Absence of Eigenvalues of Analytic Quasi-Periodic Schrödinger Operators on ${\mathbb {R}}^d$

Abstract

In this paper we study the Schrödinger operator $H=-\Delta + \lambda V(\varvec{x})$ with a real analytic quasi-periodic potential $V(\varvec{x})$ defined on ${\mathbb {R}}^d$ for arbitrary $d\ge 1$. We prove that H has no eigenvalues in the energy interval $\left[ -|\log \lambda |^{C}, |\log \lambda |^{C}\right]$ for every $C>0$ if $\lambda >0$ is sufficiently small. The proof is based on the Aubry duality and the multi-scale analysis in the momentum space.

Appendices

Appendix A

Lemma A.1

Let $H=-\Delta +V(\varvec{x})$ be a Schrödinger operator with $M=\sup _{\varvec{x}\in {\mathbb {R}}^d}|V(\varvec{x})|<\infty$. Then for any $E\ge 0$, we have

$$\begin{aligned} \sigma (H)\cap [E-M,E+M]\ne \emptyset , \end{aligned}$$

where $\sigma (H)$ denotes the spectrum of H.

Proof

Let ${\mathbb {E}}(\cdot )$ be the (projection-valued) spectral measure of $H=-\Delta +V(\varvec{x})$. Then the Spectral Theorem reads as

$$\begin{aligned} H=\int _{\sigma (H)}\lambda \mathrm{d}{\mathbb {E}}(\lambda ). \end{aligned}$$

Note that $\sigma (-\Delta )=\sigma _{ess}(-\Delta )=[0,\infty )$. From the Weyl criterion, for each $E\ge 0$, there is a sequence $\{F_n(\varvec{x})\}_{n\in {\mathbb {N}}}\subset H^2({\mathbb {R}}^d)$ such that

$$\begin{aligned}&(-\Delta -E)F_n\rightarrow 0\ \mathrm{as}\ n\rightarrow \infty ,\\&\Vert F_n\Vert _{L^2}=1\ \mathrm{for}\ \forall \ n. \end{aligned}$$

For each $n\in {\mathbb {N}}$, we know $\mu _n(\cdot )=\langle {\mathbb {E}}(\cdot )F_n,F_n\rangle$ is a positive Borel measure and $\mu _n(\sigma (H))=\Vert F_n\Vert _{L^2}^2=1$, where $\langle F,G\rangle =\int _{{\mathbb {R}}^d}F(\varvec{x})\overline{G}(\varvec{x})\mathrm{d} \varvec{x}$. Obviously, $\Vert VF_n\Vert _{L^2}\le M$. Hence

$$\begin{aligned} \inf _{\lambda \in \sigma (H)}|\lambda -E|^2\Vert F_n\Vert ^2_{L^2}&\le \int _{\sigma (H)}|\lambda -E|^2\mathrm{d}\mu _n(\lambda )\nonumber \\&=\Vert (-\Delta +V(\varvec{x})-E)F_n\Vert _{L^2}^2\nonumber \\&\le (\Vert (-\Delta -E)F_n\Vert _{L^2}+M)^2. \end{aligned}$$

(A.1)

Letting $n\rightarrow \infty$ in (A.1), we get

$$\begin{aligned} \mathrm{dist}(E, \sigma (H))\le M, \end{aligned}$$

(A.2)

which shows $[E-M,E+M]\cap \sigma (H)\ne \emptyset$. Otherwise, there must be $\mathrm{dist}(E, \sigma (H))\ge {2}M$, which contradicts (A.2). $\square$

Appendix B

We write $G_{(\cdot )}=G_{(\cdot )}(E;\varvec{\Theta })$ for simplicity.

We have the following lemmas.

Lemma B.1

(Lemma A.1, [Shi19a]). Fix ${\bar{\rho }}>0$. Let $\Lambda \subset {\mathbb {Z}}^{b}$ satisfy $\Lambda \in {\mathcal {E}}_N$ and let A, B be two linear operators on ${\mathbb {C}}^\Lambda$. We assume further

$$\begin{aligned} \Vert A^{-1}\Vert&\le e^{\sqrt{N}},\\ |A^{-1}({\varvec{n}},{\varvec{n}}')|&\le e^{-{\bar{\rho }}|{\varvec{n}}-{\varvec{n}}'|}\ \mathrm {for}\ |{\varvec{n}}-{\varvec{n}}'|\ge N/10. \end{aligned}$$

Suppose that for all ${\varvec{n}},{\varvec{n}}'\in \Lambda$,

$$\begin{aligned} |(B-A)({\varvec{n}},{\varvec{n}}')|\le e^{-3{\bar{\rho }} N-{\bar{\rho }}|{\varvec{n}}-{\varvec{n}}'|}. \end{aligned}$$

Then

$$\begin{aligned} \Vert B^{-1}\Vert&\le 2\Vert A^{-1}\Vert ,\\ |B^{-1}({\varvec{n}},{\varvec{n}}')|&\le |A^{-1}({\varvec{n}},{\varvec{n}}')|+e^{-{\bar{\rho }}|{\varvec{n}}-{\varvec{n}}'|}. \end{aligned}$$

Lemma B.2

(Lemma 3.2, [JLS20]). Let $\epsilon >0, {\bar{\rho }}\in (\epsilon ,\rho ]$, $M_1\le N$ and let $\Lambda \subset {\mathbb {Z}}^b$ with $\mathrm{diam}(\Lambda )\le 2N+1$. Suppose that for any ${\varvec{k}}\in \Lambda$, there exists some $W=W({\varvec{k}})\in {\mathcal {E}}_M$ with $M_0\le M\le M_1$ such that ${\varvec{k}}\in W\subset \Lambda$, $\mathrm{dist} ({\varvec{k}},\Lambda \backslash W)\ge {M}/{2}$ and

$$\begin{aligned} \Vert G_{W}\Vert&\le 2 e^{\sqrt{M}},\\ |G_{W}({\varvec{n}},{\varvec{n}}')|&\le 2e^{-{\bar{\rho }}|{\varvec{n}}-{\varvec{n}}'|}\ {\mathrm {for} \ |{\varvec{n}}-{\varvec{n}}'|\ge {M}/{10}}. \end{aligned}$$

We assume further that $M_0\ge M_0(\epsilon , \rho , b)>0$. Then

$$\begin{aligned} \Vert G_{\Lambda }\Vert \le 4 (2M_1+1)^{b} e^{\sqrt{M_1}}. \end{aligned}$$

Lemma B.3

(Theorem 3.3, [JLS20]). Let $\Lambda _1\subset \Lambda \subset {\mathbb {Z}}^{b}$ satisfy $\mathrm{diam}(\Lambda )\le 2N+1$, $\mathrm{diam}(\Lambda _1)\le N^{\frac{1}{3b}}$. Let $M_0\ge (\log N)^{2}$ and ${\bar{\rho }}\in \left[ \frac{\rho }{2},\rho \right]$. Suppose that for any ${\varvec{k}}\in \Lambda \backslash \Lambda _1$, there exists some $W=W({\varvec{k}})\in {\mathcal {E}}_M$ with $M_0\le M\le N^{1/3}$ such that ${\varvec{k}}\in W\subset \Lambda \backslash \Lambda _1$, $\mathrm{dist} ({\varvec{k}},\Lambda \backslash \Lambda _1\backslash W)\ge {M}/{2}$ and

$$\begin{aligned} \Vert G_{W}\Vert&\le e^{\sqrt{M}},\\ |G_{W}({\varvec{n}},{\varvec{n}}')|&\le e^{- \bar{\rho }|{\varvec{n}}-{\varvec{n}}'|}\ {\mathrm {for} \ |{\varvec{n}}-{\varvec{n}}'|\ge {M}/{10}}. \end{aligned}$$

Suppose further that

$$\begin{aligned} \Vert G_{\Lambda }\Vert \le e^{\sqrt{N}}. \end{aligned}$$

Then

$$\begin{aligned} |G_{\Lambda }({\varvec{n}},{\varvec{n}}')|\le e^{-(\bar{\rho }-\frac{C}{\sqrt{M_0}})|{\varvec{n}}-{\varvec{n}}'|}\ \mathrm {for}\ |{\varvec{n}}-{\varvec{n}}'|\ge {N}/{10}, \end{aligned}$$

where $C=C(\rho ,b)>0$ and $N\ge N_0(\rho ,b)>0$.