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\markboth{Exponential decay of two-body eigenfunctions: A review} {P. D. Hislop}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Mathematical Physics and Quantum Field Theory, \newline
Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 265--288\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Exponential decay of two-body eigenfunctions: \\ A review
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\thanks{ {\em Mathematics Subject Classifications:} 81Q10.
\hfil\break\indent
{\em Key words:} Schrodinger operator, eigenfunction, exponential decay, Dirac operator.
\hfil\break\indent
\copyright 2000 Southwest Texas State University 
and University of North Texas. \hfil\break\indent
Published November 3, 2000. \hfil\break\indent
Supported in part by NSF grant DMS 97-07049} }

\date{}
\author{ P. D. Hislop \\[12pt]
{\em Dedicated to Eyvind H. Wichmann}}

\maketitle

\begin{abstract}
We review various results on the exponential decay of the eigenfunctions of
two-body Schr\"odinger operators.  The exponential, isotropic bound results of
Slaggie and Wichmann \cite{SlaggieWichmann} for eigenfunctions of Schr\"odinger
operators corresponding to eigenvalues below the bottom of the essential
spectrum are proved.  The exponential, isotropic bounds on eigenfunctions for
nonthreshold eigenvalues due to Froese and Herbst \cite{FroeseHerbst} are
reviewed.  The exponential, nonisotropic bounds of Agmon \cite{Agmon} for
eigenfunctions corresponding to eigenvalues below the bottom of the essential
spectrum are developed, beginning with a discussion of the Agmon metric.  The
analytic method of Combes and Thomas \cite{CT}, with improvements due to
Barbaroux, Combes, and Hislop \cite{BCH}, for proving exponential decay of the
resolvent, at energies outside of the spectrum of the operator and localized
between two disjoint regions, is presented in detail.  The results are applied
to prove the exponential decay of eigenfunctions corresponding to isolated
eigenvalues of Schr\"odinger and Dirac operators.
\end{abstract}

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\section{Introduction}

The decay properties of bound state wave functions of Schr\"odinger operators
have been intensively studied for many years. We are concerned here with the 
simplest part of the theory: The decay of the wave functions for two-body
Schr\"odinger operators. We consider a Schr\"odinger operator of the form
\begin{equation}
H = - \Delta + V,
\end{equation}
acting on $L^2 ( {\mathbb R}^n )$. The potential $V$ is a real-valued function
assumed to be sufficiently regular so that $H$ is essentially
self-adjoint on $C_0^{\infty} 
( {\mathbb R}^n )$. We will assume that 
\begin{equation}
\lim_{ \|x\| \rightarrow \infty } | V ( x ) | = 0 .
\end{equation}
We also assume that $V$ is well-enough behaved so that the spectrum of $H$, 
denoted by $\sigma (H)$, has the standard form
\begin{eqnarray}
\sigma ( H ) & = & \sigma_{ess} ( H ) \cup \sigma_d ( H )  \nonumber \\
             & = & [ 0 , \infty ) \cup \{  E_j \; | \; - E_j > 0 , \;
                       j = 0,1, \ldots \}
\end{eqnarray}
For example, if $V$ is continuous and satisfies
(1.2), then the spectrum of $H$ has this form.
This is due to the invariance of the essential
spectrum under relatively-compact perturbations \cite{HislopSigal}.
In many commonly encountered cases,
for example, when $V$ has decaying derivatives,
the essential spectrum is purely absolutely continuous.
However, it is known that condition (1.2) is not sufficient to
guarantee the absolute  continuity of the spectrum. There
are examples due to Pearson \cite{Pearson} of bounded, decaying potentials
for which the Schr\"odinger operator has purely 
singular continuous spectrum.
We will not need these fine spectral results here. 

Suppose that $E < 0$ is an eigenvalue of $H$ with 
eigenfunction $\psi_E \in L^2( {\mathbb R}^n )$ so that $H \psi_E = E \psi_E$.
We will always 
assume that the eigenfuncton is normalized so that 
$$ \int_{{\mathbb R}^n} \: | \psi_E (x) |^2 \: d^nx = 1. $$
We are interested in the spatial behavior of this function
$\psi_E ( x)$, as $\| x \| \rightarrow \infty$. The well-known example
of the eigenfunctions of the
hydrogen atom Hamiltonian provides a guide. The hydrogen atom Hamiltonian
on $L^2 ( {\mathbb R}^3 )$ has the form
\begin{equation}
H= - \Delta - \frac{1}{\| x \|} .
\end{equation}
It is easy to check that the spherically symmetric function
\begin{equation}
\psi_E ( x ) = \frac{1}{ \sqrt{ 8 \pi } } 
               e^{ - \sqrt { \frac{1}{4} } \| x \| } ,
\end{equation}
is a normalized eigenfunction of $H$ with eigenvalue $E = - 1/4$, the
ground state energy. 
We note that the eigenfunction decays exponentially with a factor
given by the square root of the distance from the eigenvalue $E = - 1/4$
to the bottom of the essential spectrum $ \mbox{inf} \; \sigma_{ess}
( H ) = 0$. We will see that this is a characteristic exponential decay
behavior. 

In general, we do not know if an eigenfunction is continuous,
and so pointwise bounds are not meaningful.
For the general case,
it is convenient to describe the decay of an eigenfunction in the
$L^2$-sense. For a nonnegative function $F$, we say that $\psi$ decays
like $e^{-F}$ {\it in the $L^2$-sense} if 
\begin{equation}
\| e^F \psi \| \; \leq C_0 ,
\end{equation}
for some finite
constant $C_0 > 0$. Of course,
if we know more about the regularity of the
potential $V$, we can use a simple argument to conclude the regularity
of the eigenfunction $\psi_E$ corresponding to an eigenvalue $E < 0$.
From this regularity and an $L^2$-exponential decay estimate, we
can prove the pointwise decay of the eigenfunction. 
Simple regularity results are based on the
Sobolev Embedding Theorem, which we now state.

\vspace{.1in}
\noindent
{\bf Theorem 1.1} {\it Any function $f \in H^{s+k} ( {\mathbb R}^n )$, for $s > n / 2$
and $k \geq 0$, can be represented by a function $f \in C^k ( {\mathbb R}^n )$.}

\vspace{.1in}
\noindent
{\bf Proposition 1.2} {\it Suppose that $H$ is a self-adjoint Schr\"odinger 
operator and the potential $V$ is bounded and satisfies (1.2). 
If, additionally,
$V \in C^{2k} ( {\mathbb R}^n )$, with bounded derivatives,
then an eigenfunction $\psi_E \in L^2 ({\mathbb R}^n)$, 
corresponding to an eigenvalue $E < 0$, satisfies $\psi_E
\in H^{2k+2} ( {\mathbb R}^n )$.
If $(2k + 2 ) > n/2 + l$, then 
the eigenfunction satisfies $\psi_E \in C^l ( {\mathbb R}^n )$.}

\vspace{.1in}
\noindent
{\bf Proof.} We note that since $E < 0$, the resolvent of $H_0
= - \Delta$ exists at energy $E$. Furthermore, the resolvent 
$R_0 ( E) = (H_0 - E)^{-1}$
maps $H^s ( {\mathbb R}^n ) \rightarrow H^{s+2} ( {\mathbb R}^n )$, for all $s \in {\mathbb R}$.
The eigenvalue equation can be written as
\begin{equation}
\psi_E = - R_0 (E) V \psi_E .
\end{equation}
Since $V$ is bounded, the potential is a bounded operator
on $L^2 ({\mathbb R}^n)$. Equation (1.7) then shows that $\psi_E \in H^2 ( {\mathbb R}^n )$.
We now repeat this argument $k$-times since
$V$ is a bounded operator on $H^{s}({\mathbb R}^n)$, provided
$ s \leq k$. From this, we conclude that $\psi_E \in H^{2k + 2} ( {\mathbb R}^n )$.
The last statement follows from the Sobolev
Embedding Theorem. $\Box$  

Once regularity of the eigenfunction has been established,
the pointwise decay estimate is derived from a local estimate of the
following type (cf.\ \cite{HislopSigal} for a simple case, or
\cite{Agmon}, for a general proof). 
Let $B ( y , r )$ denote the ball of radius
$r > 0$ about the point $y \in {\mathbb R}^n$. There exists a constant
$C_{E , V}$, depending on the potential $V$, the energy $E$, and
$\mbox{inf} \; \sigma  ( H )$, but independent of $x_0 \in {\mathbb R}^n$,
so that
\begin{equation}
\displaystyle{ \mbox{max}_{x \in B( x_0 , 1/2 ) } }
| \psi_E ( x ) | \; \leq C_{E, V} \; 
\| \psi_E \|_{ L^2 ( B ( x_0 , 1 ) )} .
\end{equation}
Let us suppose that the exponential weight $F$ in (1.6) is translation
invariant and satisifes a triangle inequality: $F(x) \leq F(x-y) + F(y)$.
With this assumption and estimate (1.8),
we find
\begin{eqnarray}
\lefteqn{ \mbox{max}_{ x \in B( x_0 , 1/2 )} \; | \psi_E ( x ) e^{F(x)} |}
\nonumber \\
  & \leq & 
C_{E,V} \left( \mbox{max}_{ x \in B( x_0 , 1 )} e^{F(x)} \right)
   \;   \left( \mbox{max}_{ y \in B( x_0 , 1 )} e^{-F(y)} \right) 
        \| \psi_E e^F \|_{L^2 ( B ( x_0 , 1 ) )} \nonumber \\
  & \leq & C_1 \| \psi_E e^F \|_{ L^2 ( {\mathbb R}^n ) } \nonumber \\
   & \leq & C_2  ,
\end{eqnarray}
where we used the triangle inequality to combine the exponential terms.
This proves that $| \psi_E ( x_0 )| \leq C_1 e^{ - F(x_0 )} $, for
a constant $C_1$ independent of $x_0$. Since $x_0$ is
arbitrary, the eigenfunction satisfies a pointwise exponential bound.

There are two basic types of upper bounds
on wave functions as $\| x \| \rightarrow \infty$. 
We will present these as pointwise estimates on the eigenfunction,
although they can be formulated in the $L^2$-sense as described above.
We say that a function
$\phi$ satisfies an {\it isotropic} decay estimate if
there exists a nonnegative function $F : {\mathbb R}^+ \rightarrow {\mathbb R}^+$ 
so that
\begin{equation}
| \phi ( x ) | \leq C_\alpha e^{ - F ( \| x \| ) } .
\end{equation}
In many situations, we have $F ( \| x \| ) = \alpha \| x \| $,
for some $0 < \alpha < \infty $.
In the situation where $|V ( x ) | \rightarrow 0 $ uniformly 
with respect to $ \omega \in S^{n-1}$, as
$\|x\| \rightarrow \infty$, isotropic decay estimates are
optimal. We have seen this from the example of the hydrogen
atom ground state wave function. In the situation that the potential has
different limits at infinity, depending on the direction, or, the limit
at infinity is not achieved uniformly, as in the case of some 
nonspherically-symmetric potentials, it
is more precise to replace the isotropic exponential factor
with an anisotropic function which expresses this variation. 
These anisotropic exponents are described by
a function $\rho_E ( x ) : {\mathbb R}^n \rightarrow {\mathbb R}^+$ 
that, as we will see, depends on the
eigenvalue $E$ and the potential $V$. An anisotropic upper bound
has the form
\begin{equation}
| \phi ( x) | \leq C_0 e^{ - \rho_E (x) } .
\end{equation} 
This is the case for $N$-body Schr\"odinger operators 
when the potential is a sum of pair potentials. Agmon \cite{Agmon}
has developed an extensive theory of anisotropic decay estimates.

We will generalize the family of Hamiltonians described so far
in order to incorporate Schr\"odinger 
operators with gaps in the essential spectrum.
The situation we envision is the perturbation of a periodic
Schr\"odinger operator $H_{per} = - \Delta + V_{per}$ by a compactly
supported potential $W$. It is well-known that the spectrum of $H_{per}$
is the union of intervals $B_j$, called bands, so that 
\begin{equation}
\sigma (H_{per}) = \cup_{j \geq 0} B_j .
\end{equation}
For many periodic potentials,
there exist two consecutive bands $B_j$ and $B_{j+1}$ 
that do not overlap.
We say that there is an open spectral gap $G$ between the two bands.  
A local perturbation $W$, with compact support, 
preserves the essential spectrum. In the case that
$W$ has fixed sign, say $W \geq 0$, it is easy to show,
using the Birman-Schwinger principle, that for $\lambda > 0$
sufficiently large, the perturbed Hamiltonian, $H ( \lambda ) =
H_{per} + \lambda W$, has bound states at energies in the gap $G$. 
Suppose that $E \in G$. The existence of an eigenvalue
for $H( \lambda) = H_0 + \lambda W$ at $E$, for some $\lambda \neq 0$,
is equivalent to the existence of an eigenvalue
of the compact, self-adjoint
operator $K(E) \equiv W^{1/2} ( H_0 - E )^{-1} 
W^{1/2}$ equal to $- 1 / \lambda$.
We simply choose $\lambda \in {\mathbb R}$ so that $- 1 / \lambda$ is an
eigenvalue of the compact, self-adjoint operator $K(E)$. 
This argument can be generalized to perturbations $W$ with compact
support (or, sufficiently rapid decay) and not
necessarily fixed sign \cite{HD}. We are interested in the exponential decay
of the eigenfunctions corresponding to eigenvalues in open spectral gaps. 

The methods used to describe eigenfunction decay below cover three
main cases encountered in the study of Schr\"odinger operators:
\begin{enumerate}
\item Isolated eigenvalues below the bottom of the essential spectrum;
\item Eigenvalues embedded in the essential spectrum;
\item Eigenvalues lying in the spectral gap of an unperturbed operator.
\end{enumerate}
As we will discuss in section 3, 
embedded eigenvalues at positive energies do not occur for most Schr\"odinger
operators. The embedded eigenvalues referred to in point 2
occur at negative energies in the $N$-body case for $N \geq 3$. 
A notable exception is the Wigner-von Neumann potential in
one-dimension that has an embedded eigenvalue
at positive energy. In general,
embedded eigenvalues do occur for Schr\"odinger operators, but they are rare. 
The methods presented here, especially the Agmon technique,
can be extended so as to apply to 
resonance eigenfunctions (cf.\ \cite{HislopSigal}). 
The methods can also be applied to
the study of eigenfunctions for the Laplace-Beltrami operator
on noncompact Riemannian manifolds (cf.\ \cite{Hislop}). 
Other references on the exponential decay of eigenfunctions can be found
in \cite{RS4}.

\section{The Slaggie-Wichmann Results on Two-Body Wave Functions}

In 1962, Slaggie and Wichmann \cite{SlaggieWichmann}
published a paper in which
they studied the decay properties of the eigenfunctions of three-body
Schr\"odinger operators using integral operator methods. 
Although we will not discuss the many-body problem here,
we are interested in their proof of exponential decay of eigenfunctions
corresponding to negative energies $E_j < 0$ in the two-body situation. 
The proof of Slaggie and Wichmann is very simple and requires minimum 
regularity on the potential. The method
capitalizes on a basic fact that will be used again below: 
The  Green's function for the unperturbed
operator $H_0 = - \Delta$, in dimensions $n \geq 3$ and at negative
energies, decays exponentially in space. In three dimensions,
the kernel of the resolvent is given by 
\begin{equation}
R_0 ( z ) ( x , y ) = \{ 4 \pi \| x - y \| \}^{-1} e^{ i \sqrt{z} \| x - y \|},
\end{equation}
with the branch cut for the square root taken along the positive real axis.
In higher dimensions, the kernel of the resolvent is given by a Hankel
functions of the first kind that exhibits similar exponential decay.   
Since the spectrum of $H_0$ is the half-axis $[ 0 , \infty )$, the 
bound state energies of $H$ lie outside the spectrum of $H_0$ and, 
consequently, the free
Green's function exhibits exponential decay at those energies.
In section 5, 
we will discuss the exponential decay of the resolvent in more detail. 

The hypotheses of Slaggie and Wichmann
on the real-valued potential $V$ are rather general.

\vspace{.1in}

\noindent
{\bf Hypothesis 1.} {\it There exists a positive, continuous function
$Q ( s )$, with $s \in {\mathbb R}^+$, having the properties
\begin{eqnarray}
\lim_{s \rightarrow \infty } Q ( s ) & = & 0 \\
\lim_{s \rightarrow 0} s Q(s)        &  \leq & C_0 ,
\end{eqnarray}
and such that 
\begin{equation}
| V(x) | \leq Q ( \| x \| ) , \; \mbox{for} \; x \neq 0.
\end{equation}
We also assume that $V \in L^1_{loc} ( {\mathbb R}^3 )$,
and that $V$ is relatively Laplacian bounded with relative bound
less than one.}

These conditions allow a Coulomb singularity at the origin and slow decay
at infinity. It is not difficult to prove, using a Weyl sequence argument,
that $\sigma_{ess} ( H ) = [ 0 , \infty ) $. Alternatively, one can show that
$| V |^{1/2} ( - \Delta + 1 )^{-1}$ is a compact operator. 
It follows that $( - \Delta + 1 )^{-1} V ( - \Delta + 1 )^{-1}$ is compact
so that $\sigma_{ess} ( H ) = \sigma_{ess} ( - \Delta ) = [ 0 , \infty ).$

\begin{theorem}{}{}{\it}
Let $H = - \Delta + V$ be a two-body Schr\"odinger operator satisfying 
Hypothesis 1. Let $E_j < 0$ be a negative bound state energy
and let $\phi_j$ be any corresponding normalized
eigenfunction satisfying $H \phi_j =
E_j \phi_j $. For any $0 < \theta < \sqrt{ | E_j| }$, there is a constant
$0 < C( \theta ) < \infty$, so that 
\begin{equation}
| \phi_j ( x ) | \leq C ( \theta ) e^{ - \theta \| x \| } .
\end{equation}
\end{theorem}

Let us note that this bound is saturated for the Coulomb ground state 
wave function which is 
$\psi ( x ) = C_0 e^{ - \sqrt{ |E_0| } \; \| x \| }$, with
$E_0 =  - 1/4$.
The square root in the energy behavior comes from the dependence of the free
Green's function on the energy, as seen in (2.1). 

Simon \cite{[Simon]} proved a similar result
on the exponential decay of eigenfunctions 
corresponding to negative eigenvalues of two-body Schr\"odinger operators
on $L^2 ( {\mathbb R}^3 )$,
with $V \in L^2 ( {\mathbb R}^3)$ (and also for $V$ in the Rollick class), 
using the integral equation (2.7).
Simon used a result, proved in \cite{[Simon]},
on the solutions of certain integral equations associated
with Hilbert-Schmidt kernels.
Suppose that $K ( x , y)$ is a Hilbert-Schmidt kernel,
that is,
$$
\int_{{\mathbb R}^3} \int_{{\mathbb R}^3 } | K ( x , y )|^2 \; d^3x d^3y < \infty ,
$$
and suppose that
$G(x)$ is a nonzero, measurable function, and that $G(x)^{-1}$
exists. Consider the kernel $M(x , y ) = G ( x ) K ( x , y ) G ( y )^{-1}$, 
and suppose that
the kernel $M(x,y)$ is also Hilbert-Schmidt. 
If $\psi \in L^2 ( {\mathbb R}^n )$  satisfies
$K \psi = \psi$, then $G(x) \psi (x) \in L^2 ( {\mathbb R}^n )$.
In the application of this general result, we note that the kernel of the
integral equation (2.7) is Hilbert-Schmidt. For the function $G(x)$, we choose
$G( x) = e^{ \theta \|x\|}$, for $0 \leq \theta < \sqrt{ | E_j |}$, 
as in Theorem 2.1.
It is easy to show that the modified kernel $M ( x , y)$
is also Hilbert-Schmidt, so the exponential decay of the eigenfunction
follows. 

\vspace{.1in}

\noindent
{\bf Proof of Theorem 2.1.} 
We will repeated use one basic fact below. An $L^2$-eigenfunction
in 3-dimensions is necessarily continuous. This is a consequence
of the facts that such an eigenfunction is in the Sobolev space $H^2 ( 
{\mathbb R}^3 )$, and the Sobolev Embedding Theorem, Theorem 1.1.
The eigenvalue equation
\begin{equation}
( - \Delta + V ) \psi_j =  E_j \psi_j ,
\end{equation}
implies the integral equation for $\psi_j$,  
\begin{equation}
\psi_j ( x ) = - \int_{ {\mathbb R}^3 } \frac{ e^{ - \sqrt{ |E_j|} \| x - x' \| }
}{ 4 \pi \| x - x' \| } V( x' ) \psi_j ( x' ) \; d^3 x' .
\end{equation}
It follows immediately from Hypothesis 1 and (2.7) that
\begin{equation}
| \psi_j ( x ) | \leq \int_{ {\mathbb R}^3 } \frac{ e^{ - \sqrt{ |E_j|} \| x -
x' \| } }{ 4 \pi \| x - x' \| } Q ( \| x' \| )  | \psi_j ( x' )| \; d^3 x' . 
\end{equation}
Let us define a function $m(x)$ by
\begin{equation}
m(x) \equiv \sup_{x'} \{ \; | \psi_j ( x' ) | e^{ - \theta \| x -
x' \| } \; \} ,
\end{equation}
and, motivated by (2.8), another function $h_\theta (x)$ by
\begin{equation}
h_\theta (x) \equiv 
\int_{ {\mathbb R}^3 } \frac{ e^{ - ( \sqrt{ |E_j|} - \theta) \| x -
x' \| } }{ 4 \pi \| x - x' \| } Q ( \| x' \| ) \; d^3 x',
\end{equation}
for $0 < \theta < |E_j|$.
It is easy to check that $h_\theta$ is continuous, 
rotationally invariant, and thus a
function of $\|x\|$ only. 
These two definitions and inequality (2.8) imply that
\begin{equation}
| \psi_j ( x ) | \leq h_\theta ( \| x \| ) m ( x ) .
\end{equation}

We next prove that 
\begin{equation}
\lim_{ \|x\| \rightarrow \infty } h_\theta ( \|x\| ) = 0.
\end{equation}
We divide the region of integration
${\mathbb R}^3$ into two regions: $\| x - x' \| < \epsilon$ and
$\| x - x' \| > \epsilon $, for some $\epsilon > 0$, to be determined below. 
In the first region, we easily
show that $h_\theta ( x ) \leq C \epsilon^2  \|x\|^{-1}$. In the second region,
we use the boundedness of $Q$ and write
\begin{equation}
\int_{ \| x -x' \| > \epsilon } \frac{ e^{- \lambda \| x - x' \| }}{ 4 \pi
\| x - x' \| } \:
Q ( \|x\|) d^3 x' \leq \frac{ C_0 }{ 4 \pi \epsilon } \int_{ {\mathbb R}^3 }
e^{- \lambda \| u \| } d^3 u ,  
\end{equation}
where $\lambda = \sqrt{ | E_j | } - \theta > 0$.
If we choose $\epsilon = \|x \|^{1/4}$, for example, property
(2.12) follows from these two estimates. 

This decay of the function $h_\theta (x)$ as $\|x\| \rightarrow \infty$
controls the decay of the eigenfunction in the following sense.
Because $h_\theta$ vanishes at infinity,
there exists a region ${\cal R} \subset {\mathbb R}^3$ on
which $h_\theta(x) < 1$. 
We can simply take ${\cal R} = {\mathbb R}^3 \backslash B_R (0)$, 
for a radius $R$ sufficiently large.
We denote by ${\cal R}^c$ the complement of this
region. We have from (2.11) that $| \psi_j ( x ) |
< m ( x )$, for $x \in {\cal R}$. On the otherhand,
we have for all $x \in {\mathbb R}^3$ that 
\begin{equation}
m(x) = \mbox{max} \; \left( \; \sup_{x' \in {\cal R}} \{ | \psi_j ( x' ) | 
e^{ - \theta \| x - x' \| } \} , \; 
\sup_{x' \in {\cal R}^c } \{ | \psi_j ( x' ) | e^{ - \theta 
\| x - x' \| } \} \right)  .
\end{equation}

Our goal is to show that the maximum is obtained by 
the ${\cal R}^c$-term. This will immediately imply the 
result. To this end, we first note that 
from the definition of $m$, for any $x'' \in {\mathbb R}^3$, we have 
\begin{eqnarray}
m(x) & = &  \sup_{x'} \{ | \psi_j ( x' ) | e^{ - \theta \| x - x' \| }
              \}   \nonumber \\
     & =  & \sup_{x'} \{ | \psi_j ( x' ) | 
             ( \sup_{x''} e^{ - \theta \| x' - x'' \| } 
               e^{ - \theta \| x'' - x \| } )  \}  \nonumber  \\
     & = & \sup_{x''} \{ m ( x'' )  e^{ - \theta \| x'' - x \| } \} .
\end{eqnarray}
We have used the identity
\begin{equation}
e^{ - \theta \| x - x' \| } = 
\sup_{ x''} e^{ - \theta \| x - x'' \|} e^{ - \theta \| x'' - x' \|} ,
\end{equation}
that is proved by the triangle inequality and the definition of the
supremum.
Using (2.15), and the fact that $| \psi_j ( x ) |     
< m ( x )$, for $x \in {\cal R}$, we compute,
\begin{eqnarray}
\lefteqn{ \sup_{x' \in {\cal R}} \{ | \psi_j ( x' ) | e^{ - \theta
\| x - x' \| } \} } \nonumber \\
 & < & \sup_{x' \in {\cal R}} \{ m( x' ) e^{ - \theta
\| x - x' \| } \} \nonumber \\
  & \leq & \sup_{x' \in {\mathbb R}^3 } \{ m( x' )  e^{ - \theta \| x - x' \| } \} 
  \nonumber \\
  & = & m(x) .
\end{eqnarray}
That is, the supremum over $x' \in {\cal R}$ occurring in (2.14)
is strictly less that $m(x)$.
Hence, we have that
\begin{equation}
m( x ) = \sup_{x' \in {\cal R}^c } \{ | \psi_j ( x' ) | e^{ - \theta
\| x - x' \| } \} .
\end{equation}
We can take ${\cal R}$ to be the exterior of a ball of radius
$R$, for sufficiently large $R$, due to the vanishing of $h_\theta$.
It follows immediately from the continuity of the eigenfunction $\psi_j$
and (2.18) that 
\begin{equation}
m ( x ) \leq C( R , \theta ) e^{ - \theta \|x\| } ,
\end{equation}
for all $x \in {\mathbb R}^n$, and for
some constant depending on $R > 0$ and $\theta$.
Inequality (2.11), that $| \psi_j ( x ) | \leq  h ( \|x\| ) m ( x )$, for
all $x \in {\mathbb R}^3$,  and the boundedness of $h$,
implies that there exists a constant $C_0 > 0$ so
that $| \psi_j ( x ) | \leq C_0 m ( x )$. This, together with (2.19),
establishes the upper bound on the eigenfunction.  $\Box$

As noted by Slaggie and Wichmann, the proof requires less restrictive
conditions on the potential $V$. 
The potential must satisfy conditions (2.2)--(2.4),
and, for each $0 < \theta < \sqrt{ | E_j | }$, there must
exist an $ R_\theta > 0$ so that $h_\theta ( \| x \| ) 
< 1$ for $\| x \| > R_\theta$.  

 
\section{The Froese-Herbst Method}

We indicate the basic ideas of the Froese-Herbst method \cite{FroeseHerbst} 
for proving the decay of eigenfunctions. The authors' main motivation 
and results concern the exponential decay of the  
eigenfunctions of $N$-body Schr\"odinger operators, and the absence of
positive eigenvalues for $N$-body Schr\"odinger operators.
We will only give the simplest version of the results here.
The Froese-Herbst method does not depend upon the
explicit properties of the free Green's function, as
in the Slaggie-Wichmann method. 
Consequently,
the Froese-Herbst method
can be applied to more general classes of differential operators,
such as Laplace-Beltrami operators on noncompact manifolds,
and to the study of eigenfunctions corresponding to eigenvalues embedded in the
essential spectrum.

The Froese-Herbst method is tied to the theory of positive commutators
as developed by E.\ Mourre \cite{Mourre}. 
We will briefly review the main points of this theory below.
The Froese-Herbst method yields $L^2$-exponential bounds of the form
\begin{equation}
e^F \psi \in L^2 ( {\mathbb R}^n ) ,
\end{equation}
for some function $F$. 
Under more regularity assumptions on $V$, this
$L^2$-expo\-nen\-tial bound can be converted to a pointwise exponential
bound, as explained in section 1.

The Froese-Herbst method identifies the {\it threshhold energies} 
associated with the Hamiltonian $H$ as controlling
the rate of decay of the eigenfunctions.
This means the following.
Let $\Sigma = \; \mbox{inf} \; \sigma_{ess} ( H )$ be the bottom of
the essential spectrum. For many-body systems, this can be strictly
negative. The bound state energies of subsystems lie between $\Sigma$ and
$0$. These energies are called thresholds of the system. 
More generally, we define threshold energies as
those energies at which the Mourre estimate (3.7) fails to hold.
For many two-body Schr\"odinger operators, the only
threshold energy is zero, which is also the bottom of the
essential spectrum. We mention
that there may be an eigenvalue at zero energy, 
or at any threshold.
There is no general
method for obtaining
estimates on the decay rate of the corresponding eigenfunctions.
There are examples for which the decay rate is only
inverse polynomial, rather than exponential.

The general Froese-Herbst result states that
any eigenfunction $\psi_E \in L^2 ( {\mathbb R}^n )$,
with $H \psi_E = E \psi_E$, decays exponentially
at a rate given by the square root of the 
distance from the eigenvalue to a threshold above the eigenvalue.
That is, for some threshold energy $\tau > E$, we have the bound,  
\begin{equation}
e^{ ( \sqrt{ \tau - E } - \epsilon ) \; \|x\|} \psi_E ( x )  \in L^2 ( {\mathbb R}^n ) ,
\end{equation}
for any $\epsilon > 0$. 
Note that when $\tau = 0$, this is basically the Slaggie-Wichmann result.

In the two-body case studied here, the 
potential $V$ must satisfy the following hypothesis.
We write $R_0 (z)$ for the resolvent of the Laplacian, $R_0 (z) =
( - \Delta - z )^{-1}$.  

\vspace{.1in}

\noindent
{\bf Hypothesis 2.} {\it We assume that the potential $V \in
C^1 ( {\mathbb R}^n )$, and is relatively $- \Delta$-bounded, with relative
bound less than one. Furthermore, we assume that there exists a
constant $0 < C_0 < \infty$, so that $V$ satisfies
\begin{equation}
\| R_0 ( -1 ) ( x \cdot \nabla V ) R_0 (-1) \| \; \leq C_0 .
\end{equation}
}
We can relax the $C^1$-condition and assume that 
condition (3.3) holds in the sense of quadratic forms, cf.\ \cite{CFKS}.
The main result of the Froese-Herbst in the two-body case is the following 
theorem.

\begin{theorem}{}{}{\it}
Let $H = - \Delta + V$ be a two-body Schr\"odinger operator 
with potential $V$ satisfying
Hypothesis 2. Suppose that for $E < 0$, 
is a bound state energy with eigenfunction $\psi_E \in L^2 ( {\mathbb R}^n)$,
satisfying $H \psi_E =  E \psi_E$.
Then, we have for any $\epsilon > 0$,
\begin{equation}
e^{ ( \sqrt{|E|} - \epsilon ) \; \|x\| } \psi_E ( x ) \in L^2 ( {\mathbb R}^n ) .
\end{equation}
If, in addition, we assume that $x \cdot \nabla V$ is relatively Laplacian
bounded with relative bound less than 2, then $H$ has no positive eigenvalues.
\end{theorem}   

One of the main applications of the method of proof of this
theorem (and its counterpart in the 
$N$-body case) is to prove the nonexistence of positive eigenvalues of Schr\"odinger
operators. The idea is to prove that any $L^2$-eigenfunction $\psi_E$,
corresponding to a positive energy eigenvalue $E$,
must decay faster than any exponential. That is, for all $\theta > 0$, 
we have 
\begin{equation}
e^{ \theta \|x\| } \psi_E ( x ) \in L^2 ( {\mathbb R}^n) .
\end{equation}
Since the decay of an eigenfunction is controlled by the distance
to a threshold larger than the eigenvalue, one must
prove that an $N$-body Schr\"odinger operator has no positive thresholds.
For the two-body case, we will show that the Mourre estimate
holds at all positive energies, so there are no positive thresholds.
Consequently, we see that the eigenfunction for a positive
eigenvalue must decay faster than any exponential.
A variant of a unique continuation argument then shows that such a function
$\psi_E = 0$.  

We now give an outline of the proof of the exponential
decay part of the Froese-Herbst Theorem.
A complete textbook presentation is given in \cite{CFKS}.
We will work with the specific case of two-body operators.
We begin with the Mourre theory of positive commutators \cite{Mourre}.
Let $A = \frac{1}{2} ( x \cdot \nabla + \nabla \cdot x)$ be the 
skew-adjoint operator so that $-i A$ is the self-adjoint generator of
the dilation group on $L^2 ( {\mathbb R}^n )$. We assume that the potential $V$
satisfies Hypothesis 2.
A simple computation shows that, formally, the 
commutator $[ H , A ] \equiv HA - AH$, is
\begin{equation}
[H , A ] = 2H - 2V - 2x \cdot \nabla V  = 2H + K.
\end{equation}
The operator $K$ is relatively-Laplacian compact. Let $I = [ I_0 , I_1 ]
\subset {\mathbb R}$ be a finite, closed interval, and let $E_H (I)$ be the spectral
projection for $H$ and the interval $I $. 
Conjugating the commutator
relation (3.6) by the spectral projectors $E_H (I)$, we obtain
the Mourre estimate,
\begin{eqnarray}
E_H (I) [ H , A ] E_H (I) & = & 2 E_H (I) H + E_H (I) K E_H (I)
\nonumber \\
               & \geq & 2 I_0 E_H (I) + E_H (I) K E_H (I) .
\end{eqnarray}

This estimate implies a Virial Theorem of the following type.
Suppose that the Mourre estimate (3.7) holds in a neighborhood $I$
with compact operator $K=0$. Then, the operator $H$ cannot have an
eigenvalue $E \in I$. Since, if $E \in I$ is an eigenvalue with
an eigenfunction $\psi_E$,
we have (neglecting domain considerations),
\begin{eqnarray}
\langle \psi_E , [ H , A ] \psi_E \rangle & = & \langle \psi_E , [ H - E ,
A ] \psi_E \rangle \nonumber \\
 & = & 0 . 
\end{eqnarray}  
On the otherhand, the Mourre estimate (3.7) with $K=0$ implies
\begin{equation}
\langle \psi_E , [ H , A ] \psi_E \rangle \geq 2 I_0 > 0.
\end{equation}
This inequality clearly contradicts (3.8).
Consequently, the energy $E$ cannot be an eigenvalue for $H$.

This simple idea lies behind
the proof of the Froese-Herbst
Theorem. Suppose $E$ is an eigenvalue of $H$, and define
$\langle x \rangle \equiv ( 1 + \| x \|^2 )^{1/2}$. We
define $\tau  \equiv \mbox{sup} \: \{ E + \alpha^2 \; | \;
\alpha > 0, \mbox{and} \: e^{ \alpha \langle x \rangle } \psi_E \in
L^2 ( {\mathbb R}^n ) \}$. If $\tau = \alpha_0^2 + E$ is not a threshold of $H$, then 
there exist $\alpha_1 \geq 0 $ and $\gamma > 0$, with
$\alpha_1 < \alpha_0 < \alpha_1 + \gamma$, so that 
$e^{ \alpha_1 \langle x \rangle } \psi_E \in L^2( {\mathbb R}^n )$, but
$e^{ ( \alpha_1 + \gamma ) \langle x \rangle } \psi_E$ is not in
$L^2 ( {\mathbb R}^n )$.
Because $\tau$ is not a threshold of $H$, the Mourre estimate 
holds in a neighborhood of $\tau$. In particular, it holds
in a neighborhood of $E + \alpha_1^2$, for some $\alpha_1$ sufficiently
close to $\alpha_0$, since the set of thresholds is closed. 
We will construct a sequence of approximate eigenfunctions $\Psi_s$
for $H$ and the eigenvalue $E + \alpha_1^2 $ in the sense
that $\| ( H - E - \alpha_1^2 ) \Psi_s \| \leq C_0 \gamma$,
and $\Psi_s$ converges weakly to zero as $s \rightarrow 0$. 
The Virial Theorem then implies that the matrix element
of the $[H , A]$ in the state $\Psi_s$,
which is approximately an eigenfunction
with eigenvalue $E + \alpha_1^2$, is very small with respect to $\gamma$.
On the other hand, the Mourre estimate holds in a 
small neighborhood $E + \alpha_1^2$,
and, since $\Psi_s$ converges
weakly to zero, the matrix element $\langle \Psi_s , K \Psi_s \rangle 
\rightarrow 0$. This implies that the matrix element
of $[H , A ]$ in the state $\Psi_s$ is strictly positive.
This gives a contradiction for small $\gamma$. 

In the first step of the proof, we construct states with shifted energy.
For motivation, recall that a translation in momentum space has
the effect of shifting the classical energy.
Let $\psi_E$ be an $L^2$-eigenfunction of $H$,
and assume that $F$ is a differentiable function
such that $\psi_F = e^F \psi_E \in L^2 ( {\mathbb R}^n )$. We want to compute the
conjugated operator $e^F H e^{-F}$. To do this, we note that for any $u \in C_0^{\infty}
( {\mathbb R}^n)$, we have
$$
 e^F ( - i \nabla ) e^{-F} u = ( -i  \nabla + i \nabla F )u ,
$$
so that
$$
- e^F \Delta e^{-F} =  ( -i \nabla + i \nabla F)^2 = - \Delta + ( \nabla \cdot \nabla F + 
\nabla F \cdot \nabla )  - | \nabla F |^2 .
$$
It then follows that 
\begin{equation}
H_F \equiv e^F H e^{-F} = H + ( \nabla F \cdot \nabla + \nabla \cdot \nabla F )  -
| \nabla F |^2 .
\end{equation}
It follows, after a short calculation, that the expected value of the 
Hamiltonian $H$ in the state $\psi_F$ is
\begin{equation}
\langle \psi_F , H \psi_F \rangle = \langle \psi_F , [ E + | \nabla F |^ 2 ]
\psi_F \rangle,
\end{equation}
so the state $\psi_F$ appears as a state with energy $E + | \nabla F |^2$.

Next, we choose a family of functions
$F_s$ so that $| \nabla F_s |^2 \sim \alpha_1^2$,
and so that the sequence $\psi_{F_s} \equiv \psi_s$ converges weakly to zero.
Let $\chi_s (t)$ be a smooth function of compact support 
satisfying $\lim_{s \rightarrow 0 } \chi_s (t ) = t$. 
We now define a weight $F_s ( x ) \equiv \alpha_1 + \gamma \chi_s ( \langle
x \rangle )$ having the property that $\lim_{s \rightarrow 0} F_s (x)
= \alpha_1 + \gamma$. It then follows that $\psi_s = e^{F_s} \psi_E 
\in L^2 ( {\mathbb R}^n)$, provided $s > 0$, but that $\| \psi_s \| \rightarrow 
\infty$, as $s \rightarrow 0$. Furthermore, a calculation reveals
that $| \nabla F_s |^2 \sim \alpha_1^2$, for small
$\gamma$. We define $\Psi_s \equiv \psi_s \: \| 
\psi_s \|^{-1}$, so that $\Psi_s$ converges weakly to zero.
It is not too difficult to show that $\Psi_s$ is the sequence of
approximate eigenfunctions we desire, in the sense that
\begin{equation}
\| ( H - E - \alpha_1^2 ) \Psi_s \| \sim 0 ,
\end{equation}
for small $\gamma$.
Finally, it follows from the Virial Theorem (3.8) that the matrix element
$\langle \Psi_s , [ H , A ] \Psi_s \rangle \sim 0$. On the otherhand, since the
Mourre estimate holds in a neighborhood of $E + \alpha_1^2$, 
by the assumption that
$E + \alpha_0^2$ is not a threshold, and
the sequence $\Psi_s$ converges weakly to zero, we know that this matrix
element is bounded from below by a strictly positive constant. This
gives a contradiction, so that $\tau = E + \alpha_0^2$ must be a threshold.

Some final comments are in order. The Froese-Herbst technique
depends upon the existence of a conjugate operator $A$ for a
given Hamiltonian $H$. It is not
always easy to construct a conjugate operator,
but this has now been done in a variety of
situations. Secondly, if the energy $E$ itself is a
threshold, the method gives no information about
the rate of decay of a corresponding eigenfunction. Thirdly, the proof
indicates that the rate of decay of the eigenfunction is controlled
by the square root
of the distance to some threshold above $E$,
similar to the Slaggie-Wichmann result.
The proof, however, does not indicate that it
is always the nearest threshold above $E$ that controls the exponential decay.  


\section{Nonisotropic Agmon Decay Estimates}

The results that we have discussed so far are exponential decay
estimates of the form $e^F \psi \in L^2 ( {\mathbb R}^n)$, with $F$ a
function of $\|x\|$ alone. Hence, the resulting bounds are spatially
{\it isotropic}. For the case of a two-body potential, these are optimal when
the potential is spherically symmetric. In general, isotropic
bounds do not reflect the variation of the potential with direction.
In the many-body case, when the total potential is the sum of two-body
potentials, the behavior of $V$ at infinity
depends crucially on the direction. Hence, one is led to develop
{\it nonisotropic} bounds on the decay of the wave function that more
closely reflect the behavior of the potential in each direction. 
Such bounds and
techniques are also crucial for the estimation of the lifetime of
quantum resonances in terms of the potential barrier generating the
resonance. 

A systematic study of the decay of eigenfunctions of second-order
partial differential operators, corresponding to eigenvalues below
the bottom of the essential spectrum, was performed by Agmon
\cite{Agmon}. 
A key role is played by the Agmon metric on ${\mathbb R}^n$. This
pseudo-Riemannian metric is constructed
directly from the potential and the energy, and thus reflects the variation
of the potential with direction.  
The distance function corresponding to the Agmon metric
measures how the potential controls the decay. Explicitly solvable models in
one-dimension, and the $WKB$ approximation give some clue as to the
form of this metric.

\vspace{.1in}

\noindent
{\bf Definition 4.1}  {\it Let $V$ be a bounded, real-valued 
function on ${\mathbb R}^n$, and let $E \in {\mathbb R}$. For any $ x \in {{\mathbb R}}^n $, and
$ \xi , \eta \in T_x ( {{\mathbb R}}^n ) = {\mathbb R}^n $, the tangent space to ${\mathbb R}^n$
at $x$, we define a
(degenerate) inner product on $ T_x ( {{\mathbb R}}^n ) $ by
\begin{equation} 
\langle  \xi , \eta \rangle_x \equiv 
( V (x) - E )_+ \langle  \xi , \eta \rangle_E  ,
\end{equation} 
where $ \langle  \cdot , \cdot \rangle_E $ is the usual
Euclidean inner product and
$ f(x)_+ \equiv \max \{ f (x) , 0 \} $. The corresponding pseudo-metric
on ${\mathbb R}^n$ is called the Agmon metric induced by the potential
$V$ at energy $E$.}

\vspace{.1in}
It is important to note that the Agmon metric depends on both the potential
and the energy $E$. 
The Agmon metric on $ {{\mathbb R}}^n $ is degenerate because there may exist
nonempty turning surfaces $\{ x \in {\mathbb R}^n \: | \: V(x) = E \}$, and classically
forbidden regions $\{ x \in {\mathbb R}^n \: | \: V(x) < E \}$. 
These sets play an important role in the theory.
The turning surface marks the limits of
classical motion for a particle with energy $E$ moving under
the influence of the potential $V$, and such a particle
cannot penetrate into the classically forbidden region.  
Consequently, it is expected that the quantum mechanical wave
function is small in the classically forbidden region.

We use the structure given in Definition 4.1 to construct a distance function
(or, metric) on $ {{\mathbb R}}^n $.  Let 
$ \gamma :  [0,1] \rightarrow {{\mathbb R}}^n $ be a differentiable
path in $ {{\mathbb R}}^n $. The derivative $\dot{\gamma} (t)$ belongs to the
tangent space at the point $\gamma (t)$.
For any Riemannian metric $g$ on a manifold ${\mathbb R}^n$, 
the length of $ \gamma $ is given by the integral 
\begin{equation} 
L ( \gamma ) = \int_0^1 \parallel \dot{ \gamma } (t) 
\parallel_{ \gamma (t) } d t , 
\end{equation}
where $ \parallel \xi \parallel_x = \langle \xi , 
\xi \rangle_x^{ 1/2 } $, for $\xi \in T_x ( {\mathbb R}^n )$.
In the Agmon structure (4.1), the
length of the curve $ \gamma $ (4.2) is:
\begin{equation} 
L_A ( \gamma ) = \int_0^1 ( V ( \gamma (t) ) - E )_+^{ 1/2 }
\parallel \dot{ \gamma } (t) \parallel_E d t,
\end{equation}
where $ \parallel \cdot \parallel_E $ denotes the usual Euclidean
norm.
A path $ \gamma $ is a
{\em geodesic} if it minimizes the energy functional
$ E ( \gamma )
\equiv \frac{ 1 }{ 2 } \int_0^1 \parallel \dot{ \gamma }
(t) \parallel_{ \gamma (t) }^2 d t $.  

\vspace{.1in}

\noindent
{\bf Definition 4.2}  {\it  Given a bounded, real-valued potential $V$ and
energy $E$, the distance between $ x,y \in {{\mathbb R}}^n $ in the
Agmon metric is
\begin{equation} 
\rho_E (x,y) \equiv 
{ \displaystyle \inf_{ \gamma \in P_{ x,y } } } L_A ( \gamma ) , 
\end{equation} 
where $ P_{ x,y } \equiv \{ \gamma : [ 0,1 ] 
\rightarrow {{\mathbb R}}^n \: | \: 
\gamma (0) = x , \gamma (1) = y , ~ \gamma \in A C [0,1] \} $. 
Here, the set $ A C [0,1] $ is the space of all absolutely
continuous functions on $[0,1]$}.

\vspace{.1in}

The distance between $ x,y \in {{\mathbb R}}^n $ with
the Agmon metric is the length of the shortest geodesic
connecting $x$ to $y$.  The distance function $\rho_E$ in (4.4)
reduces to the usual WKB factor for the
1-dimensional case,
\begin{equation}
\rho_E ( x , y) = \int_x^y ( V ( s) - E )_+^{1/2} \: ds.
\end{equation} 
The Agmon metric 
has several nice properties: It satisfies the
triangle inequality, and is Lipschitz continuous.
The main result of this section is that the Agmon metric at energy $E$
controls the decay of an eigenfunction at energy 
$E$, provided $E < \: \mbox{inf} \; \sigma_{ess} (H)$.

\begin{theorem}{}{}{\it} 
Let $ H = -\Delta + V $, with $V$ 
real and continuous, be a closed operator bounded below with 
$ \sigma (H) \subset {{\mathbb R}} $.  Suppose $E$
is an eigenvalue of $H$, 
and that the support of the function
$( E - V ( x ))_{+} \equiv \mbox{max} \; ( 0 , E - V(x) ) $ 
is a compact subset of ${{\mathbb R}}^n $.
Let $ \psi \in L^2 ( {{\mathbb R}}^n ) $ be
an eigenfunction of $H$ such that $ H \psi = E \psi $.
Then, for any $ \epsilon > 0$, there exists a constant 
$ C_{ \epsilon }$, with $ 0 < C_{ \epsilon } < \infty $, such that  
\begin{equation} 
\int e^{ 2 ( 1 - \epsilon ) \rho_E (x) }
| \psi (x) |^2 d x \leq C_{ \epsilon } , 
\end{equation} 
where $ \rho_E (x) \equiv \rho_E (x,0) $.
\end{theorem}

\vspace{.1in}

We note that if $V$ satisfies (1.2) uniformly in the sense 
that $| V(x)| < \epsilon$
for $ \| x \| $ large enough, and if $E < 0$, then 
the support of the positive part of $(E - V)$ is compact.

\vspace{.1in}

\noindent
{\bf Sketch of the Proof.} We will sketch the proof here. A textbook
treatment is given in \cite{HislopSigal}, and the general cases are treated
in \cite{Agmon}. 
The main idea of the Agmon approach is to
use the strict positivity of $( V - E )$, outside a compact set,
in order to bound the quadratic form $\langle \Phi , ( H - E ) 
\Phi \rangle$ from below, for suitably chosen vectors $\Phi$.
Note that the set on which $(V - E) > 0$ is the classically forbidden
region. A classical particle with energy $E$ cannot penetrate this region.
One expects that the corresponding quantum wave function 
is small in this region.
The vector $\Phi$ has the form
$\Phi = \eta e^F \phi$, where $F$ is a distance function built from 
the Agmon metric, the function $\eta$ localizes the eigenfunction
to the classically forbidden region, and we will take $\phi = e^F \psi_E$
after some initial calculations.
Because $\Phi$ is built from an eigenfunction for $H$,
the quadratic form $\langle \Phi, ( H - E ) \Phi \rangle$
is bounded above by the norm of $\psi_E$, localized
near the support of $\nabla \eta$. Hence,
we arrive at an inequality roughly of the form, 
\begin{equation}
\| \eta e^F \psi_E \|^2 \: \leq \: C_1 \| g ( \nabla \eta ) e^F \psi_E \|^2,
\end{equation}
where $g ( \nabla \eta )$ represents some combination of 
$\nabla \eta$. 
Since $e^F$ will be bounded on the support of $\nabla \eta$, this, in turn,
implies the $L^2$-exponential bound of $\psi_E$.

We now illustrate how to implement this strategy. 
Let $F = ( 1 - \epsilon ) \rho_E$
and note that almost everywhere,
\begin{equation}
| \nabla F |^2 \leq ( 1 - \epsilon )^2 ( V - E )_+ \leq ( 1 -
\epsilon ) ( V - E )_+  .
\end{equation}
We first compute a lower bound for the quadratic form
$\langle \Phi , (H-E) \Phi \rangle$, for $\Phi = e^F \eta \phi$. 
We take $\eta \in C^2 ( {\mathbb R}^n ) $ to be a 
nonnegative function
supported in the region where $ (V - E) \geq \delta$, and $\eta = 1$, 
except near the boundary of this region. 
A key computation involves the gauge transformation, as in (3.10),
given by $ H \rightarrow  H_F \equiv e^F H e^{ -F } $.
We recall the result that   
\begin{equation}
e^F H e^{ -F } = H + ( \nabla \cdot \nabla F + \nabla F \cdot \nabla ) 
               - | \nabla F |^2 .
\end{equation}
Consequently, for any reasonable function $\phi$, we compute 
a lower bound on the quadratic form, 
\begin{eqnarray}
\lefteqn{ Re \langle e^F \eta \phi , (H - E) e^{-F} \eta \phi \rangle
 }  \nonumber \\
 & = & Re \langle  \eta \phi , (H + ( \nabla \cdot \nabla F + \nabla F
\cdot \nabla ) - | \nabla F |^2 - E ) \eta \phi \rangle
   \nonumber \\
 & \geq &  \langle  \eta \phi , ( V - E - | \nabla F |^2 ) \eta \phi
\rangle   \nonumber \\
 & \geq & \epsilon \delta \| \eta \phi \|^2 .
\end{eqnarray}
We made use of the fact that $Re \langle \phi , 
( \nabla g \cdot \nabla +
\nabla \cdot \nabla g ) \phi \rangle = 0 $, for any real-valued function $g$.
We use this lower bound by setting $\phi = e^F \psi_E$.
After some standard computations, 
the final formula is 
\begin{equation}
Re \langle e^{2F} \eta \psi_E , (H - E)  \eta \psi_E \rangle \geq
\delta \epsilon \| e^F \eta \psi_E \|^2 .
\end{equation}

We now turn to the upper bound. We control the exponentially growing
term on the left in (4.12) by the compactness of the support of the
gradient of $\eta$. Using the fact that $\psi_E$ is an eigenfunction,
we have
\begin{eqnarray}
(H-E) \eta \psi_E & = & [ -\Delta , \eta ] \psi_E \nonumber \\  
                  & = & ( - \Delta \eta - 2 \nabla \eta \cdot \nabla )
                        \psi_E .
\end{eqnarray}
Since $\| \nabla \psi_E \| \leq  C_E \| \nabla ( H + M )^{-1} \|$,
for some $M$ large enough, we obtain an upper bound of the form,
\begin{equation}
| \langle e^{2F} \eta \psi_E , (H - E)  \eta \psi_E \rangle | \leq
 C_E \left( \sup_{ x \in supp | \nabla \eta | } e^{2F(x)} \right) .
\end{equation}
By arranging the diameter of the support of $| \nabla \eta |$ small
enough, we combine (4.12) and (4.14) to obtain
\begin{equation}
\| e^{ ( 1 - \epsilon ) \rho_E } \eta \psi_E \|^2 \leq C( E , \eta) .
\end{equation}
A simple estimate on $\| e^{ ( 1 - \epsilon ) \rho_E } ( 1 - \eta )
\psi_E \|^2 $, using the compactness of the support of
$( 1 - \eta )$, completes the proof of the result. $\Box$ 

\section{Resolvent Decay Estimates and the Combes-Thomas Method}

In this final section, we examine another manner
of proving exponential decay bounds on eigenfunctions
based on the exponential decay of the resolvent at energies outside
of the spectrum of the Hamiltonian. 
The Slaggie-Wichmann proof discussed in section 2
used the decay of the free Green's function at energies
below the bottom of the essential spectrum. We will first prove a result,
due to Combes and Thomas \cite{CT}, on the exponential
decay of the resolvent of a self-adjoint Schr\"odinger operator, at energies outside
of the spectrum, when localized between two disjoint regions.
We will then use this estimate to prove the decay of eigenfunctions
corrsponding to eigenvalues outside of the essential spectrum in
certain cases.
We conclude this section with an application to the exponential decay of 
eigenfunctions of the Dirac operator corresponding to eigenvalues in the
spectral gap $( -m , m)$.

We begin with a form of the
Combes-Thomas method \cite{CT}, due to Barbaroux, Combes, and
Hislop \cite{BCH}, which allows an improvement on
the rate of decay of the resolvent.
Combes and Thomas, motivated by the work of O'Connor
\cite{Oconnor} and dilation analyticity,
emphasized the use of analytic methods in the study of the decay of
eigenfunctions. Their method is more flexible than the
Slaggie-Wichmann or O'Connor method in that it 
can be applied, for example, to Schr\"odinger operators with nonlocal
potentials, to $N$-body Schr\"odinger operators,
and to other forms of differential operators.
As a consequence, one obtains
exponential decay for eigenfunctions
corresponding to isolated eigenvalues in gaps of the essential
spectrum, not just to those below the bottom of the essential
spectrum. 

The Combes-Thomas method also applies
to $N$-body Schr\"odinger operators with dilation analytic two-body potentials.
For such Schr\"odinger operators, the Combes-Thomas method allows
one to prove the exponential decay on nonthreshold eigenfunctions.
The result is similar to that obtained by the Froese-Herbst method. 
If $\psi_E$ is
an eigenfunction of $H$ corresponding to a nonthreshold eigenvalue $E$, then
$e^{a \| x \|} \psi_E \in L^2 ( {\mathbb R}^n )$,  for any $a$ satisfying
$a^2 < 2 \: \mbox{inf} \{ | E - \mbox{Re} E_\alpha | + \mbox{Im} E_\alpha \}$.
The infimum is taken over all thresholds of the nonself-adjoint operator
$H ( i \pi / 4)$, obtained from $H$ by dilation analyticity. 
Since we will not discuss dilation analytic operators here, we
refer the reader to \cite{RS4}.
As with the works mentioned above, Combes and Thomas are mainly
concerned with the $N$-body problem. We will discuss the method as
applied to two-body problems only.

\subsection*{A Simple Proof of Resolvent Decay Estimates} 

The idea of Combes and Thomas is to study the deformation of the
Hamiltonian by a unitary representation of an abelian Lie group, and then to
analytically continue in the group parameters. Typically, one uses the
group of dilations in coordinate space, or boost transformations in momentum
space. As an example,
let the Hamiltonian $H = - \Delta + V$ be self-adjoint. It follows as
in previous sections, that for
a constant vector $\lambda \in {\mathbb R}^n$, we have 
\begin{equation}
H ( \lambda ) 
\equiv e^{i x \cdot \lambda } H e^{ - i x \cdot \lambda } = H + 2 i \lambda
\cdot \nabla + | \lambda |^2 .
\end{equation}
Provided that the operator $\nabla$ is relatively $H$-bounded,
the operator $H ( \lambda )$ extends to an analytic family of type A on
${\mathbb C}^n$. 

We next study the resolvent of this analytic type A
family of operators.
Let us suppose that $E \in \rho (H)$, the resolvent set of $H$.
Then, the operator $(H - E)$ is
boundedly invertible, and we can write, 
\begin{equation}
(H( \lambda ) - E)  =  ( 1  + 2 i \lambda \cdot \nabla (H -
E)^{-1} + | \lambda |^2 (H - E)^{-1} ) (H - E) .
\end{equation}
Let us choose $\lambda$ so that 
\begin{equation}
\|  2 i \lambda \cdot \nabla (H - E)^{-1} \| \; < \; 1/2 .
\end{equation}
We define a constant $C_{V,E} \equiv \| \nabla ( H - E )^{-1} \|$,
that we assume is finite. We require that $| \lambda |$ satisfies  
\begin{equation}
| \lambda | <  \frac{1}{4 C_{V , E} }. 
\end{equation}
Let us write $B \equiv  i   \nabla (H - E)^{-1} $.
Assuming the condition (5.4) for the moment, and returning to (5.2),
we see that
\begin{equation}
(H( \lambda ) - E)  = ( 1  + 2 \lambda  B )
( 1 + ( 1 + 2 \lambda B )^{-1} | \lambda |^2 (H - E)^{-1} ) (H - E) .
\end{equation}
Once again, in order to invert the second factor,
we demand $| \lambda|$ also satisfies  
\begin{equation}
\| ( 1 + 2 \lambda B )^{-1} | \lambda |^2 (H - E)^{-1} \| < 1/2 .
\end{equation}
Since $H$ is self-adjoint, we have $\| ( H - E )^{-1} \| \leq
\{ \mbox{dist} ( \sigma (H), E ) \}^{-1}$. Let us define $\eta$ by
$\eta \equiv \mbox{dist} ( \sigma (H) , E )$. 
To satisfy the bound (5.6), we require
\begin{equation}
| \lambda | \: \leq \: \frac{\sqrt{ \eta }}{2  }.
\end{equation}
Consequently, the inverse of $( H ( \lambda ) - E )$ satisfies the bound ,
\begin{eqnarray}
\lefteqn{ \| ( H( \lambda ) - E)^{-1} \| } \nonumber \\
       & \leq & \| ( H - E )^{-1} \| \; \| ( 1 + B )^{-1} \| 
         \; \| ( 1 + | \lambda |^2 ( 1 + B )^{-1} ( H - E )^{-1} \| \nonumber
          \\
       & \leq & 4 \; \{ \mbox{dist} \; ( \sigma (H) , E ) \}^{-1} ,
\end{eqnarray}
for $\lambda \in {\mathbb C}^n$ with  
\begin{equation}
| \lambda | \leq \; \mbox{min} \left( \frac{ 1 }{ 4 C_{V,E}} , 
\frac{\sqrt{ \eta }}{2} \right) .
\end{equation}
Let $\nu$ denote the minimum of the right side of (5.9).
Thus, we have proved that for all $\lambda \in {\mathbb C}^n$ with
$| \lambda | < \nu$, the dilated operator $(H ( \lambda ) - E )$
is invertible, and we have the bound
\begin{equation}
\| e^{ i \lambda \cdot x } ( H - E )^{-1}  e^{- i \lambda \cdot x } \| \; \leq
\; \frac{4}{ \eta } .
\end{equation}

This bound is the key to proving exponential decay of the resolvent.
For any $u \in {\mathbb R}^n$,
let $\chi_u$ be a function with compact support near $u \in {\mathbb R}^n$.
We consider a fixed vector $y \in {\mathbb R}^n$, and the origin.
First, for any fixed, nonzero unit
vector $\hat{e} \in {\mathbb R}^n$, we set $\lambda = \kappa \hat{e} \in {\mathbb R}^n$ and
find,  
\begin{eqnarray}
\chi_y ( H - E )^{-1} \chi_0 & = & e^{- i \kappa \hat{e} \cdot x} \chi_y  
  ( e^{ i \kappa \hat{e} \cdot x} ( H - E)^{-1} e^{ -i \kappa
   \hat{e} \cdot x } ) e^{ i \kappa \hat{e} \cdot x } \chi_0 \nonumber \\
 & = & e^{-i \kappa \hat{e} \cdot x } \chi_y ( H( \kappa \hat{e} ) - E )^{-1} 
    e^{ i \kappa \hat{e} \cdot x } \chi_0 .
\end{eqnarray}
Second, we analytically continue the last term in (5.11) 
to $\kappa = - i \nu $.
This is possible because 
of the type A analyticity proved above, and because the
localization functions have compact support. 
We obtain from (5.10)--(5.11),
\begin{eqnarray}
\| \chi_y (H - E)^{-1} \chi_0 \| & \leq & \| \chi_y e^{- \nu \hat{e}
\cdot x } ( H (-i \nu) - E)^{-1} e^{ \nu \hat{e} \cdot x } \chi_0 \|
\nonumber \\
 & \leq &  \frac{C_0}{ \eta } e^{- \nu \hat{e} \cdot y } .
\end{eqnarray}
Taking $\hat{e} = y \| y \|^{-1}$,
it follows that 
\begin{equation}
\| \chi_y ( H - E)^{-1} \chi_0 \| \leq \frac{C_1}{ \eta } e^{ - \nu \|y\| } .
\end{equation}
This is our first resolvent decay estimate. 
Notice that the estimate holds for any energy that is separated from the 
spectrum of $H$. The exponential rate of decay $\nu$ is given in (5.9).
It is the minimum of
$C_2 \eta$ and $C_3 \sqrt{ \eta }$. We will see that this can be improved.

We note an improvement of the above technique when the eigenvalue
$E$ satisfies $E < \; \Sigma \equiv \mbox{inf} \;
\sigma_{ess} ( H ) $.    
In this case, the operator $(H - E)$ is positive
in the sense that for all $\phi \in D ( H )$, 
\begin{equation}
\langle \phi , (H - E) \phi \rangle \geq ( \Sigma - E ) \| \phi \|^2.
\end{equation}
We are back in the case considered by Agmon. 
We see that in this case the Combes-Thomas method is
the same as an isotropic Agmon estimate 
with an exponential factor $\sqrt{ \Sigma - E }$. 

\subsection*{The Combes-Thomas Method}

We now present an optimal version of the Combes-Thomas method 
\cite{BCH} improving the presentation in section 5.1.
The basic technical result is the following.

\begin{lemma} 
Let $A$ and $B$ be two self-adjoint operators such that \\
$d_{\pm}\equiv \mathop{\rm dist}( \sigma(A) \cap {\mathbb R}^{\pm}, 0)>0$, 
and $\Vert B\Vert \leq 1$. Then,

\begin{enumerate}

\item[(i)] For $\beta\in{\mathbb R}$ such that $\vert
\beta\vert<\frac{1}{2}\sqrt{d_+d_-}$, one has $0\in\rho(A+i\beta
B)$,

\item[(ii)]
For $\beta\in{\mathbb R}$ as in (i),
$$
\left\Vert(A+i\beta B)^{-1}\right\Vert \leq 2 \sup(d_+^{-1},
d_-^{-1}).
$$
 
\end{enumerate}
\end{lemma}

\noindent
{\bf Proof.} Let $P_{\pm}$ be the spectral projectors for $A$
corresponding to the sets $\sigma(A)\cap{\mathbb R}^{\pm}$, respectively
and define $u_{\pm}\equiv P_{\pm}u$. By the Schwarz inequality one has
\begin{equation}
\begin{array}{lcl}
\Vert u \Vert\ \Vert(A+i\beta B)u\Vert&\geq& \mbox{Re} \; 
\left<(u_+-u_-),(A+i\beta B)(u_++u_-)\right>\\
&\geq& d_+\Vert u_+\Vert^2+d_-\Vert u_-\Vert^2-2\beta
\mbox{Im} \; \left<u_+,Bu_-\right>\\
&\geq &\frac{1}{2} (d_+\Vert u_+\Vert^2+d_-\Vert u_-\Vert^2),\\
\end{array}
\end{equation}
where we again used the Schwarz inequality to estimate the inner
product. It follows that
$$
\Vert(A+i\beta B) u\Vert\geq\frac{1}{2} \; \mbox{min} \; 
(d_+,d_-)\Vert u\Vert,
$$
and since this is independent of the sign of $\beta$, the lemma
follows. $\Box$  


\begin{proposition}{}{}{\it}   
Let $\widetilde H$ be a semibounded self-adjoint operator with a
spectral gap
$G\equiv(E_-,E_+)\subset\rho(\widetilde H)$. Let $W$ be a
symmetric operator such that $D(W)\supset D((\widetilde
H+C_0)^{\frac{1}{2}})$
and $\Vert(\widetilde
H+C_0)^{-\frac{1}{2}} W(\widetilde H+C_0)^{-\frac{1}{2}}\Vert<1$,
for some
$C_0$ such that $\widetilde H+C_0>1$. For any $E\in G$, let
$\Delta_{\pm}\equiv \mbox{dist} \; (E_{\pm},E)$. Then, we have

\begin{enumerate}

\item[(i)]
The energy $E\in \rho(\widetilde H+i\beta W)$ for all real
$\beta$
satisfying
$$
\vert\beta\vert<\frac{1}{2}\left\{\frac{\Delta_+\Delta_-}
{(E_++C_0)(E_-+C_0)}\right\}^{\frac{1}{2}};
$$
\item[(ii)]
for any real $\beta$ and energy $E$ as in (i),
$$
\Vert(\widetilde H+i\beta W-E)^{-1}\Vert\leq 2
\sup\left(\frac{E_++C_0}{\Delta_+},\frac{E_-+C_0}{\Delta_-}\right)\
.
$$
\end{enumerate}
\end{proposition}

\noindent
{\bf Proof.}{}{}{\it}
Let $E\in G$ and $C_0$ be as above. Define a self-adjoint
operator
$A\equiv(\widetilde H+C_0)^{-1}(\widetilde H -E)$ and $B\equiv(\widetilde H
+C_0)^{-\frac{1}{2}}W(\widetilde H
+C_0)^{-\frac{1}{2}}$.
By hypothesis, the operator $B$ is self-adjoint and satisfies
$\Vert
B\Vert<1$. Note that $0\in \rho(A)$ and
\begin{equation}
d_{\pm} \equiv \mbox{dist} \; (\sigma(A)\cap {\mathbb R}^{\pm},0)=
\Delta_{\pm}(E_{\pm}+C_0)^{-1}>0
\end{equation}
Applying Lemma 5.2 to these operators $A$ and $B$, we see that
for
$\beta$ as in (i), $0\in \rho(A+i\beta B)$ and that
$$
\Vert(A+i\beta B)^{-1}\Vert\leq
2\sup\left(\frac{E_++C_0}{\Delta_+},\frac{E_-+C_0}{\Delta_-}\right)\
.
$$
Let $P_{\pm}$ be as in the proof of Lemma 5.2. For any $w\in
D(\widetilde H)$,
$$
\begin{array}{lcl}
\Vert (\widetilde H +i\beta W-E)w\Vert  &=    &\Vert(\widetilde H
+C_0)^{\frac{1}{2}}(A+i\beta B) (\widetilde H +C_0)^{\frac{1}{2}} w\Vert\\
                              &\geq & \Vert (A+i\beta B)(\widetilde H
+C_0)^{\frac{1}{2}} w\Vert\ ,\\
\end{array}
$$
since $(\widetilde H + C_0)\geq 1$. In order to estimate the lower
bound, we now repeat estimate (5.20) taking
$u\equiv (\widetilde H +C_0)^{\frac{1}{2}}w$. This gives
\begin{equation}
\begin{array}{lcl}
\Vert (\widetilde H +i\beta W-E)w\Vert  &\geq&\frac{1}{2}\Vert(\widetilde H
+C_0)^{\frac{1}{2}} u\Vert^{-1}\left(d_+ \Vert P_+ (\widetilde H
+C_0)^{\frac{1}{2}} w\Vert^2\right.\\
&&\left.+(d_- \Vert P_- (\widetilde H
+C_0)^{\frac{1}{2}} w\Vert^2\right)\\
        &\geq&\frac{1}{2} \mbox{min} \; (d_+,d_-)\Vert (\widetilde H
+C_0)^{\frac{1}{2}}
w\Vert\ .\\
\end{array}
\end{equation}
Since $\Vert (\widetilde H +C_0)^{\frac{1}{2}}
w\Vert\geq \Vert w\Vert$ and $d_{\pm}$ are defined in (5.21),
result (ii) follows from (5.22) and Lemma 5.2. $\Box$  

We now apply these results first to the exponential decay of the localized 
resolvent, and second, to the decay of eigenfunctions. We let $\rho ( x) 
\equiv \sqrt{ 1 + \| x \|^2 }$ be the regularized distance function.
We assume that the unperturbed operator $H_0$ has the form
$H_0 = H_A + V_0$, where $H_A \equiv
( - i \nabla - A )^2$, and $V_0$ is relatively $H_A$ bounded.
The electric potential $V_0$
and the vector potential $A$ are assumed to be sufficiently well-behaved
so that $H_0$ is essentially self-adjoint on $C_0^{\infty} ( {\mathbb R}^n )$.
We also assume that the spectrum
of $H_0$ is semibounded from below. Most importantly, we
suppose that the spectrum has an open spectral
gap in the sense that there exist constants $- \infty \leq - C_0 \leq B_-
< B_+ \leq \infty$ so that
\begin{equation}
\sigma (H_0 ) \subset [ -C_0 , B_- ] \cup [ B_+ , \infty ) . 
\end{equation}
Of course, this gap might be the half line $( - \infty , \Sigma_0)$, where
$\Sigma_0 = \mbox{inf} \; \sigma ( H_0 )$. 
For a less trivial example, we can take $H_0 = - \Delta + V_{per}$, where
$V_{per}$ is a periodic potential.

Finally, we assume that the perturbation potential $V$ satisfies
the following hypothesis.

\vspace{.1in}

\noindent
{\bf Hypothesis 3.} {\it The potential $V$ is relatively $H_0$-compact 
with relative bound less than one.  
For each $\epsilon > 0$, the potential $V$ admits a decomposition 
$V = V_c + V_\epsilon$,
where $V_c$ has compact support and $\| V_\epsilon \| < \epsilon$.}
 
\vspace{.1in}

Since the essential spectrum of $H_0$ is stable
under relatively compact perturbations,
the effect of the potential $V$ is to create isolated eigenvalues 
for $H = H_0 + V$ in the spectral gap $G = ( B_- , B_+ )$. We
are interested in the decay of the corresponding eigenfunctions.
We first prove that the resolvent of $H_0$ at energies $E \in G$ decays
exponentially when locialized between two disjoint regions.
 
\begin{theorem}{}{}{\it}
Let the unperturbed operator $H_0$ satisfy the conditions
above. Then, the dilated operator
$H(\alpha)\equiv e^{i\alpha\rho}H e^{-i\alpha\rho}$,
$\alpha\in{\mathbb R}$,
admits an analytic continuation as a type A family on the strip
$S(\alpha_0)$, for any $\alpha_0 > 0$.
For any $E\in G = ( B_- ,B_+ )$,
define $\Delta_{\pm}\equiv \mbox{dist} \; (B_{\pm},E)$. Then there exist
finite constants $C_1$, $C_2>0$, depending only on $H_0$ and $E$,
such that

\begin{enumerate}
\item[(i)] for any real $\beta$ satisfying $|\beta|< \mbox{min} \; (\alpha_0,
C_1
\sqrt{\Delta_+\Delta_-},\sqrt{\Delta_+/2})$, the energy $E\in\rho
(H_0 (i\beta))$;
\item[(ii)] for any real $\beta$ as in (i),
\begin{equation}
\| (H_0 (i\beta)-E)^{-1} \| \; \leq \; C_2 \; \mbox{max} 
(\Delta^{-1}_+,\Delta^{-1}_-)\ ;
\end{equation} 
\item[(iii)] let $\chi_u$ be a function of compact support localized near $u \in
{\mathbb R}^n$, then for $\beta \in {\mathbb R}$ as in (i), 
\begin{equation}
\| \chi_y ( H_0 - E )^{-1} \chi_0 \| \; \leq \; C_3 e^{ \beta \rho (y) }.
\end{equation}
\end{enumerate}
\end{theorem}

\noindent
{\bf Proof.}{}{}{\it}
By a calculation similar to that done in section 5.1, 
we have, for $\alpha \in {\mathbb R}$,
\begin{eqnarray}  
H_0 (\alpha) &  =  & e^{ i \alpha \rho } H_0 e^{ - i \alpha \rho } \nonumber \\
             &  =  & H_0 + \alpha^2|\nabla\rho|^2 + \alpha W  ,   
\end{eqnarray}
where $W=-(\nabla\rho \cdot ( - i \nabla-A)+( -i \nabla -A)
\cdot \nabla\rho)$
is symmetric. Note that $\Vert\nabla\rho\Vert_{\infty}=1$ and
$\Vert\Delta\rho\Vert_{\infty}=1$. 
Under the assumption that $V_0$ is
relatively $H_A$-bounded, it suffices to show that
for some $z \in \rho (H_A )$, the operator
\begin{equation}
\{ \alpha^2 | \nabla \rho |^2 + W \} ( H_A - z )^{-1} ,
\end{equation}
is bounded with norm less than one. 
Let us take $z = - i \eta $, for $\eta > 0$ and sufficiently large.
It is then easy to show that the operator
in (5.22) is bounded above by $C_0 \eta^{-1/2}$, for some constant
$C_0$ depending on $|\alpha|$. 
Since this bound can be made as small as desired for any fixed
$\alpha_0$, it folows that $H_0 ( \alpha)$ 
is an analytic type A family on any strip
$S ( \alpha_0 )$.

Next, we take $\alpha=i\beta,\beta$ real and
$|\beta |<\alpha_0$, so from (5.21), we have
\begin{equation}  
H_0 (i\beta)= H_0 - \beta^2|\nabla\rho|^2+i\beta W.
\end{equation}  
We apply Proposition 5.2 to this operator taking $\widetilde H\equiv
H_0 -\beta^2|\nabla\rho|^2$. This operator has a spectral gap which
contains $(\widetilde B_-,\widetilde B_+)$, where $\widetilde
B_- = B_-$ and $\widetilde B_+= B_+ -\beta^2$. 
In order that $\widetilde{\Delta}_+\equiv \mbox{dist} \; 
(\widetilde B_+,E)>(\Delta_+/2)$, we require
$|\beta|<\sqrt{\Delta_+/2}$. (Note that
$\widetilde{\Delta}_-=\Delta_-$). It follows from Proposition
5.2 that $E\in\rho(H_0(i\beta))$ for
$|\beta|< \mbox{min} \; \left\{\alpha_0,C_1\sqrt{\Delta_+\Delta_-}\right.$,
$\left. \sqrt{\Delta_+/2}\right\}$, and that (ii) holds. Result(iii) follows
from (ii) as in section 5.1.  $\Box$ 

We now consider the perturbation of $H_0$ by $V$. Assuming Hypothesis 3
on $V$, the operator $H \equiv H_0 + V$ is self adjoint on the same domain
as $H_0$. The perturbation may introduce isolated eigenvalues of finite
multiplicity in the spectral gap $G$. We apply the resolvent bound (5.20) 
to prove the exponential decay of the corresponding eigenfunctions.

\vspace{.1in}
\noindent
{\bf Theorem 5.4} {\it We assume that $H_0$ satisfies
the hypotheses given above, and that
the potential $V$ satisfies Hypothesis 3.
Suppose that $H = H_0 + V$ has an eigenvalue $E \in G$, the
gap in the spectrum of $H_0$, with
an eigenfunction $\psi_E$. We assume that $\| \psi_E \| = 1$.
For any $\alpha \in {\mathbb R}$,
with $\alpha <  \nu \equiv \mbox{min} \: ( C_1 \sqrt{ \Delta_+ \Delta_- } , 
\sqrt{ \Delta_+ / 2 } )$,
we have
\begin{equation}
e^{ \alpha \rho } \psi_E  \in L^2 ( {\mathbb R}^n ) .
\end{equation}
}

\vspace{.1in}

\noindent
{\bf Proof.} Let us first suppose that $V$ has compact support and that
$\mbox{supp} V \subset K$, for some compact $K \subset {\mathbb R}^n$. We write
$R_0 (E) \equiv ( H_0 - E )^{-1}$.  From the
eigenvalue equation, we write, for $\lambda \in {\mathbb R}$,  
\begin{eqnarray}
e^{ i \lambda \rho } \psi_E & = & - ( e^{i \lambda \rho } R_0 ( E) 
e^{- i \lambda \rho} ) \; ( e^{ i \lambda \rho } V \psi_E ) \nonumber \\
  & = & ( H_0 ( \lambda ) - E )^{-1} \; ( e^{ i \lambda \rho } V \psi_E ).
\end{eqnarray}
Because of the type A analyticity of $H_0 ( \lambda )$, and the
compactness of the support of $V$, 
each term on the right in (5.25) admits an analytic continuation
onto any the strip $S( \alpha_0)$.  
We set $\lambda = - i \alpha $, for $\alpha \in {\mathbb R}$ with $| \alpha |
< \nu$. Taking the norm of 
of both sides of the equation, and using the bound in (ii) of 
Proposition 5.3, we find
that there exists a constant $C_{E, V} > 0$ so that
\begin{equation}
\| e^{ \alpha \rho } \psi_E \| \; \leq \; C_{E,V} .
\end{equation}

When the support of $V$ is not compact, 
we consider the operator $H_0 + V_\epsilon$,
instead of $H_0$. This operator also extends to an analytic type A 
family on the same strip as $H_0$ since $V_\epsilon$ is a bounded operator. 
We choose $\epsilon <  \eta \equiv \mbox{dist} ( E , \sigma ( H_0))$,
so that $H_0 + V_\epsilon$ 
has a spectral gap around $E$ of size at least $\epsilon$.
It follows that $( H_0 + V_\epsilon - E)$ is invertible,
and its inverse can be computed from the equation
\begin{equation}
( H_0 + V_\epsilon - E ) = ( 1 + V_\epsilon ( H_0 - E)^{-1} ) ( H_0 - E ) ,
\end{equation}
since $\| V_\epsilon R_0 (E) \| < 1/2$.
The eigenvalue equation is now written as 
\begin{equation}
\psi_E = - ( H_0 + V_\epsilon - e)^{-1} V_c \psi_E .
\end{equation}
The exponential decay bound
now follows from Proposition 5.2 applied to $H_0 + V_\epsilon$,  
and the argument given above. $\Box$

\subsection*{Application: Eigenfunction Decay for the Dirac Operator}

As an application of the method of Combes-Thomas, we prove the decay of
eigenfunctions corresponding to discrete, isolated
eigenvalues of the Dirac operator.
The free Dirac operator for a particle of mass $m > 0$ is constructed as
follows. The Dirac matrices $\gamma_\mu$, with $\mu = 0 , 1, 2, 3$ are
four $4 \times 4$ matrices that form a representation of the canonical
anticommutation relations, 
\begin{equation}
\gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu =  2 \delta_{ \mu  \nu } ,
\end{equation}
where $\delta_{ \mu \nu }$ is the Kronicker delta function. We write
$\gamma = ( \gamma_1 , \gamma_2 , \gamma_3 )$ for the three-vector, 
and $\beta = \gamma_0$. The free Dirac Hamiltonian is 
\begin{equation}
H_0 = - i \gamma \cdot \nabla + \beta m c ,
\end{equation}
where $c > 0$ is the speed of light. We set $c = 1$. The unperturbed
operator $H_0$ is a first-order, matrix-valued linear operator. It is 
self-adjoint on its natural domain in 
the Hilbert space ${\cal H } = L^2 ( {\mathbb R}^3 , {\mathbb C}^4 )$. 
A simple calculation, based on the relations (5.29), shows that
$H_0^2 = - \Delta + m^2$. 
It follows that the spectrum of $H_0$ consists
of two branches: $\sigma ( H_0 ) = ( - \infty , -m ] \cup [ m , \infty )$.
The interval $G = ( - m , m )$ is a gap in the spectrum of the free Dirac
operator. 

We now consider local perturbations $V$ of $H_0$. Let us suppose that
$V > 0$ and that $V$ has compact support. Let $I_4$ denote the
$4 \times 4$ identity matrix. An application of the Birman-Schwinger
principle shows that $H_\lambda = H_0 + \lambda V \cdot I_4 $ has an eigenvalue
in the gap $G$ provided $\lambda$ is suitably chosen. 
In general, relatively $H_0$-compact perturbations $V$ will create
eigenvalues in the spectral gap of $H_0$. We are interested in the
isotropic exponential decay of the corresponding eigenfunctions. 

The simplified argument presented at the beginning of section 5 
applies direcly to this situation.
Let $\alpha \in {\mathbb R}$ and take $\rho ( x ) \equiv \sqrt{ 1 + \| x \|^2 }$, 
as above.
We define $d_{\pm} (E) = \mbox{dist} ( E , \pm m)$.
We apply a standard boost transformation to $H_0 = - i \gamma
\cdot \nabla + \beta m$ to obtain
\begin{eqnarray}
H_0 ( \alpha ) & \equiv & e^{ i \alpha \rho  } H_0 
                   e^{ - i \alpha \rho  } \nonumber \\
        & = &  -i \gamma \cdot ( \nabla - i \alpha \nabla \rho ) + \beta m 
                  \nonumber \\
        & = & H_0 - \alpha \gamma \cdot \nabla \rho  .
\end{eqnarray}
We now apply Lemma 5.1 with $A = H_0$ and $B = \nabla \rho \cdot \gamma$.
Since $| \nabla \rho | \leq 1$, we have
$\| B \|  \leq 1 $. Hence, with $\alpha = i \eta$, for $\eta \in {\mathbb R}$,  
we require 
\begin{equation}
| \eta | \leq (1/2) \sqrt{ d_+ (E)  d_- (E) } . 
\end{equation}
Under this condition, we obtain
\begin{equation}
\| ( H_0 ( i \eta ) - E )^{-1} \| \leq 2 \; \mbox{max} \left( 
\frac{1}{ d_+ (E) } , \frac{1}{ d_-(E) } \right) .
\end{equation}

We now proceed as in Theorem 5.4. First, we suppose that the potential
$V$ has compact support, and that $H = H_0 + V$ has an 
eigenvalue $E \in G$. We write the eigenvalue equation as
\begin{equation}
\psi = - R_0 ( E) V \psi .
\end{equation}
By analytic continuation from 
$\alpha \in {\mathbb R}$ to $\alpha = -i \eta$, with
$\eta \in  {\mathbb R}$ and satisfying the bound (5.32), we have   
\begin{equation}
\| e^{ \eta \rho } \psi \| \; \leq \; \|  H_0 ( - i \eta ) - E )^{-1} \|
\; \| e^{ \eta \rho } V \| \; \leq \; C_{E, V } ,
\end{equation}
proving the $L^2$-exponential decay of the eigenfunction for any $\eta \in {\mathbb R}$ 
satisfying (5.32). When $V$ does not have compact support, but vanishes 
uniformly at infinity, 
we use the same argument as in the proof of Theorem 5.4.
The same exponential decay results hold in the presence of a magnetic field
for which $H_0 = \gamma \cdot ( -i  \nabla - A ) + \beta m$, 
for reasonable magnetic vector potentials $A$.
(A similar result was recently obtained by Breit and Cornean \cite{BC}).

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\noindent{\sc P. D. Hislop} \\
Mathematics Department \\
University of Kentucky   \\  
Lexington, KY 490506-0027 USA  \\
e-mail:  hislop@ms.uky.edu

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