Abstract
I analyze the microscopic behavior of the eigenvalues of skew-shift Schrödinger operators, and show that their statistics must resemble the one of the Anderson model rather than the one of quasi-periodic Schrödinger operators.
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H.K. was supported by a fellowship of the Simons foundation.
Appendix: CMV Operators
Appendix: CMV Operators
The goal of this section is to state some of the results of [12] and then use them to show that the conclusions of Theorem 1.1 do not hold in this context. I will first define the necessary objects in the quickest possible way. For \(x,y \in{\mathbb{T}}\), ω irrational, , and N≥1, we define Verblunsky coefficients
We then define the matrices
as acting on ℓ 2({n,n+1}) with \(\rho= \sqrt{1-|\lambda|^{2}} \in(0,1)\). Define
which factor as acting on ℓ 2({…,−1})⊕ℓ 2({0,…,N−1})⊕ℓ 2({N,…}). We denote by \(\mathcal{C}^{N}_{x,y}\) the product of the two matrices acting on ℓ 2({0,…,N−1}). Thus \(\mathcal{C}^{N}_{x,y}\) is a N×N matrix. One can show that \(\mathcal{C}^{N}_{x,y}\) is unitary. For details, I refer to Sect. 3 of [12].
Denote by \(0\leq\theta_{x,y;1}^{N} < \cdots< \theta_{x,y;N}^{N} < 1\) an increasing ordering of the arguments of the eigenvalues of \(\mathcal{C}^{N}_{x,y}\). This means that \(\mathrm{e}^{2\pi{\mathrm{i}}\theta_{x,y;j}^{N}}\) is an eigenvalue of \(\mathcal{C}^{N}_{x,y}\). The main goal of this section will be to prove
Theorem A.1
Let \(\varepsilon\in(0,\frac{1}{2})\) and ω Diophantine. Then
This theorem clearly shows that Theorem 1.1 does not hold in the context of CMV operators. In order to prove this theorem, we will first need Theorem 6.1 in [12]:
Theorem A.2
Let ω be Diophantine. There exists σ>0 such that for N sufficiently large and \(x,y\in{\mathbb{T}}\), there are \(\theta_{1}^{N}, \dots, \theta_{N}^{N}\) and ϑ N such that
and
Furthermore, Proposition 5.1 in [12] tells us that
Combining this with the following lemma, it is clear how to deduce Theorem A.1.
Lemma A.3
Let ω be irrational and denote by α 1,…,α N an increasing ordering of the numbers \(\{\omega n\ (\mbox{\emph{mod} }1)\}_{n=1}^{N}\). Then for any \(\varepsilon\in(0,\frac{1}{2})\), there exist arbitrarily large N such that
for n=1,…,N where α 0=α N −1.
More detailed information on the behavior of the sequence α n can be found in [16]. In order to prove this lemma, we will need to recall a few things about continued fractions. Given an irrational number ω∈(0,1), we can define
where ω 1=ω. Then ω has the continued fraction representation
Define p −2=q −1=0 and q −2=p −1=1 and
It is also well known that \(\frac{p_{n}}{q_{n}} \to\omega\) as n→∞. Finally, we note that
and \(\inf_{1 \leq q\leq q_{n+1} -1}\|q\omega\| = \|q_{n} \omega\|\).
Proof of Lemma A.3
As ∥nω−mω∥=∥(n−m)ω∥=∥(m−n)ω∥, it suffices to find N such that
If q n ≤N<q n+1, we have that the right-hand side is \(\|q_{n} \omega\| \geq\frac{1}{2 q_{n+1}}\). Hence, the condition becomes N≥2εq n+1, this condition can be satisfied for infinitely large N as long as \(\varepsilon\in(0,\frac{1}{2})\). □
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Krüger, H. Concentration of Eigenvalues for Skew-Shift Schrödinger Operators. J Stat Phys 149, 1096–1111 (2012). https://doi.org/10.1007/s10955-012-0650-3
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DOI: https://doi.org/10.1007/s10955-012-0650-3