662] which was used by many
later authors. In [421], Kotani and I joined forces to extend these results to
discrete strips.

(2) _Deift-Simon Theory_. I wrote a paper with Deift [144] that focused on
aspects of a.c. spectrum motivated by Kotani [416] and some results of Moser
[504] who had proven that the rotation number, $\alpha(E)=\pi k(E)$, obeys

$d\alpha^{2}(E)/dE\geq 1$ (9.26)

on the spectrum of periodic continuum Schrödinger operators and also for the
particular limit periodic potentials he studied in [504]. In [144], we noted
that (9.26) could not hold in general for all ergodic Schrödinger operators,
essentially because of the phenomena of Lifshitz tails (see Section 11) but we
proved in the continuum case, it holds on the set where $\gamma(E)=0$ and for
the discrete Schrödinger operator (i.e. Jacobi matrix with $a\equiv 1$) one
has the stronger

$2\sin(\alpha)d\alpha(E)/dE\geq 1$ (9.27)

on $A\equiv\\{E\,|\,\gamma(E)=0\\}$ which implies that $|A|\leq 4$. We also
proved that the a.c. spectrum has multiplicity 2. While we regarded these as
the most important results in the paper (as seen by our abstract), this paper
is probably best known for two more technical aspects. First we construct
$L^{2}$ (in $\omega$) eigenfunctions for energies in the a.c. spectrum, which,
for example, plays a critical role in the work in (4) below. Secondly, there
is a claim in [265] that our results imply mutual singularity of the singular
parts of the spectral measures for a.e. pair $(\omega,\omega^{\prime})$ based
on a theorem in [144] that for every real $E$, a certain set of $\omega$
associated to the singular spectrum has measure zero. Unfortunately, the
theorem in [144] is sloppily stated in that in the section it appears, there
is an implicit condition that $\gamma>0$ which the authors of [265] forgot
(despite the fact that I was a coauthor of both - blush!). Since [265] apply
this at the self-dual $\lambda$ where $\gamma=0$, that paper has one incorrect
claim. I note that the claimed mutually singularity when $\gamma=0$ is still
open although many expect that it is true.

(3) _The Maryland Model_ Dick Prange and two of his postdocs at Maryland found
and studied [195, 276, 196, 543] a fascinating exactly solvable almost
periodic model which I dubbed “the Maryland model”, a name that has stuck in
later literature. I wrote two papers [663, 671] on rigorous aspects of the
model which had some overlap with another independent rigorous analysis by
Figotin-Pastur [194]. The model is just like AMO but $\cos$ is replaced by
$\tan$, i.e.

$(H_{\alpha,\lambda,\theta}u)_{n}=u_{n+1}+u_{n-1}+\lambda\tan(\pi\alpha
n+\theta)u_{n}$ (9.28)

Since $\tan$ is unbounded, one needs to eliminate the countable set of
$\theta$’s where for some $n$, $\pi\alpha n+\theta$ is of the form
$(k+\tfrac{1}{2})\pi;\,k\in{\mathbb{Z}}$. Then one has a well defined but
unbounded operator. When $\alpha$ is Diophantine, the Maryland group found an
explicit set of eigenfunctions but didn’t prove it complete and their
computation of the density of states had other formal elements.

One surprise is that the DOS was the same as for the Lloyd model which is the
random model whose single site distribution is
$\pi^{-1}\tfrac{\lambda}{\lambda^{2}+x^{2}}\,dx$ which physicists call
Lorentzian and mathematicians the Cauchy distribution. $\lambda$ is called the
_half-width_ of the distribution. This distribution has the following weird
property. If $X_{1},X_{2}$ are two independent random variables each with a
Cauchy distribution of the same half-width, then for any $t\in[0,1]$,
$tX_{1}+(1-t)X_{2}$ is also a Cauchy random variable with half width
$\lambda$. [663] noted that this implies that the Lloyd model has the same DOS
as the free Hamiltonian with a random Cauchy constant added to it and so does
the Maryland model with the same $\lambda$ (note that if $\theta$ is uniformly
distributed on $[0,\pi]$, then $\lambda\tan(\theta)$ is Cauchy distributed
with half width $\lambda$).

In addition to a proof of completeness of those eigenfunctions that had been
found by Prange’s group, [671] studies other properties of this model
including the fact that for $\alpha$ a Liouville number the spectral measures
are purely singular continuous and the structure of the (non-normalizable)
eigenfunctions for such $\alpha$. I note that this model was one of the first
times that non-mathematical physicists had to face singular continuous
spectrum; they gave the corresponding eigenfunctions which decay in an average
sense but are not normalizable, the name “exotic states.”

(4) _Clock Spacing of Zeros for Ergodic Jacobi Matrices with Absolutely
Continuous Spectrum_ We saw above that the DOS describes the bulk properties
of the eigenvalue distribution of Schrödinger operators in large boxes in that
the eigenvalue counting distribution converges to the DOS. But there is the
issue of the fine structure, in particular the spacing between nearby
eigenvalues. The earliest results on this problem are in the random Anderson
type case where Molchanov [501] in $1D$ and Minami [499] in higher dimensions
proved the distribution is asymptotically Poisson. For random matrices, the
fine spacing in the bulk is governed by the Wigner surmise for GOE/GUE (see
Mehta [494] or Deift [139] for discussion, proof and history). Basically,
because of strong localization in the Anderson case, nearby eigenvalues don’t
impact each other and the placement of such eigenvalues are close to
independent of each other. In the random matrix case, eigenfunctions are more
spread out, there is some eigenvalue repulsion so eigenvalues are less likely
to be too close to each other.

I gave the problem of extending the Poisson results to OPUC to my then
graduate student, Mihai Stoiciu which he solved [732]. Along the way, I
suggested that he do some numeric calculations and, for comparison, suggested
he look at some rapidly decaying Verblunsky coefficients. Here is the striking
result that he found for the zeros of $\Phi_{n=20}$ when
$\alpha_{j}=\left(\tfrac{1}{2}\right)^{j+1}$:

Figure 3. Zeros of an OPUC

By a Theorem of Mhaskar-Saff [496], it was known that the counting measure for
the zeros in this case converges to a uniform distribution on the circle of
radius $\tfrac{1}{2}$ but I was amazed when I saw that the eigenvalue
repulsion was so strong they seemed to be spaced like the numerals on a clock.
I called this “clock spacing”, a name which stuck even when eventually applied
to OPRL when the spacing was locally rigid but globally not equally spaced
because the limiting DOS didn’t have a uniform density. I wrote a series (the
last with Last) [699, 698, 700, 442] for situations there Verblunsky or Jacobi
parameters converged to a constant or periodic sequence.

Motivated by this, Lubinsky [478, 477], found a new approach to clock behavior
for OPRL with a.c. spectrum $[-1,1]$ based on proving a universality result
for the Christoffel-Darboux kernel

$K_{n}(x,y)=\sum_{j=0}^{n}p_{j}(x)p_{j}(y)$ (9.29)

namely that, one says that _bulk universality_ holds at $x_{0}$ if and only if

$\frac{K(x_{0}+a/n,x_{0}+b/n)}{K_{n}(x_{0},x_{0})}\rightarrow\frac{\sin(\pi\rho(x_{0})(b-a))}{\pi\rho(x_{0})(b-a))}$
(9.30)

uniformly in bounded $a,b$ where $\rho(x)$ is the weight in the DOS which is
assumed to be absolutely continuous. Earlier, although Lubinsky and I didn’t
realize it until later, Freud [206] had studied the fine structure of zeros,
had proven (9.30) in a less general setting and realized that it implied clock
spacing in the sense that if $\ldots
x^{[n]}_{-k}<\ldots<x^{[n]}_{-1}<x^{[n]}_{0}\leq x_{0}<x^{[n]}_{1}<\ldots$ are
the zeros near $x_{0}$, then

$\lim_{n\to\infty}n(x^{[n]}_{j+1}-x^{[n]}_{j})=1/\rho(x_{0})$ (9.31)

for all $j\in{\mathbb{Z}}$. Levin [455] independently rediscovered this
connection, so the fact that bulk universality implies clock behavior is
sometimes called the Freud-Levin theorem.

Lubinsky proved bulk universality for a large class of measures supported on
$[-1,1]$ with a.e. non-vanishing a.c. weight there. Totik [753] and I [703]
were able to replace $[-1,1]$ by fairly general sets
${\mathfrak{e}}\subset{\mathbb{R}}$ but we required that ${\mathfrak{e}}$ have
a large interior, so large it was dense in ${\mathfrak{e}}$. Last and I
realized that Lubinsky’s second approach [477] might allow one to handle
various almost periodic Jacobi matrices with a.c. spectrum even though that
spectrum is nowhere dense (e.g. the AMO with $|\lambda|<2$) but we ran into a
couple of hard technical problems. Fortunately, we were able to convince Avila
to attack these issues and the three of us [28] were able to prove bulk
universality and clock behavior for a.e. $x_{0}$ in $\sigma_{ac}$ for all
ergodic Jacobi matrices.

One of the goals when I wrote my two OPUC books [696, 697] was to extend the
spectral analysis of Jacobi matrices to OPUC (i.e. replacing Jacobi parameters
by Verblunsky coefficients). Included were two sections concerning ergodic
Verblunsky coefficients, one on random [697, Section 12.6] and one on
subshifts [697, Section 12.8], a class of weakly almost periodic functions.

I end this section on almost periodic Jacobi matrices and Schrödinger
operators by emphasizing that because it has focused on my own work, there is
no discussion of some issues and limited discussion of later work on the
issues we do discuss. In particular, I have not said anything on Hausdorff
dimension of the spectrum in the case where $\sigma(H)$ has Lebesgue measure
zero nor about long time behavior of powers of the position (although I do
have two papers on the latter [546, 676]). Nor have I discussed subshifts and
substitution models except for the OPUC work just mentioned and [317]. For
more on these subjects, the reader can consult a number of books and review
articles [3, 97, 127, 128, 357, 359, 364, 439, 528] (that said, we really do
need a more recent comprehensive review of the AMO).

##  10\. Topological Methods in Condensed Matter Physics

I was a pioneer [102] in the use of topology and geometry (mathematicians
sometimes use “geometry” when there is an underlying distance and “topology”
for those geometric object that don’t rely on a distance) in NRQM. In
particular, Avron, Seiler and I [41] realized that the approach of Thouless et
al. [750] to the quantum Hall effect (for which Thouless got the Nobel prize)
was basically an expression of the homotopic invariants (aka Chern integers)
of a natural line bundle that arises in certain eigenvalue perturbation
situations, and I realized [664] that the phase that Berry [71] found in the
quantum adiabatic theorem is holonomy in this bundle and that the quantity
Berry [71] used to compute this phase (and which independently had been found
by Avron et al. [41]), now called the _Berry curvature_ , is just the
curvature in this line bundle. I emphasize that Thouless et al. [750] never
mention “topology” and that Thouless learned they’d found a topological
invariant, essentially the Chern class, from me. And the only mention of
curvature or holonomy in Berry [71] is where he remarks that _Barry Simon,
commenting on the original version of this paper, points out that the
geometrical phase factor has a mathematical interpretation in terms of
holonomy, with the phase two-form emerging naturally (in the form (7 b)) as
the curvature (first Chern class) of a Hermitian line bundle_.

As a mathematician, I am mainly an analyst and most of my training and
expertise is analytic so, as background, I should explain something about how
I came to know enough toplogy/geometry to realize its significance in NRQM. As
a freshman at Harvard, I took the celebrated Math 55 Advanced Calculus course
whose first half did differential calculus in Banach spaces and second half
integral calculus on manifolds. This was a dip into the sea of geometry but
from an analytic point of view without any discussion of Riemannian metrics or
curvature. I did some self study of general relativity but the true
topology/geometry was hidden since my study was in physics books (and before
the era of those that emphasized the geometry). A key part of my education was
a course on Algebraic Topology given my senior year by Valentin Poénaru, then
a recent refugee from Romania, who was visiting Harvard. It was a wonderful
course and I really got into the subject, so much that Poénaru took me aside
and tried to convince me to give up mathematical physics and switch to
topology. I was particularly taken with the homotopy group long exact sequence
of a fibration. (See Hatcher [299] for background on this and other
topological issues).

Let me mention one of the simplest examples of fibrations of interest in
physics, namely, the Hopf fibration, which is a natural map of $S^{3}$ to
$S^{2}$. Let $\mathbf{\sigma}_{j};j=1,2,3$ be the usual Pauli $\sigma$
matrices. If
$\mathbf{a}=(a_{0},a_{1},a_{2},a_{3})=(a_{0},\mathbf{\overrightarrow{a}})$ is
a unit vector in ${\mathbb{R}}^{4}$, then $U(\mathbf{a})\equiv
a_{0}{\boldsymbol{1}}+i\mathbf{\overrightarrow{a}}\cdot\mathbf{\sigma}$ is a
unitary matrix with determinant $1$ if and only if $\mathbf{a}\in S^{3}$. This
sets up a homeomorphism between $S^{3}$ and the group $SU(2)$ of $2\times 2$
unitary matrices of determinant $1$. In that case there is a rotation
$R(\mathbf{a})$ on $S^{2}$ defined by
$U(\mathbf{a})(\mathbf{b}\cdot\mathbf{\sigma})U(\mathbf{a})^{-1}=R(\mathbf{a})(\mathbf{b})\cdot\mathbf{\sigma}$.
This is the Cayley-Klein parametrization of rotations, a map of $SU(2)$ onto
$SO(3)$. If $e_{3}$ is a unit vector in the $z$ direction, then
$\mathbf{a}\mapsto R(\mathbf{a})e_{3}$ defines the Hopf fibration, $H$, which
maps $S^{3}$ to $S^{2}$. The point is that it is easy to see (for example, by
looking at the inverse images of the north and south poles) that inverse
images of distinct points under $H$ are circles which are linked so the map is
homotopically non-trivial proving that $\pi_{3}(S^{2})$ is non-zero (in fact,
this homotopy group is generated by $H$ and is just ${\mathbb{Z}}$).

Of course, geometry in the naive sense was present, even central, to some of
my work in 1970’s, for example the work on phase space methods in N-body NRQM
(see Section 6) and I had even mentioned that the Agmon metric was the
geodesic distance in a suitable Riemann metric but if one thinks of “real”
geometry needing curvature and “real” topology needing homology or homotopy
invariants, I’d not used them in my research in the ’70’s. In Section 8, I
mention work that was motivated by Witten’s seminal paper [771] on the
supersymmetry proof of the Morse inequalities and index theorem. This paper
has been celebrated not only for the results itself but because of the bridge
it opened up between high energy theorists studying gauge (and later string)
theories and topologists but it also impacted me in leading me to consider
certain geometric ideas that I needed in the work I’ll describe in this
Section. This is not so much in those of my papers directly motivated by
Witten [665, 669] but through other mathematics motivated by it. For Witten
motivated several reworkings of the proof of the Atiyah-Singer Index theorem,
in particular, a preprint of Getzler [246] (see [125, Chapter 12] for
additional references) which caught my attention in the period just after I
gave the Bayreuth lectures which eventually appeared as [125]. I had lectured
on Witten’s proof of the Morse inequalities there and decided to add a chapter
on this further extension (the chapter, chapter 12, was actually the only
chapter that I wrote in [125] \- the other chapters were written by my
coauthors based on and usually expanding the lectures I’d given).

For pedagogical reasons, I decided to give details only in the special,
indeed, classical case of the Gauss-Bonnet theorem where it turns out that
Getzler’s proof is essentially one found in 1971 by Patodi [530] who didn’t
know that he was speaking supersymmetry! While I’d heard of the Gauss-Bonnet
theorem, I hadn’t known exactly what it said until following up on Witten
taught me all about it. Since it will explain some of my later work, let me
say a little about this theorem (and also holonomy) in the case of $S^{2}$,
the sphere of radius $R$ embedded in ${\mathbb{R}}^{3}$. At each point, the
Gaussian curvature is $1/R^{2}$ so, if $K$ is the curvature and $d^{2}\omega$
the surface area, we have that

$\frac{1}{2\pi}\int K\,d\omega=\frac{1}{2\pi}\frac{1}{R^{2}}4\pi R^{2}=2$
(10.1)

The remarkable fact is that if you deform the sphere to another surface, say,
an ellipsoid, then the curvature is no longer constant but the integral in
(10.1) is still $2$. But this is not true for the torus. The integral is still
independent of the underlying metric needed to define $K$, but it is $0$, as
can be seen by looking at the flat torus ${\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$
with the Euclidean metric on ${\mathbb{R}}^{2}$ (which cannot be isometrically
embedded into ${\mathbb{R}}^{3}$ but can in ${\mathbb{R}}^{4}$). In fact, for
any surface in ${\mathbb{R}}^{3}$ (and for hypersurfaces in general dimension)
the integral is the Euler characteristic of the surface (Euler-Poincaré
characteristic in higher dimension). This is the Gauss-Bonnet theorem. It says
that the integral of a natural geometric quantity lies in a discrete set and
is determined by topological invariants.

To explain holonomy, consider someone carrying a spear around the earth trying
at all times to keep the spear tangent to the sphere and parallel to the
direction it was pointing (which may or may not be parallel to the direction
the person is walking). Imagine, going along the equator through one quarter
of the earth, turning left, going to the north pole, turning left and going
back to the original point. Suppose the spear is parallel to the equator at
the start. The person turns to move along a line of longitude, but being
careful not to turn the spear, it will point directly to his right. After the
next turn, the spear will point backwards. So despite having tried to keep it
parallel , upon return, it has rotated by $90^{\circ}$, i.e. $\pi/2$ radians.
This rotation after parallel transport is _holonomy_. The path encloses one
eighth of the earth, a area of $4\pi R^{2}/8=\pi R^{2}/2$ so the integral of
the curvature over the enclosed area is the holonomy!

Perhaps relevant to my work is the following amusing story. Avron and I were
talking in my office about our work on almost periodic things and Dick Feynman
burst in and exclaimed “how do you compute the homotopy groups of spheres?”
There had been several papers in the high energy literature that mentioned
those and he was puzzled why the higher homotopy groups were not trivial. I
told him about the Hopf fibration which had always struck me and then
retrieved from my memory the exact sequence of a fibration. The conclusion of
our discussion continues to amuse me. When I finished Avron looked at me and
said: “Barry, I didn’t realize you knew anything about that”. Before I could
answer, Dick with a huge grin on his face turned around waved his hands at my
rather full bookshelves and exclaimed in his trademark New York accent:
“Whadya mean? He’s a Professor, of course he knows it!” I might have recalled
all that when I needed homotopy and the exact sequence of a fibration several
months later in my work with Avron and Seiler, but it helped that I’d had this
interaction.

In early 1983, Yosi Avron told me about the paper of Thouless et al. paper
[750] which gave a novel explanation of the quantum Hall effect, a subject
that had fascinated Yosi. The striking aspect of that effect is that a
resistance was quantized. In the TKNN approach (we quickly came up with that
abbreviation, sometimes TKN2, especially TKNN integers, a name which has
stuck), this arose because, using the Kubo formula, they got the resistance
(in a certain idealized situation) was given by an integral over a torus that
turned out to be an integer (in suitable units).

We quickly realized that their integers were associated to a single band which
was assumed non-degenerate (i.e. at every point in the Brillouin zone, the
eigenstate for that band is simple) and their integrand involved the change of
eigenfunction. We also realized that since the integrand was an integer it had
to be invariant under continuous change and so an indication of a homotopy
invariant of maps from the two dimension torus $T^{2}$ to unit vectors in
Hilbert space mod phases (equivalently a continuous assignment of a one
dimensional subspace in the Hilbert space to each point in $T^{2}$). After
more thought and study, we learned that the homotopy class of maps from
$T^{2}$ could be classified by maps from $S^{1}$ and $S^{2}$ and so the
underlying homotopy groups of ${\mathbb{P}}(\infty)$, the one dimensional
subspaces of a Hilbert space. We also considered that there might be non-
trivial homotopy invariants depending on several bands so what we wanted to
consider was the homotopy groups of the set, ${\mathcal{N}}$, of compact
operators with non-degenerate eigenvalues. We got excited since if, for
example, we found a non-trivial $\pi_{3}$, there would be new topological
invariants for the physically relevant three-dimensional torus.

By a continuous deformation, we could consider maps to a fixed set of simple
eigenvalues but variable eigenspaces. Given the phase change this was the same
as the quotient of all unitary maps by the diagonal unitary maps
${\mathcal{U}}({\mathcal{H}})/D{\mathcal{U}}({\mathcal{H}})$. So these
homotopy groups might be computable via the exact sequence of the fibration
that my talk with Feynman had reminded me about! Indeed, since it was known
that the set of all unitaries ${\mathcal{U}}({\mathcal{H}})$ is contractible,
it has no homotopy, i.e. all homotopy groups are trivial and, thus, by the
exact sequence of the fibration, we knew that
$\pi_{j}({\mathcal{N}})=\pi_{j-1}(D{\mathcal{U}}({\mathcal{H}}))$. Since the
diagonal unitaries is just an infinite product of circles, $T^{1}$, and
$\pi_{j}(T^{1})$ is ${\mathbb{Z}}$ for $j=1$ and $0$ for all other $j$, we had
discovered that the _only_ non-trivial homotopy group of ${\mathcal{N}}$ was
$\pi_{2}$, that the same was true for ${\mathbb{P}}(\infty)$ and that
$\pi_{2}({\mathcal{N}})$ was just an infinite product of
$\pi_{2}({\mathbb{P}}(\infty))$’s. In other words, the only homotopy
invariants were the TKNN integers.

We added Reudi Seiler, whom Yosi had been consulting, to the authors and
published this negative result in _Physical Review Letters_ [41]. We made a
big deal of our new result that if two non-degenerate bands with TKNN integers
$n_{1}$ and $n_{2}$ went through a degeneracy as parameters were varied so
that afterwards they were again non-degenerate with TKNN integers $n_{3}$ and
$n_{4}$, then $n_{1}+n_{2}=n_{3}+n_{4}$. But there were results that were more
important although only noted in passing. Most basic was the new one that the
TKNN integers were homotopy invariants, something that would be clarified by
my work on Berry’s phase which I turn to shortly. We also found two compact
formulae for the integrand that eventually became commonly used in further
work. First if that $\psi_{j}$ is the eigenstate of band $j$, then the
corresponding TKNN integer, $n_{j}$, is given by

$n_{j}=\frac{1}{2\pi}\int_{T^{2}}K_{j};\qquad K_{j}=i\langle
d\psi_{j},d\psi_{j}\rangle$ (10.2)

We were especially fond of a second formula, that if $P_{j}$ is the projection
onto $\psi_{j}$, then

$K_{j}=i\text{\rm{Tr}}(dP_{j}P_{j}dP_{j})$ (10.3)

We liked this because while (10.2) requires a choice of phase in each space,
(10.3) is manifestly phase invariant. The operator $d$ in the last two
expressions is the exterior derivative and there is an implicit wedge product.
The reader might worry that because $df\wedge df=0$, if there were no trace
and $P_{j}$ in (10.3) was a function, the quantity _would_ be 0. But because
$P_{j}$ is operator valued, it is not $0$. Indeed,

$\displaystyle K$
$\displaystyle=i\sum_{k,\ell}\text{\rm{Tr}}\left(\frac{\partial
P_{j}}{\partial x_{k}}P_{j}\frac{\partial P_{j}}{\partial
x_{\ell}}\right)dx_{k}\wedge dx_{\ell}$
$\displaystyle=\frac{i}{2}\sum_{k,\ell}\text{\rm{Tr}}\left(P_{j}\left[\frac{\partial
P_{j}}{\partial x_{k}},\frac{\partial P_{j}}{\partial
x_{\ell}}\right]\right)dx_{k}\wedge dx_{\ell}$
$\displaystyle=\frac{i}{2}\sum_{k,\ell}\left\langle\psi_{j},\left[\frac{\partial
P_{j}}{\partial x_{k}},\frac{\partial P_{j}}{\partial
x_{\ell}}\right]\psi_{j}\right\rangle dx_{k}\wedge dx_{\ell}$ (10.4)

where $[\cdot,\cdot]$ is commutator and we used the antisymmetry of
$dx_{k}\wedge dx_{\ell}$.

The next part of this story took place in Australia, so I should mention that
trip in the summer of 1983 (well, the winter in Australia!) almost didn’t
happen. My fourth child, Aryeh, was born in Dec., 1982 and given the time to
get his birth certificate and passport, it was only the end of April that I
was able to contact the Australian consul in Los Angeles to get visas for all
of us including a work visa for me. He sent a long medical form for me
requiring a new general exam from a doctor and xray. I’d had them 3 months
before at Kaiser but was told by the consul that I had to do them over. I’ve
been raised to avoid unnecessary xrays and I wasn’t sure Kaiser would agree to
a second exam. As far as I could tell, this was a restriction put in place to
make it difficult for Asians to come and work and I tried to use my invitation
from the Australian Academy of Sciences to get a waiver. The consul was
uncooperative, almost nasty. This was not only pre-Skype but email was almost
non-existent and intercontinental phone calls were very expensive, so I sent a
telex to my host, Derek Robinson, explaining that, because of visa issues, I
would probably have to cancel my trip. The next day, he called me, which
impressed me given the cost of international calls, telling me to stay calm
and he’d fix it. I didn’t know that Derek was the secretary of the Australian
Academy of Sciences. But three days later, I get a call from the consul saying
“Sir, I am anxious to issue your visas, but I need you to return the forms I
sent you.” I replied “But what about the medical form.” “Oh, you don’t need
that, sir.” According to the current vogue, I should feel guilty for having
used my white privilege, but given how important this visit turned out to be,
I am glad.

Derek was actually away for the first two weeks of my visit but Brian Davies
had also just arrived so we collaborated together on the work on
ultracontractivity that is mentioned in Section 3. Midway through my visit, I
heard that Michael Berry, whom I’d meet several years before at Joel Lebowitz’
seminar, was visiting physics at Australian National University where Derek
was in mathematics and where I was visiting. He’d given a seminar, but before
I’d learned he was there, so I called and asked him for a private version
which he kindly agreed to. He explained to me his work on an extra phase he’d
found in the adiabatic theorem (see below) and gave me a copy of the
manuscript [71] that he’d recently submitted to Proc. Roy. Soc. He mentioned
that Bernard Souillard, when he heard about Berry’s work, told Berry that he
thought it might have something to do with the paper of Thouless on TKNN
integers but then Berry added that when he asked Thouless about it, Thouless
said that he doubted there was any connection (no pun intended). I replied I
thought there probably was and that night, I figured out all the main points
that appeared in my paper [664]!

Berry’s paper dealt with the quantum adiabatic theorem. This theorem deals
with a time dependent Hamiltonian $H(s);0\leq s\leq 1$ and considers $T$ large
and $H(s/T)$ so one is looking at very slow changes.
$\varphi_{T}(s)\equiv\widetilde{U}_{T}(s)\varphi;\,0\leq s\leq T$ solves
$\dot{\varphi}_{T}(s)=-iH(s/T)\varphi_{T}(s);\varphi_{T}(0)=\varphi$. Let
$E(s)$ be an isolated, simple eigenvalue of $H(s)$ and let $P(s)$ be the
projection onto the corresponding eigenspace. The adiabatic theorem says that
if $P(0)\varphi=\varphi$, then
$\lim_{T\to\infty}({\boldsymbol{1}}-P(s/T))\varphi_{T}(s/T)=0$, i.e. if you
start in an eigenspace you stay in it adiabatically. Berry asked and answered
the question, what happens if $H(1)=H(0)$ so you end where you start. What is
the limiting phase of $\varphi_{T}(T)$. The surprise he found (it turned out
that in 1956 Pancharatnam [524] had done the same thing, but it had been
forgotten) is that the naive guess that $\varphi_{T}(T)\sim
e^{-iT\int_{0}^{1}E(s)ds}\varphi$ is wrong but that there is an additional
phase, $e^{i\Gamma}$. In my paper, I gave $\Gamma$ the name it is now known by
- _Berry’s phase_.

Berry originally wrote $\Gamma$ as a line integral but, then, assuming that
the family $H(s)$ was a closed curve in a parameter space, he used Stokes
theorem to write $\Gamma$ as the integral over a surface, $S$, in parameter
space whose boundary was the closed curve in the form

$\Gamma=\int_{S}K(\omega)\,d\omega$ (10.5) $K=\operatorname{Im}\sum_{m\neq
0}\frac{\langle\varphi_{m}(\omega),\nabla
H(\omega)\varphi_{0}(\omega)\rangle\times\langle\varphi_{0}(\omega),\nabla
H(\omega)\varphi_{m}(\omega)\rangle}{(E_{m}(\omega)-E_{0}(\omega))^{2}}$
(10.6)

where he supposed the interpolating Hamiltonian $H(\omega)$ had a complete set
$\\{\varphi_{m}\\}_{m}$ of simple eigenfunctions with
$H(\omega)\varphi_{m}(\omega)=E_{m}(\omega)\varphi_{m}(\omega)$ and
$P(\omega)\varphi_{0}(\omega)=\varphi_{0}(\omega);\,E(\omega)=E_{0}(\omega)$.

What I did in my paper [664] is realize that what Berry was doing was simple
and standard geometry in the exact same setting as TKNN. I’d learned in the
meantime that the TKNN integers were called the Chern invariant and the
curvature $K$ was called the Chern class and used those names for the first
time in this context. The adiabatic theorem defines a connection, i.e. a way
of doing parallel transport and Berry’s phase was nothing but the holonomy in
this connection. Berry had used (10.2) as an intermediate formula in his paper
but didn’t have the phase invariant formula of Avron-Seiler-Simon. Despite the
fact that our independent work was earlier (dates of submission for our paper
is May 31, 1983 and his June 13, 1983) and that the geometric ideas were in
our paper (and more explicitly with the name curvature in [664]), $K$ is
universally known as the _Berry curvature_.

Berry also realized that in situations where the parameter space could be
interpolated into higher dimensions, that eigenvalue degeneracies were sources
of curvature, a theme I developed in [664].

In the vast literature related to these issues, I should mention two
especially illuminating points. The first involves the fact that the first
mathematically precise and, in many ways, still the best proof of the quantum
adiabatic theorem is Kato’s 1950 proof [374] (see [713, Section 17] for an
exposition). Without loss, one can suppose E(s)=0 (otherwise replace $H(s)$ by
$H(s)-E(s){\boldsymbol{1}}$). Kato constructs a comparison dynamics solving

$\frac{d}{ds}W(s)=iA(s)W(s),\,0\leq s\leq 1;\qquad W(0)={\boldsymbol{1}}$
(10.7) $iA(s)\equiv[P^{\prime}(s),P(s)]$ (10.8)

for which

$W(s)^{-1}P(s)W(s)=P(0)$ (10.9)

by an explicit calculation and he proves that

$\lVert W(s)P(0)-U_{T}(s)P(0)\rVert=\textrm{O}(1/T)$ (10.10)

The relevant point here is that $W(s)$ defines a connection whose
differential, by (10.8), is $[P,dP]$ so that its differential, the curvature,
is given by (10.4). Thus the Avron-Simon-Seiler formula for the Berry
curvature was almost in Kato’s paper nearly 35 years before!

Secondly, as noted in [664], when the Hilbert space is ${\mathbb{C}}^{n}$,
this connection appeared a 1965 paper of Bott-Chern [80]. As noted later by
Aharonov-Anadan [7], this connection is induced by a Riemannian metric going
back to Fubini [220] and Study [735] at the start of the twentieth century.

I returned to the subject of the quantum Hall effect and Berry’s phase twice.
As background, I note that from Berry’s paper onwards, a key observation was
that Berry’s phase is zero if all the $H(s)$ can be taken simultaneously real
(indeed, Berry tells the story that prior to this work, he noted a curiosity
in eigenvalue perturbation theory; if one has real matrices depending on two
parameters with an eigenvalue degeneracy only at $0\in{\mathbb{R}}^{2}$, then
going around the degeneracy causes a sign flip in the eigenvector. In this
case, because eigenvectors are chosen real, there is only a $\pm$ degeneracy
and so a unique way of continuing. He talked about this result and someone
asked him what happened in the complex case and he replied, there was no
difference. But after the talk, he realized that in the complex case, phase
ambiguity meant there was no unique way to continue under just perturbation of
parameters and then, that the adiabatic theorem did give a way of continuing
which in the complex case could lead to a non-trivial phase). Since the
curvature must be real, the $i$ in (10.4) (or the $\operatorname{Im}$ in
(10.6)) show if all the $P$’s are real then $K=0$ and there is no Berry phase.
For spinless particles, time reversal just complex conjugates the wave
function so the mantra became “time reversal invariance kills Berry’s phase”.
Magnetic fields destroy reality of the operators (and are not time reversal
invariant). Indeed, the basic example is to take a constant magnetic field,
$\mathbf{B}\in{\mathbb{R}}^{3}$ and
$H(\mathbf{B})=\mathbf{B}\cdot\mathbf{\sigma}$ where $\mathbf{\sigma}$ is a
spin $s$ spin. The curvature is then $(2s+1)\mathbf{B}/B^{3}$.

In work with Avron and two then Caltech postdocs Sadun and Seigert [39, 40], I
discovered that for fermions you could have a non-zero Berry phase even with
time reversal invariance and that there was a remarkable underlying
quaternionic structure relevant to their study. The underlying issue goes back
to a 1932 paper of Wigner [768] on time reversal invariance, $T$, in quantum
mechanics. He first proved his famous theorem that symmetries in quantum
mechanics are given by either unitary or anti-unitary operators and then
argued that $T$ was always antiunitary with $T^{2}={\boldsymbol{1}}$ for
bosons and $T^{2}=-{\boldsymbol{1}}$ for fermions. In the Bose case, that
means $T$ acts like a complex conjugate and so the argument of no Berry’s
phase applies but not in the fermion case. Instead $J\equiv T$ and, $I$, the
map of multiplication by $i$ are two anticommuting operators whose squares are
each $-{\boldsymbol{1}}$, so they and $K=IJ$ turn the underlying vector space
into one over the quaternions! It was Dyson [173] who first realized that
fermions under time reversal have a quaternionic structure (although he first
notes the relevance of sympletic groups and that the connection between such
groups and quaternions is well known in the mathematical literature on group
representations).

We worked out the details, especially for half integral spin systems. Just as
the simplest example of Berry’s phase is a spin $1/2$ magnetic dipole, our
simple example is a spin $3/2$ electric quadrupole. An interesting feature
concerns the fact that eigenspaces are never simple but always even complex
dimension. This degeneracy is known as Kramers degeneracy (after [423]) - one
point of Wigner’s paper [768] is to explain this degeneracy as a result of
time reversal symmetry for fermions. Thus one looks for simple holonomy in
systems with quaternionicly simple eigenvalues, i.e. eigenvalues of fixed
complex multiplicity $2$. The Berry phase is thus a $2\times 2$ unitary
matrix.

One noteworthy aspect of [40] is its abstract which reads in full: _Yes, but
some parts are reasonably concrete_. While I had introduced topological ideas,
I was somewhat dismayed about all the terribly fancy stuff that appeared in
the math physics literature, especially throwing around the term “fiber
bundle”. Yosi and I used to joke that some people seemed to suffer from bundle
fibrosis. So we were concerned about some of the abstruse language in [40] and
decided to work out several examples in full as a counterweight. We liked our
abstract, but getting it into the journal was not easy, an interesting story
that I’ll not include (but see
http://www.math.caltech.edu/SimonFest/stories.html#barry). Almost twenty five
years after our paper, the abstract earned a fan blog post entitled _Abstract
Snark_ [547] that declared our abstract and one other “almost Zen in their
simplicity and perfection”.

My other work in this area is three related papers that I wrote with Avron and
Seiler [42, 43, 44] that followed up on an alternate approach to the quantum
Hall effect due to Bellissard [60] in which topology entered as an index in
$C^{*}$-algebraic K-theory. We developed an index theory for the simpler case
where certain subsidiary operators were Fredholm. To me, some of the
mathematics we developed was most fascinating. In particular we proved

######  Theorem 10.1.

Let $P$ and $Q$ be two orthogonal projections so that $P-Q$ is trace class.
Then $\text{\rm{Tr}}(P-Q)$ is an integer.

######  Remarks.

1\. For discussion of trace class, see [711, Section 3.6].

2\. This is a result that begs to be proven by Goldberger’s method 111Murph
Goldberger was one of my professors at Princeton (see the remark after (8.24))
and, in his day, a famous theoretical physicist who used to joke about things
that just had to be true: _oh, you just use Goldberger’s method which is a
proof by reductio ad absurdum. Suppose it’s false; why that’s absurd!_ Murph
was the President of Caltech when I moved and played a role in my
recruitment..

3\. Our proof relied on two operators used extensively by Kato [386], $A=P-Q$
and $B=1-P-Q$ which he showed obeyed $A^{2}+B^{2}={\boldsymbol{1}}$. We noted
[44] that one also had the supersymmetry relation $AB+BA=0$. Since $A$ is
trace class and self-adjoint, using a basis of eigenfunctions and the Hilbert-
Schmidt Theorem [711, Theorem 3.2.1] shows that

$\text{\rm{Tr}}(A)=\sum_{\lambda}\lambda d_{\lambda}$ (10.11)

where we sum over eigenvalues and $d_{\lambda}=\dim({\mathcal{H}}_{\lambda})$
with ${\mathcal{H}}_{\lambda}=\\{\varphi\,\mid\,A\varphi=\lambda\varphi\\}$.
The supersymmetry implies that $\psi\in{\mathcal{H}}_{\lambda}\Rightarrow
B\psi\in{\mathcal{H}}_{-\lambda}$. Since
$B^{2}\restriction{\mathcal{H}}_{\lambda}=(1-\lambda^{2}){\boldsymbol{1}}$, we
see if $\lambda\neq\pm 1$, then $B$ is a bijection of
${\mathcal{H}}_{\lambda}$ and ${\mathcal{H}}_{-\lambda}$ so, for such
$\lambda$, we have that $d_{\lambda}=d_{-\lambda}$. Thus (10.11) implies that
$\text{\rm{Tr}}(A)=d_{1}-d_{-1}\in{\mathbb{Z}}$.

4\. Slightly earlier, this result was proven by different methods by Effros
[178]. His proof is sketched in [711, Problem 3.15.20] and our proof can also
be found [711, Example 3.15.19]. I found another proof using the Krein
spectral shift which is sketched in [711, Problem 5.9.1]. Amrein-Sinha [18]
has a fourth proof.

5\. For a review of some of the literature on pairs of projections, see [713,
Section 5]. I have several more recent papers on pairs of projections [712,
81].

##  11\. Anderson Localization: The Simon-Wolff Criterion

I have some contributions to random Schrödinger operators, especially in one
dimension. While the first of my papers in the area predates slightly the work
of the last section, I’ve placed this here because my two most significant
contributions were finished near the end of 1984, so close to each other that
there was a joint announcement [720]! One is the work with Tom Wolff [721] on
a necessary and sufficient condition for point spectrum which appears in the
title of this section and the other is work with Michael Taylor [719] on
regularity of the density of states in the Anderson model. While the Simon-
Taylor work was done first (indeed, I talked about it at the conference where
I learned of Kotani’s work that motivated Simon-Wolff), I begin with [721].

The generalized Anderson model is described in Section 9. Suppose that the
single site distribution, $d\kappa$, is acwrt Lebesgue measure. If
$H_{\omega}$ has dense point spectrum for a.e. $\omega$, then, by independence
at distinct sites, if we fix all sites but one, we will have dense point
spectrum for Lebesgue a.e. choice of the potential at the remaining point in
the a.c. support of the single site distribution. So it is natural to discuss
a family of rank one perturbations,

$A_{\lambda}=A+\lambda Q,\qquad Q=\langle\varphi,\cdot\rangle\varphi$ (11.1)

where is A is a self-adjoint operator with simple spectrum (I discuss in
remark $1$ why assuming simplicity is no loss), $Q$ the projection onto a unit
vector $\varphi$ and $\lambda\in{\mathbb{R}}$. If $d\mu_{\lambda}$ is the
spectral measure for $A_{\lambda}$ in vector, $\varphi$, a key role is played
by the function

$K(E)=\int\frac{d\mu_{0}(E^{\prime})}{(E-E^{\prime})^{2}}$ (11.2)

which is well-defined as a function with values in $(0,\infty]$ including the
possible value of $\infty$. This function will play a crucial role in the next
section also. One main theorem of Simon-Wolff [721] is

######  Theorem 11.1 (Simon-Wolff criterion [721]).

Suppose $A$ is a self adjoint operator with cyclic unit vector $\varphi$. Fix
an open interval $(\alpha,\beta)$ in $\text{\rm{spec}}(A)$. The following are
equivalent:

(a) For (Lebesgue) a.e. real $\lambda$, $A_{\lambda}$ has dense point spectrum
in $(\alpha,\beta)$

(b) For (Lebesgue) a.e. real $E\in(\alpha,\beta)$, we have that $K(E)<\infty$.

######  Remarks.

1\. We supposed that $A$ has simple spectrum, with $\varphi$ a cyclic vector.
For general $A$ and $\varphi$, we can restrict $A$ to the cyclic subspace
generated by $\varphi$ and that restriction obeys the simplicity and cyclicity
assumptions, so we can conclude something about the spectral measure
$d\mu_{\lambda}$. For an Anderson type model, if we know each
$\delta_{\beta},\beta\in{\mathbb{Z}}^{\nu}$ has a spectral measure with dense
pure point spectrum, we get the result for the full operator. We also note
that it was later shown that any single $\delta_{\beta}$ is cyclic in the
localization region; see the end of this section.

2\. Simon-Wolff [721] further note that if
$G(\beta,\gamma;z)={\langle\delta_{\beta},(H-z)^{-1}\delta_{\gamma}\rangle}$
is the Green’s function and $d\mu$ is the spectral measure for $H$ and
$\delta_{0}$, then

$\int|E-E^{\prime}|^{-2}d\mu(E^{\prime})=\lim_{\varepsilon\downarrow
0}\sum_{\beta\in{\mathbb{Z}}^{\nu}}|G(\beta,0;E+i\epsilon)|^{2}$ (11.3)

The two main approaches to the spectral analysis of multidimensional
generalized Anderson models are the multiscale analysis of Fröhlich-Spencer
(see Fröhlich-Spencer [218, 219] for the original work and [733] for a
pedagogical presentation) and the method of fractional moments of Aizenmann-
Molchanov (see Aizenmann-Molchanov [12] for the original work and [16] for a
pedagogical presentation). Both most directly prove exponential decay of
Green’s functions with some kind of uniformity as one approaches the real axis
and prove the finiteness of the right side of (11.3) for a.e. $\omega$ and
a.e. $E$ in some interval so, by Theorem 11.1, they imply dense point
spectrum.

The proof of Theorem 11.1 relies on two elements - a general analysis of the
spectral type under rank one perturbations due to Aronszajn [21] and Donoghue
[168] (Aronszajn discussed the special case of variation of boundary condition
for ODEs and Donoghue extended to general rank one perturbations; some
elements appeared earlier in their joint work [22]). We need the Stieltjes
(aka Borel, aka Cauchy) transforms

$F_{\lambda}(z)=\int\frac{d\mu_{\lambda}(E^{\prime})}{E^{\prime}-z}$ (11.4)

and we define various subsets of ${\mathbb{R}}$ using $F_{\lambda=0}$ and $K$:

$\begin{split}&S_{\lambda}=\\{x\,\mid\,\lim_{\varepsilon\downarrow
0}F_{0}(x+i\varepsilon)=-\lambda^{-1}\\}\text{ for }\lambda\neq 0;\\\
&S_{0}=\\{x\,\mid\,\lim_{\varepsilon\downarrow
0}\operatorname{Im}F_{0}(x+i\varepsilon)=\infty\\}\end{split}$ (11.5)
$\begin{split}&P=\\{x\,\mid\,K(x)<\infty\\};\qquad P_{\lambda}=S_{\lambda}\cap
P\text{ for }\lambda\neq 0;\\\
&P_{0}=\\{x\,\mid\,\limsup_{\varepsilon\downarrow
0}\varepsilon\operatorname{Im}F_{0}(x+i\varepsilon)>0\\}\end{split}$ (11.6)
$\begin{split}&L=\\{x\,\mid\,\lim_{\varepsilon\downarrow
0}\operatorname{Im}F_{0}(x+i\varepsilon)\in(0,\infty)\\};\\\
&B={\mathbb{R}}\setminus\left(\bigcup_{\lambda\in{\mathbb{R}}}S_{\lambda}\cup
L\right)\end{split}$ (11.7)

where, when we write a $\lim$ is equal to some value, it includes the
statement that the limit exists.

As preliminaries, we note first that, by the dominated convergence theorem, if
$K(x)<\infty$, we have that $\lim_{\varepsilon\downarrow
0}F_{0}(x+i\varepsilon)$ exists and lies in ${\mathbb{R}}$ so $P$ is
$\bigcup_{\\{\lambda\neq 0\\}}P_{\lambda}$ plus the set where the limit is
$0$. Secondly, the general theory of Stieltjes transforms implies that each
$S_{\lambda}$ has measure zero. Note also that the sets $P_{\lambda}$ are
disjoint from each other and from $L$. We say that $Z\subset{\mathbb{R}}$
_supports a measure_ , $\nu$, if and only if

$\nu({\mathbb{R}}\setminus Z)=0$ (11.8)

Then the work of Aronszajn-Donoghue implies

######  Theorem 11.2 (Aronszajn-Donoghue [21, 168]).

Let $A_{\lambda}$ be a family of rank one perturbations. Then

(a) The a.c. parts of the measures $d\mu_{\lambda,ac}$ are mutually absolutely
continuous for all $\lambda\in{\mathbb{R}}$ and are supported on $L$.

(b) The singular parts of the measures $d\mu_{\lambda,sing}$ are mutually
singular and for distinct $\lambda\in{\mathbb{R}}$ and each is supported on
$S_{\lambda}$.

(c) For all $\lambda\in{\mathbb{R}}$, the pure point part of the measure,
$d\mu_{\lambda,pp}$ is supported on $P_{\lambda}$ and the singular continuous
part of the measure is supported on $S_{\lambda}\setminus P_{\lambda}$.

(d) The set $B$ has Lebesgue measure zero and, for all
$\lambda\in{\mathbb{R}}$, we have that $\mu_{\lambda}(B)=0$.

######  Remarks.

1\. For proofs, see [711, Section 5.8] or [694, Section 12.2].

2\. One can say much more about $P_{\lambda}$ and $L$. First, $L$ is the
essential support of the all the $d\mu_{\lambda,a.c.}$. Secondly, for
$\lambda\neq 0$, each point, $x_{0}$ in $P_{\lambda}$ is a pure point with
$d\mu_{\lambda,pp}(x_{0})=(\lambda^{2}K(x_{0}))^{-1}$.

3\. After my introduction to rank one theory in the course of this work, I was
motivated to do a lot more in the subject. First, the work on Baire generic
singular continuous components [150, 148] discussed in Section 12. Secondly, I
worked on the natural meaning of $A_{\lambda}$ when $\lambda=\infty$ [236] and
the extension of the theory when $A$ is unbounded and $\varphi$ is very
singular [407]. Finally, I extended the theory to multiplicative rank one
perturbations of unitary operators, a subject useful in OPUC [696, 701].

4\. After those works, I wrote some lecture notes on rank one perturbations
[682]. When the AMS decided to reprint my trace ideals book, which had gone
out of print, it made sense to include those notes as some extra chapters in
the second edition [694].

The second element of the Simon-Wolff analysis was our result that has come to
be called _spectral averaging_ :

######  Theorem 11.3 (Spectral Averaging [721]).

For general rank one perturbations, one has that

$\int\left[d\mu_{\lambda}(x)\right]\,d\lambda=dx$ (11.9)

in the sense that

$\int\left[f(x)d\mu_{\lambda}(x)\right]\,d\lambda=\int f(x)\,dx$ (11.10)

for any continuous function, $f$, of compact support on ${\mathbb{R}}$.

######  Remarks.

1\. Theorems 11.2 and 11.3 immediately imply Theorem 11.1 because, by Theorem
11.2, (a) is equivalent to $d\mu_{\lambda}((\alpha,\beta)\setminus P)=0$ for
a.e. $\lambda$ and by Theorem 11.3,
$\int\left[d\mu_{\lambda}((\alpha,\beta)\setminus
P)\right]=|(\alpha,\beta)\setminus P|$.

2\. There are variants of spectral averaging that predate [721]. In 1971,
Javrjan [352] proved equivalent formulae for the special case of boundary
condition variation of Sturm-Liouville operators on $[0,\infty)$. For some
applications all that is needed is the consequence of spectral averaging that
if a set $Q\subset{\mathbb{R}}$ has Lebesgue measure zero, then for a.e.
$\lambda$ one has that $d\mu_{\lambda}(Q)=0$ for a.e. $\lambda$. This fact (or
the stronger one that some average of $d\mu_{\lambda}$ with an a.e. positive
weight is dominated by an a.c. measure) appears in the literature in several
place prior to [721]: for example Carmona [95] and Kunz-Souillard [426].

I conclude the discussion of Simon-Wolff [721] with a bit about the history of
its genesis. In the summer of 1984, Kotani reported on some interesting work
at a conference in Maine. I didn’t hear about this work until he and I attend
a conference in Bremen in November although he eventually published his work
in the Proceeding of the conference in Maine [417]. While Kotani focused, as
he often did, on continuum Schrödinger operators, I’ll discuss the discrete
case which he mentioned in passing. He looked at an ergodic discrete
Schrödinger operators on a half line ($n\geq 1$) (i.e. $a_{n}\equiv 1$,
$b_{n}(\omega)$ samples of an ergodic process) in an energy region,
$(\alpha,\beta)$ where one knew the Lyapunov exponent was positive. He
considered operators $h_{\omega}^{\theta}$ where the eigenfunction equation
$hu(n)=u(n+1)+u(n-1)+b_{n}u(n)=Eu(n),\,n\geq 1$ was supplemented by the
boundary condition $\cos(\theta)u(1)+\sin(\theta)u(0)=0$. This is equivalent
to truncating the doubly infinite matrix but replacing $b_{1}$ by
$b_{1}-\cot(\theta)$. As explained in Section 9, one has exponentially growing
or decaying solutions except for an $\omega$-dependent set of energies in
$(\alpha,\beta)$. By making explicit an argument of Carmona [95], he showed
that for Lebesgue a.e. $\theta$, the spectral measures were supported on the
set where one had this exponential dichotomy. Thus he had the shocking result
that in cases like the one where Avron and I proved there was purely singular
spectrum (AMO with large coupling and Liouville frequencies), the half line
problem had pure point spectrum for a.e. boundary condition $\theta$.

It was immediately clear to me that these ideas might say something about
dense point spectrum for the higher dimensional Anderson model where Fröhlich-
Spencer had recently announced results on exponential decay of Green’s
functions. I asked Kotani if he’d thought about such applications and when he
said no, I asked if he minded if I thought about it and he said fine. I
returned to Caltech and quickly realized the relevance of the Aronszajn-
Donoghue theory and understood the key was finding some abstract version of
the Carmona argument. I decided it was a question connected with Hilbert
transforms and so consulted Wolff and we came up with spectral averaging.

Bernard Souillard was also at the conference in Bremen and he also realized
the possible applicability of Kotani’s scheme to multidimensional localization
and he, together with Delyon and Lévy also developed an approach [153, 154,
155] to these problems. They did not phrase it in terms in general rank one
perturbations and required exponential decay (rather than only $\ell^{2}$
decay) and didn’t have a necessary and sufficient theorem so, Simon-Wolff has
been much more generally quoted. But their ideas worked more easily in some
non-rank one situations and, indeed, Delyon, Souillard and I [158] used their
approach to prove some results about random operators with so-called off-
diagonal disorder (which are rank $2$)!

After my work with Wolff, I wrote two papers, one with Kotani [674, 420]
applying these ideas to discrete Schrödinger operators in strips.

Before leaving the subject of spectral averaging, I should mention a later
work of mine [688] that extends it to trace class perturbations and averages
over finite intervals and relates it to a wonderful formula of Birman-Solomjak
[76]. It involves the Krein spectral shift (see [694, Section 11.4] for
references and the theory), $\xi_{A,B}(x)$ which, whenever $B-A$ is trace
class, can be defined by

$\text{\rm{Tr}}(f(B)-f(A))=\int f^{\prime}(x)\xi_{A,B}(x)dx$ (11.11)

Javrjan [352] actually had a local version of (11.9) which generalized to
arbitrary rank one perturbations says that

$\int_{\lambda_{0}}^{\lambda_{1}}\left[d\mu_{\lambda}(x)\right]\,d\lambda=\xi_{A+\lambda_{0}Q,A+\lambda_{1}Q}(x)dx$
(11.12)

from which (11.9) follows because $\lim_{\lambda\to\infty}\xi_{A-\lambda
C,A+\lambda C}(x)=\text{\rm{rank}}(C){\boldsymbol{1}}$ (for all $x$) if $C$ is
finite rank. The main result in [688] considers general families,
$A(s);s_{0}\leq s\leq s_{1}$, of self-adjoint operators with a weak derivative
$C(s)$ which is trace class, positive and continuous in trace norm. I defined
$d\mu_{s}(x)=\text{\rm{Tr}}(C(s)^{1/2}dE_{s}(x)C(s)^{1/2})$ (with $A(s)=\int
xdE_{s}(x)$ in the spectral resolution form of the spectral theorem [711,
Section 5.1]) and proved that

$\int_{s_{0}}^{s_{1}}\left[d\mu_{s}(x)\right]\,ds=\xi_{A(s_{0}),A(s_{1})}(x)dx$
(11.13)

I showed that this was equivalent to the formula of Birman-Solomjak that

$\frac{d}{ds}\text{\rm{Tr}}(f(A(s)))=\text{\rm{Tr}}(C(s)f^{\prime}(A(s)))$
(11.14)

and provided a half page proof of (11.14).

I turn next to my work with Taylor [719] which concerns the issue of
regularity of the IDS, $k(E)$. The most heavily quoted and used regularity
result is the estimate of Wegner [759] that for the generalized Anderson model
on ${\mathbb{Z}}^{\nu}$, if the $b_{n}$ are iidrv with distribution (9.2)
where

$d\kappa(x)=g(x)dx$ (11.15)

with $g\in L^{\infty}$, then one has the _Wegner estimate_

$|k(E)-k(E^{\prime})|\leq 2\lVert g\rVert_{\infty}|E-E^{\prime}|$ (11.16)

This estimate is easy to prove and can be deduced from spectral averaging
(although it predates it!). It (or rather its finite volume analog) is the
starting point for most variants of multiscale analysis. This estimate and
others that are known in general dimension are of the form that $k$ is as
regular as $E\mapsto\kappa(-\infty,E)$. What Taylor and I proved was the
possibility of significantly greater smoothness in one dimension (at least
once $\kappa$ has some minimal smoothness).

Why did I think this might be true? For the free case,

$k(E)=\left\\{\begin{array}[]{ll}0,&\hbox{ if }E\leq-2\\\
\tfrac{1}{\pi}\arccos\left(\tfrac{-E}{2}\right),&\hbox{ if }-2\leq E\leq 2\\\
1,&\hbox{ if }E\geq 2\end{array}\right.$ (11.17)

which implies that $k(E)$ is $C^{\infty}$ on $(-2,2)$ with
$dk/dE=\left[2\pi\sqrt{4-E^{2}}\right]^{-1}$ so there is a singularity in
$dk/dE$ at $E=\pm 2$. In general, one would expect there are singularities at
the edges of the spectrum. Indeed, this $k(E)$ is globally Hölder continuous
of order $1/2$ and no higher order. For the Anderson model, I knew there were
Lifshitz tails (see below) which implies that $k(E)$ went to zero as
$E\downarrow\Sigma_{-}$, the bottom of the spectrum faster than the inverse of
any power of $(E-\Sigma_{-})^{-1}$ consistent with $k(E)$ being $C^{\infty}$
as $E$ shifts from the spectrum to below the spectrum. I discussed this with
Tom Spencer who was dubious that $k$ was $C^{\infty}$ for the original
Anderson model, so we made a 25 cent bet on whether it was true (I would win
if someone, not necessarily me, proved it true and he would win if someone,
not necessarily him, proved it false).

Of course one expected this not merely for the Anderson model (where $g$ is
the characteristic function of an interval), but for at least some generalized
Anderson models. I found that one needed some minimal regularity on $\kappa$
because Bert Halperin [296] had shown that there were examples where $d\kappa$
was a two point measure, where $k$ was not even $C^{1}$. While Halperin went
on to become a distinguished condensed matter theorist, he wrote this paper as
a junior undergrad at Harvard! While the argument was solid, there were
missing points of mathematical clarity, so that Taylor and I, who wanted to
advertise the result, included details in an appendix [719]. The model has

$d\kappa(x)=(1-\theta)\delta(x)+\theta\delta(x-\lambda)$ (11.18)

with $0<\theta<1/2$. This model came to be called the _Bernoulli-Anderson
model_. We showed that $k(E)$ was not Hölder continuous of any order larger
than
$\alpha_{0}=2|\log(1-\theta)|/\operatorname{Arc\,cosh}(1+\tfrac{1}{2}|\lambda|)$
so by taking $\theta$ small and/or $\lambda$ large, one can assure lack of
Hölder continuous of any prescribed order. We also gave heuristics and
conjectured that for those extreme values $dk$ should have a singular
component (we recall that the Cantor function is Hölder continuous of order
$\log(2)/\log(3)$, so $dk$ can be singular continuous even though $k$ is
Hölder continuous). Motivated by this, Carmona et. al. [96] proved this
conjecture (and more importantly proved localization in Bernoulli Anderson
models) and Martinelli-Micheli [487] even proved for any fixed $\theta$, $dk$
was purely singular continuous for all large $\lambda$.

The main result of Simon-Taylor is

######  Theorem 11.4.

Let $k$ be the IDS for a generalized Anderson model in $\nu=1$ dimension with
$d\kappa$ of the form (11.15) where $g$ has compact support and for some
$\alpha>0$, one has that $(1+k^{2})^{\alpha/2}\hat{g}(k)$ is the Fourier
transform of an $L^{1}$ function. Then $k$ is $C^{\infty}$.

When $g$ is the characteristic function of an interval, the hypothesis holds
for any $\alpha<1$, so this won my 25 cent bet with Spencer! Let me say
something about the strategy and genesis of this result. Most of the early
proofs of localization in the 1D Anderson model relied on a theorem of
Furstenberg [222], who proved that, under certain circumstances, products of
iidrv ${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ matrices had positive
Lyapunov exponent. His proof relied on the action of
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ on ${\mathbb{R}}{\mathbb{P}}(1)$,
real projective space (by $(A,[\varphi])\mapsto A[\varphi]$) and the induced
natural convolution of measures on ${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$
with measures on ${\mathbb{R}}{\mathbb{P}}(1)$ to get measures on
${\mathbb{R}}{\mathbb{P}}(1)$. If $\mu$ was the probability measure on
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ describing the distribution of
individual matrices in the random product, Furstenberg showed and used that
there was a unique measure $\nu$ on ${\mathbb{R}}{\mathbb{P}}(1)$ so that
$\mu*\nu=\nu$. In the Anderson case, for each real energy, $E$, there is a
distribution of transfer matrix (9.11) and so an invariant measure, $\nu_{E}$
for each $E$. I realized that by a discrete analog of the Sturm oscillation
theorem, $k(E)$ was the weight that $\nu_{E}$ gave to those lines in
${\mathbb{R}}{\mathbb{P}}(1)$ with two coordinates of the same sign so that
smoothness of $k$ should be implied by smoothness of $\nu_{E}$ in $E$. Since
$\nu_{E}$ was also invariant for multiple
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ convolutions of $\mu_{E}$, what one
needed is that these multiple convolutions got smoother and smoother in $E$.
While I was interested in smoothness in $E$, I suspected (correctly it turns
out) that what one really needed was that these high order convolutions of
$\mu_{E}$ were a.c. wrt Haar measure on
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ with weights that were smoother and
smoother in the group parameters.

This was a question in noncommutative harmonic analysis and I assumed the
representation theory of ${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ would play
a major role, so I contacted Michael Taylor, who I’d heard was a big expert on
the topic (shortly after this, he published two books on the subject [744,
745]) and suggested that we work on it. At some point, I also spoke to Eli
Stein who was also a big expert on the representation theory of
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ and he made the suggestion that it
is often easier to control convolutions on
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ with one’s “bare hands” rather than
by using the non-commutative Fourier transform which is what we did. The
underlying $\mu_{E}$ are certainly not a.c. wrt Haar measure on
${\mathbb{S}}{\mathbb{L}}(2,{\mathbb{R}})$ since they are supported on a one
dimensional subset of the three dimensional group but we proved that under the
technical condition on $g$ in Theorem 11.4 the three fold convolution _is_
a.c. wrt Haar measure with a weight that has a tiny bit of smoothness so that
in the standard way, the higher order convolutions of that will be smoother
and smoother. While conceptually the proof was straightforward, some of the
technical details were formidable. In particular, we strongly used the compact
support hypothesis on $g$.

Our paper stimulated several others that obtained strengthening of our result
- two by Klein and others [90, 413] and one by March-Snitman [484]. Their
techniques were very different from ours and each other. In particular, [413]
only needs the weak condition on $g$ that its Fourier transform is
$C^{\infty}$ with all derivatives vanishing at $\infty$ (automatic if $g$ is
of compact support and the analog is even true if $d\kappa$ is the Cantor
measure!).

Besides these two major works on random potentials, I have papers on four
other aspects ((1) and (3) only in one dimension). Let me briefly discuss
them.

(1)_localization for slowly decaying random potentials_ I wrote a number of
papers on the model (half or whole line) where $a_{n}\equiv 1$ and

$b_{n}=(1+|n|)^{-\alpha}\omega_{n}$ (11.19)

where $\omega_{n}$ are iirdv (sometimes with restrictions on their common
distribution) [660, 156, 157, 406] and [697, Section 12.7]. The first and most
basic paper [660] showed that if $0<\alpha<1/2$, with minor assumptions on the
distribution $d\kappa$, of $\omega$ (basically, it has the form (11.15) with
$g$ bounded and of compact support), then for a.e. $\omega$, $h_{\omega}$ has
dense point spectrum in $[-2,2]$ with eigenfunctions decaying at least as fast
as $e^{-C|n|^{\beta}};\,\beta=1-2\alpha$. As noted there, the proof is an easy
adaptation of the proof of localization in the one dimensional Anderson model
by Kunz-Souillard [426]. I pointed out that one knew (by the trace class
theory) there was pure a.c. spectrum on $[-2,2]$ when $\alpha>1$ and that
while $1/2\leq\alpha\leq 1$ was open, it was likely that localization required
$\alpha<1/2$.

The transition region $\alpha=1/2$ was discussed in two papers that I wrote
with Delyon and Souillard [156, 157] which is especially interesting because
the same ideas allow the analysis of a random Kronig-Penny model in non-zero
electric field. We fixed $g$ and added a coupling constant, $\lambda$ in
$b_{n}=\lambda(1+|n|)^{-1/2}\omega_{n}$. We showed for all sufficiently large
$\lambda$, for a.e. $\omega$, the model has dense point spectrum with power
decaying eigenfunctions and for all sufficiently small $\lambda$ no point
spectrum. Subsequently Delyon [151] proved purely singular continuous spectrum
in these small $\lambda$ regions.

Kotani-Ushiroya [422] studied a closely related set of models. They studied 1D
random continuum Schrödinger operators of the type studied by the Russian
group [260] but with the potential multiplied by $(1+|x|)^{-\alpha}$. They
proved purely a.c. spectrum on $[0,\infty)$ when $1/2<\alpha$ and sharpened
the results of the last paragraph when $\alpha=1/2$. Kiselev, Last and I [406]
used discrete analogs of Prüfer variables to recover and strengthen the
results for the decaying discrete models. In particular, if $b_{n}=\lambda
n^{-1/2}\omega_{n}$ with $\omega_{n}$ as in the classical Anderson model, we
proved that the spectrum is purely dense pure point if $\lambda^{2}\geq 12$
and if $\lambda^{2}<12$, the spectrum is purely s.c. in the region
$|E|\leq\sqrt{4-\lambda^{2}/3}$ and dense pure point in the complementary part
of $[-2,2]$. We even found the local Hausdorff dimension of the spectral
measures (see [708, Section 8.2] for discussion of the local Hausdorff
dimension of a singular measure) in the singular continuous region.

Random decaying operators, as we will see (Section 12), play an important role
in discussions of singular continuous spectrum for Baire generic decaying
potentials. Gordon et al [266] (I was a coauthor of an extension of this work
to higher dimensions [264]) considered the random potentials of the form
(11.19) with $\alpha<0$. One might think that $|b_{n}|\to\infty$ so the
spectrum is discrete but if $g$ has zero in its support, it might happen that
although $\limsup|b_{n}|=\infty$, one has that $\liminf|b_{n}|=0$! Indeed,
when $\omega$ is uniformly distributed in $[0,1]$, the spectrum is discrete if
and only if $-\alpha>1$. When $0<-\alpha\leq 1$, there is a semi-infinite
interval of dense point spectrum.

(2)_Lifshitz tails_ I made some contributions to the theory of Lifshitz tails
[673, 400, 675] (I am embarrassed to say that I sometimes used the atypical
spelling Lifschitz although I do note the original is Cyrillic). There is a
huge literature, so I’ll only include my papers and the original one of
Lifshitz referring the reader to the excellent review of Kirsch-Metzger [399]
from my 60th birthday festschrift for more references and more history. Here’s
a rough heuristic argument close to Lifshitz original [473]. Consider a model

$\displaystyle H_{\omega}u(n)$ $\displaystyle=2\nu
u(n)-\sum_{|j|=1}u(n+j)+b_{\omega}(n)u(n)$
$\displaystyle\equiv[(H_{0}+b_{\omega})u](n)$ (11.20)

on ${\mathbb{Z}}^{\nu}$ with, say the $b_{\omega}(n)$ uniformly distributed in
$[0,1]$. The free term is written as a Laplacian so that $H_{0}$ is a positive
operator whose spectrum is $[0,4\nu]$ and it is easy to prove that for a.e.
$\omega$ the spectrum of $H_{\omega}$ is $[0,4\nu+1]$. Imagine putting the
system in a large box and looking for eigenvalues with energy less than
$\varepsilon$ and normalized eigenfunction $\varphi$. For
$\langle\varphi,H_{0}\varphi\rangle$ to be small, $\varphi$ must be spread out
over a region of radius $R$ at least $\varepsilon^{-1/2}$ with
$\text{O}(\varepsilon^{-\nu/2})$ sites. For $\langle\varphi,V\varphi\rangle$
to also be small, we need $V$ to be small at (most of) these sites, certainly
no less than $1/2$ so we are looking at probabilities of order
$c^{\varepsilon^{-\nu/2}}$ with $0<c<1$. (One could argue that $c$ should be
$\varepsilon$ but that only introduces a log term in the exponent and would
restrict the form of the single site probability). In any event, the
expectation is that at least

$\lim_{E\downarrow 0}\log(-\log(k(E)))/\log(E)=-\nu/2$ (11.21)

the weakest form of Lifshitz tails (and the only one that I proved). The early
rigorous results in this area used the method of large deviations. My work was
motivated by a breakthrough of Kirsch-Martinelli who found the first proof
that used bare hands rather than some fancy probabilistic methods. They only
obtained results of the form (11.21) (which was weaker than some earlier work)
but for more general models. They relied on Dirichlet-Neumann bracketing [711,
Section 7.5] and treated continuum models. I wrote [673] mainly to advertise
their work but also to extend it to the discrete case. The most important
contribution of that paper was the use of Temple’s inequality which was often
used in later works. In [400], Kirsch and I proved results like (11.21) for
random perturbations of periodic problems near the bottom and top of the
spectrum. We could not handle the issue of showing there are also Lifshitz
tails near the internal gap edges, a problem that, so far as I know, remains
open, but I did handle the case of interior gaps in an Anderson model where
there are gaps due to gaps in the support of $d\kappa$ [675]. I shouldn’t
leave this subject without mentioning there are interesting issues involving
Lifshitz tails in random alloys with long range potentials and in magnetic
fields which are discussed in [399].

(3) _the notion of semi-uniform localization of eigenfunctions (SULE)_ Del
Rio, Jitomirskaya, Last and I [147, 148] illuminated what exponential
localization in random systems means (this work also discussed Hausdorff
dimension of singular continuous spectrum, so I will return to it in Section
12). To use the title of our paper aimed to physicists we dealt with the
question, “What is localization?”. At the time we wrote it, given the
acceptance of Anderson’s picture, many theoretical physicists would tell you
that a system on ${\mathbb{Z}}^{\nu}$ is localized (at all energies) means
there is a complete set of eigenfunctions,
$\\{\varphi_{\omega,m}\\}_{m=1}^{\infty}$, each obeying

$|\varphi_{\omega,m}(n)|\leq C_{\omega,m}e^{-A|n-n_{\omega,m}|}$ (11.22)

where $n_{\omega,m}$ is the center of localization of the $m$th eigenfunction.
Physically though, localization means that a function which at time zero lives
on a finite set should remain not too spread out uniformly in time. The
natural estimate is to expect that

${\mathbb{E}}\left(\sup_{t}|e^{-itH_{\omega}}(n,\ell)|\right)\leq
Ce^{-\tilde{A}|n-\ell|}$ (11.23)

Indeed, Delyon et al [152] proved this for 1D Anderson models and Aizenman [9]
proved this in high dimension for large coupling Anderson models. One point of