\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \TagsOnRight \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 $a0$ 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 \Refs \widestnumber\key{8} \ref\key 1 \by F.~Atkinson \paper The asymptotic solutions of second 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 integration \jour J.~Funct.~Anal. \vol 28 \yr 1978 \pages 309--354 \endref \ref\key 3 \bysame \paper Product integrals and the Schr\"odinger equation \jour J.~Math.~Phys. \vol 18 \yr 1977 \pages 1598--1607 \endref \ref\key 4 \by M.S.P.~Eastham and H.~Kalf \book Schr\"odinger-type Operators with Continuous Spectra \bookinfo Research Notes in 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 Mathematical Physics, III. Scattering Theory \publ Academic Press \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