\input amstex \documentstyle{amsppt} \TagsOnRight \topmatter \title Operators with Singular Continuous Spectrum:\\ III. Almost Periodic Schr\"odinger Operators \endtitle \rightheadtext{S.C.~Spectrum: Almost Periodic Schr\"odinger Operators} \author S. Jitomirskaya and B. Simon$^{*}$ \endauthor \thanks{$^{*}$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 To appear in {\it Commun.~Math.~Phys.} \endthanks \leftheadtext{S. Jitomirskaya and B. Simon} \address Department of Mathematics, University of California, Irvine, CA 92717 \endaddress \address Division of Physics, Mathematics and Astronomy, California Institute of Technology 253-37, Pasadena, CA 91125 \endaddress \abstract We prove that one-dimensional Schr\"odinger operators with even almost periodic potential have no point spectrum for a dense $G_\delta$ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than $2$ and a dense $G_\delta$ in $\theta$ even if the frequency is an irrational with good Diophantine properties. \endabstract \endtopmatter \document \bigpagebreak \flushpar {\bf \S 1. Introduction} \medpagebreak This is a paper that provides yet another place where singular continuous spectrum occurs in the theory of Schr\"odinger operators and Jacobi matrices (see \linebreak[5,6,2,10,3]). It is especially interesting because it will provide examples where a non-resonance condition in a {\smc KAM} argument is not merely needed for technical reasons but necessary. Our main results, proven in \S 2, do not deal directly with singular continuous spectrum but only with continuous spectrum. \proclaim{Theorem 1S} Let $V$ be an even almost periodic function on $(-\infty, \infty)$ and let $\Omega$ be the hull of $V$ and $V_{\omega}(x)$ the corresponding function for $\omega\in\Omega$. Then there is a dense $G_{\delta}, U$ in $\Omega$ \rom(in the natural metric topology\rom), so that if $\omega\in U$, then $H_{\omega}\equiv\frac{-d^{2}}{dx^{2}}+V_{\omega}(x)$ has no eigenvalues as an operator on $L^{2}(\Bbb R)$. \endproclaim For the Jacobi case, we let $h_0$ be the operator on $\ell^{2} (\Bbb Z)$ defined by $(h_{0}u)(n)=u(n+1)+u(n-1)$. \proclaim{Theorem 1J} Let $V$ be an even almost periodic function on $\Bbb Z$, $\Omega$ its hull, and $V_{\omega}(n)$ the function associated to $\omega\in\Omega$. Then there is a dense $G_{\delta}, U$ in $\Omega$ so that if $\omega\in U$, then $H_{\omega}=h_{0}+ V_{\omega}(n)$ has no eigenvalues as an operator on $\ell^{2} (\Bbb Z)$. \endproclaim The $G_\delta$ set $U$ will be rather explicit---see \S 2. By combining this with the machinery of [10], we can sometimes get singular continuous spectrum. \proclaim{Theorem 2} In the context of Thm. 1, suppose there is a single $\omega\in\Omega$ so that $H_\omega$ has no absolutely continuous spectrum. Then for a dense $G_{\delta}, \tilde{U}$, $H_\omega$ has purely singular continuous spectrum. \endproclaim \demo{Proof} Let $U_{1}=\{\omega\in\Omega\mid H_{\omega} \ \text{has no a.c.~spectrum}\}$. By [10], $U_1$ is a $G_\delta$. By hypothesis, $\omega_0$ and its translates lie in $U_1$, so $U_1$ is a dense $G_\delta$. Thus, $\tilde{U}=U_{1}\cap U$ is a dense $G_\delta$. \qed \enddemo \example{Example 1} Consider the Jacobi matrix with $$V_{\theta}(n)=\lambda\cos(\pi\beta n+\theta). \tag 1 $$ If $\lambda >2$, the Lyapunov exponent is positive ([1,7]) so if $\beta$ is irrational, there is no a.c.~spectrum for Lebesgue a.e.~ $\theta$ (see e.g.~[1]), so $h_\theta$ has purely singular continuous spectrum for a dense $G_\delta$ of $\theta$. \endexample Sinai [11] and Fr\"ohlich-Spencer-Wittwer [4] have proven for $\lambda$ large and $\beta$ having good Diophantine properties, a.e.~$\theta$ has pure point spectrum, and Jitomirskaya [8] has proven that for $\lambda\geq 15$. In that case there are intertwined locally uncountable sets of $\theta$ with only pure point and with only singular continuous spectrum. For $\lambda =2$, spec$(h_{\theta})$ has zero measure for many irrational $\beta$'s [9] and so no a.c.~spectrum. We conclude \proclaim{Theorem 3} For the example \rom{(1)}, $h_\theta$ has purely singular continuous spectrum for a dense $G_\delta$ of $\theta$'s if $\beta$ is irrational and $\lambda >2$ or if the continued fraction expansion of $\beta$ has unbounded integers and $\lambda =2$. \endproclaim \example{Example 2} Consider the Schr\"odinger case with $V_{\theta}(x)=-k[\cos(2\pi x)+ \cos (2\pi \beta x+\theta)]$. Then, Fr\"ohlich-Spencer-Wittwer [4] have proven for a.e.~$\theta$ ($k$ large enough), there is pure point spectrum for low energies. Sorets-Spencer [12] have proven positivity of the Lyapunov exponent for a wider area of low energy. We conclude that for a dense $G_\delta$ of $\theta$, there is purely singular continuous spectrum for low energies. \endexample \bigpagebreak \flushpar {\bf \S 2. Proof of Theorem 1} \medpagebreak We'll consider the Jacobi case in detail and then discuss the changes for the Schr\"odinger case. Let $V_{\omega_0}$ be the even almost periodic function on $\Bbb Z$: $$V_{\omega_0}(-n)=V_{\omega_0}(n).$$ Fix once and for all a number $B$ so $$B>4 \ln (3+2 \sup_{n} |V_{\omega_0}(n)|)\equiv 4\ln\alpha. \tag 2.1 $$ $\alpha$ is chosen so that the matrix $\left(\smallmatrix {E-V(u)} & 1 \\ 1 & 0 \endsmallmatrix\right)$ has norm bounded by $\alpha$ if $|E|\leq 2+\sup\limits_{n}|V_{\omega_0}(n)|$. Let $\Omega$ be the hull of $V$, that is, the closure in $\|\cdot\|_{\infty}$ of translates of $V$; it is compact by hypothesis. Define $\rho$ on $\Omega$ by $$\rho(\omega, \omega')\equiv\sup_{n}(|V_{\omega}(n)-V_{\omega'}(n)|)$$ and define maps $R$ and $T$ on $\Omega$ by $$V_{R\omega}(n)=V_{\omega}(-n)\qquad V_{T\omega}(n)=V_{\omega}(n- 1).$$ \proclaim{Lemma 2.1} Let $U_{n}=\operatornamewithlimits{\cup}\limits_{|m|>n} \{\omega\mid\rho(RT^{2m}\omega, \omega)< e^{-B|m|}\}$ and let $U=\operatornamewithlimits{\cap}\limits^{\infty}_{n=1} U_{n}$. Then $U_n$ is a dense open set and $U$ is a dense $G_\delta$ in $\Omega$. \endproclaim \demo{Proof} Let $\omega_{m}=T^{-m}\omega_0$. Then $RT^{2m}\omega_{m}=\omega_{m}$ since $R\omega_{0}=\omega_0$, so $\omega_{m}\in U_n$ if $|m|>n$. It is easy to see the set of $\{\omega_{m}\mid |m|>n\}$ is dense in $\Omega$, so $U_n$ is dense. It is clearly open and so $U=\cap U_n$ is a dense $G_\delta$ by the Baire category theorem. \qed \enddemo $U$ is the set of $\omega$'s for which there exists an infinite sequence $m_i$ with $|m_{i}|\to\infty$ with $\rho (RT^{2m_i}\omega, \omega)< e^{-B|m_{i}|}$. For a subsequence, either $m_{i}\to\infty$ or $m_{i}\to -\infty$ and by reflection invariance, we can suppose $m_{i}\to\infty$. Thus, Thm. 1J follows from \proclaim{Theorem 2.2} Suppose that $V$ is a function obeying $$|V(2m_{i}-n)-V(n)|\leq e^{-Bm_{i}} \tag 2.2 $$ for a sequence $m_{i}\to\infty$ where $B$ is given by \rom{(2.1)}. Then $$u(n+1)+u(n-1)+V(n)u(n)=Eu(n) \tag2.3 $$ has no $\ell^2$ solutions for any $E$. \endproclaim \remark{Remark} The intuition behind the proof is that any $u$ obeying (2.3) has to be close to being even or odd about $m_i$ so $u(n)\nrightarrow 0$. \endremark \demo{Proof} Suppose not. Then we can find a solution $u$ of (2.3) in $\ell^2$ which we normalize, so that $$\sum_{n}|u(n)|^{2}=1. \tag 2.4 $$ We let $u_{i}(n)\equiv u(2m_{i}-n)$. Let $W(f, g)(n)=f(n+1)g(n)- f(n)g(n+1)$ be the Wronskian as usual, and let $$\Phi(n)=\binom{u(n+1)}{u(n)}; \qquad \Phi_{i}(n)=\binom{u_{i}(n+1)}{u_{i}(n)}$$ as two component vectors. \medpagebreak \flushpar {\bf Step 1.} {\it Almost constancy of $W(u, u_{i})$} \smallpagebreak By a standard calculation using (2.3) $$ \align |W(u, u_{i})(n)-W(u, u_{i})(n-1)| &\leq |V(n)-V(2m_{i}-n)| |u(n)u_{i}(n)| \\ &\leq e^{-Bm_{i}} \tag 2.5 \endalign $$ by (2.2) and (2.4). \medpagebreak \flushpar {\bf Step 2.} {\it Smallness of $W(u, u_{i})$ for $m_{i}$ large} \smallpagebreak Since $u$ and $u_i$ are in $\ell^2$ with $\ell^2$ norm 1, the Schwarz inequality implies that $\sum\limits_{n}|W(n)|\leq 2$. Thus for some $n$ with $|n|\leq e^{Bm_{i}/2}$, we must have that $|W(n)|\leq e^{-Bm_{i}/2}$. By (2.5) we se that for $|n|\leq e^{Bm_{i}/2}$, we have that $$|W(n)|\leq 3e^{-Bm_{i}/2} \tag 2.6 $$ and in particular for $n=m_i$. Now define $u^{\pm}_{i}=u\pm u_i$, $\Phi^{\pm}_{i}=\Phi\pm\Phi_i$. \medpagebreak \flushpar {\bf Step 3.} {\it Smallness of $\Phi^{+}_{i}(m_{i})$ or $\Phi^{-}_{i}(m_{i})$} \smallpagebreak Since $W(u^{-}_{i}, u^{+}_{i})=2W(u, u_{i})$ and $u^{-}_{i}(m_{i})=0$, we see that $$|u^{+}_{i}(m_{i})u^{-}_{i}(m_{i}+1)|\leq 6e^{-Bm_{i}/2}$$ so either $$|u^{+}(m_{i})|\leq\sqrt{6} e^{-Bm_{i}/4} \tag 2.7 $$ or $$|u^{-}_{i}(m_{i}+1)|\leq\sqrt{6} e^{-Bm_{i}/4}. \tag 2.8 $$ We claim that this means either $$\|\Phi^{\pm}_{i}(m_{i})\|\leq Ce^{-Bm_{i}/4} \quad\text{(for one of $+$ {\it or }$-$)}. \tag 2.9 $$ If (2.8) holds, (2.9) is immediate since $u^{-}_{i}(m_{i})=0$. If (2.7) holds, note that by (2.3) $$u^{+}(m_{i}+1)+\frac12 (V(m_{i})-E)u^{+}(m_{i})=0$$ so (2.9) holds for $\Phi^{+}_{i}$. \medpagebreak \flushpar {\bf Step 4.} {\it Smallness of $\Phi^{\pm}_{i}(0)$} \smallpagebreak Let $T^{(1)}_{i}$ be the transfer matrix for (2.3), taking $\Phi(m_{i})$ to $\Phi(0)$ and let $T^{(2)}_{i}$ be the same with $V(2m_{i}-n)$ so $$\gather T^{(1)}_{i}\Phi(m_{i})=\Phi(0) \\ T^{(2)}_{i}\Phi_{i}(m_{i})=\Phi_{i}(0). \endgather $$ Writing out $T_i$ as a product and using the definition of $\alpha$ and (2.2), we have that $$\|T^{(1)}_{i}-T^{(2)}_{i}\|\leq 2m_{i}\alpha^{m_{i}-1} e^{-Bm_{i}} \leq 2m_{i}e^{-3Bm_{i}/4}.$$ Writing $$ \align \Phi^{\pm}_{i}(0) &= T^{(1)}_{i}\Phi(m_{i})\pm T^{(2)}_{i}\Phi_{i}(m_{i})\\ &= T^{(1)}_{i}(\Phi^{\pm}_{i} (m_{i}))\mp (T^{(1)}_{i}-T^{(2)}_{i}) \Phi_{i}(m_{i}) \endalign $$ we see that $$\|\Phi^{\pm}_{i}(0)\|\leq m_{i}e^{-3Bm_{i}/4}+C(\alpha e^{-B/4})^{m_i}$$ goes to zero as $m_{i}\to\infty$. \medpagebreak \flushpar {\bf Step 5.} {\it Completion of the proof} \smallpagebreak By the last fact, $\|\Phi(0)\|-\|\Phi(2m_{i})\|\to 0$ which is only consistent with $u\in\ell^2$ if $\|\Phi(0)\|=0$ which implies that $u=0$. \qed \enddemo \bigpagebreak For the continuum (Schr\"odinger case), here are the changes: We can suppose (2.2) holds, but with $e^{-Bm_{i}}$ replaced by $e^{-m^{2}_{i}}$ (any $f(m)$ with $\lim\limits_{i\to\infty} m^{-1}_{i} \ln f(m)^{-1}=\infty$ will do). We normalize $u$ so that $$\int \left[u(x)^{2}+ u'(x)^{2}\right]dx=1. \tag 2.10 $$ \medpagebreak \flushpar {\bf Step 1.} By (2.10) and a Sobolev estimate, $u$ and $u'$ are uniformly bounded so $\left|\frac{dW}{dx} (u, u_{i})(x)\right|\leq Ce^{-m^{2}_{i}}$ for some $C$. \medpagebreak \flushpar {\bf Step 2.} $\int |W(u, u_{i})|dx\leq 2$, so, by the same argument $$|W(u, u_{i})|(x)\leq (2C+1)e^{-m^{2}_{i}} \qquad \text{if $|x|\leq e^{m^{2}_{i}/2}$}.$$ \medpagebreak \flushpar {\bf Step 3.} This is actually easier since $(u^{+}_{i})'(m_{i})=0$ and $u^{-}_{i}(m_{i})=0$. \medpagebreak \flushpar {\bf Step 4.} This is similar. The transfer matrix is bounded by $e^{Cm_{i}}$ where $C$ is $E$-dependent (and goes to infinity as $E\to\infty$) which is always beaten out by $e^{-m^{2}_{i}/2})$. \medpagebreak \flushpar {\bf Step 5} is unchanged. \bigpagebreak \Refs \widestnumber\key{11} \medpagebreak \ref \key 1 \by J. Avron and B. Simon \paper Almost periodic Schr\"odinger operators, II. 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