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 \(\delta >1\).
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\( {\hat{P}}_n-\widetilde{P}_n=\sum _{i=0}^{n-1}{\hat{A}}_{n-1}\dots {\hat{A}}_{i+1}({\hat{A}}_i-\widetilde{A}_i)\widetilde{A}_{i-1}\dots \widetilde{A}_0\).
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Acknowledgements
I am grateful to Svetlana Jitomirskaya for sparking my interest in the spectral theory of Schrödinger operators and for her hospitality during my visit to the University of California Irvine. This material is based upon work supported in part by the National Science Foundation EPSCoR RII Track-4 # 1738834 and the University of Mississippi College of Liberal Arts Summer Research Grant.
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Proof of Lemma 3.2
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
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
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
holds, for every \(i\ge 0\). Moreover, there exists \(D_2>0\) such that
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 \(0\le i\le N\),
where \(a_0=\tan ^{-1}(s_0/\sqrt{\kappa })\). Moreover, there exists \(D_4>0\) such that
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
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 \(i_l\le i< i_c\),
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
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 \(k-j_r< i\le k\). \(\square \)
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Kocić, S. Singular continuous phase for Schrödinger operators over circle maps. Math. Ann. 389, 1545–1573 (2024). https://doi.org/10.1007/s00208-023-02646-2
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DOI: https://doi.org/10.1007/s00208-023-02646-2