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\topmatter
\title Stability of Spectral Types for Sturm-Liouville Operators
\endtitle
\author R.~del Rio$^{1,4}$, B.~Simon$^{2,4}$, and G.~Stolz$^{3}$
\endauthor
\leftheadtext{R.~del Rio, B.~Simon, and G.~Stolz}
\thanks $^{1}$ IIMAS-UNAM, Apdo.~Postal 20-726, Admon.~No.~20,
01000 Mexico D.F., Mexico. Research partially supported by DGAPA-UNAM
and CONACYT.
\endthanks
\thanks $^{2}$ Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, 253-37, Pasa-dena, CA 91125.
\endthanks
\thanks $^{3}$ Fachbereich Mathematik, Johann Wolfgang
Goethe--Universit\"at, D-60054 Frankfurt am Main, Germany.
\endthanks
\thanks $^{4}$ 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
\thanks Appeared in {\it{Math.~Research Lett.}}, {\bf{1}} (1994),
437--450
\endthanks
\abstract For Sturm-Liouville operators on the half line, we show that
the property of having singular, singular continuous, or pure point
spectrum for a set of boundary conditions of positive measure depends
only on the behavior of the potential at infinity. We also prove that
existence of recurrent spectrum implies that of singular spectrum and
that ``almost sure'' existence of $L_2$-solutions implies pure point
spectrum for almost every boundary condition. The same results hold
for Jacobi matrices on the discrete half line.
\endabstract
\endtopmatter
\bigpagebreak
\document
\flushpar {\bf \S1 Introduction}
For Sturm-Liouville operators generated by $-\frac{d^2}{dx^2}+q$ on
the half line $[0, \infty)$, we study the dependence of spectral types
on the boundary condition at 0 and on compactly supported
perturbations of the potential. In Weidmann [17], it was conjectured
that the existence of singular spectrum depends only on the behavior
of the potential close to infinity. This, strictly speaking, is not
true (see [4,5]). However, we now prove that existence of singular,
singular continuous, or pure point spectrum for a set of boundary
conditions of positive measure does not depend on the local behavior
of the potential (\S5).
Our proof of this result is prepared in \S2--3 and relies mainly on:
\roster
\item"{(i)}" The identification of Aronszajn [1] and Donoghue [7] of
the various parts of the spectrum under variation of boundary
condition or rank one perturbation.
\item"{(ii)}" A result on the average of the spectral measure with
respect to boundary condition that goes back to Javrjan [9]; we use it
in a form rediscovered by Kotani [12].
\item"{(iii)}" The invariance of the absolutely continuous spectrum
under local perturbations.
\item"{(iv)}" Invariance of the set of energies with solutions $L^2$
at infinity under local perturbations.
\endroster
There are two other interesting consequences of these basic results:
\roster
\item"{(i)}" A further subdivision of the subspace of absolute
continuity of a self-adjoint operator was proposed by Avron-Simon [2],
where the definitions of transient and recurrent subspaces were
introduced. It was noticed in [2] that the recurrent spectrum is, in
some sense, close to the singular continuous spectrum. In fact, as we
shall see in Corollary 4.3, the presence of recurrent spectrum for some
boundary condition implies the presence of singular spectrum for more
than a null set of other boundary conditions.
\item"{(ii)}" The existence of $L_2$-solutions for (Lebesgue-) almost
every value of the eigenvalue parameter implies pure point spectrum
for almost every boundary condition (Corollary 3.2). This was noted
already in [6, Theorem 5.1] and generalizes a result of Kirsch, Molchanov,
and Pastur [11, Theorem 2], who applied this to potentials with an infinite
number of (high or wide) barriers [10,11]. Further applications will be
given in [16].
\endroster
As noted in [8], there is a unified approach to rank one
perturbations and variation of boundary condition if we consider
perturbations
$$
A_{\alpha}=A+\alpha B \tag 1.1a
$$
with
$$
B\psi=(\varphi, \psi)\varphi \tag 1.1b
$$
where $\varphi\in\Cal H_{-1}(A)$, the quadratic form dual in the scale
of spaces associated to $A$. $\varphi$ is always assumed cyclic.
(1.1) is then interpreted as a form sum. Variation of boundary
condition is then obtained by defining $A$ to have Neumann boundary
condition, $\varphi$ to be $\delta(x)$. If $\alpha=-\cot(\theta)$,
then $A_\alpha$ has boundary condition
$$
u(0)\cos\theta+u'(0)\sin\theta=0. \tag 1.2
$$
Details of this relationship, including the connection between the
functions $F(E)$ and the Weyl $m$-functions, and other results we'll
need are reviewed in Simon [14]. $\theta=0$, that is, $\alpha=\infty$
is discussed in Gesztesy-Simon [8].
Note that this identification of variation of boundary condition with
rank one perturbation of type (1.1) needs that the negative part of
$q_{-}$ of the potential is infinitesimally form bounded with respect
to $-\frac{d^2}{dx^2}$ on $L^{2}(0, \infty)$ with Neumann boundary
condition at $0$; compare [14]. Therefore, the application of our
general results below to Sturm-Liouville operators needs that
$q_{+}\in L^{1}_{\text{\rom{loc}}}(0, \infty)$ and, for example,
$q_{-}$ is bounded (more generally, $q_{-}$ which are locally
uniformly integrable are included). However, all our results on
variation of the boundary condition or local perturbations for
Sturm-Liouville operators are true under the weaker assumption that $-
\frac{d^2}{dx^2}+q$ is limit point at infinity. In particular, this
includes many operators which are not bounded from below. The proof
uses the Weyl $m$-function and the Weyl spectral measure instead of
the function $F_\alpha$ and measure $\mu_\alpha$ introduced below (for
their relation, see [14]) and proceeds in almost complete analogy.
Nevertheless, we use the rank one perturbation approach, which is more
general in other respects. For example, it immediately shows that all
our results for Sturn-Liouville operators have analogs for Jacobi
matrices on the discrete half line and extends to the case where $q$
is singular at $x=0$, but not so singular that $0$ stops being limit
circle.
\bigpagebreak
\flushpar {\bf \S2. Rank One Spectral Theory}
In the context of (1.1a), let $d\mu_\alpha$ be the spectral measure
for $\varphi$ and $A_\alpha$, so
$$
F_{\alpha}(z)\equiv (\varphi, (A_{\alpha}-z)^{-1}\varphi)=\int
\frac{d\mu_{\alpha}(x)}{(x-z)}.
$$
We set $F(z)\equiv F_{\alpha=0}(z)$.
It is also convenient to define
$$
d\rho_{\alpha}=(1+\alpha^{2})\, d\mu_{\alpha}
$$
since $d\rho_{\infty}=\lim\,d\rho_{\alpha}$ then exists and [8]
$$
d\rho_{\infty}=\lim\limits_{\epsilon\downarrow 0} [\pi^{-1}\text{Im}(-
F(x+i\epsilon))^{-1}\, dx].
$$
$A_\alpha$ converges to a self-adjoint operator $A_\infty$ in norm
resolvent sense [8]. In the half-line Sturm-Liouville case,
$A_\infty$ is $-\frac{d^2}{dx^2}+q$ with Dirichlet boundary condition.
Moreover, since $\varphi$ is assumed cyclic, $A_\alpha$ is unitarily
equivalent to multiplication by $x$ on $L^{2}(\Bbb R, d\rho_{\alpha})$
if $\alpha\neq\infty$ and $A_\infty$ is equivalent to multiplication
by $x$ on $L^{2}(\Bbb R, d\rho_{\infty})$ [8].
General principles imply $F(z)$ has boundary values $F(E+i0)$ for
a.e.~$E$. Set $G(x)=\int\frac{d\mu_{0}(y)}{|x-y|^{2}}$. Note that
$G(E)<\infty$ implies that $F(E+i0)$ exists and is real [14].
The following is essentially in Aronszajn [1] and Donoghue [7]; see
[14] for a short proof:
\proclaim{Theorem 2.1} \rom{([1, 7])} For $\alpha\neq 0$
\rom($\alpha=\infty$ allowed with $\infty^{-1}=0$\rom), define
$$\align
S_{\alpha} &= \{x\in\Bbb R\mid F(x+i0)=-\alpha^{-1}; G(x)=\infty\} \\
P_{\alpha} &= \{x\in\Bbb R\mid F(x+i0)=-\alpha^{-1}; G(x)<\infty\} \\
L &= \{x\in\Bbb R\mid F(x+i0) \text{\rom{ exists and Im }} F(x+i0)\neq 0\}.
\endalign
$$
Then
\roster
\item"\rom{(i)}" $\{S_{\alpha}\}_{\alpha\neq 0; |\alpha|\leq\infty}$,
$\{P_{\alpha}\}_{\alpha\neq 0; |\alpha|\leq\infty}$ and $L$ are mutually
disjoint.
\item"\rom{(ii)}" $P_\alpha$ is the set of eigenvalues of $A_\alpha$.
In fact
$$\alignat2
(d\rho_{\alpha})_{\text{\rom{pp}}}(x) &=\sum\limits_{x_{n}\in P_{\alpha}}
\frac{1+\alpha^2}{\alpha^{2}G(x_{n})}\, \delta (x-x_{n}) &&\qquad
\alpha <\infty \\
(d\rho_{\infty})_{\text{\rom{pp}}}(x) &= \sum\limits_{x_{n}\in
P_{\infty}} \frac{1}{G(x_{n})}\, \delta (x-x_{n}) &&\qquad
\alpha=\infty.
\endalignat
$$
\item"\rom{(iii)}" $(d\rho_{\alpha})_{\text{\rom{ac}}}$ is supported
on $L$, $(d\rho_{\alpha})_{\text{\rom{sc}}}$ is supported on
$S_\alpha$.
\item"\rom{(iv)}" For $\alpha\neq\beta$, $(d\rho_{\alpha})_{\text
{\rom{sing}}}$ and $(d\rho_{\beta})_{\text{\rom{sing}}}$ are mutually
singular.
\endroster
\endproclaim
In the above, we say $S$ ``supports a measure $d\nu$'' if $\nu(\Bbb
R\backslash S)=0$. And for any measure $d\nu$, we use
$d\nu_{\text{pp}}, d\nu_{\text{ac}}, d\nu_{\text{sc}},
d\nu_{\text{sing}}\equiv d\nu_{\text{pp}}+d\nu_{\text{sc}}$ for the
pure point, absolutely continuous, singular continuous, and singular
parts of $d\nu$.
One can say more than that $L$ supports $(d\rho_{\alpha})_{\text{ac}}$.
Recall that any a.c.~measure $d\nu(E)$ is of the form $f(E)\, dE$ and
that $\{E\mid f(E)\neq 0\}$ which is a.e.~defined is called the {\it{essential
support}} of $d\nu$ (also called a minimal support). Since (see, e.g.,
[14]):
$$\align
d\rho_{\alpha, \text{\rom{ac}}} &=\frac{1+\alpha^2}{\pi} \text{\rom{ Im }}
F_{\alpha}(E+i0)\, dE \qquad \alpha<\infty \tag 2.1a \\
\text{\rom{ Im }}F_{\alpha}(z) &=\text{\rom{ Im }}F(z)/|(1+\alpha
F(z)|^{2} \tag 2.1b \\
d\rho_{\infty, \text{\rom{ac}}}(E) &=-\frac{1}{\pi}\, \text{\rom{Im }}
F_{0}(E+i0)^{-1}\, dE, \tag 2.2
\endalign
$$
we see
\proclaim{Theorem 2.2} The set $L$ of Theorem \rom{2.1} is the
essential support of each $(d\rho_{\alpha})_{\text{\rom{ac}}}$.
\endproclaim
It is useful to have $\alpha$ independent sets:
\proclaim{Corollary 2.3} Let
$$\align
P &=\{x\mid G(x)<\infty\}\cup\{x\mid x\text{\rom{ is an eigenvalue
of }}A\} \\
L &=\{x\mid F(x+i0) \text{\rom{ exists and Im }} F(x+i0)\neq 0\} \\
S &= \Bbb R\backslash (P\cup L).
\endalign
$$
Then for $\alpha$ \rom(including $\alpha=0$ and $\infty$\rom):
$$
(d\rho_{\alpha})_{\text{ac}} = \chi_{L}\, d\rho_{\alpha}; \quad
(d\rho_{\alpha})_{\text{pp}} = \chi_{P}\, d\rho_{\alpha}; \quad
(d\rho_{\alpha})_{\text{sc}} = \chi_{S}\, d\rho_{\alpha}.
$$
\endproclaim
\demo{Proof} For $\alpha\neq 0$, this follows immediately from
$P_{\alpha}\subset P$, $S_{\alpha}\subset S$ and Theorem 2.1. For
$\alpha=0$, $P$ contains the eigenvalues by construction, $L$ supports
$d\rho_{\alpha=0, \text{ac}}$ by (2.1), and $S$\linebreak
supports $d\rho_{\alpha=0, \text{sc}}$. Since $S$ contains $\{E\mid\lim
\limits_{\epsilon\downarrow 0} |F(E+i\epsilon)|=\infty\}\backslash
\{E\mid E \text{ is an eigenvalue of }\mathbreak A\}.$ \qed
\enddemo
\proclaim{Proposition 2.4} The sets, $P, S, L$ are $\alpha$
independent, that is, one obtains the same sets starting from any
$A_\alpha$, $|\alpha|<\infty$.
\endproclaim
\demo{Proof} (2.1b) and the related $F_{\alpha}(E)=F(E)/1+\alpha F(E)$
show that $L$ is independent of $\alpha$. If $G(E)<\infty$, then
$F(E+i0)$ has a real value, so $E$ is actually an eigenvalue of
$\{A_\alpha\}_{\alpha\neq 0}$ with $\alpha=\infty$ allowed (if
$F(E+i0)=0$). Using also Theorem 2.1(ii), we get that $P=\cup\{E\mid
E \text{ is an eigenvalue of } A_{\alpha}; \, \alpha\in\Bbb R\cup
\{\infty\}\}$. Since $\{A_{\beta+\alpha}\}=\{A_{\beta}\}$ for any fixed
$\alpha$, $P$ is $\alpha$ independent. Thus, $S=\Bbb R\backslash (L\cup P)$
is also $\alpha$ independent. \qed
\enddemo
The following integral relation is a result of Javrjan [9]; see also
Kotani [12] and Simon-Wolff [15].
\proclaim{Theorem 2.5} For any Borel set $M$, we have that
$$\align
\int \mu_{\alpha}(M)\, d\alpha &= |M| \\
\int \rho_{\cot(\theta)}(M)\, d\theta &= |M|
\endalign
$$
where $|M|$ is the Lebesgue measure of $M$.
\endproclaim
It follows from the fact that the sets in Corollary 2.3 are $\alpha$
independent that
\proclaim{Theorem 2.6} For any Borel set $M$, we have that
$$\align
\int \mu_{\alpha, \text{pp}}(M)\, d\alpha &= |M\cap P| \\
\int \mu_{\alpha, \text{sc}}(M)\, d\alpha &= |M\cap S| \\
\int \mu_{\alpha, \text{sing}}(M)\, d\alpha &= |M\cap (S\cup P)| \\
\int \mu_{\alpha, \text{ac}}(M)\, d\alpha &= |M\cap L|
\endalign
$$
\endproclaim
Thus, we immediately have:
\proclaim{Corollary 2.7} For any Borel set $M$, $\mu_{\alpha,
\text{\rom{pp}}}(M)\neq 0$ for some set of $\alpha$'s of positive
measure if and only if $|M\cap P|\neq 0$ and similarly for
$\mu_{\alpha, \text{\rom{sc}}}(M)$ and $|M\cap S|$, and for
$\mu_{\alpha, \text{\rom{sing}}}(M)$ and $M\cap(S\cup P)$.
In particular, $\mu_{\alpha, \text{\rom{sc}}}\not\equiv 0$ for a set of
$\alpha$'s of positive measure if and only if $|S|\neq 0$.
\endproclaim
If one wants to state this theorem in terms of spectrum, one has to
face the fact that the spectrum is a poor invariant for measures, so
one is restricted to open sets.
\proclaim{Corollary 2.8} For any open set $I$, $I\cap\sigma_{\alpha,
\text{\rom{sc}}}\neq\emptyset$ for a set of $\alpha$'s of positive measure if
and only if $|I\cap S|\neq 0$ and similarly for $\sigma_{\alpha,
\text{\rom{pp}}}$ and $\sigma_{\alpha, \text{\rom{sing}}}$.
\endproclaim
\demo{Proof} For an open set $I$, and arbitrary measure $d\nu$,
$I\cap\text{supp}(d\nu)\neq\emptyset$ if and only if $\nu(I)\neq
0$. \qed
\enddemo
\example{Example} Suppose $A$ has only point spectrum in $(-\infty,
0)$ and an infinity of eigenvalues $e_{1}0$,
$$
\lim\limits_{|t|\to\infty}\,|t|^{N}(\varphi, e^{-itH}\varphi)=0.
$$
The transient subspace $\Cal H_{\text{tac}}$ is the closure of the set
of transient vectors. We have $\Cal H_{\text{tac}}\subset\Cal
H_{\text{ac}}$. The recurrent space $\Cal H_{\text{rac}}$ is the
orthogonal complement of $\Cal H_{\text{tac}}$ in $\Cal
H_{\text{ac}}$, that is, $\Cal H_{\text{rac}}=\Cal H_{\text{ac}}\cap
\Cal H^{\perp}_{\text{tac}}$.
\enddefinition
\proclaim{Lemma 4.1} Let $I$ be an open subset of $\Bbb R$. Suppose
that $A$ is multiplication by $x$ on $L^2(\Bbb R, d\mu)$ and
$$
d\mu_{\text{\rom{ac}}}(x)=f(x)\, dx
$$
with $f>0$ a.e.~on $I$. Then $E_{I}\Cal H_{\text{\rom{rac}}}=0$ where
$E_I$ is the spectral measure for $A$.
\endproclaim
\demo{Proof} Let $U:L^{2}(I, dx)\to L^{2}(I, f(x)\,dx)$ by
$$
(Ug)(x)=f^{-1/2}(x)\, gx.
$$
$U$ sets up a unitary equivalence between multiplication by $x$ on
$L^{2}(I, dx)$ and $A\restriction E_{I}\Cal H_{\text{ac}}$. That all
vectors are transient follows from Proposition 3.3 and Example 3.8 in
[2]. \qed
\enddemo
\proclaim{Theorem 4.2} In the context of rank one perturbations,
suppose $I\subset\Bbb R$ is open and $A_\alpha$ has only absolutely
continuous spectrum in $I$ for a.e.~$\alpha$. Then for any $\alpha$,
the spectrum is purely transient in $I$.
\endproclaim
\demo{Proof} By hypothesis and Theorem 2.6, $|I\cap P|=|I\cap S|=0$
so $|I\triangle L|=0$; that is, Lemma 4.1 applies for any $A_\alpha$
and so $E_{I}\Cal H_{\text{rac}}=0$. \qed
\enddemo
\proclaim{Corollary 4.3} Suppose $A$ is self-adjoint and
$I\subset\Bbb R$ is open. Suppose $\varphi$ is cyclic for $A$ and
that $E_{I}\Cal H_{\text{\rom{rac}}}\neq 0$. Then for a set of
$\alpha$'s of positive measure, $A_{\alpha}$ has singular spectrum in
$I$.
\endproclaim
Note that this result does not say if the singular spectrum is point
or singular continuous. As we'll see in \S 6, either can occur.
\bigpagebreak
\flushpar {\bf \S5. Local Perturbations}
In this section, we prove our main new result that the occurrence of
any specific spectral type for a set of positive measures of boundary
conditions for a Sturm-Liouville operator is invariant under local
perturbations of potential.
Let $q$ be locally in $L^1$ on $[0, \infty)$ be such that $-
\frac{d^2}{dx^2}+q$ is limit point at $\infty$ and let $H_{\theta}(q)$
be the operator $-\frac{d^2}{dx^2}+q$ with boundary condition (1.2) at
0 (see [3] for the precise definition). For $\theta\neq 0$, let
$S_{\theta}(q), P_{\theta}(q), L_{\theta}(q)$ be the set defined in Corollary
2.3 for $\varphi=\delta_{0}$ and $A=H_{\theta}(q)$. We already know
(Proposition 2.4) that these sets are $\theta$ independent. Here we
note that
\proclaim{Theorem 5.1} Let $v$ be in $L^1$ with compact support. Then
$$\gather
|S_{\theta}(q+v)\triangle S_{\beta}(q)|=0 \tag 5.1 \\
P_{\theta}(q+v)=P_{\beta}(q) \tag 5.2 \\
|L_{\theta}(q+v)\triangle L_{\beta}(q)|=0 \tag 5.3
\endgather
$$
for any $\theta, \beta$.
\endproclaim
\demo{Proof} As already noted, the sets are $\theta$ independent and
since $S=\Bbb R\backslash (P\cup L)$, we need only prove the results
(5.2), (5.3). (5.2) follows immediately from Theorem 3.1 and the fact
that solutions of $-u''+qu=eu$ and $-w''+(q+v)w=ew$ agree near
infinity.
To prove (5.3), let $\text{supp}\,v\subset [0, c]$. Let $A=-
\frac{d^2}{dx^2}+q$ with Neumann boundary condition at 0; let
$B=A+v$; let $\tilde A$ be $A$ with an additional Dirichlet boundary
at $c$, and similarly for $\tilde B$. Then $(\tilde A +i)^{-1}-
(A+i)^{-1}$ and $(\tilde B+i)^{-1}-(B+i)^{-1}$ are rank one.
Moreover, $(\tilde A+i)^{-1}=(\tilde A_{\text{in}}+i)^{-1}\oplus
(\tilde A_{\text{out}}+i)^{-1}$ on $L^{2}(0, c)\oplus L^{2}(c,
\infty)$ with $(\tilde A_{\text{in}}+i)^{-1}$ compact, and similarly for $B$.
Moreover, $\tilde A_{\text{out}}=\tilde B_{\text{out}}$. Thus, $A$
and $B$ have unitarily equivalent a.c.~subspaces (by using the Kuroda-Birman
theorem (see, e.g., [13])). Since $L$ is an essential support of the
a.c.~part of the spectral measure, we see that $L_{\theta}(q)$ and
$L_{\theta}(q+v)$ agree up to sets of measure zero. \qed
\enddemo
\proclaim{Corollary 5.2} Let $I\subset\Bbb R$ be open. If
$L_{\theta}(q)$ has singular continuous spectrum in $I$ for a set of
positive measure of $\theta$'s, the same is true for $q+v$.
\endproclaim
\demo{Proof} Follows from Corollary 2.8 and Theorem 5.1. \qed
\enddemo
Similar results hold for point spectrum and singular spectrum.
\bigpagebreak
\flushpar {\bf \S6. Examples}
Here are five examples closely related to Examples 1--3 in
Simon-Wolff [15] and Appendix 2 of [2]. They illustrate the kinds of
spectrum that can appear under rank one perturbations $A+\alpha
(\varphi, \cdot)\varphi$ when $A$ either has recurrent a.c.~spectrum
(Examples 1, 4, 5) or transient a.c.~spectrum (Examples 2, 3). In
Example 1 the singular spectrum appearing under rank one perturbations
is just discrete eigenvalues, but lying outside the a.c.~spectrum.
Examples 4 and 5 are somewhat more interesting in providing situations
where either s.c.~spectrum or eigenvalues appear embedded in the
recurrent a.c.~spectrum. Examples 2 and 3 show that there is no
converse to Corollary 4.3; namely, that the existence of singular
spectrum for a set of positive measure $\alpha$'s does not imply that
$A$ has any recurrent spectrum, even if the a.c.~spectrum has full
support.
In all cases we can totally describe the example by giving the spectral
measure $d\mu^{\varphi}_{A}$, which we'll call $d\mu$. In each case
we'll take $d\mu(x)=\chi_{B}(x)\, dx$ where $B$ is a Lebesgue measurable set.
\example{Example 1} Take $B$ to be a positive measure Cantor-type
set. For example, start with $[0, 1]$, remove the middle
$(\frac{1}{n_j})$, the fraction at step $j$ with $n_{j}=j^{2}$. As
usual, $B$ is a closed nowhere dense set. Let $F(z)=\int\limits_{B}
(x-z)^{-1}\, dx$. Then it is easy to see that $\varlimsup\limits_{\epsilon
\downarrow 0}\,\text{Im }F(x+i0)>0$ on $B$; indeed, we believe
$\lim\limits_{\epsilon\downarrow 0}\, \text{Im }F(x+i\epsilon)$ is
$\frac12$ if $x$ is a boundary point of a connected component of $[0,
1]\backslash B$ and is 1 otherwise. Because $\varlimsup\,\text{Im }
F>0$ for all $x$ in $B$, $d\mu_{\alpha,\text{sing}}(B)=0$ for all
$\alpha$. $A_\alpha$ for $\alpha\neq 0$ has a.c.~spectrum $B$, and a
single eigenvalue in each component of $[0, 1]\backslash B$. In this case
the singular spectrum guaranteed by Corollary 4.3 is just discrete
eigenvalues.
\endexample
\example{Example 2} Let $\{q_{n}\}^{\infty}_{n=1}$ be a counting of
the rationals. Let $a<\frac{1}{2}$ and let $B=[0, 1]\cap
[\operatornamewithlimits{\cup}\limits^{\infty}_{n=1} (q_{n}-
\frac {a^{n}}{2}, q_{n}+ \frac {a^{n}}{2})]$. Then $|[0, 1]\backslash B| > 1-
(a+a^{2}+a^{3}\dots)=(1-2a)/(1-a)$ is a closed nowhere dense set of
positive Lebesgue measure. It is easy to see that $G(x)<\infty$ for
a.e.~$x$ in $[0, 1]\backslash B$ by the argument in Example 3 in [15].
Thus, since $|[0, 1]\backslash B|>0$, we know that for a set of
$\alpha$'s of positive measure, $A_\alpha$ has eigenvalues in $[0, 1]$.
Of course, $\sigma_{\text{ac}}=\bar B=[0, 1]$.
\endexample
\example{Example 3} Let $B=\bigl[\operatornamewithlimits{\cup}
\limits^{\infty}_{n=1}\,\operatornamewithlimits{\cup}\limits^{2^n}_
{j=1} [\frac{1}{2^n} (1-\frac{1}{4}n^{-2}), \frac{1}{2^n} (1+\frac
{1}{4}n^{-2})]\bigr]\cap [0, 1]$. Then $|[0, 1]\backslash B|\geq 1-
\frac{1}{2} \sum\limits^{\infty}_{n=1} n^{-2}>0$ so $[0, 1]\backslash
B$ is a closed nowhere dense set of positive Lebesgue measure. As in
Example 2 in [15], $G(x)=\infty$ on all of $[0, 1]$ so no $A_\alpha$
has point spectrum in $[0, 1]$. By Theorem 2.6, for a set of
$\alpha$'s of positive measure, $A_\alpha$ has some singular continuous
spectrum embedded in $\sigma_{\text{ac}}(A_{\alpha})=[0, 1]$.
\endexample
For Examples 4 and 5, let $n_{j}=2\ell_{j}+1$ be a sequence of odd
integers with
$$
\sum^{\infty}_{j=1} n^{-1}_{j}<\infty. \tag 6.1
$$
As in Appendix 2 of [2], we can define functions $a_{j}(x)$ for
$x\in [-\frac12, -\frac12]$ with $a_{j}\in\{-\ell_{j}, -\ell_{j}+1,
\dots, \ell_{j}-1, \ell_{j}\}$ by using a variable base expansion:
$$
x=\sum^{\infty}_{j=1} \frac{a_{j}(x)}{n_{1}\dots n_{j}}. \tag 6.2
$$
Lebesgue measure corresponds to taking the $a_j$'s independent with
uniform distribution among the $n_j$ values. By (6.1) and the
Borel-Cantelli lemma, $|\{x\mid a_{j}(x)=0$ for infinitely many
$j$'s$\}|=0$. As in [2], define
$$\align
B &= \{x\mid a_{j}(x)=0 \text{ for an odd number of $j$'s}\} \\
C &= \{x\mid a_{j}(x)=0 \text{ for an even number or for
infinitely many $j$'s}\}.
\endalign
$$
We'll also define
$$\align
D &= \{x\mid |a_{j}(x)|\geq 2 \text{ all }j\} \\
B_{j} &= \{x\mid a_{j}(x)=0\} \\
S_{j}(y) &=\biggl\{x\mid |x-y|\leq \frac{1}{n_{1}\dots n_{j}}\biggr\}.
\endalign
$$
We note first that $|D|>0$ if all $n_{j}\geq 5$ since
$$
|D|=\prod\limits^{\infty}_{j=1} \biggl(1-\frac{3}{n_j}\biggr)>0
\tag 6.3
$$
by (6.1).
Next (following the argument in [2]):
$$\align
|S_{j}(y)\cap B| &\geq \gamma |S_{j}(y)| (n_{j+1})^{-1} \tag 6.4a \\
|S_{j}(y)\cap C| &\geq \gamma |S_{j}(y)| (n_{j+1})^{-1} \tag 6.4b
\endalign
$$
so long as $y\neq\pm\frac 12$ and $j$ is so large that $S_{j}(y)\subset
(-1, 1)$. In (6.4) $\gamma$ is the fixed constant
$$
\gamma=\frac12 \,\prod\limits^{\infty}_{\ell=1} \biggl(1-\frac{1}{n_\ell}
\biggr) >0.
$$
To prove (6.4), note first that
$$
a_{\ell}(x)=a_{\ell}(y); \quad \ell=1,\dots, j \Longrightarrow
x\in S_{j}(y).
$$
Suppose $\#\{\ell\leq j\mid a_{\ell}(y)=0\}$ is odd. Then
$$
S_{j}(y)\cap B \supset \{x\mid a_{\ell}(x)=a_{\ell}(y),\,
\ell=1,\dots, j; \quad a_{\ell}(x)\neq 0, \, \ell >j\}
$$
which has measure $(n_{1}\dots n_{j})^{-1}\prod\limits^{\infty}_{\ell=
j+1}\bigl(1-\frac{1}{n_\ell}\bigr)\geq\frac12 |S_{j}(y)|
\prod\limits^{\infty}_{\ell=1} \bigl(1-\frac{1}{n_\ell}\bigr)$ while
$$
S_{j}(y)\cap C\supset \{x\mid a_{\ell}(x)=a_{\ell}(y),\, \ell=1,\dots,
j; \quad a_{j+1}(x)=0, \, a_{\ell}(x)\neq 0, \, \ell > j+1\}
$$
which has measure $(n_{1}\dots n_{j})^{-1} n^{-1}_{j+1}
\prod\limits^{\infty}_{\ell=j+1} \bigl(1-\frac{1}{n_\ell}\bigr)\geq
\gamma |S_{j}(y)|/n_{j+1}$. A similar argument applies if $\#\{\ell\leq j
\mid a_{\ell}(y)=0\}$ is even.
As a final preliminary we need that
$$
\inf \{|x-y|\mid x\in D,\, y\in B_{j}\}=(n_{1}\dots n_{j})^{-1}.
\tag 6.5
$$
\example{Example 4} Pick $n_j$ so that (6.1) holds and
$$
\lim\limits_{j\to\infty} \frac{n_{1}\dots n_{j}}{n_{j+1}}=\infty,
\tag 6.6
$$
for example, $\ell_{j}=2^j$. Let $d\mu=\chi_{B}\, dx$. Then we claim
$G(y)=\infty$ for all $y$. For
$$\align
G(y) &\geq |S_{j}(y)\cap B| (n_{1}\dots n_{j})^{2} \\
&\geq 2\gamma\,\frac{n_{1}\dots n_{j}}{n_{j+1}}
\endalign
$$
by (6.4a). Moreover by (6.4a,b) the essential closure of $B$ is
$[-\frac{1}{2},\frac{1}{2}]$ and the essential closure of
$[-\frac{1}{2},\frac{1}{2}]\backslash B$ is also $[-\frac{1}{2},
\frac{1}{2}]$. Thus, the operator $A$ has recurrent spectrum $[-\frac{1}{2},
\frac{1}{2}]$ and no $A_\alpha$ has point spectrum. Since $\bigl|[-\frac{1}
{2}, \frac{1}{2}]\backslash B\bigr|=|C|>0$ for a positive measure set of
$\alpha$'s, $A_\alpha$ has singular continuous spectrum embedded in
$[-\frac{1}{2}, \frac{1}{2}]=\sigma_{\text{ac}}(A_{\alpha})$.
\endexample
\example{Example 5} Pick $n_j$ so that
$$
\sum^{\infty}_{j=1} \biggl(\frac{n_{1}\dots n_{j}}{n_{j+1}}\biggr)^{2}
<\infty \tag 6.7
$$
and that
$$
\sum^{\infty}_{j=k+1} \frac{1}{n_j}\leq\frac{1}{n_k}, \tag 6.8
$$
for example, $n_{j}=2^{j!}$. Define $\tilde B$ analogously to $B$ but
with
$$
\tilde B =\{x\mid\text{ the number of $j$ with } a_{j}(x)=0
\text{ lies in }\{3,5,7,\dots\}\}
$$
and let $d\mu=\chi_{\tilde B}\, dx$. As above, $A$ has
recurrent spectrum with essential support $\tilde B$ but closed
support $[-\frac{1}{2}, \frac{1}{2}]$. We claim that $G(x)<\infty$ on
$D$. Since $|D|>0$, $A_\alpha$ has point spectrum embedded in $[-\frac{1}{2},
\frac{1}{2}]=\sigma_{\text{ac}}$ for a set of $\alpha$'s of positive
measure. Note that this does not exclude the occurrence of singular
spectrum in addition.
To see that $G(x)<\infty$, define
$$
B_{j, k, \ell}=\{x\mid a_{j}(x)=a_{k}(x)=a_{\ell}(x)=0\}.
$$
Thus by (6.8),
$$
\sum \Sb k,\ell>j \\ k\neq\ell \endSb |B_{j, k,\ell}| \leq\biggl(
\sum^{\infty}_{m=j+1} \frac{1}{n_m}\biggr)^{2}\frac{1}{n_j}\leq
\frac{4}{n_{j}(n_{j+1})^{2}} \tag 6.9
$$
since (6.8) implies that
$$
\sum^{\infty}_{j=k+1} \frac{1}{n_j}\leq\frac{2}{n_{k+1}}.
$$
On the other hand, by (6.5),
$$
\inf \{|x-y|\mid x\in D,\, y\in B_{j,k,\ell}\}\geq (n_{1}\dots n_{j})^{-1}
\qquad k,\ell >j. \tag 6.10
$$
Since $\tilde B\subset\operatornamewithlimits{\cup}\limits\Sb j,k,\ell \\
\text{all unequal} \endSb B_{j,k,\ell}$, we have by (6.9--6.10) that if
$x\in D$
$$
G(x)\leq 4 \sum^{\infty}_{j=1} \frac{(n_{1}\dots
n_{j})^{2}}{n_{j}(n_{j+1})^{2}} <\infty
$$
by (6.7).
\endexample
\bigpagebreak
\example{Acknowledgments} R.~del Rio would like to thank J.~Weidmann
for having initiated him in the study of stability of spectral types.
R.~del Rio and G.~Stolz would also like to thank C.~Peck and M.~Aschbacher
for the hospitality at Caltech.
\endexample
\vskip .5in
\Refs
\widestnumber\key{14}
\ref\key 1 \by N.~Aronszajn \paper On a problem of Weyl in the theory
of singular Sturm-Liouville equations \jour Amer.~J. Math. \vol 79 \yr
1957 \pages 597--616
\endref
\ref\key 2 \by J.~Avron and B.~Simon \paper Transient and recurrent
spectrum \jour J.~Funct.~Anal. \vol 43 \yr 1981 \pages 1--31
\endref
\ref\key 3 \by B.A.~Coddington and N.~Levinson \book Theory of
Ordinary Differential Equations \publ McGraw-Hill \yr 1955
\endref
\ref\key 4 \by R.~del Rio \paper Instability of the absolutely
continuous spectrum of ordinary differential operators under local
perturbations \jour J.~Math.~Anal.~Appl. \vol 142 \yr 1989
\pages 591--604
\endref
\ref\key 5 \by R.~del Rio \paper Ein Gegenbeispiel zur Stabilit\"at
des absolut stetigen Spektrums gew\"ohnlicher Differentialoperatoren
\jour Math.~Z \vol 197 \yr 1988 \pages 61--68
\endref
\ref\key 6 \by R.~del Rio, N.~Makarov, and B.~Simon \paper Operators
with singular continuous spectrum, II. Rank one operators
\jour Commun.~Math.~Phys. \vol 165 \yr 1994 \pages 59--67
\endref
\ref\key 7 \by W.~Donoghue \paper On the perturbation of the spectra
\jour Commun.~Pure Appl.~Math. \vol 18 \yr 1965 \pages 559--579
\endref
\ref\key 8 \by F.~Gesztesy and B.~Simon \paper Rank one perturbations
at infinite coupling \jour J.~Funct. Anal. \vol 128 \yr 1995
\pages 245--252
\endref
\ref\key 9 \by V.A.~Javrjan \paper A certain inverse problem for
Sturm-Liouville operators \jour Izv.~Akad.~Nauk Armjan.~SSR Ser.~Mat.
\vol 6 \yr 1971 \pages 246--251
\endref
\ref\key 10 \by W.~Kirsch, S.~Molchanov, and L.~Pastur
\paper One-dimensional Schr\"odinger operators with unbounded potential:
The pure point spectrum \jour Funct.~Anal.~Appl. \vol 24 \yr 1990
\pages 176--186
\endref
\ref\key 11 \bysame \paper One-dimensional Schr\"odinger operators with
high potential barriers \inbook Operator Theory: Advances and
Applications \vol 57 \publ Birkh\"auser \yr 1992 \pages 163--170
\endref
\ref\key 12 \by S.~Kotani \paper Lyapunov exponents and spectra for
one-dimensional random Schr\"odinger operators \jour Contemp. Math.
\vol 50 \yr 1986 \pages 277--286
\endref
\ref\key 13 \by M.~Reed and B.~Simon \book Methods of Modern
Mathematical Analysis, III. Scattering Theory \publ Academic Press \yr
1979
\endref
\ref\key 14 \by B.~Simon \paper Spectral analysis of rank one
perturbations and applications \jour Proc.~1993 Vancouver Summer
School in Mathematical Physics \toappear
\endref
\ref\key 15 \by B.~Simon and T.~Wolff \paper Singular continuous
spectrum under rank one perturbations and localization for random
Hamiltonians \jour Commun.~Pure Appl.~Math. \vol 39 \yr 1986 \pages
75--90
\endref
\ref\key 16 \by G.~Stolz \paperinfo In preparation
\endref
\ref\key 17 \by J.~Weidmann \paper Absolut stetiges Spektrum bei
Sturm-Liouville Operatoren und Dirac Systemen \jour Math. Z. \vol 180
\yr 1982 \pages 423--426
\endref
\endRefs
\enddocument