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\begin{document}
\title[Schr\"odinger Operators in the Twentieth Century]{Schr\"odinger Operators \\
in the Twentieth Century}
\author[B. Simon]{Barry Simon}
\address{Division of Physics, Mathematics, and Astronomy, 253-37\\
California Institute of Technology\\
Pasadena, CA~91125, USA}
\email{bsimon@caltech.edu}
\thanks{This material is based upon work supported by the National Science
Foundation under Grant No.~DMS-9707661. The Government has certain rights
in this material.}
\thanks{To appear in {\it Journal of Mathematical Physics}}
\dedicatory{Dedicated to Tosio Kato (1917--1999), \\
father of the modern theory of Schr\"odinger operators}
\date{February 3, 2000}
\maketitle
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} \lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The twentieth century is the century of science. In a century that has seen special
and general relativity, quantum electrodynamics and chromodynamics, a total revamping
of our understanding of molecules and of the cosmos, plate tectonics, and the rise of
microbiology, one can make the case that the most spectacular scientific development
was the discovery of nonrelativistic quantum mechanics in the first quarter of the
century. Its aftermath not only changed the physicist's view of matter but it set
the stage for the revolutions in chemistry, our understanding of stars, biology, and
practical electronics.
In what is one of the more striking cases of serendipity, just as Heisenberg and
Schr\"odinger were discovering the ``new" quantum theory, von Neumann was developing
the theory of unbounded self-adjoint operators and Weyl the representations of
compact Lie groups---two subjects of great relevance to the mathematics underlying
nonrelativistic quantum mechanics. In short order they produced books (von Neumann
\cite{Neu} and Weyl \cite{Weyl2}) that used this mathematics to give a mathematical
foundation to the framework of quantum mechanics. With later additions, notably by
Bargmann, Wigner, and Mackey, the basic foundations are mathematically firm.
This is analogous to having formulated classical mechanics as Hamiltonian flows on
symplectic manifolds. What remains is what might be called the second-level
foundations---existence of solutions of the time-dependent Schr\"odinger equation
(which is equivalent to self-adjointness of these operators) and general qualitative
issues in dynamics. It is this subject, essentially born fifty years ago, that
I will review here. The subject matter is vast with hundreds of contributors and
thousands of papers. Each section of this paper is a proxy for what deserves a
book or at least a very long review article. In attempting to overview such a vast
area in a few pages, I have had to focus on the high points. No proofs are given
and I have settled for usually quoting the initial or especially significant papers.
I have no doubt that I have left out some important papers, and if so, I ask the
forgiveness of the reader (and their authors!).
To keep this paper a reasonable size, I have focused almost entirely on the general
basics of Schr\"odinger operators and some simple applications to atomic and
molecular Hamiltonians. That means, among other areas, I haven't considered
general second-order operators on $\bbR^n$ and on general manifolds (but see
Davies-Safarov \cite{DaS}, Davies \cite{Dav2}, and Kenig \cite{Ken}) nor have
I considered some of the detailed papers on perturbations of Hamiltonians with
periodic potential (see, e.g., Deift-Hempel \cite{DH} and Gesztesy-Simon \cite{GS93})
nor the extensive literature on Dirac operators nor the considerable work on
Schr\"odinger operators in a bounded region with some boundary conditions
including subtle results on what happens at irregular boundary points (see
Davies \cite{Dav2}) nor the results on phenomena like the quantum Hall effect
that apply and extend the general theory to results in condensed matter physics.
While there are a few results about $-\Delta +V$ for cases where $V(x)\to
\infty$ as $|x|\to\infty$, again there is a large literature we won't attempt
to review. While Section~\ref{s10} has a brief discussion of constant magnetic
field, we have not attempted to discuss the recent extensive literature on
nonconstant magnetic fields.
There is a companion piece to this one on open problems (Simon \cite{Sinew}).
\medskip
I would like to thank Michael Aizenman, Brian Davies, Percy Deift, Fritz Gesztesy,
Dirk Hundertmark, Walter Hunziker, and Rowan Killip for useful input.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mathematical Tools and Issues} \lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The mathematics most relevant to the modern theory of Schr\"odinger operators is
functional, real, harmonic, and complex analysis. In this section, we will briefly
set the stage to fix notation. For more details, see Reed-Simon \cite{RS1,RS2}.
Quantum Hamiltonians are unbounded operators, defined on a dense subspace rather
than the whole Hilbert space. Physics books tend to emphasize the symmetry
(``Hermiticity") of the Hamiltonian; that is, that $\langle H\varphi,\psi\rangle =
\langle\varphi, H\psi\rangle$ for all $\varphi,\psi$ in $D(H)$. But more important is
a property called self-adjointness. The adjoint $H^*$ of an operator $H$ is defined
to be the maximal operator so that $\langle H^*\varphi,\psi\rangle = \langle \varphi,
H\psi\rangle$ for all $\psi\in D(H)$, $\varphi\in D(H^*)$. Hermiticity says only that
$H^*$ is an extension of $H$.
We say $H$ is self-adjoint if $H=H^*$, $H$ is called essentially self-adjoint if and
only if $H$ is symmetric and has a unique self-adjoint extension. This holds if and
only if $H^*$ is self-adjoint. Self-adjointness is important in the first place because
if $H$ is self-adjoint, one can form the unitary group $e^{-itH}$ and so solve
$i\, \dot{\varphi}_t = H\varphi_t$ (as $\varphi_t = e^{-itH}\varphi$) for initial
conditions $\varphi\in D(H)$. Indeed, Stone's theorem says that any one-parameter
continuous unitary group is associated to a self-adjoint operator. Secondly,
self-adjointness implies the spectral theorem. There is for each Borel set
$A\subset\bbR$, a projection, $E_A(H)$, so that $H=\int \lambda\, dE_\lambda$
and $e^{-itH} = \int e^{-itH}\, dE_\lambda$. One defines spectral measures
$d\mu^H_\varphi$ by
\begin{equation} \lb{2.1}
\mu^H_\varphi (A) = (\varphi, E_A(H)\varphi)
\end{equation}
so that
\begin{equation} \lb{2.2}
\int e^{-it\lambda} d\mu^H_\varphi(\lambda) = (\varphi, e^{-itH}\varphi)
\end{equation}
and
\begin{equation} \lb{2.3}
\int \f{d\mu^H_\varphi (\lambda)}{\lambda -z} = (\varphi, (H-z)^{-1}\varphi).
\end{equation}
$\sigma(H)$, the spectrum of $H$ is precisely, $\cup_\varphi\, \supp(d\mu^H_\varphi)$.
Much of what we discuss in this paper involves two distinct decompositions of
the spectrum of $H$. The first is
\begin{align*}
\sigma_{\disc}(H) &= \{\lambda \mid \lambda \text{ is an eigenvalue of finite
multiplicity} \\
&\qquad\qquad\text{ and an isolated point of $\sigma(H)$}\} \\
\sigma_{\ess}(H) &= \sigma(H) \backslash \sigma_{\disc}(H).
\end{align*}
Equivalently, $\lambda\in\sigma_{\disc} (H)$ if and only if for some $\veps >0$,
$\dim E_{(\lambda -\veps, \lambda +\veps)}(H)$ is finite and for all $\veps >0$,
$E_{(\lambda-\veps, \lambda +\veps )}(H)\neq 0$. $\sigma_{\disc}(H)$ captures the
notion of bound states.
The second breakup involves the fact that any measure $d\mu$ on $\bbR$ has a
decomposition
\[
d\mu = d\mu_{\pp} + d\mu_{\ac} + d\mu_{\singc},
\]
where $d\mu_{\pp}$ is a pure point measure (sum of delta functions), $d\mu_{\ac}$ is
$F(\lambda)\, d\lambda$, with $F$ a nonnegative locally integrable density, and
$d\mu_{\singc}$ is a singular continuous measure (like the Cantor measure).
I will define $\sigma_{\pp}(H)$ to be the set of eigenvalues of $H$; it is not
the union of the supports of $\mu_{\pp}$ because it may not be closed
\begin{align*}
\sigma_{\ac}(H) &= \bigcup_\varphi \, \supp (d\mu^H_\varphi)_{\ac} \\
\sigma_{\singc}(H) &= \bigcup_\varphi \, \supp (d\mu^H_\varphi)_{\singc}.
\end{align*}
One often defines a refined set $\Sigma_{\ac}$ with $\overline{\Sigma}_{\ac} =
\sigma_{\ac}(H)$, the essential support of the a.c.~measure. Basically, the essential
support of the a.c.~measure $F(\lambda)\, d\lambda$ is $\{\lambda \mid F(\lambda) \neq
0\}$. It is defined modulo sets of Lebesgue measure zero. $\Sigma_{\ac}$ is the union of
the essential support of $(d\mu^H_\varphi)_{\ac}$ over a countable dense set of
$\varphi$'s.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Self-Adjointness} \lb{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The theory of Schr\"odinger operators was born with Kato's famous self-adjointness
theorem for atomic Hamiltonians. His theorem abstracted says the following:
\begin{theorem}\lb{t3.1} \mbox{\rm (Kato \cite{Kato51a})} Let $\calH = L^2 (\bbR^{3N})$
where $x\in \bbR^{3N}$ is written $(x_1, \dots, x_N)$ with $x_i \in \bbR^3$. Let
$\Delta_i$ be the Laplacian in $x_i$ and let $V_i, V_{ij}$ be functions on $\bbR^3$
in $L^2 (\bbR^3) + L^\infty (\bbR^3)$. Let
\begin{align}
H_0 &= -\sum_{i=1}^N (2\mu_i)^{-1} \Delta_i \lb{3.1} \\
V &= \sum_{i=1}^N V_i (x_i) + \sum_{i0$ and all $\varphi \in D(A)$, that
\begin{equation} \lb{3.3}
\| B\varphi \| \leq \alpha \|A\varphi\| + \beta \|\varphi\|,
\end{equation}
then $A+B$ is self-adjoint on $D(A)$ and essentially self-adjoint on any domain
of essential self-adjointness for $A$. If \eqref{3.3} holds, we will say $B$ is
$A$-bounded. The infimum over all $\alpha$ is called the relative bound of $B$
with respect to $A$.
\smallskip
3. If one looks at a general bound of type \eqref{3.3} with $\alpha <1$ where
$A=-\Delta$ on $L^2 (\bbR^k)$ and $B$ is multiplication by $V$\!, then in terms of
requirements that $V\in L^p_{\loc} (\bbR^k)$, one needs
\begin{subequations}\lb{3.4}
\begin{alignat}{2}
p &\geq 2 & &\qquad k=1,2,3 \lb{3.4a} \\
p &> 2 & &\qquad k=4 \lb{3.4b} \\
p & \geq\tfrac{k}{2} & &\qquad k\geq 5 \lb{3.4c}
\end{alignat}
\end{subequations}
by using Sobolev estimates (see, e.g., Cycon et al. \cite{CFKS})
\smallskip
4. If $k=3N$ and we use {\it only} the $L^p$ requirements of Remark~3, Coulomb
potentials stop working already at $N=2$. Thus, for Kato's theorem, it is
critical to use Sobolev estimates in subsets of variables as Kato did.
\medskip
An industry developed in understanding when $-\Delta + V$ is essentially self-adjoint
on $C^\infty_0 (\bbR^n)$. An illustrative example is
\begin{example} \lb{e3.1} Let $H=-\Delta - c|x|^{-2}$ on $C^\infty_0 (\bbR^k)$ with
$n\geq 5$ (needed for $H\varphi \in L^2$ for all $\varphi\in C^\infty_0 (\bbR^k)$).
Then it can be seen (Reed-Simon \cite[Example 4 in Section X.2]{RS2}) that if
$c>c_0 = \f{(n-4)n}{4}$, then $H$ is not self-adjoint on $C^\infty_0$. This is a
quantum analog of the classical fact that if $V=-c|x|^{-2}$ for any $c>0$, a set
of initial conditions of positive measure falls into $x=0$ in finite time ($c_0 >0$
is a reflection of an uncertainly principle repulsion).
\end{example}
This example shows that for pure $L^p$ requirements, one cannot do better than
\eqref{3.4} since $|x|^{-2}\in L^p + L^\infty$ if $p<\f{k}{2}$. But it turns out
this is only so if $V$ is allowed to have any sign. For $V\geq 0$, one can do much
better. The best result of this genre is
\begin{theorem} \lb{t3.2} \mbox{\rm (Leinfelder-Simader \cite{LeS})} Let $V\geq 0$,
$V\in L^2_{\loc}(\bbR^k)$, \linebreak $\{a_j\}^k_{j=1} \in L^4_{\loc}(\bbR^k)$ with
$\nabla \cdot a\in L^2_{\loc}(\bbR^k)$ \rm{(}distributional derivatives\rm{)}. Then
\begin{equation} \lb{3.6}
H=\sum_{j=1}^k (i\partial_j - a_j)^2 + V
\end{equation}
is essentially self-adjoint on $C^\infty_0 (\bbR^k)$.
\end{theorem}
\smallskip
\noindent{\it Remarks.} 1. For a proof, see Cycon et al. \cite{CFKS}.
\smallskip
2. This is essentially a best possible result. If $a=0$, $H$ is defined on $C^\infty_0$
if and only if $V\in L^2_{\loc}$; so the result says for positive $V$\!, we have essential
self-adjointness if and only if $H$ is defined. Similarly, unless there are cancellations,
$a_j\in L^4_{\loc}$ and $\nabla\cdot a \in L^2_{\loc}$ is required for $H$ to be defined
on $C^\infty_0$.
\smallskip
3. It was Simon \cite{Si73ma} who first realized that for $V\geq 0$, there only needed to
be local $L^2$ conditions. However, he required a global condition $\int |V(x)|^2 e^{-bx^2}\,
dx<\infty$ for some $b>0$. It was Kato \cite{Kato78} who proved the general $a=0$ result
(and also allowed for smooth $a$'s). Kato's paper used the distributional inequality,
now called Kato's inequality
\begin{equation} \lb{3.7}
\Delta |u| \geq \Real (\sgn u \Delta u)
\end{equation}
that is also critical to the Leinfelder-Simader proof.
\smallskip
4. \eqref{3.7} is essentially equivalent to the fact that $e^{t\Delta}$ is
positivity preserving. The version of \eqref{3.7} with magnetic fields is equivalent
to diamagnetic inequalities:
\begin{equation} \lb{3.8}
|(e^{-tH}\varphi)(x)| \leq (e^{t\Delta} |\varphi|) (x)
\end{equation}
for the $H$ of \eqref{3.6} (with $V\geq 0$). These ideas were discovered by
Nelson \cite{Nel}, Simon \cite{Si76prl,Si79jfa}, and Hess-Schrader-Uhlenbrock \cite{HSU}.
\medskip
While there are best possible self-adjointness results for magnetic fields and positive
potentials, the results for $V$'s which can be negative are not in such a definitive
form. All the basic principles are understood but I'm not aware of a single result
that puts them all together (one of the best results is in Kato's paper \cite{Kato72}
although, as we will see, it is not quite optimal with regard to local singularities).
So I will present the general principles that are understood in this case.
\medskip
\noindent (a) $-|x|^2$ {\it borderline for behavior at infinity}. Negative potentials
$V$ of compact support for which $H=-\Delta + V$ is essentially self-adjoint on
$C^\infty_0$ normally obey a global estimate of the form \eqref{3.3} (with $A=-\Delta$,
$B=V$) and, in particular, $H$ is bounded from below. However, if $V$ is not of compact
support, it can go to minus infinity at infinity without destroying self-adjointness.
More or less, the borderline for keeping self-adjointness is $-|x|^2$. For example,
it can be proven (see, e.g., Reed-Simon \cite[Theorem X.9]{RS2}) that $-\f{d^2}{dx^2}
- |x|^\alpha$ on $L^2 (-\infty, \infty)$ is essentially self-adjoint on $C^\infty_0
(-\infty, \infty)$ if and only if $\alpha \leq 2$. This is attractive since a
classical particle with the same potential reaches infinity in finite time if and
only if $\alpha >2$. Nelson has examples (see Reed-Simon \cite{RS2}, p.~156) of
$V(x)$ with $V(x) \leq -cx^4$ so $-\f{d^2}{dx^2} + V(x)$ is still essentially
self-adjoint and thus, the borderline won't be if and only if, but the general
version of this is that if $V(x) \geq -cx^2$ in some averaged sense, then $-\Delta
+ V(x)$ will be essentially self-adjoint on $C^\infty_0$. The earliest
version of this is Ikebe-Kato \cite{IK}. My favorite theorem of this genre is due
to Faris and Lavine \cite{FL} (see Reed-Simon \cite[Theorem X.38]{RS2}). In particular,
Stark Hamiltonians where $V=\vec c\dott \vec x + V_0$ are essentially self-adjoint
for suitable $V_0$. In any event, I will focus henceforth on cases where $-\Delta
+ V$ is not unbounded from below.
\bigskip
\noindent (b) {\it Stability of relative boundedness under adding $V\geq 0$ or a
magnetic field}. Suppose $A\geq 0$. Then \eqref{3.3} holds for some $\alpha <1$
if and only if
\[
\lim_{\gamma\to\infty} \|B(A + \gamma)^{-1}\| <1.
\]
On the other hand, \eqref{3.8} implies that for $V\geq 0$, any $\vec a$ and any
multiplication operator $W$:
\[
\| W(H+\gamma)^{-1}\| \leq \| W(-\Delta + \gamma)^{-1}\|
\]
and so the second principle is that in studying the negative part of $V$\!, one can
assume $V$ is negative and then add back an arbitrary positive $L^2_{\loc}$
positive $V$\!. While this is true, it ignores situations where there are cancellations
between the positive and negative parts which can occur (see, e.g.,
Combescure-Ginibre \cite{CG}).
\bigskip
\noindent (c) {\it Relative bounds need only hold uniformly locally}. The following
proposition holds:
\begin{proposition} \lb{p3.3} Suppose $V$ is a function on $\bbR^d$ so that for some
$\alpha,\beta$ and every $y$,
\begin{equation} \lb{3.9}
\| V\chi (\dott -y) \varphi \| \leq \alpha \| -\Delta\varphi\| + \beta\|\varphi\|,
\end{equation}
where $\chi$ is the characteristic function of the unit cube. Then for any $\tilde\alpha
> \alpha$, there is some $\tilde\beta$ so that
\begin{equation} \lb{3.10}
\| V\varphi\| \leq \tilde\alpha \| -\Delta \varphi\| + \tilde\beta\|\varphi\|.
\end{equation}
\end{proposition}
This result is proven by a variant of an idea of Sigal \cite{Sig82}. Find a
``partition of unity" $\{j_\mu\}_\mu$ so that $\sum j^2_\mu = 1$, each $j_\mu$ is
supported in some unit cube (so $j_\mu \chi(\dott -y_\mu) = j_\mu$ for some $j_\mu$),
and the $j_\mu$'s are locally finite, $\sum(\vec\nabla j_\mu)^2$ is uniformly bounded
(the $j_\mu$'s can be translates of a single $j_\mu$) and $\sum |\Delta j_\mu|$ is
uniformly bounded. If $H_0 = -\Delta$, we have (where $C$ is related to $\|\sum
(\nabla j_\mu)^2\|_\infty$ and $\|\sum(\Delta j_\mu)\|_\infty$)
\[
\sum_\mu [j_\mu, [j_\mu, H^2_0]] \leq C(H_0 +1)
\]
and from this that
\begin{equation} \lb{3.11}
\sum \| H_0 j_\mu \varphi\|^2 \leq (1+\veps) \|H_0 \varphi\|^2 + C_\veps
\|\varphi\|^2.
\end{equation}
Thus
\begin{alignat*}{2}
\|V \varphi\|^2 &= \sum_\mu \|V\chi (\dott -y_\mu) j_\mu \varphi\|^2 \\
&\leq (1+\veps) \alpha^2 \sum_\mu \|H_0 j_\alpha \varphi\|^2 + (1+\veps^{-1})\beta^2
\|\varphi\|^2 &&\quad \text{(by \eqref{3.9}} \\
&\leq (1+\veps)^2 \alpha^2 \|H_0 \varphi\|^2 + ((1+\veps^{-1}) \beta^2 + C_\veps)
\|\varphi\|^2 &&\quad \text{(by \eqref{3.11}}
\end{alignat*}
which yields \eqref{3.10}.
Proposition~\ref{p3.3} says that the proper condition on $V$ to yield a $-\Delta$
bound is a uniform local condition.
\medskip
\noindent (d) {\it Convolution results are the proper local condition.} As discussed earlier,
$L^p$ conditions on $V$ do not properly control functions on subspaces. Explicitly, let
$\pi:\bbR^k \to\bbR^\ell$ be a projection and $V(x) = W(\pi(x))$. Then for $V$ to be
$-\Delta$ bounded (assuming $k\geq\ell\geq 5$), we need $W\in L^p_{\loc} (\bbR^\ell)$ for
$p \geq \f{\ell}{2}$ and so $V\in L^p_{\loc}(\bbR^k)$ with $p \geq \f{\ell}{2}$. But if $V$
is not a function of a subset of variables, in general we need $p \geq \f{k}{2}$. It is a
discovery of Stummel \cite{Stu} that by stating conditions in terms of convolution estimates,
one can find conditions that respect subsets of variables. In particular, the following
is a space, $S_\nu$, introduced in Stummel \cite{Stu}: Let $V$ be a function on $\bbR^\nu$;
we say $V\in S_\nu$ if and only if
\begin{gather*}
\lim_{\alpha\downarrow 0} \biggl[ \sup_x \int_{|x-y|\leq \alpha} |x-y|^{4-\nu}
|V(y)^2| \, d^\nu y\biggr] = 0 \quad \text{if } \nu\geq 5 \\
\lim_{\alpha\downarrow 0} \biggl[ \sup_x \int_{|x-y|\leq\alpha} \ln (|x-y|^{-1})
|V(y)|^2 \, d^\nu y\biggr] =0 \quad\text{if } \nu=4 \\
\sup_x \int_{|x-y|\leq 1} |V(y)|^2 \, d^\nu y < \infty \quad\text{if }\nu \leq 3.
\end{gather*}
This class respects functions of subvariables in the sense that if $\pi:\bbR^k\to
\bbR^\ell$ is a projection, $V(x) = W(\pi(x))$ and $W\in S_\ell$, then $V\in S_k$.
Moreover, it is not hard to show (see, e.g., Cycon et al. \cite{CFKS}) that if $V\in S_\nu$,
then $V$ is $-\Delta$-bounded with relative bound zero. Moreover (see Cycon et al.
\cite{CFKS}, Thm.~1.9), if for some $a,b>0$ and $\delta$ with $0<\delta <1$ and all
$0 < \veps <1$ and $\varphi\in D(H_0)$
\begin{equation} \lb{3.12}
\| V\varphi\|^2 \leq \veps\|\Delta\varphi\|^2 + a\exp(b\veps^{-\delta}) \|\varphi\|^2,
\end{equation}
then $V$ is in $S_\nu$. See Schechter \cite{Sche} for more on Stummel conditions.
\medskip
\noindent (3) {\it The Kato class and going beyond relative boundedness.} In his
inequality paper \cite{Kato78}, Kato introduced a form analog, $K_\nu$ of $S_\nu$:
Let $V$ be a function on $\bbR^\nu$; we say $V\in K_\nu$ if and only if
\begin{subequations} \lb{3.13}
\begin{gather}
\lim_{\alpha\downarrow 0} \biggl[ \sup_x \int_{|x-y|\leq\alpha} |x-y|^{2-\nu}
|V(y)|\, d^\nu y \biggr] = 0 \quad\text{if } \nu \geq 3 \lb{3.13a} \\
\lim_{\alpha\downarrow 0} \biggl[ \sup_x \int_{|x-y|<\alpha} \ln (|x-y|^{-1})
|V(y)|\, d^\nu y\biggr] =0 \quad\text{if } \nu=2 \lb{3.13b} \\
\sup_x \int_{|x-y|\leq 1} |V(y)|\, d^\nu y < \infty \quad\text{if } \nu=1. \lb{3.13c}
\end{gather}
\end{subequations}
Then Kato \cite{Kato72} proved if $\max(-V,0)\in K_\nu$ and $V\in L^2_{\loc}(\bbR)$, then
$-\Delta +V$ is essentially self-adjoint on $C^\infty_0 (\bbR^\nu)$. While it is
not Kato's proof, this is intimately connected with the semigroup result discussed
in the next section. Defining the form sum, $H$, one knows $\exp(-tH):L^2\to
L^\infty$ so $L^\infty \cap L^2 \cap D(H)$ is a domain of essential self-adjointness.
It is not hard to then show $L^\infty_0 \cap D(H)$, the $L^\infty$ functions of compact
support are a domain of essential self-adjointness. Then convolution allows one to get
$C^\infty_0$ approximations.
\medskip
\noindent (f) {\it Logarithmic improvements.} Neither $S_\nu$ nor $K_\nu$ is quite the
ideal space for essential self-adjointness. For example, if $\nu\geq 5$ and $V(x)=
|x|^{-2} (1+ | \log |x| \,| )^{-\alpha}$, $V$ is in $K_\nu$ only if $\alpha >1$, in
$S_\nu$ only if $\alpha > \f{1}{2}$, but $-\Delta$-bounded with relative bound zero
if $\alpha >0$.
\bigskip
Analogous to the issue of self-adjointness is a question of whether maximal and minimal
forms agree. This is discussed in Kato \cite{Kato78} and Simon \cite{Si79op}
(see Thm.~1.13 in Cycon et al. \cite{CFKS}).
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Properties of Eigenfunctions, Green's Functions, Semigroups, and All That} \lb{s4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I wrote a long review of these subjects twenty years ago (Simon \cite{Siss}) and the
situation has hardly changed since then, although there has been extensive interesting
work on what happens for general elliptic operators and for bounded regions (see, e.g.,
Davies \cite{Dav2}). So it will suffice to hit a few major themes.
The basic theorem is
\begin{theorem} \lb{t4.1} Let $V_+ \in L^1_{\loc} (\bbR^\nu)$ and $V_- \in K_\nu$, the
space of \eqref{3.13}. Let $H=-\Delta +V$ as a form sum. Then for any $p\leq q$, $e^{-tH}$
maps $L^p$ to $L^q$ and for $t\leq 1$,
\begin{equation}\lb{4.1}
\| e^{-tH}\|_{p,q} \leq C\, t^{-\alpha},
\end{equation}
where
\begin{equation}\lb{4.2}
\alpha = \f{\nu}{2} \left(\f{1}{p} - \f{1}{q}\right)\, .
\end{equation}
\end{theorem}
\noindent{\it Remarks.} 1. Semigroup $L^p$ bounds were first found by Davies \cite{Dav},
Herbst-Sloan \cite{HSl}, and Kovalenko-Semenov \cite{KoS} with further developments by Carmona
\cite{Car74}, Simon \cite{Sifi}, and Aizenman-Simon \cite{AiS}.
\smallskip
2. In particular, it was Aizenman-Simon \cite{AiS} who found that $K_\nu$ is the natural class
for $L^p$ bounds. Indeed, they not only proved Theorem~\ref{t4.1} in this form but also
showed that if $V\leq 0$ and $\exp(-tH)$ maps $L^\infty$ to itself with
$\lim_{t\downarrow 0} \|e^{-tH}\|_{\infty, \infty}=1$, then $V\in K_\nu$.
\smallskip
3. The result holds when magnetic fields are added (by a diamagnetic inequality).
\smallskip
4. Most of these authors use a combination of path integral estimates and
$L^p$-interpolation theory. In particular, the Feynman-Kac and Feynman-Kac-It\^{o}
formulae (see Simon \cite{Sifi} for extensive discussion) are useful tools in studying
Schr\"odinger operators. See Simon \cite{Silei} for an extension to cases when $V(x)
\geq -cx^2$.
\smallskip
5. In fact, $e^{-tH}$ takes $L^p$ not only into $L^\infty$ but into the continuous
functions (see Simon \cite[Thm.~B.3.1]{Siss}).
\smallskip
6. \eqref{4.1}/\eqref{4.2} are precisely the best results for $H=-\Delta$.
\smallskip
7. This theorem says that $H$ can be defined as the generator of a semigroup on
each $L^p$ space. The spectrum has been shown to be $L^p$ independent in
Hempel-Voight \cite{HV}. For a general discussion of $L^p$ Schr\"odinger operators,
see Davies \cite{Dav3}.
\medskip
Once one has these estimates, they can be used to derive:
\begin{enumerate}
\item[(a)] {\it Sobolev estimates}: As in the free case if $V$ obeys the conditions
of Theorem~\ref{t4.1}, then $(H-z)^{-\nu}$ takes $L^p$ to $L^q$ if
\begin{equation}\lb{4.2a}
p^{-1} - q^{-1} < \left( \f{2\alpha}{\nu}\right)\,.
\end{equation}
The result (see Simon \cite[Thm.~B.2.1]{Siss}, ) is obtained by integrating the
semigroup bound. \eqref{4.2a} comes from \eqref{4.2} and the requirement of
integrability at $t=0$.
\smallskip
\item[(b)] {\it Integral kernels}: Bounded operators from $L^1$ to $L^\infty$ have
bounded integral kernels and so Theorem~\ref{t4.1} can be used (see Simon
\cite[Thm.~B.7.1]{Siss}) to prove $e^{-tH}$, $(H-z)^{-\alpha}$ ($\alpha >
\f{\nu}{2}$) are integral operators with continuous integral kernels. One can also
show (Simon \cite[Thm.~B.7.2]{Siss}) that for $0<\alpha<\f{\nu}{2}$, $(H-z)^{-\alpha}$
is an integral operator with an integral kernel that is continuous away from $x=y$
with a precise singularity at $x=y$.
\smallskip
\item[(c)] {\it Eigenfunctions}: Since global eigenfunctions (i.e., $\varphi\in L^2$
that obey $H\varphi =E\varphi$) are in $\Ran(e^{-tH})$, Theorem~\ref{t4.1} implies
such eigenfunctions are in $L^\infty$. In fact, all this can be done locally. Any
eigenfunction (distributional solution of $H\varphi = E\varphi)$ is automatically
continuous and one can prove Harnack inequalities and subsolution estimates. This is
discussed in detail in Aizenman-Simon \cite{AiS} and Simon \cite{Siss}.
\end{enumerate}
\medskip
We end this section with a discussion of some issues involving eigenfunctions. There
is a huge literature on when Schr\"odinger operators have positive solutions. This was
begun by Allegretto \cite{All} and Piepenbrink \cite{Pie} with later results by
Agmon \cite{Ag84} and Pinchover \cite{Pin}.
Here is a typical theorem (Simon \cite[Thm.~C.8.1]{Siss}):
\begin{theorem}\lb{t4.2} Let $V_-\in K_\nu$ and $K_+\in K^{\loc}_\nu$. Then $Hu = Eu$
has a nonzero distributional solution which is everywhere positive if and only if
$\inf\spec (H)\geq E$.
\end{theorem}
\smallskip
There is also a huge literature on the issue of exponential decay of eigenfunctions. One
result (see Simon \cite[Thm.~C.3.1]{Siss}) says that any $L^2$ eigenfunction actually
goes to zero pointwise---of interest only for eigenfunctions of embedded eigenvalues.
For discrete spectrum, the decay is at least exponential under minimal regularity
hypothesis on $V$. The original key papers on this theme are by O'Connor \cite{Oco}
and Combes-Thomas \cite{CT}. From their ideas, one obtains (see Section~C.3 of
Simon \cite{Siss});
\begin{theorem}\lb{t4.3} Let $V_-\in K_\nu$, $V_+ \in K^{\loc}_\nu$ and let $H =
-\Delta + V$ and let $Hu=Eu$ with $u\in L^2$. Then
\begin{equation}\lb{4.3}
|u(x)| \leq C\, e^{-A|x|},
\end{equation}
where:
\begin{enumerate}
\item[{\rm (i)}] For general $E$ in the discrete spectrum, \eqref{4.3} holds for some
$A>0$ and $C>0$.
\item[{\rm (ii)}] If $H$ has compact resolvent, then \eqref{4.3} holds in the sense
for any $A>0$, there is a suitable $C>0$.
\item[{\rm (iii)}] If $\Sigma_{\ess} = \inf \sigma_{\ess}(H)$ and $E<\Sigma_{\ess}$,
then \eqref{4.3} holds in the sense that for any $A\leq \sqrt{E-\Sigma_{\ess}}$, there
is a suitable $C>0$.
\end{enumerate}
\end{theorem}
One can go beyond this to get fairly detailed behavior on decay in cases when $H$ has
compact resolvent or for $N$-body potentials. In one dimension, one can justify under
some regularity conditions the WKB formula that says when $V(x)\to\infty$, eigenfunctions
decay like
\begin{equation} \lb{4.4}
V(x)^{-1/4} \exp \biggl( -\int_a^x \sqrt{V(y) -E}\, dy\biggr).
\end{equation}
It was Agmon \cite{Ag82} who realized the proper higher-dimensional analog for this involves
what is now called the Agmon metric: $\rho(x)$ is the geodesic distance of $x$ to $0$
in the Riemannian metric $\rho_{ij}(x) = \delta_{ij}(V(x) - E)_+\, d^2 x$. There is
a related but more subtle definition for $N$-body systems. See Agmon \cite{Ag82} and
Deift et al. \cite{DHSV} for further discussions. See Simon \cite{Si84} and
Helffer-Sj\"ostrand \cite{HS1} for an application to tunnelling probabilities.
Eigenfunctions play a critical role in explicit spectral representations of
Schr\"odinger operators. The basic ideas go back to work of Browder \cite{Bro},
Garding \cite{Gar}, Gel'fand \cite{Gel}, Kac \cite{Kacg}, and especially
Berezanskii \cite{Ber56,Ber68}. See Section~C.5 of Simon \cite{Siss} and
Last-Simon \cite{LaS} for some addditional one-dimensional results.
Finally, we mention issues of cusps and nodes of eigenfunctions. Kato \cite{Kato57cpam}
has a famous paper on cusps at Coulomb singularities for atomic eigenfunctions.
See Hoffmann-Ostenhof et al. {\cite{HHHO,HHN,HHS} for recent developments in this area.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{One-Dimensional Decaying Potentials} \lb{s5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One-dimensional Schr\"odinger operators
\begin{equation} \lb{5.1}
-\f{d^2}{dx^2} + V(x)
\end{equation}
on $L^2 (-\infty, \infty)$ and $L^2 (0,\infty)$ and their discrete analogs
\begin{equation} \lb{5.2}
hu(n) = u(n+1) + u(n-1) + V(u) u(n)
\end{equation}
on $\ell^2 (-\infty, \infty)$ and $\ell^2 [0,\infty)$ have been heavily studied for
two reasons. First, ODE/difference equation methods allow one to study them in much
greater detail than one can the higher-dimensional analogs. Second, if $V(x) = V(|x|)$
is a spherically symmetric function on $\bbR^\nu$, then $-\Delta + V$ is unitarily
equivalent to a direct sum of operators on $L^2 (0, \infty)$ or the form \eqref{5.1}
where the effective $V$'s have the form $V_\ell (x) = \kappa_\ell |x|^{-2} + V(x)$
for suitable $\kappa_\ell$'s. The details can be found, for example, in Reed-Simon
\cite{RS2}, Example~4 to the Appendix for X.1.
The one-dimensional theory has been in and out of vogue. It was extensively studied
from 1930--1950 with important contributions by Titchmarsh, Kodaira, Gel'fand,
Hartman-Wintner, Levinson, Coddington-Levinson, and Jost. Significant developments
during the next twenty-five years were mainly in the area of inverse spectral theory
(a major exception was Weidmann's work \cite{Wei}, to be discussed shortly) which will
be discussed in Section~\ref{s6} below. From about 1975 starting with work of
Goldsheid, Molchanov, and Pastur \cite{GMP} and Pearson \cite{Pea}, this has been
an active area with extensive study of the one-dimensional case, especially with
long-range and with ergodic potentials.
One special feature of one dimension is that one can limit spectral multiplicities
under very general conditions on $V$:
\begin{theorem} \lb{t5.1} {\rm (a)} Let $H = -\f{d^2}{dx^2} + V(x)$ on $L^2 (0,\infty)$
with fixed $hu(0) + u'(0)=0$ boundary conditions and suppose $H$ is essentially
self-adjoint on $C^\infty_0 [0,\infty)$. Then $H$ has simple spectrum
{\rm(}multiplicity $1${\rm)}.
{\rm (b)} Let $H = -\f{d^2}{dx^2} + V(x)$ on $L^2 (-\infty,\infty)$ and suppose $H$
is essentially self-adjoint on $C^\infty_0 (-\infty, \infty)$. Then
\begin{enumerate}
\item[{\rm (i)}] The a.c. spectrum of $H$ is of multiplicty at most $2$.
\item[{\rm (ii)}] The singular spectrum of $H$ is of multiplicity $1$.
\end{enumerate}
\end{theorem}
\noindent{\it Remarks.} 1. All one needs for local regularity of $V$ is $V\in
L^1 [0,R]$ for all $R>0$ or $L^1_{\loc}(-\infty, \infty)$.
\smallskip
2. The result holds even if $H$ is not essentially self-adjoint ($V$ limit circle at
$\pm\infty$) so long as a boundary condition is imposed at $\infty$ or at $-\infty$.
\smallskip
3. The only subtle part of the result is that the s.c.~spectrum is simple on the
real line. This is a theorem of Kac \cite{Kacis1,Kacis2}; see also Berezanskii
\cite{Ber56,Ber68}. My preferred proof is due to Gilbert \cite{Gil,Gil2}.
\medskip
In this section, we will discuss the case where $V(x)\to 0$ at infinity. In the next
section, we will discuss inverse spectral theory, and in the section after that, we will
discuss ergodic potentials. (These two subjects are mainly one-dimensional.) The issue of
the asymptotic eigenvalue distribution when $V\to\infty$ as $\pm\infty$ is discussed in
Section~\ref{s14} on the quasiclassical limit.
This section will discuss \eqref{5.1}/\eqref{5.2} in situations where $V(x)$ (or $V(n)$)
goes to zero (at least in an average sense) as $x\to\infty$ (or $n\to\infty$). The
interesting thing is that there are three natural breaks in behavior. Expressed in terms
of $|x|^{-\alpha}$ behavior, they are
\begin{enumerate}
\item[(i)] At $\alpha=2$, we shift between a finite number of bound states ($\alpha >2$)
or an infinite number ($\alpha <2$) at least if $V(x) <0$.
\item[(ii)] At $\alpha =1$ ($V\in L^1$), we shift between a pure scattering situation
for positive energies ($\alpha >1$) and the possibility of positive energy bound states
($\alpha <1$).
\item[(iii)] At $\alpha =\f{1}{2}$, ($V\in L^2$), we shift from there being a.c.~spectrum
for a.e.~positive energy ($\alpha > \f12$) to at least the possibility of very different
spectrum.
\end{enumerate}
(i) and (ii) have been known since the earliest days of quantum mechanics. The $\alpha =
\f12$ borderline first occurred in Simon \cite{Si82} who found that random decay potentials
had point spectrum when $\alpha <\f12$. Delyon, Simon, and Souillard \cite{DSS} then showed
if $\alpha = \f12$, there may be some nonpoint spectrum. As we will see, subsequent results
confirmed this borderline.
The negative spectrum for decaying potentials is easy: So long as \linebreak
$\int_x^{x+1} |V(y)| \, dy\to 0$, $H$ is bounded below and has $[0,\infty)$ as
essential spectrum by Weyl's criterion (see, e.g., Reed-Simon \cite[Section~XIII.4]{RS3}),
which means that $(-\infty, 0)$ has only discrete eigenvalues of finite multiplicity, which
can only accumulate at energy $0$. Indeed, by Theorem~\ref{t5.1}, the point spectrum is of
multiplicity $1$. Once these basics are established for the discrete spectrum, a number of
detailed questions about it arise:
\smallskip
\noindent (a) {\it Is $\sigma_{\disc}$ finite or infinite?} The borderline, as mentioned
above, is $r^{-2}$ decay. Explicitly, one has Bargmann's bound \cite{Barg} that the number
of eigenvalues on a half-line with $u(0)=0$ boundary conditions is bounded by $\int x
|V(x)|\, dx$ and on a whole line by $1 +\int_{-\infty}^\infty |x| \, |V(x)| \, dx$
(see Simon \cite{Sivb} for a review of bounds on the number of bound states). On the
other hand, if $\varlimsup_{x\to\infty} |x|^2 V(x) \leq -\f14$, one can prove that $H$
has an infinity of bound states (see, e.g., Reed-Simon \cite[Thm.~XIII.6]{RS3}).
\smallskip
\noindent (b) {\it If $\sigma_{\disc}$ is infinite, how does $\lim_{\lambda\uparrow 0}
\dim E_{(-\infty, \lambda)}(H)$ diverge?} This is a quasiclassical limit and discussed
in Section~\ref{s14} below.
\smallskip
\noindent (c) {\it Bounds on moments of eigenvalues.} Lieb and Thirring \cite{LT76},
motivated in part by their work on the stability of matter \cite{LT75}, initiated
extensive study on the best constant $L_{\gamma, 1}$ in
\[
\sum_j |e_j|^\gamma \leq L_{\gamma, 1} \int |V(x)|^{\gamma + 1/2} \, dx,
\]
which holds if $\gamma\geq \f12$. Here $\{e_j\}$ are the negative eigenvalues of $H$.
For $\gamma \geq \f32$, the constant $L_{\gamma, 1}$ is known to be quasiclassical
(Aizenman-Lieb \cite{AL}). For $\gamma\in [\f12, \f32)$, it is known that $L_{\gamma, 1}$
is strictly larger than the quasiclassical result \cite{LT76}. It is conjectured to be
the optimal value for a single bound state, as explained in Lieb-Thirring \cite{LT76},
but this is still open (except at $\gamma = \f12$ (Hundertmark-Lieb-Thomas \cite{HLT})).
\smallskip
\noindent (d) {\it Is there a bound state for weak coupling?} In one (and two)
dimensions, $H$ has bound states even for very weak coupling. The result
(Simon \cite{Si76ap}) is that if $\int |x|\, |V(x)|\, dx < \infty$ and $\int V(x)\,
dx \leq 0$ and $V\neq 0$, then $H$ always has a bound state and the binding energy of
$-\Delta +\mu V$ is $\sim c\mu^2$ as $\mu\downarrow 0$ (if $\int V(x)\, dx <0$; it is
$\sim c\mu^4$ if $\int V(x) =0$).
\medskip
As for positive energies, the situation is simple if $V\in L^1$:
\begin{theorem} \lb{t5.2} Let $V\in L^1 (-\infty, \infty)$ or $L^1 (0, \infty)$. Then
$HE_{(0,\infty)}(H)$ is unitarily equivalent to $-\f{d^2}{dx^2}$ {\rm (}on $L^2 (-\infty,
\infty)$ or $L^2 (0, \infty)$ with $u(0)=0$ boundary conditions{\rm )}.
\end{theorem}
\noindent{\it Remarks.} 1. This result is essentially due to Titchmarsh \cite{Tibook}.
\smallskip
2. In terms of $r^{-\alpha}$ falloff, $V\in L^1$ means $\alpha >1$.
\smallskip
3. Using scattering theoretic ideas, one can prove wave operators exist and are complete
(see Section~\ref{s8}).
\smallskip
4. This says there is no point of singular continuous spectrum at positive energies and
that the a.c.~spectrum has essential support $(0,\infty)$ with multiplicity $2$ or $1$.
\smallskip
5. We have stated the result for $u(0)=0$ boundary condition for simplicity; it holds for
all boundary conditions at $0$.
\medskip
As for slower decay than $L^1$, if one has control of derivatives, one can still conclude
the positive spectrum is purely absolutely continuous. The simplest result of this genre
is
\begin{theorem} \lb{t5.3} \mbox{\rm (Weidmann \cite{Wei})} Let $V=V_1 + V_2$ where $V_1$ is in
$L^1$, $V_2(x)\to 0$ as $x\to\pm\infty$, and $V_2$ is of bounded variation. Then,
$HE_{(0,\infty)}(H)$ is unitarily equivalent to $-\f{d^2}{dx^2}$ {\rm (}on $L^2 (-\infty,
\infty)$ or on $L^2 (0, \infty)$ with $u(0)=0$ boundary conditions{\rm )}.
\end{theorem}
\noindent{\it Remarks.} 1. $V_2$ of bounded variation with $V_2\to 0$ at infinity
essentially says that $-\f{dV_2}{dx}\in L^1$; in fact, any $V_2$ of bounded variation
can be written $V_3 + V_4$ with $V_3 \in L^1$ and $V_4$ a $C^1$ function with $\f{dV_4}
{dx} \in L^1$.
\smallskip
2. Pure power potentials $r^{-\alpha}$ for any $\alpha >0$ are included in this
theorem; indeed, any monotone function $V(x)$ with $V(x)\to 0$ as $x\to\infty$
is of bounded variation.
\medskip
For a short proof of Theorems~\ref{t5.2}/\ref{5.3}, see Simon \cite{Si96}. Both theorems
can be understood as coming from the fact that all solutions of $-u''+Vu =\lambda u$
with $\lambda >0$ are bounded. That such a conclusion implies the spectrum is purely
absolutely continuous was first indicated by Carmona \cite{Car83} (who required some kind of
uniformity in $\lambda$). Important later developments that capture this idea are
due to Gilbert-Pearson \cite{GP}, Last-Simon \cite{LaS}, and Jitomirskaya-Last \cite{JL}.
The tools in those papers are also important for the proofs of the results of Section~\ref{s7}.
Once one allows decay slower than $r^{-1-\veps}$ for both $V$ and $V'$, the
conclusion of Theorems~\ref{5.2}/\ref{5.3} can fail because of embedded point
spectrum. The original examples of this were found by von Neumann-Wigner \cite{VNW}.
Basically, if $V(x)=\gamma |x|^{-1} \sin(x)$ for $x$ large and $\gamma >1$, then
$-u'' + Vu = \f14 u$ has a solution which is $L^2$ at infinity (see, e.g., Theorem~XI.67
in Reed-Simon \cite{RS3}). By adjusting $V$ at finite $x$, one can arrange for any boundary
condition one wants at $x=0$. In fact, if one allows slightly slower decay than
$|x|^{-1}$, one can arrange dense point spectrum. Naboko \cite{Nab} and Simon
\cite{Si97} have shown that for any sequence $\{\lambda_n\}_{n=1}^\infty$ of energies
in $(0,\infty)$ (Naboko has a mild restriction on the $\lambda$'s) and any $g(r)$
obeying $\lim_{r\to\infty} r\, g(r) = \infty$, there is a $V(x)$ obeying:
\begin{enumerate}
\item[(i)] $|V(x)| \leq g(|x|)$ for $x$ large
\item[(ii)] $-u'' + Vu = \lambda_n u$ has a solution $L^2$ at infinity
\end{enumerate}
and obeying a prescribed boundary condition at $x=0$.
\smallskip
\noindent{\it Remark.} It is an interesting open question about whether there exist
potentials decaying faster than $|x|^{-1/2-\veps}$ with dense singular continuous
spectrum (rather than dense point spectrum).
\smallskip
The interesting fact is that even though potentials of Naboko-Simon type have dense
point spectrum, they may also have lots of a.c.~spectrum. The best result is:
\begin{theorem} \lb{t5.4} \mbox{\rm (Deift-Killip \cite{DK})} Let $V\in L^2$. Then the
essential support of the a.c.~spectrum of $H=-\f{d^2}{dx^2} + V$ is $[0,\infty)$.
\end{theorem}
\noindent{\it Remarks.} 1. In terms of $r^{-\alpha}$ decay, this result requires
$\alpha > \f12$.
\smallskip
2. This result is optimal in that it is known for any Orlicz space strictly larger
than $L^2$ in terms of behavior at infinity, there are $V$'s whose associated $H$
has no a.c.~spectrum.
\smallskip
3. The first result of this genre was found by Kiselev \cite{Kis96} who proved the
conclusion of this theorem for $|V(x)| \leq C \, x^{-3/4 - \veps}$. There were
subsequent improvements of this by Kiselev \cite{Kis98}, Christ-Kiselev \cite{CK},
and Remling \cite{Rem98}.
\smallskip
4. Killip \cite{Kil} has a partially alternate proof of Theorem~\ref{t5.4}.
\medskip
Once the decay is allowed to be slower than $r^{-1/2}$, one can have much different
spectrum in $[0, \infty)$:
\begin{enumerate}
\item[(i)] If $W$ is a suitable family of random homogeneous potentials and $V(x) =
|x|^{-\alpha} W(x)$ with $\alpha <\f12$, then $H$ has only dense point spectrum in
$(0,\infty)$. This was first proven in the discrete case by Simon \cite{Si82}
and later in the continuum case by Kotani-Ushiroya \cite{KU}.
\item[(ii)] Generic potentials decaying like $|x|^{-\alpha}$ ($\f12 > \alpha >0$)
produce singular continuous spectrum as discovered by Simon \cite{Si95}. For
example, in $\{V\in\bbC(\bbR) \mid \sup_x |x|^\alpha |V(x)| \equiv \| V\|_\alpha \}$
viewed as a complete metric space in $\| \dott\|_\alpha$, a dense $G_\delta$ of $V$'s
are such that $-\f{d^2}{dx^2} + V(x)$ has purely singular continuous spectrum on
$[0,\infty)$.
\item[(iii)] Much more is known in the borderline $\alpha=\f12$ case, at least for
the discrete Schr\"odinger operator \eqref{5.2}. For example, if $a_n$ are
independent, identically distributed random variables uniformly distributed in
$[-1,1]$ and $V(n) = \mu n^{-1/2} a_n$, then for suitable coupling constants $\mu$
and energies $E$ in $[-2,2]$, the spectral measures have fractional Hausdorff
dimension with an exactly computable local dimension. This is discussed in
Kiselev-Last-Simon \cite{KLS}. There are earlier results on this model by
Delyon-Simon-Souillard \cite{DSS} and Delyon \cite{Del}.
\item[(iv)] A very different class of decaying potentials was studied by
Pearson \cite{Pea}. His potentials are of the form
\begin{equation} \lb{5.3}
V(x) = \sum_{n=1}^\infty a_n \, W(x-x_n),
\end{equation}
where $W\geq 0$, $a_n\to 0$, and $x_n\to \infty$ very rapidly so the bumps are sparse.
He showed that for suitable $a_n, x_n$, the corresponding $H$ has purely singular
spectrum---providing the first explicit examples of such spectrum. Strong versions
of his results were found by Remling \cite{Rem97} and Kiselev-Last-Simon \cite{KLS}.
In particular, the latter authors proved if $\f{x_{n+1}}{x_n}\to\infty$ (e.g., $x_n = n!$),
then potentials of the form \eqref{5.3} lead to $H$'s with purely singular spectrum if
$\sum a^2_n = \infty$ and to ones with purely a.c.~spectrum if $\sum a^2_n < \infty$.
\end{enumerate}
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Inverse Spectral Theory} \lb{s6}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One area related to Schr\"odinger operators, especially in one dimension, is the question
of inverse theory: How does one go from spectral or scattering information to the
potential. There is a huge literature, including three books I would like to refer the
reader to: Chadan-Sabatier \cite{CS}, Levitan \cite{Lev87}, and Marchenko \cite{Mar86}.
I will only touch some noteworthy ideas here.
In one dimension, a key role is played by the Weyl $m$-function and the associated
spectral measure, $d\rho$. Given a potential $V$ so that $H$ is self-adjoint with
$u(0)=0$ boundary conditions, for each $z$ with $\Ima z>0$, there is a solution
$u(x;z)$ of $-u'' + Vu =zu$ which is $L^2$ at infinity. The $m$-function is defined
by
\begin{equation} \lb{6.1}
m(z) = \f{u'(0;z)}{u(0,z)}.
\end{equation}
$\Ima m(z) >0$ in $\Ima z>0$ so by the Herglotz representation theorem
\begin{equation} \lb{6.2}
m(z) = B + \int d\rho(\lambda) \biggl[ \f{1}{\lambda - z} - \f{\lambda}{1 + \lambda^2}
\biggr]
\end{equation}
for a suitable constant $B$. $d\rho$ is called the spectral measure for $H$. One can
recover $d\rho$ from $m$ by
\begin{equation} \lb{6.3}
\f{1}{\pi}\, \Ima m(\lambda + i\veps)\, d\lambda \to d\rho(\lambda)
\end{equation}
weakly as $\veps\downarrow 0$ and \eqref{6.2} allows the recovery of $m$ from $d\rho$
given the known asymptotics (Atkinson \cite{Atk}, Gesztesy-Simon \cite{GS})
\begin{equation} \lb{6.3*}
m(-\kappa^2) = -\kappa + o(1)
\end{equation}
as $|\kappa| \to \infty$ with $\delta < \Arg \kappa < \f{\pi}{2} - \delta$. $d\rho$ really
is a spectral measure for let $\tilde\varphi (x,\lambda)$ solve $-\tilde\varphi''+ V
\tilde\varphi = \lambda\tilde\varphi$ with boundary conditions $\tilde\varphi (0, \lambda)
=0$, $\tilde\varphi'(0,\lambda) =1$, and define for $f\in C^\infty_0 (0,\infty)$
\begin{equation} \lb{6.4}
(Uf)(\lambda) = \int \tilde\varphi (x,\lambda) f(x)\, dx.
\end{equation}
Then $U$ is a unitary map of $L^2 (0,\infty, dx)$ to $L^2 (\bbR, d\rho(\lambda))$;
in particular,
\begin{equation} \lb{6.5}
\int |(U(f)(\lambda)|^2\, d\rho(\lambda) = \int |f(x)|^2 \, dx
\end{equation}
or formally
\begin{equation} \lb{6.6}
\int \varphi(x,\lambda) \varphi (y,\lambda)\, d\rho(\lambda) = \delta(x-y).
\end{equation}
Moreover, $(UHf)(\lambda) = \lambda (Uf)(\lambda)$. $d\rho$ and its equivalent function $m$
is therefore close to spectral information. One way of seeing this explicitly is if $V(x)\to
\infty$. In that case, $m$ is meromorphic, the poles of $m$ are precisely the eigenvalues
of $H$ with $u(0)=0$ boundary conditions and by definition of $m$, the zeros are precisely
the eigenvalues with $u'(0)=0$ boundary conditions. $m$ is uniquely determined by these
two sets of eigenvalues.
In many ways, the fundamental result in inverse theory is the following one:
\begin{theorem} \lb{t6.1} \mbox{\rm (Borg \cite{Borg}-Marchenko \cite{Mar50})} $m$
determines $q$, that is, if $q_1$ and $q_2$ have equal $m$'s, then $q_1 = q_2$.
\end{theorem}
Recently, the following local version of the Borg-Marchenko theorem was proven
\begin{theorem} \lb{t6.2} Let $q_1$ and $q_2$ be potentials and $m_1$ and $m_2$
their $m$-functions. Then $q_1 = q_2$ on $[0,a]$ if and only if
\[
|m_1 (-\kappa^2) - m_2 (-\kappa^2)| = O(e^{-2a\kappa})
\]
as $\kappa\to\infty$ for $\kappa$ obeying $\delta \leq \arg \kappa \leq \f{\pi}{2}
-\delta$.
\end{theorem}
\noindent{\it Remarks.} 1. This result was first proven by Simon \cite{Si99} when
$q_1$ and $q_2$ are bounded from below.
\smallskip
2. The general result which even allows $q_i$ to be limit circle at infinity was first
obtained by Gesztesy-Simon \cite{GS}.
\smallskip
3. A simple proof of Theorem~\ref{t6.2} was subsequently obtained by Gesztesy-Simon
\cite{GSnew}.
\medskip
Given the uniqueness result, it is natural to ask about concrete methods of determining
$q$ given $m$. There are two approaches for the general case. The first is due to
Gel'fand-Levitan \cite{GL} and depends on the orthogonality relation \eqref{6.6}, while
the other, due to Simon \cite{Si99}, is a kind of continuum analog of the continued
fraction approach to solving the moment problem.
The Gel'fand-Levitan approach depends on a representation of the solutions $\varphi$
due to Povzner \cite{Pov48} and Levitan \cite{Lev73}:
\begin{equation} \lb{6.7}
\varphi(x,\lambda) = \f{\sin (kx)}{k} + \int_0^x K(x,y) \, \f{\sin (ky)}{k} \, dy,
\end{equation}
where $\lambda = k^2$. In essence, \eqref{6.6} leads to a linear Volterra integral
equation for $K$ whose kernel is determined by $\rho$. Once one has $K$, one can
determine $V$ from \eqref{6.7} and $-\varphi'' + V\varphi = \lambda\varphi$ or from
more direct relations of $K$ to $V$.
The approach of Simon depends on a representation of $m$ as a Laplace transform:
\begin{equation} \lb{6.8}
m(-\kappa^2) = -\kappa - \int_0^a A(\lambda) \, e^{-2\kappa a}\, d\alpha +
O(e^{-2a\kappa}),
\end{equation}
which determines $A$ given $m$ (there is also a direct relation of $A$ to $\rho$
given in Gesztesy-Simon \cite{GS}). One can introduce a second variable and function
$A(x,\alpha)$ so $A(x=0,\alpha) \equiv A(\alpha)$. $A$ obeys
\begin{equation} \lb{6.9}
\f{\partial A}{\partial x} = \f{\partial A}{\partial\alpha} + \int_0^\beta A(x,\beta)
A(x, \alpha -\beta)\, d\beta
\end{equation}
and
\begin{equation} \lb{6.10}
\lim_{\alpha\downarrow 0} A(x, \alpha) \equiv V(x).
\end{equation}
In this approach, $m$ determines $A(x=0, \dott)$ by \eqref{6.8}; the differential equation
\eqref{6.9} determines $A(x,\alpha)$, and then \eqref{6.10} determines $V$.
Inverse spectral theory is connected to inverse scattering for short-range potentials
since $d\rho$ on $[0,\infty)$ is determined by scattering data. Scattering data also
determine the positions of the negative eigenvalues. One needs to supplement that with
the weight of the pure points at these negative eigenvalues known as norming constants.
Marchenko \cite{Mar86,Mar73} has an approach to inverse scattering related to the
Gel'fand-Levitan approach by using a different representation than \eqref{6.7}.
When $\int_0^\infty x |V(x)|\, dx < \infty$, Levin \cite{Levin} has proven that in
$\Ima k>0$, there is a solution $\psi (x,k)$ of $-\psi'' + V\psi = k^2 \psi$ given by
\[
\psi(x,k) = e^{ixk} + \int_x^\infty \tilde K(x,y)\, e^{iky}\, dy.
\]
Krein \cite{Kre1,Kre2,Kre3} also developed an approach to inverse problems.
A different approach to inverse scattering is due to Deift-Trubowitz \cite{DT}.
For another approach to inverse problems, see Melin \cite{Mel}. Inverse
theory for periodic potentials also has an extensive literature starting with
Dubrovin-Matveev-Novikov \cite{DMN}, Its-Matveev \cite{IM}, McKean-van Moerbeke
\cite{MVM}, McKean-Trubowitz \cite{MT}, and Trubowitz \cite{Tru}.
As for higher-dimensional inverse scattering, these scattering data overdetermine the
potential. For example, for short-range $V$'s, the scattering amplitude at fixed
momentum transfer approaches the Fourier transform of $V$ at large energy, so the
large energy asymptotics of scattering determine $V$. There is considerable
literature on recovering $V$ from partial scattering data, which we will not try
to summarize here.
One reason for the interest in inverse theory is the connection it sets up
between spectral theory of Schr\"odinger operators and the analysis of certain
nonlinear PDEs like KdV (see Dodd et al. \cite{DEGM}, Novikov et al. \cite{NMPZ},
and Belokolos \cite{BBE}).
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ergodic Potentials} \lb{s7}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\Omega$ be a compact metric space with probability measure $d\gamma$ and $T_t$
with $t\in R^\nu$ or $T_n$ with $n\in\bbZ^\nu$ be an ergodic family of
measure-preserving transformations. Let $f:\Omega\to\bbR$ be continuous. For
$\omega\in\Omega$, define
\begin{equation} \lb{7.1}
V_\omega (x) = f(T_x \omega)
\end{equation}
and
\begin{equation} \lb{7.2}
H_\omega = -\Delta + V_\omega .
\end{equation}
{\it Note}: To allow unbounded $V$'s as seen, for example, in Gaussian random potentials,
one wants to extend this picture to either allow $f$ to be discontinuous and/or take
values in $\bbR\cup \{\infty\}$, and/or allow $\Omega$ to be noncompact; for simplicity,
we will discuss this model for motivation.
$H_\omega$ is a family of Schr\"odinger operators, not a single one, but by the
ergodicity and an obvious translation covariance $V_{T_y\omega}(x) = V_\omega (x+y)$,
many spectral properties occur with probability one. So one can speak of typical
properties. In particular, it is known that the full spectrum $\Sigma$, the essential
support of the absolutely continuous spectrum $\Sigma_{\ac}$, the closure of the point
spectrum $\overline{\Sigma}_{\pp}$, and the singular continuous spectrum $\Sigma_{\singc}$
are a.e.~constant in $\omega$ (see, e.g., Theorems~9.2 and 9.4 in Cycon et al.
\cite{CFKS} for proofs; the result for $\Sigma$ is due to Pastur \cite{Pas} and the
other results to Kunz-Souillard \cite{KuS}). Note only $\overline{\Sigma}_{\pp}$ is
a.e.~constant; $\Sigma_{\pp}$, the actual set of eigenvalues is not.
\smallskip
\noindent{\bf Examples.} 1. Let $\Omega = [a,b]^{\bbZ^\nu}$ and let $d\gamma$ be the
infinite product of normalized Lebesgue measure on $[a,b]$. Let $(T_m\omega)_n =
\omega_{n+m}$. The corresponding discrete Schr\"odinger operator is called the Anderson
model and is typical of random potential models.
\smallskip
2. If $\Omega$ is a compact Abelian group with $\bbZ^\nu$ or $\bbR^\nu$ as dense
subgroup, $d\gamma$ is Haar measure and $T_x$ is group translate, then $V$ is a periodic
or almost periodic function. A frequently discussed example is
\begin{equation} \lb{7.3}
V(n) = \lambda \cos (\pi \alpha n + \theta),
\end{equation}
where $\alpha$ is irrational, $\theta$ runs in $[0,2\pi)$ (which is $\Omega$), and
$\lambda$ is a parameter. The corresponding discrete Schr\"odinger operator is called
the almost Mathieu model.
\medskip
The simplest example of this framework---which is atypical in many ways---is the
periodic potential. The basic facts in this case go back to the physics literature at
the start of quantum mechanics (Bloch, Brillouin, Kramer, and Wigner) and, in one
dimension, to work on Hill's equation (Floquet, Lyapunov, Hamel, and Haupt). A critical
early mathematical paper on the multidimensional case is Gel'fand \cite{Gel}. The key
result is that for periodic $V$'s with a mild local regularity condition, $H=-\Delta +V$
has purely absolutely continuous spectrum. This result is discussed in detail in
Reed-Simon \cite[Section XIII.16]{RS4}. The only subtle part of the argument is to
eliminate the possibility of what are called flat bands, a result of Thomas \cite{Tho}.
In the mathematical physics literature, the period from 1975 onwards has seen enormous
interest in the study of almost periodic and random models and special cases thereof.
Three books that discuss this are part of Carmona-Lacroix \cite{CaL}, Cycon et al.
\cite{CFKS}, and Pastur-Figotin \cite{PF}. We will only touch some of the general
principles, leaving the details---especially of detailed models---to the books and
the vast literature. We will make references to the Lyapunov exponent without defining
it; see Cycon et al. \cite{CFKS}, Section 9.3.
For random potentials, the most interesting results concern localization. While
the spectrum is typically an interval (e.g., for the Anderson model in $\nu$-dimensions,
it is $[a-2\nu, b+2\nu]$), the spectrum is pure point with eigenvalues dense in the
interval and exponentially decaying eigenfunctions.
In one dimension, localization was first rigorously proven by Goldsheid, Molchanov, and
Pastur \cite{GMP} with a later alternative by Kunz-Souillard \cite{KuS}. Following an
idea of Kotani \cite{Kot}, Simon-Wolff \cite{SW} and Delyon-Levy-Souillard \cite{DLS}
found another proof. Typical is
\begin{theorem} \lb{t7.1} For the one-dimensional Anderson model, the spectrum is
$[a-2, b+2]$ and is pure point with probability one with eigenfunctions decaying
at the Lyapunov rate.
\end{theorem}
Carmona-Klein-Martinelli \cite{CKM} and Shubin-Vakilian-Wolff \cite{SVW} have approaches
that work if the single site distribution is discrete (the other quoted approaches
require an absolutely continuous component for this distribution).
In higher dimensions, the two main approaches to localization are due to
Fr\"ohlich-Spencer \cite{FS} (see also von Dreifus-Klein \cite{VDK}) and to
Aizenman-Molchanov \cite{AM}. (See also Aizenman-Graf \cite{AG} and Aizenman
et al. \cite{ASFH}.) Basically, these authors and the many papers that extend
their ideas prove dense point spectrum in regimes where the coupling constant is
large or one is near the edge of the spectrum. It is believed---but not proven---that
in suitable regimes when $\nu\geq 3$, there is absolutely continuous spectrum.
For almost periodic models, one can have any kind of spectral type. The almost
Mathieu model has been almost entirely analyzed and the spectral type shows a
great variety. Recall this is the discrete model with potential
\[
V_{\alpha,\lambda,\theta}(n) = \lambda \cos(\pi\alpha n + \theta),
\]
where $\lambda,\alpha$ are fixed parameters and $\theta$ runs through $\Omega$. Then
\begin{enumerate}
\item[(i)] If $\lambda <2$, there is always (i.e., for any irrational $\alpha$) lots
of a.c.~spectrum and it is known for some $\alpha$ and believed for all $\alpha$ that
is all there is (see Last \cite{Last}, Gesztesy-Simon \cite{GSxi}, Gordon et al.
\cite{GJLS}, Jitomirskaya \cite{Jito}; the earliest results of this genre are due
to Dinaburg-Sinai \cite{DS}).
\item[(ii)] If $\lambda =2$ and $\alpha$ is an irrational whose continued
fraction integers are unbounded (almost all $\alpha$ have this property), then
the spectrum is known to be purely singular continuous for almost all $\theta$ (see
Gordon et al. \cite{GJLS}).
\item[(iii)] If $\lambda >2$ and $\alpha$ is an irrational with good Diophantine
properties ($|\alpha - \f{p}{q}| \geq C\, q^{-\ell}$ for some $C,\ell$ and all
$p,q,\in\bbZ$), then for a.e.~$\theta$, the spectrum is dense pure point
(Jitomirskaya \cite{Jito}; see also Bourgain-Goldstein \cite{BG}.
\item[(iv)] If $\lambda >2$ and $\alpha$ is irrational, there are always lots of
$\theta$ (a dense $G_\delta$) for which the spectrum is purely singular continuous
(Jitomirskaya-Simon \cite{JS}). For some $\alpha$, like those in (iii), the set while
a dense $G_\delta$ has measure $0$. For Liouville $\alpha$ (irrational $\alpha$'s
with $\varliminf \f{1}{q} \ln |\sin\pi\alpha q| =-\infty$), the spectrum is purely
singular continuous (Avron-Simon \cite{AvS} using results of Gordon \cite{Gor}).
\end{enumerate}
In general, for almost periodic models, the spectral type is dependent on the number
theoretic properties of the frequencies.
Among the general spectral results known for almost periodic models is that the
spectrum is everywhere constant on $\Omega$ (rather than only almost everywhere
constant; Avron-Simon \cite{AvS}) and that the essential support of the a.c.~spectrum
is everywhere constant (Last-Simon \cite{LaS}). It is known (see (iv) above) that
$\bar\sigma_{\pp}$ and $\sigma_{\singc}$ may only be almost everywhere constant
and fail to be constant on all of $\Omega$.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two-Body Hamiltonians} \lb{s8}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hamiltonians of the form $-\Delta + V$ where $V(x)\to 0$ at infinity are often
referred to as two-body Hamiltonians since the Hamiltonian of two particles with
a potential $W(\vec r_1 - \vec r_2)$ reduces to $-\Delta + V$ (where $V$ is a
multiple of $W$ depending on the masses) after removal of the center of mass. The
issues are essentially the same as for one-dimensional decaying potentials as
discussed in Section~\ref{s5}.
With regard to the negative spectrum, again Weyl's criterion easily shows that
$\sigma_{\ess}(H)=[0,\infty)$ so that $H$ has only discrete spectrum of finite
multiplicity in $(-\infty, 0)$ and only $0$ can be an accumulation point.
Typical is:
\begin{theorem} \lb{t8.1} For $\alpha\in \bbZ^\nu$, let $\chi_\alpha$ be the
characteristic function of the unit cube about $\alpha$. Let $V:\bbR\to\bbR$.
Suppose $V\in K_\nu$ and that as $\alpha\to\infty$, $\|\chi_\alpha V\|_{K_\nu}
\to 0$. Then $\sigma_{\ess}(-\Delta + V) = [0,\infty)$.
\end{theorem}
As for whether $N(V)$, the number of negative bound states (counting multiplicity,
i.e., $N(V)=\dim E_{(-\infty, 0)}(H)$), is finite or infinite, there is a considerable
literature. The earliest bound is due to Birman \cite{Bir} and Schwinger \cite{Schw}
for $\nu =3$. It says
\begin{equation} \lb{8.1}
N(V) \leq \f{1}{(4\pi)^2} \int \f{|V(x)|\, |V(y)|}{|x-y|^2}\, dx\,dy \qquad (\nu=3).
\end{equation}
Perhaps the most famous bound is that of Cwickel \cite{Cwi}, Lieb \cite{Lieb76},
and Rosenbljum \cite{Rosc}:
\begin{equation} \lb{8.2}
N(V) \leq L_{0, \nu} \int |V(x)|^{\nu/2}\, dx \qquad (\nu\geq 3).
\end{equation}
One reason this is of special interest is that for nice $V$'s as $\lambda\to\infty$, \linebreak
$N(\lambda V) / \int |\lambda V|^{\nu/2}\, dx$ converges to a universal constant
(see Section~\ref{s14}). In particular, \eqref{8.1} has the wrong large $\lambda$
behavior while \eqref{8.2} has the right such behavior. (Simon \cite{Si76t} had
the first bounds with the right large $\lambda$ behavior for nice enough $V$'s;
he also conjectured \eqref{8.2}.)
As in the one-dimensional case, there are Lieb--Thirring-type bounds on the moments
of the negative eigenvalues $e_j$ of $-\Delta + V$
\[
\sum_j |e_j|^\gamma \leq L_{\gamma,\nu} \int dx \, |V(x)|^{\gamma + \nu/2}
\, d^\nu x
\]
for $\gamma >0$ if $\nu=2$ and $\gamma \geq 0$ if $\nu\geq 3$. These were proven
first in Lieb-Thirring \cite{LT75}. There has been considerable literature on best
values of $L_{\gamma,\nu}$. In particular, a recent pair of papers of
Laptev-Weidl \cite{LW} and Hundertmark-Laptev-Weidl \cite{HLW} has obtained
a breakthrough in understanding the $\nu$-dependence of $L_{\gamma,\nu}$. In
particular, they show that for $\gamma\geq \f32$, $L_{\gamma,\nu}$ is given by
the quasiclassical value. On the other hand, it is known that $L_{\gamma=0,\nu} >
L^{\qc}_{\gamma =0,\nu}$, the quasiclassical value for all $\nu$ (Helffer-Robert
\cite{HR1,HR2}).
For a review of the literature on bounds on the number of eigenvalues, especially
the subtle two-dimensional case, see Birman-Solomyak \cite{BS}.
The absence of eigenvalues at positive energies is a specialized issue largely
independent of the rest of the analysis of positive spectrum. Given the examples of
Wigner-von Neumann and related ones of Naboko and Simon discussed in Section~\ref{s5},
one needs some condition on the falloff or lack of oscillations. Here is a simple
result:
\begin{theorem} \lb{t8.2} Let $V(x) = V_1 (x) + V_2 (x)$ where $|x|\, |V_1(x)|\to 0$
and $|(x\dott \nabla) V_2 (x)| \to 0$. Then $-\Delta + V$ has no eigenvalues in
$[0,\infty)$.
\end{theorem}
\noindent{\it Remarks.} 1. The stated theorem requires local regularity ($V_1$ bounded
near infinity and $V_2$ is $C^1$), but there are extensions that allow local singularities.
\smallskip
2. Rellich \cite{Rel43} proved that if $V$ has compact support, there are no positive
energy eigenvalues. Theorem~\ref{8.2} when $V_2 =0$ is due to Kato \cite{Kato59}
and the full result to Agmon \cite{Ag69} and Simon \cite{Si69}.
\smallskip
3. See Froese et al. \cite{FHH} for another result of this genre; we will discuss their
result further in Section~\ref{s9}.
\medskip
The methods we will discuss below typically show that $\sigma_{\pp} \cap (0,\infty)$
is finite; one can then usually use Theorem~\ref{8.2} to prove that the set is
actually empty.
As for positive spectrum, it is intimately related to scattering theory. Given two
self-adjoint operators $A,B$, one says the wave operators exist if
\[
\Omega^\pm (A,B) = \slim_{t\to\mp\infty} e^{itA} e^{-itB} P_{\ac}(B)
\]
exists where $P_{\ac}$ is the projection onto the a.c.~subspace for $B$. We say
they are complete if $\Ran\, \Omega^\pm (A,B)=\Ran\, P_{\ac}(A)$, in which case
$\Omega^\pm (A,B)$ are unitary maps of $\Ran\, P_{\ac}(B)$ to $\Ran\, P_{\ac}(A)$
which intertwine $A$ and $B$. See Reed-Simon \cite{RS3}, Baumg\"artel-Wollenberg
\cite{BaW}, or Yafaev \cite{Yafaev} (or many other books) for a discussion of the
physics involved.
The development of abstract scattering theory is closely intertwined (pun intended)
to its applications to Schr\"odinger operators. Fundamental work was done by
Jauch \cite{Jau}, Cook \cite{Cook}, Rosenblum \cite{Rosm}, Kato \cite{Kato57},
Birman \cite{Bir2}, and Birman-Krein \cite{BK}.
The basic result for positive spectrum for ``short-range" potentials is:
\begin{theorem} \lb{t8.3} Let $V$ be such that $(1+|x|)^{1+\veps}\, V(x)\in L^p
+L^\infty (\bbR^\nu)$ for $\max(2, \f{\nu}{2})< p<\infty$ and let $H=-\Delta + V$
and $H_0 = -\Delta$. Then $\Omega^\pm (H, H_0)$ exist and are complete. Moreover,
$H$ has no singular continuous spectrum and any eigenvalues in $(0,\infty)$ are
isolated {\rm (}from other eigenvalues{\rm )} and of finite multiplicity.
\end{theorem}
\noindent{\it Remarks.} 1. The first results on absence of singular continuous
spectrum depended on eigenfunction expansions and were obtained by Povzner \cite{Pov53}
($V$'s of compact support) and Ikebe \cite{Ike} ($V$'s which were $O(|x|^{-2-\veps})$
at infinity). The earliest results on completeness of wave operators depended on the
trace class theory of scattering (of Rosenblum \cite{Rosm} and Kato \cite{Kato57})
and were obtained by Kuroda \cite{Kur1,Kur2}. From 1960 to 1972, the decay was
successively improved until Agmon \cite{Ag75} obtained the $O(|x|^{-1-\veps})$ result
quoted.
\smallskip
2. Enss \cite{Enss78} has a different, quite physical, approach to this result. Enss'
work depends in part on an earlier geometric characterization of the continuous
subspace of a Schr\"odinger operator by Ruelle \cite{Rue} and Amrein-Georgescu
\cite{AGe}. This is sometimes called the RAGE theorem after the initials of the
authors.
\smallskip
3. It is known (e.g., Dollard \cite{Dol}) that if $V(x) = O(|x|^{-1})$, $\Omega^\pm
(H, H_0)$ may not exist.
\medskip
For long-range behavior decaying slower than $O(|x|^{-1})$, there are results if
$\nabla V$ decays faster than $O(|x|^{-1-\veps})$. Basically, there is only a.c.~spectrum
at positive energy if $V=V_1 + V_2$ with $V_1 =O(|x|^{-1-\veps})$ and $x \dott
\nabla V_2 = O(|x|^{-\veps})$. For details, see Lavine \cite{Lav2}, Agmon-H\"ormander
\cite{AgH}, and H\"ormander \cite{Hor}. These works use modified wave operators as
introduced by Dollard \cite{Dol}.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$N$-Body Hamiltonians} \lb{s9}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\wti H$ be the Hamiltonian of $N$ particles in $\bbR^\nu$. Explicitly,
$\wti H$ is an operator on $L^2 (\bbR^{\nu N})$ given by $\wti H = \wti H_0 + V$
where
\[
\wti H_0 = - \sum_{j=1}^N \f{1}{2m_j}\, \Delta _{x_j}
\]
with $x=(x_1, \dots, x_N)$ a point in $\bbR^{\nu N} = \bbR^\nu \times \bbR^\nu
\times \cdots \times \bbR^\nu$ ($N$ times) and
\[
V=\sum_{i
1.2Z$ for $Z$ large.
\item[{\rm (d)}] \mbox{\rm (Lieb \cite{Lieb84})} $N_0(Z) \leq 2Z$.
\end{enumerate}
\end{theorem}
\noindent{\it Remarks.} 1. If $N\geq N_0$, then $\inf \spec(H(N,Z))=\inf\spec
(H(N_0,Z))<\inf\spec (H(N_0 -1, Z))$.
\smallskip
2. Some of these results hold if $M<\infty$.
\medskip
With short-range potentials, the situation is simple if the bottom of the essential
spectrum is two body. Define
\[
\Sigma_3 = \min_{\# (a)\geq 3} (\Sigma(a)).
\]
Then (see Cycon et al. \cite[Section~3.9]{CFKS}),
\begin{theorem} \lb{t9.5} \mbox{\rm (Sigal \cite{Sig82})} Suppose $\Sigma_3 > \Sigma$,
$\nu\geq 3$, and each $V_{ij}$ lies in $L^{\nu/2}(\bbR^\nu)$. Then
$\dim E_{(-\infty, \Sigma)} (H)<\infty$.
\end{theorem}
On the other hand, if $\Sigma_3 = \Sigma$, there can be an infinite number of bound
states even if the $V_{ij}$'s have compact support (in $x_{ij}$). In particular,
if $N=3$, $V_{12} = V_{23} = V_{13} = -c\chi_1$ with $\chi$ the characteristic
function of a unit ball and $c$ chosen so that $\inf\spec(H)=0$ but $\inf\spec(H +
\veps V)<0$ for all $\veps >0$, it is known that $\dim E_{(-\infty, 0)}(H)=\infty$.
This is known as the Efimov effect after work of Efimov \cite{Efi1,Efi2}. For
proofs of this phenomenon, see Yafaev \cite{Ya1} and Ovchinnikov-Sigal \cite{OS}.
In analyzing the spectrum of $H$ on $[\Sigma,\infty)$, a particular class of physically
significant energies occurs, the thresholds. For each partition $a$ of $\{1, \dots, N\}$
with $\# a\geq 2$, there is a natural decomposition of $L^2 (\bbR^{\nu(N-1)}) = \calH_a
\otimes \calH^a$ where $\calH_a$ are functions of $x_i - x_j$ with $i$ and $j$
in the same cluster of $a$ and $\calH^a$ are functions of $R_\alpha - R_\beta$ where
$R_\alpha$ is the center of mass of a cluster (see \cite{HuS} for an elegant way of
doing this kinematics). Under the decomposition $H(a) = H_a \otimes I + I \otimes T^a$.
$H_a$ is the internal energy of the cluster and $T^a$ the kinetic energy of the
cluster centers of mass. $\calI (a)$ is the set of eigenvalues of $H_a$ (with
the condition that if $\#(a) = N$, so $H_a$ is $0$ on $\bbC$, then $\calI(a) =
\{0\}$. The set of thresholds is defined to be
\[
\calI = \bigcup_a \calI(a).
\]
{\it Note}: An energy in $\calI(a)$ is a sum of eigenvalues of individual cluster
Hamiltonians. In particular, the statement in the theorems below that the set
of thresholds is a closed countable set follows by induction from the other
statement that eigenvalues can only accumulate at thresholds.
\medskip
The three-body problem turns out to have some aspects that make it simpler
than the general $N$-body problem, and Faddeev \cite{Fad} and later Enss \cite{Enss83}
(using very different methods) have fairly complete results on spectral and
scattering theory for $N=3$. We will focus here on results that apply for
all $N$.
Historically, the first aspect of the continuous spectrum for general $N$-body
systems controlled was the absence of singular continuous spectrum. The earliest
result required analyticity of the potentials but included atoms:
\begin{theorem} \lb{t9.6} \mbox{\rm (Balslev-Combes \cite{BaC})} Suppose each $V_{ij}
(x) = f_{ij}(x_i - x_j)$ where $f_{ij}$ is a function on $\bbR^\nu\backslash
\{0\}$ that obeys
\[
A(\theta) = V(e^\theta x) (-\Delta + 1)^{-1}
\]
is compact and has an analytic continuation from $\theta \in \bbR$ to $\{\theta\mid
|\Ima\theta| < \veps\}$ for some $\veps >0$. Then $\sigma_{\singc}(H)=\emptyset$.
Moreover,
\begin{enumerate}
\item[{\rm (i)}] Any eigenvalue of $H$ in $\bbR\backslash\calI$ is of finite
multiplicity, and eigenvalues can only accumulate at thresholds.
\item[{\rm (ii)}] The set of eigenvalues union thresholds is a closed countable set.
\end{enumerate}
\end{theorem}
\noindent{\it Remarks.} 1. Such potentials are called dilation analytic.
\smallskip
2. This result was first proven for two-body systems by Aguilar-Combes \cite{AC}.
\smallskip
3. See Simon \cite{Si72,Si73am} for extensions of this result.
\medskip
The most general results on absence of singular continuous spectrum depend on ideas
of Mourre \cite{Mou}.
\begin{theorem} \lb{t9.7} Suppose $V_{ij}(x) = f_{ij}(x_i - x_j)$ where $f_{ij}$
is a function on $\bbR^\nu$ that obeys {\rm (}as operators on $L^2(\bbR^\nu)${\rm )}
\begin{enumerate}
\item[{\rm (i)}] $f_{ij}(x) (-\Delta + 1)^{-1}$ is compact
\item[{\rm (ii)}] $(-\Delta + 1)^{-1} x\dott \nabla f_{ij} (-\Delta + 1)^{-1}$ is
compact.
\end{enumerate}
Then $\sigma_{\ess}(H)$ is empty. Moreover, any eigenvalue in $\bbR\backslash\calI$
is discrete, eigenvalues can only accumulate at thresholds, and the set of eigenvalues
and thresholds is a closed countable set.
\end{theorem}
\noindent{\it Remarks.} 1. This theorem was proven for $N=3$ by Mourre \cite{Mou}. His
methods were extended and elucidated by Perry-Sigal-Simon \cite{PSS} who obtained the
general $N$-body result. Substantial simplifications of the proof were found by
Froese-Herbst \cite{FH}.
\smallskip
2. Condition (ii) does not require that $f_{ij}$ be smooth because $\nabla f_{ij} =
[\nabla, f_{ij}]$ and $\nabla (-\Delta + 1)^{-1}$ is bounded. Basically, (i), (ii)
hold if $f_{ij} = f^{(1)}_{ij} + f^{(2)}_{ij}$ where $xf^{(1)}_{ij} (-\Delta + 1)^{-1}$
is compact and $f^{(2)}_{ij}$ is smooth with $(x \cdot \nabla) f^{(2)}_{ij}
(-\Delta + 1)^{-1}$ and $f^{(2)}_{ij} (-\Delta + 1)^{-1}$ compact.
\smallskip
3. Froese-Herbst \cite{FH} have some general results that imply that $\calI \, \cap
(0,\infty) =\emptyset$ (see Thm.~4.19 in Cycon et al. \cite{CFKS}).
\medskip
Finally, there has been extensive study of scattering theory and completeness. For each
cluster with $\#(a) \geq 2$, let $P_a$ on $\calH_a$ be the projection onto the point
spectrum of $H_a$ and let $P(a) = P_a \otimes I$, the projection onto vectors which
are bound within the clusters and arbitrary for the centers of mass coordinates.
The cluster wave operators are defined by
\begin{equation} \lb{9.3}
\Omega^\pm (a) = \slim_{t\to \mp\infty} e^{+itH} e^{-itH(a)} P(a).
\end{equation}
$\Ran (\Omega^+(a))$ are those states which in the distant past look like bound
clusters (coresponding to the partition $a$) moving freely relative to one another.
The existence of cluster wave operators \eqref{9.3} was proven first by Hack \cite{Hack}.
It is not hard to see (e.g., Theorem XI.36 in Reed-Simon \cite{RS3}) that for $a\neq b$,
$\Ran\, \Omega^+(a)$ is orthogonal to $\Ran\, \Omega^+(b)$. Asymptotic completeness
is the statement that
\[
\bigoplus_{\#(a)\geq 2} \Ran (\Omega^+(a)) = \calH_{\ac}(H),
\]
where $\calH_{\ac}(H)$ is the absolutely continuous subspace for $H$. After fairly
general results for $N=3$ (Faddeev \cite{Fad} and Enss \cite{Enss83}) and for
general $N$ with weak coupling (Iorio-O'Carroll \cite{IOC}) and repulsive potentials
(Lavine \cite{Lav1}), Sigal and Soffer \cite{SS87} solved the general result. Their
theorem is
\begin{theorem} \lb{t9.8} \mbox {\rm (Sigal-Soffer \cite{SS87})} If each $V_{ij}(x) =
f_{ij}(x_i- x_j)$ where \linebreak $|(D^\alpha f_{ij}(x)| \leq C(1+|x|)^{-|\alpha|-\veps-1}$
for all multi-indices with $|\alpha|\leq 2$, then asymptotic completeness holds.
\end{theorem}
Extensions and clarifications of this work are due to Graf \cite{Graf}, Hunziker
\cite{Hun98}, and Yafaev \cite{Ya2}. Long-range potentials are discussed in
Derezinski \cite{Der}, Sigal-Soffer \cite{SS94}, and Derezinski-Gerard \cite{DG}.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Constant Electric and Magnetic Fields} \lb{s10}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Quantum mechanics with a potential and constant electric or magnetic field played
a critical role experimentally and theoretically in the earliest days of the
subject, and there has been considerable mathematical literature on the spectral
properties of these operators. The basic Stark Hamiltonian on $L^2 (\bbR^\nu)$ is
\begin{equation} \lb{10.1}
H=-\Delta + Ex_1 + V(x),
\end{equation}
where $V$ is short range. A key role has been played by an explicit formula of
Avron-Herbst \cite{AvH} for the operator when $V=0$, viz.,
\begin{equation} \lb{10.2}
\exp(-it (-\Delta + x_1)) = \exp (-it^3/3) \exp(-itx_1) \exp(-it\Delta +
ip_1 t^2),
\end{equation}
where $p_1 = \f{1}{i} \f{\partial}{\partial x_1}$. Classically in an electric
field, a particle has $x_1 = N - ct^2$ as $t\to\infty$ and \eqref{10.2} realizes this
with the $p_1 t^2$ term. It means the borderline for short range is $|x|^{-1/2-\veps}$
rather than $|x|^{-1-\veps}$. The result is
\begin{theorem} \lb{t10.1} Suppose $|V(x)| \leq C (1+|x|)^{-\veps} (1+|x_1|)^{-1/2-\veps}$.
Then $H$ given by \eqref{10.1} has complete wave operators and empty singular continuous
spectrum. Eigenvalues are isolated and of finite multiplicity.
\end{theorem}
This result and ones similar to it are discussed by Herbst \cite{Her1}, Yajima
\cite{Yaj}, and Simon \cite{Si79d}. Multiparticle completeness in electric fields has
been studied by Herbst-M{\o}ller-Skibsted \cite{HMS}, and Adachi-Tamura \cite{AT}.
There is a large literature on both constant and variable magnetic fields but an
extensive review of it is beyond the scope of this article. One can begin looking
at the literature by consulting a series by Avron, Herbst, and Simon
\cite{AHS1,AHS2,AHS3} and Chapter~6 of Cycon et al. \cite{CFKS} and references therein.
See also Section~\ref{s12}.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Coulomb Energies} \lb{s11}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
While much of the mathematical theory of nonrelativistic quantum mechanics has
focused on general potentials, nature uses the Coulomb potential and there is
considerable literature on binding energies of Coulomb systems, especially as
some parameter goes to infinity. Section~\ref{s9} (see Theorem~\ref{t9.4})
already discussed one such result. We will only introduce some seminal themes;
consult Lieb \cite{Lieb90} for a review of the subject.
The most famous of these results is the stability of matter. In its simplest form,
it concerns the Hamiltonian
\begin{multline*}
H(N,M; R_1, \dots, R_M) = \\
-\sum_{i=1}^N \Delta_i - \sum_{i,\alpha} \f{1}{|x_i - R_\alpha|} + \sum_{i0$ has an asymptotic series
\[
E_n (\beta) \sim \sum_{n=0}^\infty a_n \beta^n
\]
even though this series can be divergent (and is for the case \eqref{12.2}, as shown
by Bender-Wu \cite{BW}). See Herbst-Simon \cite{HS} for an example where an asymptotic
series converges but to the wrong answer! See Simon \cite{Si83} for a study
of multiwell problems.
In some cases, including \eqref{12.2}, it is known that the divergent perturbation
series can be made to give the right eigenvalue with a summability method, either
Pad\'e approximation (Loeffel et al. \cite{LMSW}) or Borel summation (Graffi et al.
\cite{GGS}). Borel summability is also known to work for the Zeeman series for
hydrogen-hydrogen perturbed by turning on a constant magnetic field; see
Avron-Herbst-Simon \cite{AHS3} and Avron et al. \cite{AAC}.
In certain cases, eigenvalues are perturbed into resonances, the subject of
Section~\ref{s13}. For eigenvalues embedded in continuous spectrum under regular
perturbations (like \eqref{12.1}), the convergence of the perturbation series for a
resonance and its related time-dependent perturbation theory and the Fermi golden rule
is discussed in Simon \cite{Si72,Si73am}. For Stark Hamiltonians, the basic paper
is Herbst \cite{Her2}. Harrell-Simon \cite{HaS} found the leading resonance
asymptotics in this case.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resonances} \lb{s13}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Almost everything we have discussed so far has involved a single operator and properties
invariant under unitary transformations. The notion of resonances has got to involve
additional structure. For example, the operators $-\Delta - |x|^{-1} - Fx = H(F)$ are
unitarily equivalent for all $F\neq 0$. But according to the physics lore, there is a
resonance with an $F$-dependent position. We will not emphasize the extra structure,
but it is there. We will focus on two definitions of resonances: one suitable for
potentials that decay very rapidly (see Zworski \cite{Zw94,Zw99} for reviews) and
the method of complex scaling already discussed in a different context in
Section~\ref{s9}. (See Reed-Simon \cite{RS4} and Simon \cite{Si78} for reviews.)
Let $\nu$ be an odd dimension, let $V$ be a bounded potential of compact support on
$\bbR^\nu$, and for $\Real\kappa>0$, define
\[
B(\kappa) = |V|^{1/2} (-\Delta + \kappa^2)^{-1} V^{1/2},
\]
where $V^{1/2} = |V|^{1/2} \sgn (V)$. Then $-\kappa^2$ is an eigenvalue of $-\Delta
+ V$ if and only if $-1$ is an eigenvalue of $B(\kappa)$. Since $\nu$ is odd,
$B(\kappa)$ has an analytic continuation as a compact operator-valued function of
$\kappa$ to all of $\bbC$ (when $\nu =1$, there is a simple pole at $\kappa =0$
but $\kappa B(\kappa)$ is entire). If $\Real \kappa <0$ and $-1$ is an eigenvalue
of $B(\kappa)$, we say $-\kappa^2$ is a resonance of $-\Delta + V$.
Froese \cite{Fro97} has a lovely formula that relates resonances defined by this
method to scattering theory. For all $\kappa$, $B(\kappa) - B(-\kappa)$ is trace
class so $(1+ B(-\kappa))(1+B(\kappa))^{-1}$ is $1$ plus trace class and has a
determinant as an operator on $L^2(\bbR^\nu)$. For $k$ real and $S(k)$, the
$S$-matrix on $L^2 (S^{\nu-1})$,
\[
\det(S(k)) = \det((1+B(-ik))(1+B(ik))^{-1}),
\]
so resonances are related to poles of the analytic continuation of $S$.
There has been considerable literature on the number of resonances. Let $N(R)$ be the
number of resonances with energy $E$ obeying $|E|0$, $\sigma_{\ess}(H(\theta)) \cap\bbR$ consists precisely of
$\calI$. Basically as we increase $\Ima \theta$ from $0$, the essential spectrum
rotates about the thresholds. In doing that, it can uncover resonances.
\medskip
Resonances defined by this method have been used by quantum chemists for numerical
calculations as well as a theoretical tool. Simon \cite{Si72,Si73am} used it to
study the Fermi golden rule and Harrell-Simon \cite{HaS} to prove various
one-dimensional tunnelling estimates.
Avron \cite{Avr} used these ideas to study large-order perturbation theory for Hydrogen
in a magnetic field; a rigorous proof of his results was obtained by Helffer-Sj\"ostrand
\cite{HS2}.
Herbst \cite{Her2} has extended the ideas to Hamiltonians with constant electric field.
Among his results is the surprising one that if $0<\Ima \theta <\f{\pi}{3}$, then
$-e^{-2\theta} \Delta + e^\theta x$ has empty spectrum!
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Quasiclassical Limit} \lb{s14}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There has been considerable literature on the connection between quantum and classical
mechanics. Much of it has focused on what happens as $\hbar\to 0$, but there are
other limiting situations where a classical or semiclassical picture is appropriate---for
example, the large $Z$ limit of atoms. We will touch on some of the subjects considered,
but the literature is vast. Robert \cite{Rob} has an excellent review of those results
obtained for very smooth potentials using the Fourier integral operator methods
pioneered by H\"ormander and Maslov. Therefore, I will not try to cover these results
here. We note that in Section~\ref{s11}, we referenced the Thomas-Fermi limit,
which is quasiclassical.
Consider first the $\hbar\downarrow 0$ limit. Let $H_\hbar = -\f{\hbar^2}{2m} \Delta + V$.
Kac \cite{Kac1,Kac2} had the idea that the small $\hbar$ limit of $\exp(-sH_\hbar)$ was
the same as the zero time limit in Brownian motion. This allows one to prove under great
generality that the quantum partition function $\Tr (\exp(-sH_\hbar))$ approaches a
classical partition function as $\hbar\downarrow 0$; see, for example, Theorem~10.1
in Simon \cite{Sifi}. The earliest results I know of on this subject are due to
Berezin \cite{Ber}.
Quantum dynamics, $e^{-isH_\hbar/\hbar} \psi_\hbar$, on suitable states $\psi_\hbar$ make
an elegant classical limit---one takes $\psi_\hbar$ to be a coherent state which collapses
to a single point in phase space as $\hbar\downarrow 0$. Such results were found by
Hagedorn \cite{Hag1,Hag2,Hag3} (similar methods were developed independently by
Ralston \cite{Ral}).
Since $-\hbar^2 \Delta + V = \hbar^2 [-\Delta + \hbar^{-2}V]$, the small $\hbar$ limit
is the same as the large coupling constant limit for $-\Delta + \lambda V$. In particular,
if $N(V) = \dim E_{(-\infty, 0)}(-\Delta + V)$, the quantity discussed in Section~\ref{s8},
one has
\begin{theorem} \lb{t14.1} Let $\nu\geq 3$ and $V\in L^{\nu/2}(\bbR^\nu)$. Then
$\lim_{\lambda \to\infty} N(\lambda V)/\lambda^{\nu/2} = (2\pi)^{-\nu} \tau_\nu
\int_{V\leq 0} (-V(x)^{\nu/2} \, d^\nu x$ where $\tau_\nu$ is the volume of a unit ball
in $\bbR^\nu$.
\end{theorem}
\noindent{\it Remarks.} 1. This theorem is quasiclassical since the right side is
$(2\pi)^{-\nu}$ times the volume of the classical phase space region where $p^2 + V(x)
\leq 0$.
\smallskip
2. The historical thread for this theorem goes back to a celebrated paper of Weyl
\cite{Weyl1} on Dirichlet Laplacians. Theorems like \ref{t14.1} with stronger conditions
on $V$ are due to Birman-Borzov \cite{BB}, Kac \cite{Kac2}, Martin \cite{Mart}, and
Tamura \cite{Tam}. See Reed-Simon \cite[Theorem~XIII.80]{RS4} for the proof under
the stated assumptions.
\medskip
Let $V(x)\to\infty$ as $|x|\to\infty$ in a fairly regular way (e.g., suppose $V$ is an
elliptic polynomial). Then $-\Delta + V$ has discrete spectrum and the asymptotics of
the number of eigenvalues $\dim E_{(-\infty, \alpha]} (-\Delta + V)$ as $\alpha\to\infty$
is determined by phase space. Results of this type go back to Titchmarsh \cite{Tibook};
see also Reed-Simon \cite[Theorem~XIII.81]{RS4}. Similarly, if $V(x)\to 0$ but
so slowly that $N(V) = \infty$, for example, $V(x)\sim -|x|^{-\beta}$ with $0<\beta <2$,
then the divergence of $\dim E_{(-\infty, \alpha]} (-\Delta + V)$ as $\alpha\uparrow 0$
is sometimes given by quasiclassical considerations; see Brownell-Clark \cite{BrC},
McLeod \cite{McL}, and Reed-Simon \cite[Theorem~XIII.82]{RS4}.
\vskip 0.3in
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\end{thebibliography}
\end{document}