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\topmatter
\title Operators with Singular Continuous Spectrum:\\ II. Rank One
Operators
\endtitle
\rightheadtext{Singular Continuous Spectrum: Rank One Operators}
\author R. del Rio$^{1,3}$,
\thanks $^{1}$ Permanent address: IIMAS-UNAM,
Apdo. Postal 20-726, Admon. No. 20, 01000 Mexico D.F., Mexico.
Research partially supported by DGAPA-UNAM and CONACYT.
\endthanks
N. Makarov$^2$
\thanks $^2$ This material is based upon work
supported by the National Science Foundation under Grant No. DMS-9207071.
The Government has certain rights in this material.
\endthanks
and B. Simon$^3$
\thanks $^3$ This material is based upon work
supported by the National Science Foundation under Grant No.
DMS-9101715. The Government has certain rights in this material.
\endthanks
\leftheadtext{R. del Rio, N. Makarov and B. Simon}
\endauthor
\thanks To appear in {\it Commun.~Math.~Phys.}
\endthanks
\affil Division of Physics, Astronomy and Mathematics\\California
Institute of Technology, 253-37\\Pasadena, CA 91125
\endaffil
\abstract For an operator, $A$, with cyclic vector $\varphi$, we study
$A+\lambda P$ where $P$ is the rank one projection onto multiples of
$\varphi$. If $[\alpha,\beta] \subset \text{spec} (A)$ and $A$ has no
a.c. spectrum, we prove that $A+\lambda P$ has purely singular
continuous spectrum on $(\alpha,\beta)$ for a dense $G_\delta$ of
$\lambda$\!'s.
\endabstract
\endtopmatter
\document
\flushpar{\bf \S 1. Introduction}
\smallpagebreak
The subject of rank one perturbations of self-adjoint operators and
the closely related issue of the boundary condition dependence of
Sturm-Liouville operators on $[0,\infty)$ has a long history. We're
interested here in the connection with Borel-Stieltjes transforms of
measures $(\text{Im}\ z>0)$:
$$F(z)=\int \frac {d \rho (x)}{x-z}\tag1.1$$
where $\rho$ is a measure with
$$\int (|x|+1)^{-1} d \rho(x)<\infty .\tag1.2$$
In two fundamental papers Aronszajn [1] and Donoghue [5] related $F$ to
spectral theory with important later input by Simon-Wolff [13]. In all
three works, as in ours, the function ($y$ real)
$$G(y)=\int \frac {d \rho (x)}{(x-y)^2}$$
plays an important role. Note we define $G$ to be $+\infty$ if the
integral diverges. Note too if $G(y)<\infty$, then the integral
defining $F$ is finite at $z=y$ and so we can and will talk about
$F(y)$.
Donoghue studied the situation
$$A_{\lambda}=A_{0}+\lambda P$$
where $P \psi =(\varphi, \psi)\varphi$ with $\varphi$ a unit vector
cyclic for $A$. $d \rho$ is then taken to be spectral measure for
$\varphi$, that is,
$$(\varphi, e^{isA_{0}}\varphi)=\int e^{isx}d \rho(x).$$
Aronszajn studied the situation
$$H_{\text{formal}}=-\frac {d^2}{dx^2} +V(x)$$
on $[0,\infty)$ where $V$ is such that the operator is limit point at
$\infty$. Then, there is a one-parameter family of operators,
$H_\theta$, with boundary condition
$$u(0) \cos\theta +u'(0) \sin\theta =0.$$
$\rho$ is the conventional Weyl-Titchmarsh-Kodaira spectral measure
for a fixed boundary condition, $\theta_0$.
An important result of the Aronszajn-Donoghue theory is
\proclaim{Theorem 1} $E$ is an eigenvalue of $A_\lambda$ \rom(resp.
$H_\theta$\rom) if and only if
\roster
\item"\rom{(i)}" $G(E)<\infty$
\item"\rom{(ii)}" $F(E)=-\lambda^{-1}$\qquad \rom(resp. $\cot(\theta -
\theta_0)$\rom).
\endroster
\endproclaim
Our goal here is to the prove the following pair of theorems:
\proclaim{Theorem 2} $\{E|G(E)=\infty\}$ is a dense $G_\delta$ in
\rom{spec}$(A_0)$ \rom(resp. $H_{\theta_0}$\rom).
\endproclaim
\proclaim{Theorem 3} $\{\lambda |A_\lambda$ has no eigenvalues in
\rom{spec}$(A_0)\}$ \rom(resp. $\{\theta|H_\theta$ has no eigenvalues in
\rom{spec}$(H_{\theta_0}\}$ is a dense $G_\delta$ in $\Bbb R$ \rom(resp.
$[0,2\pi]$\rom {))}.
\endproclaim
\bigpagebreak
While not stated precisely in those terms, Thm.~2 is a generalization
of del Rio [4]. Gordon [8,9] has independently obtained these results
by different methods.
Thm.~2 is quite easy and appears in \S2. Thm.~3 is deeper and depends
on some subtle estimates of $F$ found in \S3 and applied in \S4 to
prove Thm.~3.
The interesting applications of Thm.~3 found in \S5 concern singular
continuous spectrum. For example, suppose $A_0$ has spectrum $[0,1]$
but has no a.c.~spectrum. By general principles, $A_\lambda$ has no
a.c.~spectrum either. Then, Thm.~3 says that for a dense $G_\delta$
of $\lambda$, $A_\lambda$ has purely singular continuous spectrum.
This is especially interesting because there are examples where the
Simon-Wolff theory implies that for Lebesgue a.e.~$\lambda$, $A_\lambda$ has
only point spectrum. However, this is not always the case. There
exist $A_0$ and $P$ so that spec$(A_{0}+\lambda P)$ is purely singular
continuous on $(0, 1)$ for {\it all} $\lambda$. However, Thm.~3
implies that it {\it cannot} happen that the spectrum is {\it always}
pure point.
\bigpagebreak
\flushpar {\bf \S 2. Forbidden Energies}
\smallpagebreak
In this section we'll essentially prove Thm.~2.
\proclaim{Theorem 2.1} Let $d \rho$ be a measure obeying \rom{(1.2)}.
Let
$$G(y)=\int \frac{d \rho (x)}{(x-y)^2}.$$
Then, $\{ y|G(y)=\infty \}$ is a dense $G_\delta$ in
\rom{supp}\rom($d\rho$\rom), the support of $d\rho$.
\endproclaim
\remark{Remarks \rom{1}} By the Thm.~1, only $E$'s with
$G(E)<\infty$ can be eigenvalues of $A_\lambda$ or $H_\theta$, so
$E$'s with $G(y)=\infty$ are ``forbidden energies,'' that is, energies
which cannot be eigenvalues.
2. If supp$(d \rho)$ ($=\text{spec}(A)$) is perfect (no isolated
points), then the theorem says the forbidden energies are locally
uncountable in supp$(d \rho)$.
3. Obviously, $\{y|G(y)=\infty\}\subset\text{supp}(d \rho)$.
4. This says that $\{y\in\text{supp}(d \rho)|G(y)<\infty\}$ has
interior empty in supp$(d \rho)$. Even more so is the interior empty in
$\Bbb R$. The theorem is a stronger result than interior empty in
$\Bbb R$ since supp$(d \rho)$ might itself have interior empty in $\Bbb
R$.
\endremark
\demo{Proof} The following are fundamental facts about Borel-Stieltjes
transforms and their relation to $d \rho$ (see [3]):
\roster
\item $\lim\limits_{\epsilon\downarrow 0}F(E+i\epsilon)\equiv F(E+i0)$
exists and is finite for Lebesgue a.e.~$E$.
\item $d \rho_{\text{ac}}$ is supported on $\{E|\text{Im}\
F(E+i0)>0\}$.
\item $d \rho_{\text{sing}}$ is supported on
$\{E|\lim\limits_{\epsilon\downarrow 0}\ \text{Im}\ F(E+i0)=\infty\}$.
\endroster
\flushpar If $G(y)<\infty$, it is easy to see that
$\lim\limits_{\epsilon\downarrow 0}\ F(E+i0)$ exists is finite and real.
Thus, if $G(y)<\infty$ on an interval $(\alpha, \beta)\subset\Bbb R$,
then by (2), (3), $d \rho(\alpha, \beta)=0$, that is, $(\alpha,
\beta)\cap \text{supp}(d \rho)=\emptyset$. Thus, $\{ y|G(y)=\infty\}$
is dense in supp$(d \rho)$.
That $\{y|G(y)=\infty\}$ is a $G_\delta$ follows from the fact that
$G$ is lower semicontinuous. To be explicit, let
$$G_{m}(y)=\int \frac {d \rho(x)}{(x-y)^{2}+(m^{-1})^2}$$
which is a $C^\infty$ function by (1.2) and $G(y)=\sup\limits_m G_{m}(y)$.
Thus
$$
\align
\{y|G(y)=\infty\}&=\{y|\forall n, \exists mG_{m}(y)>n\}\\
&=\bigcap_{n} \bigcup_{m} \{y|G_{m}(y)>n\}
\endalign
$$
is a $G_\delta$. \qed
\enddemo
\bigpagebreak
\flushpar {\bf \S 3. The Main Technical Lemma}
\smallpagebreak
In this section we'll prove
\proclaim{Lemma 3.1} Let $d \rho$ obey \rom{(1.2)}. Then
$$\{F(y)|G(y)<\infty \ \text{\rm and}\ y\in\text{\rm supp}(d \rho)\}$$
is a countable union of nowhere dense subsets of \thinspace $\Bbb R$.
\endproclaim
Note that $G(y)<\infty$ implies the integral defining $F(y)$ is
absolutely convergent and $F(y)$ is real. The proof will depend
critically on the fact that $F$ is the boundary value of an analytic
function. That such considerations must enter is seen by
\example{Example} Let $A\subset[0, 1]$ be a nowhere dense set of
positive measure (e.g., remove the middle open $\frac {1}{4}$ from
$[0, 1]$, the middle $\frac {1}{9}$ from the remaining two pieces, the
middle $\frac {1}{16}$ \dots , $\frac {1}{n^2}$ at the $(n-1)$st
step). Let
$$\tilde F(y)=\left|A\cap[0, y]\right|$$
where $|\ \cdot \ |$ is Lebesgue measure. Then $\tilde F$ is Lipschitz; indeed,
if $x0$. Then $F[B]$ is nowhere dense.
\endproclaim
\demo{Proof} By (3.1) $F$ has a unique continuum extension to $\bar
B$ obeying (3.1). $\Bbb R\setminus\bar B$ is a union of intervals
$(x_{i}, y_{i})$ with $x_{i}, y_{i}\in\bar B$. Extend $F$ to the
interval by linear interpolation using slope $\frac {1}{2}(\alpha +
\beta)$ on any semi-infinite subintervals of $\Bbb R\setminus\bar B$.
The extended $F$ also obeys (3.1) and so defines a homeomorphism of
$\Bbb R$ to $\Bbb R$. As a homeomorphism, it takes nowhere dense sets
to nowhere dense sets. \qed
\enddemo
\demo{Proof of Lemma 3.1} We first break $A=\{y\in\text{supp}(d
\rho)|G(y)<\infty\}$ into a countable family of sets $A_n$ so that for
each $n$, there is $a_{n}>0$, $\delta_{n}>0$ so that
\roster
\item"\rom {(i)}" for $y\in A_n$, $\frac {8a_n}{9}x$
implies that
$$\frac {1}{3}\ a_{n}(y-x)