\input amstex \documentstyle{amsppt} \TagsOnRight \NoBlackBoxes \magnification = 1100 \hsize= 6.5 truein \vsize = 8.8 truein \def\cntl {\centerline} \def\noi{\noindent} \def\ep{\epsilon_\nu} %%%% as a fixed small number \def\diam{\text{diam}} %%%% diameter \def\im{\text{Im}} %%%% imaginary part \def\re{\text{Re}} %%%% real part \def\cosh{\text{cosh}} %%%% cosh function %%%%%%%%%%%%%%%%%% REFERENCE NUMBER %%%%%%%%%%%%%%%%%%%%% \def\as {1} %%%% Avron Simon \def\behncke{2} %%%% H. Behncke:absolute continuity of ... \def\cl {3} %%%% Coddington and Levinson: ODE \def\cfks {4} %%%% Schrodinger operator \def\das {5} %%%% Physical review \def\djms {6} %%%% Del Rio, Jitomirskaya, Makarov, Simon, %%%% Bull. Am. Math. Soc. \def\dss {7} %%%% Del Rio, Simon and Stolz \def\ff {8} %%%% Froman and Froman: JWKB \def\gp {9} %%%% Galindo and Pascual:Quantum Mechanics \def\hs {10} %%%% Harrell and Simon: resonance \def\kmpI {11} %%%% Kirsch, Molchanov and Pastur: Jocabi \def\kmpII {12} %%%% Kirsch, Molchanov and Pastur: Schrodinger \def\ko {13} %%%% Kotani \def\rsII {14} %%%% Reed and Simon Volume 2 \def\rsIV {15} %%%% Reed and Simon Volume 4 \def\simon {16} %%%% Simon: Vancouver Lecture Notes \def\sw {17} %%%% Simon and Wolff \def\stein {18} %%%% Stein: Singular integral \def\stolzj {19} %%%% Stolz: Jacobi case \def\stolzs {20} %%%% Stolz: Schrodinger case \def\stolzt {21} %%%% Stolz: thesis \def\weid {22} %%%% Weidmann (spectral theory for ODE) \def\measure{23} %%%% Wheeden: Hausdorff dimension \topmatter \title {The Lyapunov Exponents for Schr\"odinger Operators \\ with Slowly Oscillating Potentials} \endtitle \author{Barry Simon and Yunfeng Zhu} \endauthor \affil{Department of Mathematics, 253-37\\ California Institute of Technology\\ Pasadena, CA 91125} \endaffil %%%\email{yunfeng@cco.caltech.edu (Yunfeng Zhu)}\endemail \leftheadtext{Lyapunov Exponents for Schr\"odinger Operators} \rightheadtext{B. Simon and Y. Zhu} \abstract{By studying the integrated density of states, we prove the existence of Lyapunov exponents and the Thouless formula for the Schr\"odinger operator $-d^2/ dx^2 + \cos x^{\nu}$ with $0< \nu < 1$ on $L^2[0,\infty)$. This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences. }\endabstract \endtopmatter \document \baselineskip = 18 truept %%\parindent = 22 truept \font\eightteenbf=cmbx10 at 18 truept \font\fourteenbf=cmbx10 at 14 truept \head{1. Introduction}\endhead Our goal in this paper is to prove Lyapunov behavior and compute a Lyapunov exponent for the one-dimensional half-line Schr\"odinger operator $$H_\nu = -\frac{d^2}{dx^2} + \cos x^{\nu} \qquad x \in [0, \infty) \tag 1.1$$ with $0< \nu < 1$. It is clear that $H_\nu$ is regular at $0$ and is limit point at infinity. (For the definition of limit point, see [\weid] or [\rsII].) Therefore, for each $\theta \in [0, \pi)$, $H_\nu$ has a unique self-adjoint realization on $L^2[0, \infty)$ with boundary condition at $0$ given by $$u(0) \cos \theta +u'(0) \sin \theta = 0$$ which will be denoted by $H_\nu^{\theta}$. In the spectral theory of Schr\"odinger operators, most work has concentrated on the potential $V(x)$, either $V(x) \to 0$ as $|x| \to \infty$ or $V(x)$ is periodic or almost periodic. Such models have been investigated particularly well. Comparatively new are the models with oscillating but not periodic nor almost periodic potentials. Due to recent discoveries of H. Behncke ([\behncke]), W. Kirsch, S.A. Molchanov and L.A. Pastur ([\kmpII]) and G. Stolz ([\stolzj], [\stolzs]), it is clear that some such models may yield very interesting spectrum. As one of his particular examples, Stolz has studied the spectral properties for (1.1) in [\stolzs]. Let $\sigma(H)$, $\sigma_{\text{\rom{ac}}}(H)$, $\sigma_{\text{\rom{sing}}}(H)$, $\sigma_{\text{\rom{sc}}}(H)$ and $\sigma_{\text{\rom{pp}}}(H)$ denote the spectrum, absolutely continuous spectrum, singular spectrum, singular continuous spectrum and pure point spectrum resp.~for $H$. Then from Stolz's paper, we have known that $\sigma (H_\nu)=[-1,\infty)$, $\sigma_{\text{\rom{ac}}}(H_\nu)=[1,\infty)$, and $\sigma_{\text{\rom{sing}}} (H_\nu) = [-1, 1]$. In fact, from an unpublished result of Kirsch and Stolz (see [\kmpII]), we have also know that $H_\nu^\theta$ has pure point spectrum in $[-1,1]$ for almost all boundary conditions $\theta$. We already see that this model has some subtle and fascinating spectral properties, especially for $E \in (-1,1)$. We'll continue working on this model. In particular, we will prove Lyapunov behavior and compute a Lyapunov exponent formula. We know that the Lyapunov exponent is an important tool in the spectral theory for one-dimensional Schr\"odinger operators with almost periodic or random potentials. In [\sw, \simon], the rank one perturbation theory shows that Lyapunov behavior can also be used to study Schr\"odinger operators with deterministic potentials. For almost periodic or random potentials, we have the subadditive ergodic theorem to guarantee the existence of the Lyapunov exponent, but for deterministic potentials, it's often difficult to prove Lyapunov behavior. In this paper, we first study the integrated density of states in detail, then we directly study the existence of the Lyapunov exponent and prove the Thouless formula for a.e.~$E$ (with Lebesgue measure). Now, our formula for $\gamma(E)$, $E \in (-1,1)$, which we prove off an explicitly given set of measure 0, is strictly positive. It is known (see [\dss]) that since $(-1,1) \subset \sigma(H_\nu)$, the complement of $\{\, E \mid \gamma(E) \text{ exists and is } >0 \,\}$ is a dense $G_\delta$ in $[-1,1]$. By our construction, this dense $G_\delta$ has measure zero; indeed, it has Hausdorff dimension zero. We are unaware of any other explicit (non-random) Schr\"odinger operators with a computable positive Lyapunov exponent. The explicit formula (3.22) is quasi-classical. %%%%%%%%%%%%%%%%%%%%%%%%% SECTION 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \head{2. The integrated density of states}\endhead To prove the Thouless formula, we need to study the integrated density of states, $k(E)$, and the existence of the Lyapunov exponent. Also, we need information on how rapidly $k^{(\ell)}(E)$ converges to $k(E)$ to establish the existence of the Lyapunov exponent. So, we first study the main technical object, the integrated density of states for equation (1.1). We will prove a formula for the integrated density states, and more importantly, we will estimate how fast $k^{(\ell)}(E)$ converges to $k(E)$. The basic idea to compute the integrated density of states uses the standard Dirichlet-Neumann bracketing technique. Since the potentials in our problem are slowly oscillating, Dirichlet-Neumann bracketing works perfectly. First, let us introduce some notation and definitions. In the following, when we write $H_\nu$, we always mean the Schr\"odinger operator given by (1.1). Define $$L = S_{\nu}(\ell)=(2 \pi \ell)^{\frac 1{\nu}}, \qquad \Omega_\ell =[ S_{\nu}(\ell-1), S_{\nu}(\ell)], \qquad \text{for \ell = 1,2, \cdots}.$$ $\Omega_{\ell}$ is the $l$th potential well for the potential $V(x)= \cos x^{\nu}$ $(0< \nu <1)$. Let $H_D(\Omega)$, (resp.~$H_N(\Omega)$) denote the self-adjoint operator $H_0 +V(x)$ on $L^2(\Omega)$ with Dirichlet (resp.~Neumann) boundary conditions, where $H_0 = - \Delta$. When $\Omega = (0, L)$, we use $H_D(L)$, (resp.~$H_N(L)$) to denote $H_D(\Omega)$, (resp.~$H_N(\Omega)$). In this case, we use $H_{DN}(L)$, (resp.~$H_{ND}(L)$) to denote the self-adjoint operator $H_0 + V(x)$ on $L^2(0,L)$ with Dirichlet (resp.~Neumann) boundary condition at $0$ and Neumann (resp.~Dirichlet) boundary condition at $L$. \definition{Definition} For any self-adjoint operator $A$, define $$N(E,A)= \dim P_{(-\infty, E)} (A) = \sum_{E_k < E} 1$$ where $P_\Omega (A)$ is the spectral projection for the operator $A$, and $\{ E_k \}$ are the eigenvalues of $A$ with $E_1 \le E_2 \le E_3 \le \cdots$. \enddefinition Now, let $H_{\text{\rom{bc}}}(S_\nu(\ell))$ be any self-adjoint realization of $H_\nu$ on $L^2(0,S(\ell))$ with some given boundary conditions at $0$ and $S_\nu(\ell)$. Let $N_{\text{\rom{bc}}}(E, \ell) = N(E, H_{\text{\rom{bc}}}(S_\nu(\ell)))$. \definition{Definition} Let $N_{\text{\rom{bc}}}(E, \ell)$ be as above, then we define $$k^{(\ell)}(E) = \frac 1{S_\nu(\ell)} N_{\text{\rom{bc}}}(E,\ell) \qquad \text{and \ \ k(E) = \lim_{\ell \to \infty} k^{(\ell)}(E).}$$ $k(E)$ is called the integrated density of states for (1.1). \enddefinition We will show that in the above definition, the limit $k(E)$ exists and is independent of the choice of boundary conditions. By standard Dirichlet-Neumann bracketing (see [\rsIV]), $$\sum_{j=1}^\ell {N(E, H_D(\Omega_j))} \le N(E, H_D(L)) \le N(E, H_N(L)) \le \sum_{j=1}^{\ell} {N(E, H_N(\Omega_j))}. \tag 2.1$$ By explicit construction and counting in boxes, we have \proclaim{Lemma 2.1} If we let $N_D(E;a,b)$ \rom(resp.~$N_N(E;a,b)$ \rom) denote the dimension of the spectral projection $P_{(-\infty,E]}$ for $-\Delta_D$ \rom(resp.~$-\Delta_N$\rom) on $L^2(a,b)$. Then for $E<0$, we have $$N_D(E;a,b) = N_N(E;a,b) = 0 \tag 2.2$$ and for $E \ge 0$, we have $$\biggl|N_D(E;a,b) - \frac{\sqrt E }{\pi}(b-a)\biggr| \le 1 \tag 2.3$$ $$\biggl|N_N(E;a,b) - \frac{\sqrt E}{\pi}(b-a)\biggr| \le 1. \tag 2.4$$ \endproclaim \def\ijk{I_k^{(j)}}%%% this definition is used in the following %%% First, let us estimate $N(E,H_D(\Omega_j))$ and $N(E,H_N(\Omega_j))$. Let $a_k \in \Omega_j$ and $b_k= a_{k+1}$ such that $\cup_{k} [a_k, b_k] = \Omega_j$ and $b_k-a_k=j^{\alpha}$, where $\alpha >0$ (depending on $\nu$) will be determined later. Let $\ijk=(a_k,b_k)$ and $$V^D_k = \sup{ \{ V(x) \ |x \in [a_k, b_k] \} }, \qquad V^N_k = \inf{ \{ V(x) \ |x \in [a_k, b_k] \} }.$$ Define $B_D(\ijk) = -\Delta_D(\ijk) + V^D_k$ and $B_N(\ijk) = -\Delta_N(\ijk) + V^N_k$, then $$0 \le H_D(\ijk) \equiv -\Delta_D(\ijk) + V(x) \le B_D(\ijk)$$ and $$0 \le B_N(\ijk) \le -\Delta_N(\ijk) + V(x) \equiv H_N(\ijk).$$ Obviously, $$N(E,B_D(\ijk)) \le N(E,H_D(\ijk)), \qquad N(E,H_N(\ijk)) \le N(E,B_N(\ijk))$$ and by Dirichlet-Neumann bracketing, $$N(E,H_D(\Omega_j)) \ge N(E,H_D(\cup \ijk)) = \sum_k{N(E,H_D(\ijk))} \ge \sum_k{N(E,B_D(\ijk))} \tag 2.5$$ and $$N(E,H_N(\Omega_j)) \le N(E,H_N(\cup \ijk)) =\sum_k{N(E,H_N(\ijk))} \le \sum_k{N(E,B_N(\ijk))}. \tag2.6$$ So, we only need to estimate $N(E,B_N(\ijk))$ and $N(E,B_D(\ijk))$. But by (2.2) and (2.4), \align N(E,B_N(\ijk)) &=N_N(E-V_k^N; a_k, b_k) \\ &= \cases \frac{\sqrt{E-V_k^N}}{\pi} (b_k-a_k) +C_0(k), &\text {if E \ge V_k^N,} \\ 0, &\text {if E < V_k^N} \endcases \endalign where $|C_0(k)| \le 1$. Thus, if we use the notation that $[f(x)]_+=\max \{0, f(x) \}$, then we have $$\biggl| N(E,B_N(\ijk)) - \frac{[E-V_k^N]_+^\frac12}{\pi} (b_k-a_k)\biggr| \le 1. \tag2.7$$ But $$\multline \frac 1 \pi [E-V_k^N]_+^\frac12 (b_k-a_k) - \frac 1 \pi\int_{a_k}^{b_k} [E-V(x)]_+^\frac 12\,dx \\ = \frac 1 \pi\int_{a_k}^{b_k} \{ [E-V_k^N]_+^{\frac 12} -[E-V(x)]_+^\frac 12 \} \,dx \overset\text{def}\to= J. \qquad {} \endmultline \tag2.8$$ Since \align \bigl\{ [E-V_k^N]_+^\frac 12 - &[E-V(x)]_+^\frac 12 \bigr\}^2 \\ &\le \bigl|[E-V_k^N]_+^\frac 12 - [E-V(x)]_+^\frac 12 \bigr| \bigl\{[E-V_k^N]_+^\frac 12 + [E-V(x)]_+^\frac 12 \bigr\} \\ &\le \frac \nu{a_k^{1-\nu}} (b_k-a_k) \qquad {\text{\rom{for }}} x \in \ijk. \endalign By Schwartz inequality, we have, \align |J| &\le \frac1{\pi} (b_k-a_k)^{\frac 12} \biggl[\int_{a_k}^{b_k} {\{[E-V(x)]_+^{\frac 12}- [E-V_k^N]_+^{\frac 12}\}^2\, dx\biggr]^{\frac 12}} \\ &\le \frac{\sqrt \nu}{\pi} a_k^{-\frac 12 (1-\nu)} (b_k-a_k)^\frac32 \\ &\le j^{\frac32 \alpha -\frac 12 \frac {1-\nu}{\nu}}. \tag2.9 \endalign Therefore, by (2.7)--(2.9), we have $$\biggl|N(E,B_N(\ijk)) - \frac 1{\pi} \int_{a_k}^{b_k} [E-V(x)]_+^ \frac 12 \,dx \biggr| \le j^{\frac32 \alpha - \frac 12 \frac {1-\nu}{\nu}} + 1.$$ Thus, by summing over $k$ and using (2.6), we have $$N(E,H_N(\Omega_j)) \le \frac 1{\pi} \int_{S_\nu(\ell-1)}^{S_\nu(\ell)} [E-V(x)]_+^\frac 12 \,dx + C_1 j^{\frac 12 (\alpha+\frac{1-\nu}{\nu})} + C_2 j^{\frac {1-\nu}{\nu}- \alpha} \tag2.10$$ where $C_1$ and $C_2$ are independent of $j$. Similarly, if we use (2.3) and (2.5) instead of (2.4) and (2.6), then we have $$N(E,H_D(\Omega_j)) \ge \frac 1{\pi} \int_{S_\nu(\ell-1)}^{S_\nu(\ell)} [E-V(x)]_+^\frac 12 \,dx - C_1 j^{\frac 12 (\alpha+\frac{1-\nu}{\nu})} - C_2 j^{\frac {1-\nu}{\nu}- \alpha}. \tag2.11$$ Now, by summing over $j$ in (2.10), (2.11) and using (2.1), we have $$\multline \frac 1{\pi}\int_0^{S_\nu(\ell)} [E- V(x)]_+^\frac 12 \,dx - C_1 j^{\frac 12 (\alpha+\frac{1-\nu}{\nu})+1} - C_2 j^{\frac {1-\nu}{\nu}- \alpha +1} \le N(E, H_D(L)) \\ \le N(E, H_N(L)) \le \frac 1{\pi} \int_0^{S_\nu(\ell)} [E-V(x)]_+^\frac 12 \,dx + C_1 j^{\frac 12 (\alpha+\frac{1-\nu}{\nu})+1} + C_2 j^{\frac {1-\nu}{\nu}- \alpha +1}. \endmultline$$ So, if we take $\alpha=\frac 13 \frac {1-\nu}{\nu}$, then we have $$\multline \frac 1{\pi} \int_0^{S(\ell)} [E-V(x)]_+^{\frac 12}\,dx -C \ell^{\frac 23 \frac {1-\nu}{\nu}+1} \le N(E, H_D(L)) \\ \le N(E, H_N(L)) \le \frac 1{\pi} \int_0^{S(\ell)}[E-V(x)]_+^{\frac 12}\,dx +C \ell^{\frac 23 \frac {1-\nu}{\nu}+1} \qquad {} \endmultline \tag2.12$$ where $C = C_1+C_2$. Also, we have the following estimation \align \frac 1{S_\nu(\ell)} \int_0^{S_\nu(\ell)}[E-V(x)]_+^{\frac 12}\,dx &=\frac 1{\nu(2 \pi \ell)^{\frac 1\nu}} \int_0^{2 \pi \ell} y^{\frac{1-\nu}{\nu}} [E-\cos y]_+^{\frac 12}\,dy \qquad (x^\nu = y) \\ &=\frac1{\nu(2 \pi \ell)^2} \sum_{k=1}^\ell \int_{-2\pi}^0 (z+2k\pi)^{\frac {1-\nu}{\nu}} [E-\cos z]_+^{\frac 12}\,dz \qquad (y=z+2k\pi) \\ &=\frac 1{2\pi} \int_{-\pi}^{\pi} [E-\cos x]_+^\frac 12 \,dx + O(\ell^{-1}). \endalign Thus, if we denote $$k_N^{(\ell)}(E) \equiv \frac 1{S(\ell)} N(E,H_N(L)),$$ then by the above estimations and (2.12), we have $$\biggl| k_N^{(\ell)}(E) - \frac 1{2\pi^2} \int_{-\pi}^{\pi} [E-\cos x]_+^\frac 12 \,dx \biggr| = O(\ell^{- \frac 13\frac {1-\nu}{\nu}}) +O(\ell^{-1}). \tag 2.13$$ Since variations of boundary condition are rank one perturbations (see [\simon]), $$|N(E,H_N(L)) - N(E,H_{bc}(L)) | \le 2 \tag 2.14$$ where $H_{\text{\rom{bc}}}(L)$ is defined by any other self-adjoint boundary condition. Thus, by (2.13) and (2.14), we have proved the following \proclaim{Theorem 2.1} The integrated density of states for the Schr\"odinger operator \rom(2.1\rom) exists, which is independent of the boundary conditions, and is given by $$k(E) = \frac 1{2\pi^2} \int_{-\pi}^{\pi} [E-\cos x]_+^\frac 12 \,dx.$$ Moreover, we have the following estimation $$|k^{(\ell)}(E) - k(E)| = O(\ell^{-\kappa(\nu)}) \tag 2.15$$ where $$\kappa(\nu) = \min \biggl\{ \frac 13 \frac {(1-\nu)}{\nu}, 1 \biggr\}. \tag 2.16$$ \endproclaim %%%%%%%%%%%%%%%%%%%%%%%%%% SECTION 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \head{3. The Thouless Formula and Lyapunov Exponent} \endhead Now, we begin to study the Lyapunov exponent by first proving the Thouless formula which relates the Lyapunov exponent to the integrated density of states. In [\as], the Thouless formula is proved for almost periodic potentials and random potentials. To prove the Thouless formula in our case, we can closely follow the proof given in [\as] for Schr\"odinger operators. However, we will prove the existence of the Lyapunov exponent by using the information on how fast $k^{\ell}(E)$ converges to $k(E)$ which is given in Theorem 2.1. First, we define the transfer matrix for the Schr\"odinger operator (1.1) as follows. Let $u(x,a,E)$, $v(x,a,E)$ $(x \ge 0, a \ge 0)$ solve the equation $- \phi '' +(V(x)-E) \phi =0$ with the boundary conditions given by $u(a)=0$, $u'(a)=1$; $v(a)=1$, $v'(a)=0$. Then the transfer matrix is defined by $$T_{a,x}(E) = \left( \matrix v(x,a,E) & u(x,a,E) \\ \frac {\partial v(x,a,E)}{\partial x} & \frac {\partial u(x,a,E)}{\partial x} \\ \endmatrix \right). \tag 3.1$$ In particular, when $a=0$, we use $T_x(E)$ to denote $T_{0,x}(E)$. \definition{Definition} For a given $E$, if $\gamma(E) = \lim_{x \to \infty} x^{-1} \ln \| T_x(E)\|$ exists, then we say that for the energy $E$, $H$ has Lyapunov behavior, and $\gamma(E)$ is called the Lyapunov exponent. \enddefinition To give the Thouless formula, we need to define the resonance set first. In Section 2, we defined the operators $H_D(L)$, $H_N(L)$, $H_{DN}(L)$ and $H_{ND}(L)$. Now, let $\{E_k(\ell,D)\}$, $\{E_k(\ell,N)\}$, $\{E_k(\ell,DN)\}$, and $\{E_k(\ell,ND)\}$ be the corresponding eigenvalues. \definition{Definition} For each given $\nu \in (0,1)$, let $\ep$ be a fixed small number such that $\ep < \kappa(\nu)$, where $\kappa(\nu)$ is defined by (2.16). Then the resonance set, $R_{\nu}$, for the operator $H_{\nu}$ is defined by $$R_{\nu} = R_D \cup R_N \cup R_{DN} \cup R_{ND} \tag 3.2$$ where $$R_D = \bigcup_{d=1}^{\infty} \bigcap_{m=1}^{\infty} \bigcup_{n=m}^{\infty} \bigcup_{k} \bigl\{\, E \in [-d,d] \bigm| |E-E_k(n, D)| < \exp(-n^{\kappa(\nu)-\ep})\, \bigr\}. \tag 3.3$$ $R_N$, $R_{DN}$ and $R_{ND}$ are defined by replacing $\{E_k(\ell,D)\}$ in (3.3) by $\{E_k(\ell,N)\}$,$\{E_k(\ell,DN)\}$ and $\{E_k(\ell,ND)\}$ resp. \enddefinition \remark{Remark} We conjecture that instead of (3.2) and (3.3), the resonance set in $[-1,1]$ can be defined by $$R_{\nu}= \bigcap_{m=1}^{\infty} \bigcup_{n=m}^{\infty} \bigcup_{k} \bigl\{ \, E \in [-1,1] \bigm| |E-E_k^{(n)}| <\exp(-n^{\min\{\frac {1-\nu}{2\nu}, \, \frac 12 \}} ) \,\bigr\}$$ where $\{ E_k^{(n)} \}$ are the eigenvalues of $H_\nu=H_0+V(x)$ on the $n$th potential well, $[(2n \pi -2 \pi)^{\frac 1\nu}, (2n \pi)^{\frac 1\nu}]$, with Dirichlet boundary conditions. We believe that this is the reasonable definition of the resonance set. However, in our proof of the Thouless formula, we need to use the resonance set defined by (3.2) and (3.3). \endremark From the definition, it is easy to show that \proclaim{Theorem 3.1} Let $R_{\nu}$ be the resonance set for $H_\nu$ which is defined by \rom(3.2\rom) and \rom(3.3\rom) and let $\dim_H$ denote the Hausdorff dimension. Then $$|R_{\nu}| = \dim_H R_{\nu} =0.$$ \endproclaim Now, we are ready to prove one of our main results. \proclaim{Theorem 3.2 (Thouless formula)} Let $H_\nu$ be the Schr\"odinger operator given by \rom(1.1\rom). Let $\gamma_0 (E)=[\max(0,-E)]^{\frac 12}$ and $k_0(E)=\pi ^{-1}[\max(0,E)]^{\frac 12}$. Then for any $E \notin R_{\nu}$, where $R_{\nu}$ is defined by \rom(3.2\rom) and \rom(3.3\rom), we have $$\gamma (E)=\gamma_0(E)+\int_{-\infty}^{\infty} \ln |E-E'|\, d(k-k_0)(E') \tag 3.4$$ where $\gamma (E)$ is the Lyapunov exponent for $H_\nu$, and $k(E)$ is the integrated density of states for $H_\nu$. \endproclaim We prove this theorem by proving the following series of lemmas. The first three lemmas are already given in [\as], so we will not give a proof for these results here. \proclaim {Lemma 3.3 \rm{[\as]}} For a.e. $E$, $$\lim_{\ell \to \infty} \ell^{-1} \ln |u_0(\ell,E) | = \gamma_0(E) \tag 3.5$$ the limit being through the integers. \endproclaim \proclaim {Lemma 3.4 \rm{[\as]}} Let $E_k(\ell)$ be the eigenvalue of $H_\nu$ on $L^2[0, S_\nu(\ell)]$ with vanishing boundary conditions, and let $E_k^{(0)}(\ell) = (\pi k/{S_\nu(\ell)})^2$ be the corresponding eigenvalue of $H_0$. Then $$|E_k(\ell) - E_k^{(0)} | \le \|V\|_{\infty} = 1. \tag 3.6$$ \endproclaim \proclaim{Lemma 3.5 \rm{[\as]}} For fixed $\ell$, we have that $$\frac{u(S_\nu(\ell), E)}{u_0(S_\nu(\ell), E)} = \prod_{k=1}^{\infty} \biggl [\frac{E-E_k(\ell)} {E-E_k^{(0)}(\ell)} \biggr]. \tag 3.7$$ \endproclaim >From [\as], we also know that \align \lim_{M \to \infty} &\biggl[ \int_{k(E') \le M} \ln |E-E'|\, dk(E') - \int_{k_0(E') \le M} \ln |E-E'|\, dk_0(E') \biggr] \\ &= \int_{-\infty}^{\infty} \ln |E-E'|\, d(k - k_0)(E'). \tag 3.8 \endalign \proclaim {Lemma 3.6} For $E \notin R_D$, we have $$\lim_{\ell \to \infty} \frac 1{S_\nu(\ell)} \ln \prod_{k=1}^{\infty} \biggl |\frac{E-E_k(\ell)}{E-E_k^{(0)}(\ell)} \biggr | = \int_{-\infty}^{\infty} \ln |E-E'| d(k-k_0)(E'). \tag 3.9$$ \endproclaim \demo{Proof} For a given $E \notin R_D$, without loss of generality, we can also suppose that $E \notin R_D^{(0)}$, where $R_D^{(0)}$ is the corresponding resonance set for $H_0$ with Dirichlet boundary condition. From now on, we always suppose that $E$ is fixed and $E \notin R_D \cup R_D^{(0)}$. For each fixed $E$, we can choose $M(\ell)$ such that $M(\ell) \to \infty$ as $\ell \to \infty$ and $a_i(\ell) > E+1$ ($i=0, 1$), where $$a_0(\ell) = \sup\{ E' \mid k_0^{(\ell)}(E') \le M(\ell) \}, \qquad a_1(\ell) = \sup\{ E' \mid k^{(\ell)}(E') \le M(\ell) \}.$$ For convenience, we define $$f_\ell (E) = \frac 1{S_\nu(\ell)} \ln \prod_{k=1}^\infty \biggl| \frac{E-E_k(\ell)}{E-E_k^{(0)}(\ell)} \biggr|, \qquad f(E) = \int_{-\infty}^{\infty} \ln |E-E'|\, d(k-k_0)(E').$$ Then we have \align | f_\ell(E) -f(E) | = &\biggl| \frac 1{S_\nu(\ell)} \ln \prod_{k \le M(\ell) S_\nu(\ell)} \bigl| [E-E_k(\ell)]/[E-E_k^{(0)}(\ell)] \bigr| \\ & \quad + \frac 1{S_\nu(\ell)} \ln \prod_{k > M(\ell) S_\nu(\ell)} \bigl| [E-E_k(\ell)]/[E-E_k^{(0)}(\ell)] \bigr| - f(E) \biggr| \\ \le &\biggl|\int_{-\infty}^{a_1(\ell)} \ln |E-E'|\, d(k^{(\ell)}- k)(E') - \int_{-\infty}^{a_0(\ell)} \ln |E-E'|\, d(k_0^{(\ell)}-k_0)(E') \biggr| \\ &\quad + \biggl| \int_{a_1(\ell)}^{\infty} \ln |E-E'|\,dk(E') - \int_{a_0(\ell)}^{\infty} \ln |E-E'|\,dk_0(E') \biggr| \\ &\quad +\biggl| \frac 1{S_\nu(\ell)} \ln \prod_{k > M(\ell)S_\nu(\ell)} \bigl| [E-E_k(\ell)]/[E-E_k^{(0)}(\ell)] \bigr| \biggr|. \tag 3.10 \endalign By (3.8), we have $$\lim_{\ell \to \infty} \biggl| \int_{a_1(\ell)}^{\infty} \ln |E-E'|\,dk(E') - \int_{a_0(\ell)}^{\infty} \ln |E-E'|\,dk_0(E') \biggr| = 0. \tag 3.11$$ Since $E_k^{(0)}(\ell) = \bigr(\pi k/{S_\nu(\ell)}\bigr)^2$, by using lemma 3.4, we have \align \ln \prod_{k > M(\ell)S_\nu(\ell)} \biggl| \frac{E-E_k(\ell)}{E-E_k^{(0)}(\ell)} \biggr| &= \sum_{k > M(\ell)S_\nu(\ell)} \ln \biggl| 1+ \frac{E_k(\ell)-E_k^{(0)}}{E_k^{(0)}(\ell)-E} \biggr| \\ &\le \sum_{k > M(\ell)S_\nu(\ell)} S_\nu^2(\ell)/[\pi^2 k^2 - S_\nu^2(\ell)E] \\ &\le S_\nu(\ell)\int_{M(\ell)}^\infty \frac{dx}{\pi^2 x^2 - E}. \endalign Therefore, $$\biggl| \frac 1{S_\nu(\ell)} \ln \prod_{k > M(\ell)S_\nu(\ell)} \bigl| [E-E_k(\ell)]/[E-E_k^{(0)}(\ell)] \bigr| \biggr| = O\biggl( \frac 1{M(\ell)} \biggr). \tag 3.12$$ So, it remains to estimate $$J_\ell \equiv \biggl|\int_{-\infty}^{a_1(\ell)} \ln |E-E'|\, d(k^{(\ell)}- k)(E') - \int_{-\infty}^{a_0(\ell)} \ln |E-E'|\,d(k_0^{(\ell)}-k_0)(E') \biggr|.$$ We define $$I_\ell(E) = \bigl[ E - \delta_\ell, E + \delta_\ell\bigr], \qquad \delta_\ell = \frac 13 \exp(-\ell^{\kappa(\nu)-\ep}) \tag 3.13$$ where $\kappa(\nu)$ is defined by (2.16) and $\ep$ is given in definition of the resonance set. Since $E \notin R_D \cup R_D^{(0)}$, there are no eigenvalues of $H_D(L)$ and ${H_0}_D(L)$ on the interval $I_{\ell}(E)$ which is defined by (3.13). Thus, $k^{(\ell)}(E)$, $k_0^{(\ell)}(E)$ are constant on the interval $I_{\ell}(E)$. Also, we notice that $$\biggl| \int_{I_\ell(E)} \ln |E-E'|\, dk(E') \biggr| \le C_E [|I_\ell(E) ]^\frac 12 \tag 3.14$$ where $C_E$ is a constant for a given $E$. So, we have \align J_\ell &= \biggl|\int_{(-\infty, a_1(\ell)]\setminus I_{\ell}(E)} \ln|E-E'| \,d(k^{(\ell)}-k)(E') + \int_{I_{\ell}(E)} \ln|E-E'| \,d(k^{(\ell)}-k)(E') \\ &\quad - \int_{(-\infty, a_0(\ell)]\setminus I_{\ell}(E)} \ln|E-E'| \,d(k_0^{(\ell)}-k_0)(E') - \int_{I_{\ell}(E)} \ln|E-E'| \,d(k_0^{(\ell)}-k_0)(E') \biggr| \\ &\le \biggl|\int_{(-\infty, a_1(\ell)]\setminus I_{\ell}(E)} \ln|E-E'| \,d(k^{(\ell)}-k)(E') \biggr| + \biggl| \int_{I_{\ell}(E)} \ln|E-E'| \,dk(E') \biggr| \\ &\quad + \biggl| \int_{(-\infty, a_0(\ell)]\setminus I_{\ell}(E)} \ln|E-E'|\,d(k_0^{(\ell)}-k_0)(E') \biggr| + \biggl| \int_{I_{\ell}(E)} \ln|E-E'| \,dk_0(E') \biggr|. \tag 3.15 \endalign By (3.14), we know that $$\lim_{\ell \to \infty} \int_{I_{\ell}(E)} \ln|E-E'| \,dk(E') = 0. \tag 3.16$$ Similarly, $$\lim_{\ell \to \infty} \int_{I_{\ell}(E)} \ln|E-E'| \,dk_0(E') =0. \tag 3.17$$ Using integration by parts, we have \align \biggl| &\int_{(-\infty, a_1(\ell)]\setminus I_{\ell}(E)} \ln|E-E'| \,d(k^{(\ell)}-k)(E') \biggr| \\ &\quad\le (k^{(\ell)}-k)(a_1(\ell)) \ln|E-a_1(\ell)| +\{ (k^{(\ell)}-k)(E+\delta_\ell) - (k^{(\ell)}-k)(E-\delta_\ell) \} \ln \delta_\ell \\ & \quad \qquad +\biggl| \int_{(-\infty, a_1(\ell)]\setminus I_{\ell}(E)} \frac {(k^{(\ell)}-k)(E') }{E'-E}\, dE' \biggr|. \endalign By theorem 2.1 and (3.13), we know that \align &\lim_{\ell \to \infty}(k^{(\ell)}-k)(a_1(\ell)) \ln|E-a_1(\ell)| =0 \\ &\lim_{\ell \to \infty}\{ (k^{(\ell)}-k)(E+\delta_\ell) -(k^{(\ell)}-k)(E-\delta_\ell) \} \ln \delta_\ell = 0 \endalign and \align \biggl| \int_{(-\infty, a_1(\ell)]\setminus I_{\ell}(E)} \frac {(k^{(\ell)}-k)(E') }{E'-E}\, dE' \biggr| & \le C_1 \ell^{-\kappa(\nu)} \biggl| \int_{(-\infty, a_1(\ell)] \setminus I_{\ell}(E)} \frac 1{E'-E}\, dE' \biggr| \\ & \le \ell^{-\kappa(\nu)} \{ C_2 \ln \delta_\ell + C_3 \ln |a_1(\ell) - E| \} \\ & \to 0 \qquad \text{ as \ell \to \infty. } \endalign Thus, $$\lim_{\ell\to \infty} \biggl| \int_{(-\infty, a_1(\ell)] \setminus I_{\ell}(E)}\ln|E-E'| \,d(k^{(\ell)}-k)(E') \biggr| = 0. \tag 3.18$$ Similarly, $$\lim_{\ell\to \infty} \biggl| \int_{(-\infty, a_0(\ell)] \setminus I_{\ell}(E)} \ln|E-E'|\,d(k_0^{(\ell)}-k_0)(E') \biggr|. \tag 3.19$$ So, by (3.15)--(3.19), $$\lim_{\ell\to \infty}J_\ell = 0. \tag 3.20$$ Now, by (3.10)--(3.12) and (3.20), we have proved that $\lim_{\ell\to \infty}|f_\ell(E) -f(E)| = 0$. Therefore, Lemma 3.6 is proved. \qed \enddemo Now, by combining the results of Lemma 3.5 and Lemma 3.6, we have proved the following result. For $E \notin R_D$, then we have that $$\lim_{\ell \to \infty} \frac 1{S_\nu(\ell)} \ln \biggl | \frac{u(S_\nu(\ell), E)}{u_0(S_\nu(\ell), E)} \biggr | = \int_{-\infty}^{\infty} \ln |E-E'| d(k-k_0)(E').$$ By using Lemma 3.3, we obtain the following control on the limit $$\lim_{\ell \to \infty} \frac 1{S_\nu(\ell)} \ln | u(S_\nu(\ell), E) | = \gamma_0(E) + \int_{-\infty}^{\infty} \ln |E-E'|\, d(k-k_0)(E').$$ By using different boundary conditions, we can obtain similar control of $\frac 1{S_\nu(\ell)} \ln \bigl| v(S_\nu(\ell), E) \bigr|$, $\frac 1{S_\nu(\ell)} \ln \bigl| \frac {\partial u(S_\nu(\ell), E)} {\partial x} \bigr|$ and $\frac 1{S_\nu(\ell)} \ln \bigl| \frac {\partial v(S_\nu(\ell), E)}{\partial x} \bigr|$. Therefore, we obtain control of $\frac 1{S(\ell)} \ln \| T_{S(\ell)}(E) \|$, namely \proclaim{Lemma 3.7} For $E \notin R_{\nu}$, where $R_{\nu}$ is the resonance set defined by \rom(3.2\rom) and \rom(3.3\rom), then $$\lim_{\ell \to \infty} \frac 1{S(\ell)}\ln \| T_{S(\ell)}(E) \| = \gamma_0(E)+\int_{-\infty}^{\infty} \ln |E-E'|\,d(k-k_0)(E') \tag 3.21$$ where $\| \cdot \|$ denotes the matrix norm, and $T_x (E)$ is defined by \rom(3.1\rom). \endproclaim Now, Theorem 3.2 follows from Lemma 3.7 and definition of the Lyapunov exponent. Next, we want to compute an explicit formula for the Lyapunov exponent by using the Thouless formula and the formula for integrated density of states. First, (3.4) asserts that $\pi k +i \gamma$ is the boundary value of an analytic function in the upper half plane. Let $F(z) = \pi k(z) +i \gamma(z)$ for $\text{Im}\,z \ge 0$, and define $\tilde F(z) = \frac 1{2\pi}\int_{-\pi}^{\pi} \sqrt{z-\cos x}\,dx$ with branch cut from $-1$ to $\infty$ along the real axis. Then $\tilde F(z)$ is analytic for $\text{Im}\,z>0$ and by Theorem 2.1, $\text{Re}\,\tilde F(z) \to \pi k(E)$ as $z \to E$ ($\text{Im}\,z>0$, $E \in \Bbb R$). Therefore, $$\gamma(E) = \lim_{\text{Im}\,z>0, z \to E}\text{Im}\,\tilde F(z) + C$$ where $C$ is a real constant. That is, $$\gamma(E) = \frac 1{2\pi}\int_{-\pi}^{\pi} [\cos x - E]_+^{\frac 12}\,dx + C.$$ Notice that for $E>1$, $\gamma(E) = 0$ and the integral in the right-hand side is also zero, so $C=0$. Therefore, we have \proclaim{Theorem 3.3} For all $E \notin R_{\nu}$, where $R_{\nu}$ is defined by \rom(3.2\rom) and \rom(3.3\rom), the operator $H_\nu$ in \rom(1.1\rom) has Lyapunov behavior with the Lyapunov exponent given by $$\gamma (E) = \frac 1{2 \pi} \int_{-\pi}^{\pi} [ \cos x-E]_+^{\frac 12}\,dx \tag3.22$$ where $[f(x)]_+ = \max \{ 0, f(x) \}$. \endproclaim \remark{Remarks} 1. In fact, there is no mystery for this beautiful Lyapunov exponent formula if we use the WKB (see [\ff],[\gp]) heuristic argument. However, it's not easy to justify the WKB solutions. 2. Note that while $R_\nu$ is $\nu$-dependent, the right-hand side of (3.22) is $\nu$-independent! \endremark %%%%%%%%%%%%%%%%%%%%%%%% SECTION 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \head{4. Some Spectral Consequences} \endhead We have already proved that for a.e.~$E \in [-1, 1]$, $H_\nu$ has positive Lyapunov exponent. By simply applying the Kotani argument (see [\ko]) or rank one spectral theory (see [\dss, \simon, \sw ]), we can get dense pure point spectrum on $(-1,1)$ for almost all boundary conditions. Also, we can show that the eigenfunctions are exponentially decaying. The result on pure point spectrum is an unpublished result by Kirsch and Stolz which is stated in [\kmpII] by Kirsch, Molchanov and Pastur, and the result on exponentially decaying is proved by Stolz in [\stolzt]. Now, we can give an explicit decaying rate of eigenfunctions. \proclaim{Theorem 4.1} Let $H_\nu^\theta$ be the operator $H_\nu$ given by \rom(1.1\rom) with the $\theta$ boundary condition at 0, $u(0) \cos \theta + u'(0) \sin \theta = 0$. Then for a.e. $\theta \in [0, \pi)$ \rom(with respect to Lebesgue measure\rom), $H_\nu^{\theta}$ has dense pure point spectrum on $(-1,1)$, and the eigenfunctions of $H_\nu^{\theta}$ to all eigenvalues $E \in (-1,1)$ decay like $e^{-\gamma (E) x}$ at $\infty$ for almost every $\theta$, where $\gamma(E)$ is given by \rom(3.22\rom). \endproclaim Next, as we have shown that the resonance set has Hausdorff dimension zero, by applying rank one perturbation theory, we get a new result on singular continuous spectral. \proclaim{Theorem 4.2} Let $H_\nu^\theta$ be the operator $H_\nu$ given by \rom(1.1\rom) with the boundary condition at $0$ given by $u(0) \cos \theta + u'(0) \sin \theta = 0$ for $\theta \in [0, \pi)$. Then for $\theta \neq \frac \pi 2$, the singular continuous part, $(d\mu_\theta)_{\text{sc}}$, of the spectral measure $d\mu_\theta$ for $H_\nu^\theta$ is supported on a Hausdorff dimension zero set. \endproclaim \Refs \widestnumber\key{999} \refstyle{A} \ref \key \as \by J. 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