Singular continuous phase for Schrödinger operators over circle maps

Abstract

We consider a class of Schrödinger operators—referred to as Schrödinger operators over circle maps—that generalize one-frequency quasiperiodic Schrödinger operators, with a base dynamics given by an orientation-preserving homeomorphism of a circle ${\mathbb {T}}^1={\mathbb {R}}/{\mathbb {Z}}$, instead of a circle rotation. In particular, we consider Schrödinger operators over circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity (circle maps with a break) or vanishes (critical circle maps). We show that in a two-parameter region—determined by the geometry of dynamical partitions and $\alpha$—the spectrum of Schrödinger operators over every sufficiently smooth such map, is purely singular continuous, for every $\alpha$-Hölder-continuous potential V. As a corollary, we obtain that for every sufficiently smooth such map, with an invariant measure $\mu$ and with rotation number in a set ${\mathcal {S}}$ depending on the class of the considered maps, and $\mu$-almost all $x\in {\mathbb {T}}^1$, the corresponding Schrödinger operator has a purely continuous spectrum, for every Hölder-continuous potential V. For circle maps with a break, this set includes some Diophantine numbers with a Diophantine exponent $\delta$, for any $\delta >1$.

Proof of Lemma 3.2

Let $\zeta ^*$ be a point such that $f'(\zeta ^*)=1$. Such a point exists, for sufficiently large k, since, by assumption, the first and the last intervals are of the same order, and on the interval $B_{K'}$ (which is non-empty for sufficiently large k), the function is convex. We will perform an affine orientation-preserving change of variables

$$\begin{aligned} y=h(z)=\frac{1}{2}f''(\zeta ^*)(z-\zeta ^*) \end{aligned}$$

(A.1)

that maps $\zeta ^*$ into 0 and normalizes the second derivative of f there. Under this change of variables f is transformed into $g=h\circ f\circ h^{-1}$ which satisfies $g'(0)=1$ and $g''(0)=2$. Let $\kappa :=g(0)=\min _y\{g(y)-y\}$. Since f is $C^{2+\alpha }$-smooth, so is g, and from (A.1), we have

$$\begin{aligned} |g(y)-(\kappa +y+y^2)|\le \mathfrak {C}|y|^{2+\alpha }, \qquad y\in h[-1,0], \end{aligned}$$

(A.2)

where $\mathfrak {C}>0$.

Proof of Lemma 3.2 uses some estimates proved in [18].

Lemma A.1

([18]) Suppose that, for a sequence of real numbers $\{s_i\}_{i\ge 0}$, there exist $\mathfrak {C}_1>0$ and $\alpha \in (0,1)$ such that $|s_{i+1}-(s_i-s_i^2)|\le \mathfrak {C}_1|s_i|^{2+\alpha }$, for every $i\ge 0$. Then, there exist constants $D_1>0$ and $d_1\in (0,1)$ such that, as long as $s_0\in (0,d_1]$, the estimate

$$\begin{aligned} \left| s_i-\frac{1}{i+s_0^{-1}}\right| \le \frac{D_1}{(i+s_0^{-1})^{1+\alpha }} \end{aligned}$$

(A.3)

holds, for every $i\ge 0$. Moreover, there exists $D_2>0$ such that

$$\begin{aligned} s_i-s_{i+1}=\frac{1}{(i+s_0^{-1})^2}(1+\delta _i), \end{aligned}$$

(A.4)

where $|\delta _i|\le D_2s_0^{\alpha }$, for all $i\ge 0$, as long as $s_0\in (0,d_1]$.

Lemma A.2

([18]) Suppose that, for a sequence of real numbers $\{s_i\}_{i\ge 0}$, there exist $\mathfrak {C}_2, \mathfrak {C}_3>0$ and $\kappa ,\alpha \in (0,1)$ such that

• 1. $|s_0|\le \mathfrak {C}_2\kappa$,

• 2. $|s_{i+1}-(\kappa +s_i+s_i^2)|\le \mathfrak {C}_3|s_i|^{2+\alpha }$, for every $i\ge 0$.

Fix arbitrary $\mathfrak {C}_4>0$ and define $N=\kappa ^{-1/2}\tan ^{-1}(\mathfrak {C}_4\kappa ^{-\frac{\alpha }{2(2+\alpha )}})$. Then, there exist constants $D_3>0$ and $d_2\in (0,1)$ such that, as long as $\kappa \in (0,d_2]$, the following estimate holds for every $0\le i\le N$,

$$\begin{aligned} |s_i-\sqrt{\kappa }\tan (\sqrt{\kappa }i+a_0)|\le D_3(\sqrt{\kappa }\tan \sqrt{\kappa }i)^{1+\frac{\alpha (\alpha +1)}{2}}, \end{aligned}$$

(A.5)

where $a_0=\tan ^{-1}(s_0/\sqrt{\kappa })$. Moreover, there exists $D_4>0$ such that

$$\begin{aligned} s_{i+1}-s_i=\frac{\kappa }{(\cos \sqrt{\kappa } i)^2}(1+\delta _i), \end{aligned}$$

(A.6)

where $|\delta _i|\le D_4\kappa ^{\frac{\alpha (\alpha +1)}{2(2+\alpha )}}$, for all $0\le i<N$, as long as $\kappa \in (0,d_2]$.

Proof of Lemma 3.2

Let a and b be the left and right end points of I. Let $t_0=h(a)$ and $t_i=g^i(t_0)$, i.e., $t_i=h(f^i(a))$.

Since $\kappa =g(0)$, there exists a unique number $i_c$ satisfying $0<i_c<k$ such that $t_{i_c}\in [0,\kappa )$. Let $i_l=i_c-[\kappa ^{-1/2}\tan ^{-1}\kappa ^{-\frac{\alpha }{2(2+\alpha )}}]$ and $i_r=i_c+[\kappa ^{-1/2}\tan ^{-1}\kappa ^{-\frac{\alpha }{2(2+\alpha )}}]$. Combining $\tan ^{-1}\frac{1}{x}=\frac{\pi }{2}-\tan ^{-1}x$ with $\tan ^{-1}x=x+{\mathcal {O}}(x^3)$, $x\rightarrow 0$, it is easy to derive the following asymptotic formula

$$\begin{aligned} \kappa ^{-\frac{1}{2}}\tan ^{-1}\kappa ^{-\frac{\alpha }{2(2+\alpha )}}=\frac{\pi }{2}\kappa ^{-\frac{1}{2}}-\kappa ^{-\frac{1}{2+\alpha }}+{\mathcal {O}}(\kappa ^{\frac{-1+\alpha }{2+\alpha }}),\quad \kappa \rightarrow 0. \end{aligned}$$

(A.7)

To obtain the desired estimates for $i_l\le i\le i_r$, we can apply Lemma A.2. To obtain the estimates for $i_l\le i< i_c$, we can apply this lemma to $s_i=-(t_{i_c-i}-\kappa )$, where $0\le i\le i_c-i_l$. It immediately follows from this lemma that, for $i_l\le i< i_c$,

$$\begin{aligned} t_{i+1}-t_i=s_{i_c-i}-s_{i_c-i-1}=\frac{\kappa i^2}{i^2(\cos (\sqrt{\kappa } (i_c-i-1)))^2}(1+\delta _{i_c-i-1}). \end{aligned}$$

(A.8)

It is not difficult to check that the function $\chi (\sqrt{\kappa } i)=\frac{\sqrt{\kappa }i}{\cos (\sqrt{\kappa }(i_c-i-1))}$ is monotonically increasing on $i_l\le i< i_c$. This follows from the fact that the function $\sqrt{\kappa } i\tan (\sqrt{\kappa } (i_c-i)-1)$ has maximum when $\sqrt{\kappa } i=\frac{\tan (\sqrt{\kappa } (i_c-i-1))}{1+\tan ^2(\sqrt{\kappa } (i_c-i)-1)}$ and, therefore, $\chi '(\sqrt{\kappa } i)=\frac{1-\sqrt{\kappa }i\tan (\sqrt{\kappa }(i_c-i-1))}{\cos (\sqrt{\kappa }(i_c-i-1))}\ge (\cos (\sqrt{\kappa }(i_c-i-1)))^{-1}(1+\tan ^2 (\sqrt{\kappa }(i_c-i-1))^{-1}>0$, for $i_l\le i<i_c$. Since $i_c=\frac{k}{2}+{\mathcal {O}}(\kappa ^{-\frac{1-\alpha }{2}})= \frac{\pi }{2}\kappa ^{-\frac{1}{2}}+{\mathcal {O}}(\kappa ^{-\frac{1-\alpha }{2}})$ as $\kappa \rightarrow 0$ (Lemma 3.19 in [17]) and, from asymptotic formula (A.7), $i_l=\kappa ^{-\frac{1}{2+\alpha }}+{\mathcal {O}}(\kappa ^{-\frac{1-\alpha }{2}})$ and

$$\begin{aligned} \frac{\kappa i_l^2}{\cos (\sqrt{\kappa }(i_c-i_l-1))}\rightarrow 1, \qquad \text{ as } \qquad \kappa \rightarrow 0, \end{aligned}$$

(A.9)

the function $\frac{\kappa i^2}{i^2(\cos (\sqrt{\kappa } (i_c-i-1)))^2}$ is bounded and the claim follows for $i_l\le i< i_c$. Here, we have also used the fact that, since the second derivative of f is bounded both from above and from below by positive constants, the lengths of the intervals $[t_{i-1},t_{i}]$ and $\Delta _i$ are of the same order. Similarly, we can obtain the desired estimates for $i_c\le i\le i_r$, by applying Lemma A.2 to $s_i=t_{i_c+i}$, where $0\le i\le i_r-i_c$.

For $0\le i\le i_l$ and $i_r< i\le k$, we can obtain the desired estimates by applying Lemma A.1. This is a consequence of the convexity and the fact that it follows from (A.5), using the (A.7), that $t_{i_l}=\kappa ^{\frac{1}{2+\alpha }}+{\mathcal {O}}(\kappa ^{\frac{1}{2+\alpha }+\frac{\alpha (\alpha +1)}{2(2+\alpha )}})$ and, similarly, $t_{i_r}=\kappa ^{\frac{1}{2+\alpha }}+{\mathcal {O}}(\kappa ^{\frac{1}{2+\alpha }+\frac{\alpha (\alpha +1)}{2(2+\alpha )}})$. We first obtain the estimates for $0\le i<i_l$. For $0\le i< i_l-j$, let $s_i=-t_{i+j}$. For sufficiently large k, and some fixed large j, $s_0\in (0,d_1]$. Since, for such i’s, $\kappa < \mathrm{const.} |t_{i+j}|^{2+\alpha }$, it follows from (A.2) that $s_i$ satisfy the assumptions of Lemma A.1. We can apply this lemma for $0\le i< i_l-j$. The estimate (A.4) immediately gives us the desired bounds for $1\le i< i_l$. Similarly, by defining $s_i=t_{k-j-i}$, for $0\le i\le i_r-j$, for some large j, we again have $s_0\in (0,d_1]$, for sufficiently large k. Since $\kappa < \mathrm{const.} |t_{k-j-i}|^{2+\alpha }$, it again follows from (A.2) that $s_i$ satisfy the assumptions of Lemma A.1. The estimate (A.4) of Lemma A.1 immediately gives us the desired estimates for $k-j_r< i\le k$. $\square$