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\topmatter
\title Operators with Singular Continuous Spectrum, VII. \\
Examples with Borderline Time Decay
\endtitle
\rightheadtext{Examples with Borderline Time Decay}
\author Barry Simon$^{*}$
\endauthor
\leftheadtext{B.~Simon}
\affil Division of Physics, Mathematics, and Astronomy \\ California
Institute of Technology, 253-37 \\ Pasadena, CA 91125
\endaffil
\thanks $^{*}$ This material is based upon work supported by the
National Science Foundation under Grant No.~DMS-9401491. The
Government has certain rights in this material.
\endthanks
\thanks To appear in {\it{Commun.~Math.~Phys.}}
\endthanks
\abstract We construct one-dimensional potentials $V(x)$ so that if
$H=-\frac{d^2}{dx^2}+V(x)$ on $L^{2}(\Bbb R)$, then $H$ has purely
singular spectrum; but for a dense set $D$, $\varphi\in D$ implies
that $|(\varphi, e^{-itH}\varphi)|\leq C_{\varphi} |t|^{-1/2}\ln (|t|)$
for $|t|>2$. This implies the spectral measures have Hausdorff dimension
one and also, following an idea of Malozemov-Molchanov, provides
counterexamples to the direct extension of the theorem of
Simon-Spencer on one-dimensional infinity high barriers.
\endabstract
\endtopmatter
\document
\flushpar {\bf \S 1. Introduction}
\vskip 0.1in
This is a continuation of my series of papers (some joint) exploring
singular continuous spectrum especially in suitable Schr\"odinger
operators and Jacobi matrices [3,15,4,8,2,19,17,\linebreak 5,7,16].
Our main goal here is to construct potentials $V(x)$ on $\Bbb R$ so that if
$H=-\frac{d^2}{dx^2}+V(x)$, then $\sigma(H)=[0,\infty)$,
$\sigma_{\text{ac}}(H)=\sigma_{\text{pp}}(H)=\emptyset$, and there is
a dense set $D\subset L^{2}(\Bbb R)$ so that if $\varphi\in D$, then
$$
|(\varphi, e^{itH}\varphi)| \leq C_{\varphi} t^{-1/2} \ln(|t|)
\tag 1.1
$$
for $|t|>2$. (We say $|t|>2$ because of the behavior of $\ln(|t|)$ for
$|t|\leq 1$; note all matrix elements are bounded by $1$, so control
in $|t|\leq 2$ is trivial.)
(1.1) is interesting because the stated bound on $F_{\varphi}(t)
\equiv (\varphi, e^{-itH}\varphi)$ is just at the borderline for
operators with singular continuous spectrum. Indeed, if $t^{-1/2}$
in (1.1) were replaced by $t^{-\alpha}$ for any $\alpha > \frac{1}{2}$,
then $F_{\varphi}(t)$ would be in $L^2$ and so the spectral measures
$d\mu_{\varphi}(E)=F(E)\,dE$ for $F\in L^{2}$; that is, $d\mu_{\varphi}$
would be a.c.~and so $\sigma_{\text{ac}}(H)\neq \emptyset$.
As an indication of the borderline nature of (1.1), we note that
by Falconer [6], (1.1) implies $d\mu_\varphi$ is a measure carried on
set of Hausdorff dimension 1 in the sense that it gives zero weight to
any set of Hausdorff dimension strictly less than 1.
The potentials $V$ are sparse potentials in the sense that they are
mainly zero. They are examples of the type already studied in [19]. We
will have examples where $V\to 0$ at infinity but also examples where
$\varlimsup\limits_{x\to\pm\infty} V(x)=\infty$. The latter are of some
interest because of an idea of Malozemov-Molchanov [13], which was the
starting point of our work here.
This idea is related to results of Simon-Spencer [18]. To describe it,
we need some notions. Call a barrier a compact subset $B\subset\Bbb
R^n$ so that $\Bbb R^{n}\backslash B$ has exactly one bounded
component and so that $\Bbb R^{n}\backslash B$ has two components
if $n\geq 2$ and three if $n=1$. If $B_1$ and $B_2$ are barriers, we
say $B_2$ surrounds $B_1$ if $B_1$ is contained in the bounded
component of $\Bbb R^{n}\backslash B_2$.
By the width of a barrier $B$, we mean the distance between the
bounded component of $\Bbb R^{n}\backslash B$ and the unbounded
component (in case $n=1$, the union of the two unbounded
components). By the diameter of $B$, we mean $\max\{|x-y|\mid x,y\in
B\}$.
We say a potential $V$ on $\Bbb R^n$ has a sequence of high barriers
if
\roster
\item"{(1)}" $V$ is globally bounded from below and locally bounded.
\item"{(ii)}" There is a sequence $B_{1}, B_{2},\dots$ of barriers so
$B_{k+1}$ surrounds $B_k$.
\item"{(iii)}" The width of each barrier is at least $1$.
\item"{(iv)}" There exists $a_{k}\to \infty$ so $V(x)\geq a_k$ if
$x\in B_k$.
\endroster
Then Simon-Spencer proved:
\proclaim{Theorem 1.1 [18]} If $n=1$, $H=-\frac{d^2}{dx^2}+V(x)$, and
$V$ has a sequence of high barriers, then $\sigma_{\text{\rom{ac}}}(H)=
\emptyset$.
\endproclaim
Malozemov-Molchanov [13] have studied extensions of this result to
higher dimensions, which require some relations between the size of
$a_k$ and diameter of $B_k$. It is clearly expected that the result
does not extend without restriction to $n\geq 2$ but it is unclear how
to make counterexamples. Malozemov-Molchanov noted that there exist
purely singular measures $d\nu$ on $\Bbb R$ so that the convolution
$d\nu * d\nu$ is absolutely continuous. Moreover, if $V_1$ is a
potential on $\Bbb R$ with such a spectral measure $d\nu$ and
$$
V(x,y)=V_{1}(x)+V_{1}(y)
$$
is a potential $V$ on $\Bbb R^2$, then $-\Delta +V$ has $d\nu * d\nu$
as spectral measure (specifically, if $\varphi(x)$ has spectral
measure $d\nu$, then $\tilde{\varphi}(x,y)=\varphi(x)\varphi(y)$ has
spectral measure $d\nu * d\nu$). Finally, if $V_1$ has a sequence of
high barriers, so does $V$.
Our examples in obeying (1.1) will let us implement this strategy and
so prove:
\proclaim{Theorem 1.2} If $n\geq 2$, there exist potentials $V$ with a
sequence of high barriers so that $-\Delta +V$ has purely absolutely
continuous spectrum. If $n\geq 3$, there are such $V$'s for which the
spectrum is purely transient.
\endproclaim
We'll discuss transient and recurrent spectrum further below. It was
in thinking of how to implement this Malozemov-Molchanov strategy that
I was led to think of time decay and (1.1).
The potentials $V$ which implement (1.1) will be chosen even, so we
may as well consider half-line problems with Dirichlet or Neumann
boundary conditions at $x=0$. The half-line potentials will have the
form
$$
V(x)=\sum^{\infty}_{n=1} V_{n}(x-C_{n}) \tag 1.2
$$
where $V_n$ is a potential of compact support and the $C_n$'s are
sufficiently large. In principle, our constructions let us determine
how large the $C_n$'s must be, but since the main point of this
construction is existence, we won't completely track the restrictions
on $C_n$.
Section 4 is the technical core of the paper where we prove a critical
lemma about half-line potentials $V(x)$ of the form
$$
V_{L}(x)=V_{\infty}(x)+W(x-L) \tag 1.3
$$
with $V_{\infty},W$ bounded non-negative of compact support. We obtain
some uniform in $L$ bounds on the time decay of $|(\varphi, e^{-itH}
\varphi)|$. This lemma is used in Section 2 to make the construction
of $V$ obeying (1.1). The application to Theorem 1.2 is found in
Section 3. Finally, Section 5 contains some remarks about how big the
$C_n$'s in (1.2) need to be.
While Section 4 is somewhat technical, it is technicality with an
elegant physical interpretation and technology we expect will be
useful in other contexts.
It is a pleasure to thank Y.~Last, L.~Malozemov, and S.~Molchanov for useful
discussions.
\vskip 0.3in
\flushpar {\bf \S 2. The Construction Modulo the Main Technical Lemma}
\vskip 0.1in
In this section, we'll construct potentials $V$ on $\Bbb R$ so that
$-\frac{d^2}{dx^2}+V(x)$ has purely singular continuous spectrum, but
(1.1) holds for a dense set of $\varphi$'s. The construction will
depend on a lemma only proven in Section 4.
Our $V$'s will obey
$$
V(-x)=V(x),
$$
so $-\frac{d^2}{dx^2}+V(x)$ is a direct sum of two operators,
unitarily equivalent to the half line with Neumann and Dirichlet
boundary conditions. We'll prove the result for the Neumann boundary
condition case. The argument for the Dirichlet boundary condition case
is similar: One replaces the Neumann $m$-function $m_{N}(E)$ by
$m_{D}(E)=-m_{N}(E)^{-1}$ and the ``vector'' $\delta(x)$ by
$\delta'(x)$ ($\delta(x)$ lies in $\Cal H_{-1}$ for the Neumann case
but $\delta'(x)$ is only in $\Cal H_{-2}$ (e.g., [9] but this
doesn't change the analysis in any essential way).
Suppose $V$ is bounded below and let $H$ be the Neumann b.c.~operator
$-\frac{d^2}{dx^2}+V(x)$ on $L^{2}(0, \infty)$. Let $\Cal H_{s}$ be
the usual scale of spaces associated to $H$ [14] (so, e.g., $\Cal
H_{+1}$ is the form domain of $H$). Then $\delta(x)$, the delta
function at 0, lies in $\Cal H_{-1}$; so, in particular, $f(H)\delta
\in L^2$ for any function $f\in C^{\infty}_{0}(\Bbb R)$.
The technical lemma we will prove in Section 4 is
\proclaim{Theorem 2.1} Suppose $V_L$ has the form {\rom{(1.3)}} with
$V_\infty,W$ fixed bounded non-negative functions of compact support.
Let $f,g\in C^{\infty}_{0}(0)$ with support in $(0, \infty)$. Then
\roster
\item"\rom{(i)}" $\lim\limits_{L\to\infty}\, (f(H_{L})\delta,
e^{-itH_{L}}g(H_{L})\delta)=(f(H)\delta, e^{-itH}g(H)\delta)$
uniformly for $t$ in compact subsets of $(-\infty, \infty)$.
\item"\rom{(ii)}" There exist $C$ independent of $L$ and $t$ so that
$$
|(f(H_{L})\delta, e^{-itH_{L}}g(H_{L})\delta)|\leq Ct^{-1/2} \tag 2.1.
$$
\endroster
\endproclaim
\remark{Remark} This is in essence a diffusion bound. For each fixed
$L$, eventually $(f(H_{L})\delta,\mathbreak e^{-itH_{L}}g(H_{L})\delta)$
decays faster than any power of $t$. However, suppose $f=g$ is supported
very near energy $E=k^2$. Then at time $t=\pm L/k$ we should expect a
bump in $(f(H_{L})\delta, \mathbreak e^{-itH_{L}}g(H_{L})\delta)$ due to
return of a reflected wave (the distance traveled there and back is
$2L$ but since the free energy is $p^2$, not $\frac{1}{2}p^2$, the velocity
is near $2k$). Because of diffusion, this reflected bump will decay but only
as $t^{-1/2}$ for this particular $t$. Similarly, there will be multiple
reflection bumps at times $t=\pm nL/k$. Our proof in Section 4 will
essentially invoke a rigorous multi-reflection expansion.
\endremark
A sequence $V_n$ non-negative potentials of compact support will be
called trapping if
$$
-\frac{d^2}{dx^2}+\sum^{\infty}_{n=1} [V_{n}(x-L_{n})+V_{n}(-x-L_{n}))
\tag 2.2
$$
has no a.c.~spectrum if the $L_n$'s are sufficiently large. Trapping
potentials are constructed in Simon-Spencer [18] and Last-Simon [11].
They are of three types:
\roster
\item"{1)}" High barriers: What we have called sequence of high
barriers where $V_{n}(x)\geq a_{n}$ on $(-\frac{1}{2}, \frac{1}{2})$
and $a_{n}\to\infty$.
\item"{2)}" Long random barriers: If $V_{n}(x)$ is the sample of a
random potential on the interval $\big(\frac{n(n-1)}{2},
\frac{n(n+1)}{2}\big)$, there is no a.c.~spectrum so long as $L_n$ is
large enough.
\item"{3)}" Very long decaying barriers: If $V_n$ is the sample of
$|x|^{-\alpha}W(x)$ (with $W(x)$ random and $\alpha<\frac{1}{2}$) on
$(a_{n-1}, a_{n})$ and $a_n$ is large enough, then for $L_n$ large,
there is no a.c.~spectrum.
\endroster
Potentials of type 1,2 are discussed in [18]. [11] has a method that
handles all these cases. In all cases, the $L_n$'s need only be so
large that the support of $V_{n}(x-L_{n})$ is to the right of the support
of $V_{n-1}(x-L_{n-1})$. Our main theorem in this paper is
\proclaim{Theorem 2.2} Let $\{V_{n}\}$ be a sequence of trapping
potentials and let $V$ be defined by {\rom{(2.2)}}. Then the $L_n$'s
can be chosen so that
\roster
\item"\rom{(i)}" $H=-\frac{d^2}{dx^2}+V(x)$ has purely singular
continous spectrum.
\item"\rom{(ii)}" For a dense set $D\subset L^{2}(\Bbb R)$, and all
$\varphi,\psi\in D$,
$$
|(\varphi, e^{-itH}\psi)|\leq C_{\varphi,\psi}\ln(|t|)\big/|t|^{1/2}
$$
for $|t|\geq 2$.
\endroster
\endproclaim
\demo{Proof} Without loss, we'll restrict to the half-line Neumann
problem as explained. We'll make the argument for $\varphi=f(H)\delta$
for a single $f$ and then explain the modifications needed to get a
dense set of $\varphi$.
Theorem 2.1 implies that
$$
\lim\limits_{L\to\infty}\,\sup\limits_{|t|>2} \big[(\ln|t|)^{-1} |t|^{1/2}
|(f(H_{L})\delta, e^{-itH_{L}}f(H_{L})\delta) - (f(H_{\infty})\delta,
e^{-itH_{\infty}}f(H_{\infty})\delta)|\big]=0.
$$
Thus in adding in $V_n$, we can choose $L_n$ so the change in
$$
\sup\limits_{|t|>2}\,\big[(\ln|t|)^{-1} |t|^{1/2} (f(H^{(n)})\delta,
e^{-itH^{(n)}} f(H^{(n)})\delta)\big] \tag 2.3
$$
from the same quantity for $n-1$ is at most $\frac{1}{2^n}$. Here
$$
H^{(n)}=-\frac{d^2}{dx^2}+\sum^{n}_{m=1} V_{m}(x-L_{m})
$$
(on $(L^{2}(0, \infty)$).
Since $H^{(n)}\to H$ in strong resolvent sense, we have the result for
$H$ by taking $n\to\infty$ and noting $\sum\limits^{\infty}_{n=1}
2^{-n}<\infty$. To get a dense set of vectors, choose $f_k, C^{\infty}$
functions on $(0,\infty)$ so the $f_k$'s are dense in $\|\cdot\|_{\infty}$
norm in the continuous functions on $[0,\infty)$ vanishing at zero and
infinity. Then $\{f_{k}(H)\delta\}$ is a dense set in $L^{2}(0, \infty)$.
At step $n$, arrange for the change in (2.3) to be no more than $2^{-n}$
for $f=f_k$ with $k=1,\dots,n$. Then each of
$$
|(f_{k}(H)\delta, e^{-itH}f_{k}(H)\delta)|\leq C_{k}|t|^{-1/2}\ln(|t|)
$$
for $|t|>2$. \qed
\enddemo
\remark{Note} $\ln(|t|)$ plays no special role in the proof or
statement of the theorem. It could be replaced by any function
$\ell(|t|)$ so long as $\lim\limits_{\alpha\to\infty}\,\ell(\alpha)=
\infty$.
\endremark
\proclaim{Corollary 2.3} For any potential $V$ of the form given in
Theorem {\rom{2.2}}, $H$ has singular continuous spectrum of Hausdorff
dimension $1$ in the sense that its spectral measures $E_\Delta$ have
$E_{S}=0$ if $S$ is a Borel set of Hausdorff dimension $\alpha <1$.
\endproclaim
\demo{Proof} Follows from Falconer [6], pg.~67.
\enddemo
\proclaim{Corollary 2.4} For any potential $V$ of the form of Theorem
{\rom{2.2}}, we have
$$
\lim\limits_{|t|\to\infty}\,\frac{1}{t^{2-\epsilon}}\, \|xe^{-itH}
\delta_{0}\|^{2}=\infty
$$
for any $\epsilon >0$.
\endproclaim
\demo{Proof} Follows from the results of Last [10].
\enddemo
\vskip 0.3 in
\flushpar {\bf \S 3. High Barriers in Dimension Two or More}
\vskip 0.1in
In this section, we carry through the strategy of Malozemov-Molchanov
described in the introduction.
For this section, we'll fix once and for all a function $V_1$ on $\Bbb
R$ so that
\roster
\item"{(i)}" $V_{1}(-x)=V_{1}(x)$
\item"{(ii)}" There is $a_{n}\to\infty$ so $V_{1}(x)\geq n$ on
$[a_{n}, a_{n}+1]$.
\item"{(iii)}" $\sigma(H_{1})=[0,\infty)$ and is purely singular
continuous where $H_{1}=-\frac{d^2}{dx^2}+V_{1}(x)$.
\item"{(iv)}" For a dense set $D_{1}\subset L^{2}(\Bbb R)$,
$|(\varphi, e^{-itH_{1}}\psi)|\leq C_{\varphi,\psi} |t|^{-1/2}\ln(|t|)$
for $|t|\geq 2$ and any $\varphi,\psi\in D_1$.
\endroster
On $\Bbb R^n$ define
$$
V_{n}(x_{1}, x_{2},\dots, x_{n}) = V_{1}(x_{1})+V_{1}(x_{2})+\cdots
+ V_{1}(x_{n})
$$
and on $L^{2}(\Bbb R^{n})$,
$$
H_{n}=-\Delta +V_{n}.
$$
\proclaim{Theorem 3.1} {\rom{(a)}} If $n\geq 2$, there is a dense set
$D_n$ in $L^{2}(\Bbb R^{n})$ so that for $\varphi,\psi\in D_{n}$,
$(\varphi, e^{-itH_{n}}\psi)\in L^{p}$ for all $p>1$.
{\rom{(b)}} If $n\geq 3$, there is a dense set $D_n$ in $L^{2}(\Bbb
R^{n})$ so that for $\varphi,\psi\in D_{n}$, $(\varphi, e^{-itH}\psi)
\in L^{1}\cap L^{\infty}$.
\endproclaim
\demo{Proof} If $\varphi=\varphi_{1}\otimes\cdots\otimes\varphi_{n}$,
$\psi=\psi_{1}\otimes\cdots\otimes\psi_{n}$ with $\varphi_{i},\psi_{i}
\in D_{1}$, then $(\varphi, e^{-itH_{n}}\psi)=\prod\limits^{n}_{j=1}
(\varphi_{j}, e^{-itH_{1}}\psi_{j})$ so for $|t|\geq 2$,
$$
|(\varphi, e^{-itH_{n}}\psi)|\leq C_{\varphi, \psi} |t|^{-n/2}
(\ln|t|)^{n}.
$$
Since it is also bounded, we have the $L^p$ results. Linear
combinations of those $\varphi$'s are dense. \qed
\enddemo
\proclaim{Corollary 3.2} If $n\geq 2$, $H_n$ has purely a.c.~spectrum.
\endproclaim
\demo{Proof} If $d\mu$ is a measure and $F_{\mu}(t)\equiv \int
e^{-iEt} d\mu(E)$ is in $L^p$ with $p<2$, then by the Hausdorff-Young
inequality, $d\mu(E)=g(E)\,d(E)$ with $g\in L^q$ ($q=p/p-1$). \qed
\enddemo
In [1], Avron-Simon introduced the notion of transient and recurrent
a.c.~spectrum. $\varphi\in\Cal H_{\text{ac}}(A)$ is transient if it is
a limit of $\varphi_n$'s where each $(\varphi_{n}, e^{-itH}\varphi_{n})$
decays faster than any inverse polynomial in $t$. If $\Cal H_{\text{tac}}$ is
the set of such $\varphi$'s, then $\Cal H_{\text{rac}}=\Cal H_{\text{ac}}
\cap\Cal H^{\perp}_{\text{tac}}$ is called the set of recurrent a.c.~vectors.
It is proven in [1] that if $F(t)=(\varphi, e^{-itH}\varphi)$ lies in
$L^1$, then $\varphi$ is in $\Cal H_{\text{tac}}$. Thus,
\proclaim{Corollary 3.3} If $n\geq 3$, $H_n$ has purely transient
a.c.~spectrum.
\endproclaim
Thus, if $n=2$, it is possible that there is a weakened form of the
result of Simon-Spencer [18], that is,
\remark{Open Question} Are there examples of $n=2$ with a sequence of
barriers with transient a.c.~spectrum or is any a.c.~spectrum in such
cases of necessity recurrent?
\endremark
\vskip 0.3 in
\flushpar {\bf \S 4. The Main Technical Lemma}
\vskip 0.1 in
Our goal in this section is to prove Theorem 2.1. Since $-\frac{d^2}
{dx^2}+V_L$ converges to $-\frac{d^2}{dx^2}+V_\infty$ in strong
resolvent sense, and $\delta$ is in the common form domain, (i) is
elementary but also follows from the discussion below.
Our analysis depends on the Weyl-Titchmarsh theory of spectral
measures for the Neumann problem (see [12]); explicitly, we'll use the
form:
\proclaim{Proposition 4.1} Suppose $V$ is bounded and non-negative
with compact support in $[0,\infty)$ and $H=-\frac{d^2}{dx^2}+V(x)$
with $u'(0)=0$ boundary conditions. For any $E>0$, let $k=\sqrt{E}$ and
let $u_{+}(x, E)$ be the solution of $-u''+Vu=Eu$ which is equal to
$e^{ikx}$ for $x$ large. Define $m(E)=-u_{+}(0,E)/u^{\prime}_{+}(0,E)$, the
Neumann $m$-function. Then for $f$, a smooth function of compact
support,
$$
(\delta, f(H)\delta)=\frac{1}{\pi}\int f(E)[\text{\rom{Im}}\,m(E)]\,
dE. \tag 4.1
$$
\endproclaim
Because of (4.1), we'll need to estimate integrals of the form:
\proclaim{Lemma 4.2} Let $g$ be a $C^\infty$ function of compact
support on $(0, \infty)$, and let
$$
Q(y,t)=\int\limits^{\infty}_{0} e^{iky-ik^{2}t} g(k^{2})\, d(k^{2}).
$$
Then
$$
|Q(y,t)| \leq Ct^{-1/2} \biggl[\int\limits^{\infty}_{0} \{|g(k^{2})|^{2}
+ k^{2}|g'(k^{2})|^{2}\} k^{2}\, dk \biggr]^{1/2}.
$$
\endproclaim
\demo{Proof} Let $H_0$ be the operator $-\frac{d^2}{dx^2}$ on
$L^{2}(\Bbb R)$. Let $h(y)$ be the function $Q(y,0)$. Then, using the
explicit integral kernel of $H_0$:
$$\align
Q(y,t) &= (e^{-itH_{0}}h)(y) \\
&= (4\pi t)^{-1/2} \int e^{i|x-y|^{2}/4t} h(y)\, dy
\endalign
$$
so
$$\align
|Q(y,t)| &\leq (4\pi t)^{-1/2} \int |h(y)|\, dy \\
&\leq (4\pi t)^{-1/2} \biggl( \int |h(y)|^{2} (1+y^{2})\, dy
\biggr)^{1/2} \biggl[\int (1+y^{2})^{-1}\, dy\biggr] ^{1/2}
\endalign
$$
by the Schwartz inequality, so by the Plancherel theorem,
$$
|Q(y,t)| \leq (2t)^{-1/2} \biggl[\int |f(k)|^{2} + |f'(k)|^{2}
\, dk \biggr]^{1/2}
$$
where $f(k)=2kg(k^{2})$ and we are done. \qed
\enddemo
For the remainder of this section, we'll fix $V_\infty$ and $W$ and
always take $L$ so large that $W_{L}(\,\cdot\,)\equiv W(\cdot-L)$ has its
support to the right of the support of $V_\infty$. Thus, there are
$a**0$, $M(r,E)$ is anlytic in
the complex disc $\{r\mid |r|<1\}$. Similarly, $\frac{\partial M}
{\partial E}$ is analytic there too. Moreover, both functions are
uniformly bounded as $E$ run through compact subsets of $(0, \infty)$
and $r$ through compact subsets of $\{r\mid |r|<1\}$. In particular, for
any $R_{0}<1$ and compact $K\subset (0, \infty)$, there is a $C$ with
$$\align
M(r,E) &= \sum^{\infty}_{n=0} a_{n}(E) r^{n} \\
|a_{n}(E)| &\leq CR^{-n}_{0} \tag 4.2a \\
\biggl|\frac{da_n}{dE}\biggr| &\leq CR^{-n}_{0} \tag 4.2b
\endalign
$$
if $E\in K$.
\endproclaim
\remark{Remark} The proof actually shows more, as we'll note in the
next section; namely, $|a_{n}(E)| \mathbreak \leq C$, $\big|\frac{da_n}
{dE}\big| \leq C(n+1)$.
\endremark
\demo{Proof} Let $\left(\smallmatrix \alpha & \beta \\
\gamma & \delta \endsmallmatrix\right)$ be the transfer matrix from
$a$ to zero, that is,
$$
\pmatrix w(0) \\ w'(0) \endpmatrix =
\pmatrix \alpha & \beta \\ \gamma & \delta \endpmatrix
\pmatrix w(a) \\ w'(a) \endpmatrix
\qquad \text{for solutions of $-w''+(V-E)w=0$}.
$$
Then $M(E,r)$ is the fractional linear transformation
$$
M(r) =-\frac{\alpha(\omega+r\omega^{-1}) + ik\beta(\omega-r\omega^{-1})}
{\gamma(\omega+r\omega^{-1}) + ik\delta (\omega-r\omega^{-1})}
$$
where $\omega=e^{ika}$. The denominator vanishes exactly if
$r_{0}=\omega^{2}(-\gamma +ik\delta)/(\gamma+ik\delta)$. Notice that
since $\gamma,\delta$ are real, $|r_{0}|=1$, so as claimed, $M$ is
analytic in $|r|<1$. The uniform bounds on $M$ follow by noting that
$\alpha,\beta,\gamma,\delta$ are uniformly bounded. Similarly,
$\frac{\partial M}{\partial E}$ has a second order pole on the unit
circle and we get its uniform bounds. \qed
\enddemo
1. A convenient way to write $M$ is in terms the zeros $r_0$ and $r_1$
of the denominator and numerator of $M$. As in the proof, $|r_{0}|=
|r_{1}|=1$ and
$$
M(r)=M(0) \biggl(\frac{\bar{r}_{1}}{\bar{r}_{0}}\biggr)\,
\frac{r-r_{1}}{r-r_{0}}
$$
so, in fact $|a_{n}(E)|\leq 2|M(0)|$ (just expand the geometric series
and multiply out). Similarly, we can control $\bigl|\frac{da_n}{dE}
\bigr|$.
2. That the zero of the denominator has $|r_{0}|=1$ just happens in
the proof. But one can understand it from two factors. First, every
$r$ with $|r|<1$ occurs with some $W_L$ as we run through all possible
$W$'s. Thus, since $m$ is finite for any $V_{\infty}+W_L$ of compact
support, $M$ must be analytic in $|r|<1$. Moreover, $M(\bar{r}^{-1})=
\overline{M(r)}$, so we have analyticity also in $|r|>1$. \qed
\demo{Proof of Theorem {\rom{(2.1)}}} Let $r(E)$ be the reflection
coefficient on the whole line for $-\frac{d^2}{dx^2}+W(x)$. Then by
translation covariance, $r_{L}(E)$, the reflection coefficient for
$W(x-L)$, is
$$
r_{L}(k^{2})=e^{2ikL} r(k^{2}).
$$
Thus, in terms of the expansion above equation (4.2):
$$
S(L,t)\equiv (f(H_{L})\delta, e^{-itH_{L}}g(H_{L})\delta)=
\sum^{\infty}_{n=-\infty} S_{n}(L,t)
$$
where
$$
S_{n}(L,t)=\cases \tfrac{1}{2i} \tsize{\int} \overline{f(k^{2})}\,
g(k^{2}) e^{-ik^{2}t+2ikLn} a_{n}(k^{2}) r(k^{2})^{n}\, d(k^{2});
& n\geq 1 \\
\tsize{\int} \overline{f(k^{2})}\, g(k^{2}) e^{-ik^{2}t} \text{Im}\,
m_{\infty}(k^{2})\,dk^{2}; & n=0 \\
-\tfrac{1}{2i} \tsize{\int} \overline{f(k^{2})}\, g(k^{2}) e^{-ik^{2}
t-2ikLn} \overline{a_{n}(k^{2})}\, \overline{r(k^{2})}^{n}\,
d(k^{2}); & n\leq 1
\endcases
$$
where $m_\infty$ is the Neumann $m$-function for $V_\infty$. Since
$\text{supp}(\bar{f}g)\subset (0, \infty)$, we know that on that
support $\sup|r(k^{2})|$ is some $\alpha <1$. So in (4.2), take
$R_{0}>\alpha$ and use Lemma 4.2 to be able to sum up the $t^{-1/2}$
contributions and so obtain the theorem. \qed
\enddemo
\vskip 0.3 in
\flushpar {\bf \S 5. Towards Explicit Estimates of the
$\boldkey L_{\boldkey{n}}$}
\vskip 0.1 in
Our goal here is to explain why for the $\ln(t)/t^{1/2}$ bound we
believe that one needs to take $L_{n}\sim\exp(\exp(Cn^{3/2}))$ for the
case where, say, $V_{n}=n\chi_{(-1/2, 1/2)}$. If we only wanted $t^{-1/2
+\epsilon}$ behavior for fixed $\epsilon$, these same considerations
would only require $L_{n}\sim\exp(C_{\epsilon}n^{3/2})$ (consistent
with the behavior needed in [19]).
As noted in the remark after Theorem 4.3, we have $|a_{n}(E)|\leq
2|M(0)|$, $\bigl|\frac{da_n}{dE}(E)\bigr|\leq \bigl|2M(0)n\frac{dr_{0}}
{dE}\bigr|$. Thus, if
$$
A=\inf(1-|r|)
$$
on the support in question, $|M(r)|\leq 2|M(0)|A^{-1}$ and $\bigl|
\frac{dM}{dr}\bigr|\leq 2QA^{-2}$ with $Q$ bounded by $\bigl|M(0)
\frac{dM}{dE}(0)\bigr|$. Because of the definition of the transfer
matrix, in adding bump $n$, the transfer matrix for $V_\infty$ is of
order $\prod\limits^{n}_{j=1} e^{C\sqrt{j}}\sim\exp(C_{1}n^{3/2})$, so
$M$ and $\frac{dM}{dE}$ are bounded by $\exp(C_{1}n^{3/2})$. On the
other hand, $|r|$ for $n$th bump is of order $1-e^{-\sqrt{n}}$ by
tunneling estimates, so the $A^{-1}$ term in $|M(0)|A^{-1}$ is much
smaller than the $|M(0)|$ bound. Thus, the change in $(f, e^{-itH}g)$
is of order
$$
\exp(C_{1}n^{3/2}) t^{-1/2}
$$
and only for $t$'s of order at least $L^{1-\delta}_{n}$ for any
$\delta>0$. Thus, to get a $\ln(t)/t^{1/2}$ bound, we need only
arrange
$$
\ln(L_{n})^{1/2}\geq n^{+2}\exp(C_{1}n^{3/2}),
$$
and to get $t^{-1/2+\epsilon}$, we can have
$$
L^{\epsilon}_{n}\geq n^{2}\exp(C_{1}n^{3/2}),
$$
as claimed at the start of the section.
\vskip 0.3in
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\enddocument
**