Concentration of Eigenvalues for Skew-Shift Schrödinger Operators

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.

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

(A.1)

We then define the matrices

as acting on ℓ ²({n,n+1}) with $\rho= \sqrt{1-|\lambda|^{2}} \in(0,1)$. Define

$$ \mathcal{L} = \bigoplus_{n\ \mathrm{even}} \varTheta_n, \qquad \mathcal{M} = \bigoplus_{n\ \mathrm{odd}} \varTheta_n $$

(A.2)

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

(A.3)

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

$$ \sigma\bigl(\mathcal{C}^{N}_{x,y}\bigr) = \bigl\{ \mathrm{e}^{2\pi{\mathrm {i}}\theta_{1}^{N}} \dots\mathrm{e}^{2\pi{\mathrm{i}} \theta_{N}^{N}}\bigr\} $$

(A.4)

and

(A.5)

Furthermore, Proposition 5.1 in [12] tells us that

$$ \sigma\bigl(\mathcal{C}^{N}_{\tilde{x},y}\bigr)= \mathrm{e}^{2\pi{\mathrm{i}}(x-\tilde{x})} \sigma\bigl(\mathcal{C}^{N}_{x,y} \bigr). $$

(A.6)

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

$$ \alpha_{n} - \alpha_{n-1} \geq\frac{\varepsilon}{N} $$

(A.7)

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

$$ a_n = \biggl\lfloor\frac{1}{\omega_n} \biggr\rfloor,\qquad \omega_{n+1} = \frac{1}{\omega_n} - a_n, $$

(A.8)

where ω ₁=ω. Then ω has the continued fraction representation

$$ \omega= \frac{1}{a_1 + \displaystyle\frac{1}{a_2 + \ddots}}. $$

(A.9)

Define p ₋₂=q ₋₁=0 and q ₋₂=p ₋₁=1 and

$$ p_{n} = p_{n-2} + a_n p_{n-1},\qquad q_{n} = q_{n-2} + a_n q_{n-1}. $$

(A.10)

It is also well known that $\frac{p_{n}}{q_{n}} \to\omega$ as n→∞. Finally, we note that

$$ \frac{1}{2q_n} < \|q_{n-1} \omega\| < \frac{1}{q_n} $$

(A.11)

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

$$\inf_{1\leq n\leq N} \|n \omega\| \geq\frac{\varepsilon}{N}. $$

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 $\varepsilon\in(0,\frac{1}{2})$. □