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\topmatter
\title Some Schr\"odinger Operators with Dense Point Spectrum
\endtitle
\author Barry Simon$^{*}$
\endauthor
\leftheadtext{B.~Simon}
\affil Division of Physics, Mathematics, and Astronomy \\
California Institute of Technology, 253-37 \\
Pasadena, CA 91125
\endaffil
\date May 16, 1995
\enddate
\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 be submitted to {\it{Proc.~Amer.~Math.~Soc.}}
\endthanks
\abstract Given any sequence $\{E_n\}^\infty_{n-1}$ of positive
energies and any monotone function $g(r)$ on $(0,\infty)$ with
$g(0)=1$, $\lim\limits_{r\to\infty} g(r)=\infty$, we can find a
potential $V(x)$ on $(-\infty,\infty)$ so that $\{E_n\}^\infty_{n=1}$
are eigenvalues of $-\frac{d^2}{dx^2}+V(x)$ and $|V(x)|\leq
(|x|+1)^{-1}g(|x|)$.
\endabstract
\endtopmatter
\document
\vskip 0.2in
In [7], Naboko proved the following:
\proclaim{Theorem 1} Let $\{\kappa_n\}^\infty_{n=1}$ be a sequence of
rationally independent positive reals. Let $g(r)$ be a monotone
function on $[0,\infty)$ with $g(0)=1$, $\lim\limits_{r\to\infty}
g(r)=\infty$. Then there exists a potential $V(x)$ on $[0,\infty)$ so
that
\roster
\item"\rom{(1)}" $\{\kappa^2_n\}^\infty_{n=1}$ are eigenvalues of
$-\frac{d^2}{dx^2}+V(x)$ on $[0,\infty)$ with $u(0)=0$ boundary
conditions.
\item"\rom{(2)}" $|V(x)|\leq \frac{g(x)}{(|x|+1)}$.
\endroster
\endproclaim
Our goal here is to construct $V$'s that allow the proof of the
following theorem:
\proclaim{Theorem 2} Let $\{\kappa_n\}^\infty_{n=1}$ be a sequence of
arbitrary distinct positive reals. Let $g(r)$ be a monotone function on
$[0,\infty)$ with $g(0)=1$ and $\lim\limits_{r\to\infty} g(r)=\infty$.
Let $\{\theta_n\}^\infty_{n=1}$ be a sequence of angles in $[0,\pi)$.
Then there exists a potential $V(x)$ on $[0,\infty)$ so that
\roster
\item"\rom{(1)}" For each $n$, $(-\frac{d^2}{dx^2}+V(x))u=\kappa^2_n u$
has a solution which is $L^2$ at infinity and
$$
\frac{u'(0)}{u(0)}=\cot(\theta_n). \tag 1
$$
\item"\rom{(2)}" $|V(x)|\leq \frac{g(x)}{|x|+1}$.
\endroster
\endproclaim
\remark{Remarks} 1. These results are especially interesting because
Kiselev [6] has shown that if $|V(x)|\leq C(|x|+1)^{-\alpha}$ with
$\alpha >\frac{3}{4}$, then $(0,\infty)$ is the essential support of
$\sigma_{\text{\rom{ac}}}(-\frac{d^2}{dx^2}+V(x))$, so these examples
include ones with dense point spectrum, dense inside absolutely
continuous spectrum.
2. For whole line problems, we can take each $\theta_n =0$ or
$\frac{\pi}{2}$ and let $V_\infty(x)=V(|x|)$ and specify even and odd
eigenvalues.
3. For our construction, we'll have $|u_n (x)|\leq C_n (1+|x|)^{-1}$.
By the same method, we could also specify $\{m_n\}^\infty_{n=1}$ so
$|u_n(x)|\leq C_n(1+|x|)^{-m_n}$.
4. By the same method, if $\sum\limits^\infty_{n=1}|\kappa_n|<\infty$,
we can actually take $|V(x)|\leq C(1+|x|)^{-1}$, providing an answer to
an open question of Eastham-Kalf [4], page 95. If one takes our
construction really seriously, one might conjecture that if
$V(x)=0(|x|^{-1})$, then zero is the only possible limit point of the
eigenvalues $E_n$ and, indeed, even that
$$
\sum^\infty_{n=1}\sqrt{E_n}<\infty.
$$
5. One can probably extend Naboko's method to allow $\theta$'s so
from a technical point of view, our result goes beyond his in that we
show the rational independence condition is an artifact of his
proof. The real point is to provide a different construction where the
interesting examples of the phenomena can be found.
\endremark
\vskip 0.2in
Our construction is based on examples of the Wigner-von Neumann
type [9]. They found a potential $V(x)=\frac{8\sin(2r)}{r}+0(r^{-2})$ at
infinity and so that $-u''+Vu=u$ has a solution of the form $\frac
{\sin(r)}{r^2}+0(r^{-3})$ at infinity. In fact, our potentials will be
of the form
$$
V(x)=W(x)+\sum^\infty_{n=0} 4\kappa_n \chi_n (x) \,
\frac{\sin(2\kappa_n x + \varphi_n)}{x} \tag 2
$$
where $\chi_n (x)$ is the characteristic function of the region
$x>R_n$ for suitable large $R_n \to\infty$. Since $R_n$ goes to infinity,
the sum in (2) is finite for each $x$ and there is no convergence issue.
In (2), $W$ will be a carefully constructed function on $[0,1]$ arranged
to make sure that the phases $\theta_n$ at $x=0$ come out right. We'll
construct $V$ as a limit of approximations
$$
V_m (x)=W_m (x) +\sum^m_{n=0} 4\kappa_n \chi_n (x)\,
\frac{\sin(2\kappa_n x+\varphi_n)}{x} \tag 3
$$
where $W_m$ is supported on $[2^{-m}, 1]$ and equals $W$ there. We'll
make this construction so that:
\roster
\item"\rom{(a)}" For $n\leq m$, $(-\frac{d^2}{dx^2}+V_m (x))u(x)=
\kappa^2_n u(x)$ has a solution $u^{(m)}_n (x)$ obeying $u\in L^2$ and
condition (1).
\item"\rom{(b)}"
$$\left|u^{(m)}_n (x)-\frac{\sin(\kappa_n x+\tfrac{1}{2}
\varphi_n)}{1+x}\right|\leq C_n (1+x)^{-2} \tag 4
$$
for $C_n$ uniformly bounded (in $m$ but not in $n$!). Note in (4), the
fact that $1/1+x$ appears (multiplying the $\sin$) rather than, say,
$1/(1+x)^2$ comes from the choice of $4$ in $4\kappa_n$ in (3) (in
general, if $4\kappa_n$ is replaced by $\gamma x_n$, the decay is
$r^{-\gamma/4}$).
\endroster
Central to our construction is a standard oscillation result that can
be easily proven using the method of Harris-Lutz [5] or the
Dollard-Friedman method [2,3] (see [8], problem 98 in Chapter XI);
results of this genre go back to Atkinson [1]. It will be convenient
to introduce the norm
$$
|\!|\!|f|\!|\!|=\|(1+x^2)f\|_{\infty} +\left\|(1+x^2)\frac{df}{dx}
\right\|_\infty
$$
for functions on $[0,\infty)$.
\proclaim{Theorem 3} Fix $x>0$. Let $V_0$ be a continuous function on
$[0,\infty)$ so that
$$
V_0 (x)=4\kappa \sin(2\kappa x+\varphi_0)\big/|x|
$$
for $x>R_0$ for some $R_0$. Let $V_1, V_2$ be two other continuous
functions which obey
\roster
\item"\rom{(i)}" $|V_i (x)|\leq C_1 |x|^{-1}$
\item"\rom{(ii)}" $V_i (x)=\frac{dW_i}{dx}$ where $|W_i (x)|\leq
C_2 |x|^{-1}$
\item"\rom{(iii)}" $e^{\pm 2i\kappa x}V_i (x)=\frac{dW^{(\pm)}_i}{dx}$
where $|W^{\pm}_i (x)|\leq C_3 |x|^{-1}$.
\endroster
Let
$$
V^{(R)}=\cases V_0 (x)+V_1 (x) & |x|< R \\
V_0 (x)+V_1 (x)+V_2 (x) & |x|>R
\endcases
$$
with $V^{(\infty)}(x)=\lim\limits_{R\to\infty} V^{(R)}(x)$. Then there
exists a unique function $u^{(R)}(x)$ for $R\in [0,\infty]$
\rom(including $\infty$\rom) with \rom($u\equiv u^{(R)}$\rom)
\roster
\item"\rom{(a)}" $-u''+V^{(R)}u=\kappa^2 u$
\item"\rom{(b)}" $|u(x)-\frac{\sin(\kappa x+\tfrac{1}{2}\varphi_0)}
{1+|x|}|\leq C_4 (1+x)^{-2}$ and $|u'(x)-\frac{\kappa\cos(\kappa x
+\tfrac{1}{2}\varphi_0)}{1+|x|}|\leq C_5 (1+x)^{-2}$.
\endroster
In addition,
$$
|\!|\!|u^{(R)}-u^{(\infty)}|\!|\!|\to 0 \tag 5
$$
as $R\to\infty$. Moreover, $C_4$, $C_5$, and the rate convergence in
{\rom{(5)}} only depend on $R_0$, $C_1$, $C_2$, and $C_3$.
\endproclaim
Since this is a straightforward application of the methods of [5,3],
we omit the details.
\vskip 0.2in
The second input we'll need is the ability to undo small changes of
Pr\"ufer angles with small changes of potential. We'll need the
following lemma:
\proclaim{Lemma 4} Fix $k_1, \dots, k_n >0$ distinct and
$\theta^{(0)}_1,\dots,\theta^{(0)}_n$. Let
$$
f_j (x)=\sin^2 (k_i x+\theta^{(0)}_i).
$$
Fix $a**0$ and distinct, and angles $\theta^{(0)}_1,\dots,\theta^{(0)}_n$.
Define $F:C[a,b]\to T^n$ \rom(with $T^n$ the $n$-torus\rom) to be the
generalized Pr\"ufer angles $\varphi_i (b)$ solving {\rom{(7)}}
\rom(with $k=k_i$ and $V(x)=0$ on $[0,a)$ and the argument of $F$ on
$[a,b]$\rom) with $\varphi_i (0)=\theta^{(0)}_i$. Then for any
$\epsilon$, there is a $\delta$ so that for any $\theta^{(1)}_1,
\dots, \theta^{(1)}_n$ with
$$
|\theta^{(1)}_i - k_i b -\theta^{(0)}_i |<\delta,
$$
there is a $V\in C[a,b]$ with $\|V\|_\infty <\epsilon$ and
$$
F(V)=(\theta^{(1)}_1, \dots, \theta^{(1)}_n).
$$
\endproclaim
\demo{Proof} $F(V=0)$ is $(\theta^{(0)}_1 +k_1 b,\dots, \theta^{(0)}_n
+k_n b)$ by (8), so this theorem merely asserts that $F$ takes a
neighborhood of $V=0$ onto a neighborhood of $F(V=0)$. By the implicit
function theorem, it suffices that the differential is surjective. But
$$
\left. \frac{\delta F_i}{\delta V(x)}\right|_{V\equiv 0} = -\frac{1}
{k_i}\,\sin^2 (k_i x+\theta^{(0)}_i)
$$
by (7) and (8). By the lemma, this derivative is surjective. \qed
\enddemo
\vskip 0.2in
We now turn to the proof of Theorem 2. The overall strategy will be to
use an inductive construction. We'll write
$$
W(x)=\sum^\infty_{m=1} (\delta W_m)(x) \tag 9
$$
with $\delta W_m$ supported on $[2^{-m}, 2^{-(m-1)}]$ so that the
$W_m$ of equation (3) is $W_m=\sum\limits^m_{k=1}\delta W_k$. Then
assuming we have $V_{m-1}$, we'll choose $R_m$, $\varphi_m$, $\delta
W_m$ in successive order, so
\roster
\item"\rom{(1)}" $R_m$ is so large that
$$
|8\kappa_m \chi_m(x)|\leq 2^{-m}g(x) \tag 10
$$
on all $(0,\infty)$, that is, $g(R_m)\geq 2^m (8\kappa_m)$.
\item"\rom{(2)}" $R_m$ is chosen so large that steps (3), (4) work.
\item"\rom{(3)}" Let $u^{(0)}(x)$ solve $-u''+V_{m-1}u=\kappa^2_m u$
with $u'(0)/u(0)=\cot(\theta_m)$. We show that (so long as $R_m$ is
chosen large enough) we can pick $\varphi_m$ so this $u$ matches to
the decaying solution guaranteed by Theorem 3.
\item"\rom{(4)}" By choosing $R_m$ large, we can be sure that
$|\!|\!|u^{(m-1)}_n -\tilde{u}^{(m)}_n |\!|\!| \leq 2^{-m-1}$ where
$\tilde{u}^{(m)}_n$ obeys the equation for $V_m -\delta W_m$ and that
the modified Pr\"ufer angles for $\tilde{u}^{(m)}_n$ at $b_m =
2^{-m+1}$ are within a range that can apply Theorem 5 with
$$
[a,b]=[2^{-m}, 2^{-m+1}]
$$
and $\epsilon <\frac{1}{2}$. By applying Theorem 5, we'll get $\delta
W_{m+1}$ to assure $u^{(m)}_n$ obeys the boundary conditions at zero.
\endroster
\vskip 0.2in
Here are the formal details:
\demo{Proof of Theorem 2} Let
$$
(\delta V_n)(x)=4\kappa_n \chi_n (x)\,\frac{\sin(2\kappa_n x+\varphi_n)}
{x} \tag 11
$$
where $\chi_n$ is the characteristic function of $[R_n, \infty)$ and
$\varphi_n, R_n$ are parameters we'll pick below. $R_n$ will be picked
to have many properties, among them
$$
R_n \to\infty, \, R_n \geq 1, \qquad g(R_n)\geq 2^n (8\kappa_n).
\tag 12
$$
$\delta W_n$ will be a function supported on $[2^{-n}, 2^{-n+1})$
chosen later but obeying
$$
\|\delta W_n\|_\infty \leq\frac{1}{2}. \tag 13
$$
We'll let
$$
V_m (x)=\sum^m_{n=1} (\delta V_n +\delta W_n)(x)
$$
and
$$
V(x)=\lim\limits_{m\to\infty} V_m (x)
$$
where the limit exists since $V_m (x)$ is eventually constant
for any $x$.
By (12), (13), we have
$$
|V_m (x)|\leq g(x)\big/ (|x|+1) \qquad m=1,2,\dots,\infty. \tag 14
$$
For each $m$ and each $n=1,\dots, m$, we have by Theorem 3 a unique
function $u^{(m)}_n (x)$ obeying
$$\gather
-u''+V_m u=\kappa^2_n u \tag 15 \\
|\!|\!|u-\sin((\kappa_n +\tfrac{1}{2}\varphi_n )\,\cdot\,)
(1+ |\cdot|)^{-1}|\!|\!| <\infty. \tag 16
\endgather
$$
We will choose $\delta V_n, \delta W_m$ so that
$$\alignat2
&|\!|\!| u^{(m)}_n -u^{(m-1)}_n |\!|\!| \leq 2^{-m} && \qquad n=1,2,
\dots, m-1 \tag 17 \\
& u^{(m)}_n \text{ obeys eqn.~(1)} && \qquad n=1,\dots, m. \tag 18
\endalignat
$$
Let $u_n =\operatornamewithlimits{|\!|\!|\cdot|\!|\!|\text{-lim}}
\limits_{m\to\infty} u^{(m)}_n$. Writing the differential equation as
an integral equation, we see that $u_n$ obeys $-u''+V(u)=\kappa^2_n u$.
By (18), $u_n$ obeys equation (1) and by $|\!|\!|\cdot|\!|\!|$
convergence, $u_n$ obey (16) and so lies in $L^2$. Thus as claimed,
$-\frac{d^2}{dx^2}+V$ has $\{\kappa^2_n\}^\infty_{n=1}$ as eigenvalues.
Thus we are reduced to showing that $\delta V_m, \delta W_m$ can be
chosen so that (17), (18) hold.
Let $\theta^{(0)}_i$ be defined by $\kappa_i \cot(\theta^{(0)}_i) =
\cot(\theta_i)$ so $\theta^{(0)}_i$ are the generalized Pr\"ufer
angles associated to the originally specified Pr\"ufer angles. Look at
the solutions $u^{(n-1)}_i$, $i=1,\dots, m-1$. These match to the
generalized Pr\"ufer angles $\kappa_i 2^{-m+1}+\theta^{(0)}_i$ at
$x=2^{-m+1}$.
We'll choose $\delta V_m$ so that the new solutions
$\tilde{u}^{(m)}_i$ ($i=1,\dots, m-1$) with $\delta V_m$ added obey
$|\!|\!|\tilde{u}^{(m)}_i -u^{(m-1)}_i |\!|\!| <2^{-m-1}$. We can find
$\epsilon_m$ so that if $\|\delta W_m\|<\epsilon_m$, then the new
solutions $u^{(m)}_i$ obey $|\!|\!|u^{(m)}_i -\tilde{u}^{(m)}_i |\!|\!|
<2^{-m-1}$. So using Theorem 5, pick $\delta$ so small that the
resulting $V$ given is that theorem with $a=2^{-m}, b=2^{-m+1}$ has
$\|\cdot\|$ bounded by $\min(\frac{1}{2},\epsilon_n )$. In that
theorem, use $\kappa_1, \dots, \kappa_m$ and $\theta^{(0)}_i$,
$i=1,\dots, m$.
According to Theorem 3, we can take $R_m$ so large that uniformly in
$\varphi_m$ (in $[0,2\pi/2\kappa_m]$), we have $|\!|\!|u^{(m-1)}_i -
\tilde{u}^{(m)}_i |\!|\!| < 2^{-m-1}$ for $i=1,\dots, m-1$ and so large
that again uniformly in $\varphi_m$, the generalized Pr\"ufer angles
$\theta^{(0)}_i$ for $\tilde{u}^{(m)}_i$ at $b_m \equiv 2^{-m+1}$
obeys $|\theta^{(1)}_i -\theta^{(0)}_i -\kappa_i b_i| <\delta$ for
$i=1, \dots, m-1$.
Thus, if we can pick the angle $\varphi_m$ in (11) so that
$\tilde{u}^{(m)}_m$ obeys the boundary condition at zero (and so
$\theta^{(1)}_m -\theta^{(0)}_m -\kappa_m b_m =0$), then the construction
is done.
By condition (b) of Theorem 3, for $|x|$ large, as $\varphi_m$ runs
from $0$ to $2\pi /2\kappa_m$, $(|x|u(x),\mathbreak |x|u'(x))$ runs
through a complete half-circle. Thus, by taking $R_m$ at least that
large and choosing $\varphi_m$ appropriately, we can match the angle
of the solution of $u''+V_{m-1}u=\kappa^2_m u$ which obeys the boundary
condition at $x=0$. \qed
\enddemo
\vskip 0.3in
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order differential equations \jour Ann.~Math.~Pura Appl. \vol 37
\yr 1954 \pages 347--378
\endref
\ref\key 2 \by J.~Dollard and C.~Friedman \paper On strong product
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\endref
\ref\key 3 \bysame \paper Product integrals and the Schr\"odinger
equation \jour J.~Math.~Phys. \vol 18 \yr 1977 \pages 1598--1607
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\ref\key 4 \by M.S.P.~Eastham and H.~Kalf \book Schr\"odinger-type
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Mathematics 65 \publ Pitman Books Ltd. \publaddr London \yr 1982
\endref
\ref\key 5 \by W.A.~Harris and D.A.~Lutz \paper Asymptotic integration
of adiabatic oscillator \jour J.~Math.~Anal.~Appl. \vol 51 \yr 1975
\pages 76--93
\endref
\ref\key 6 \by A.~Kiselev \paper Absolutely continuous spectrum of
one-dimensional Schr\"odinger operators and Jacobi matrices with
slowly decreasing potentials \paperinfo preprint
\endref
\ref\key 7 \by S.N.~Naboko \paper Dense point spectra of Schr\"odinger
and Dirac operators \jour Theor.-math. \vol 68 \yr 1986 \pages 18--28
\endref
\ref\key 8 \by M.~Reed and B.~Simon \book Methods of Modern
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\publaddr New York \yr 1979
\endref
\ref\key 9 \by J.~von Neumann and E.P.~Wigner \paper \"Uber
merkw\"urdige diskrete Eigenwerte \jour Z.~Phys. \vol 30 \yr 1929
\pages 465--467
\endref
\endRefs
\enddocument
**