Let \( A:\mathcal {H}\rightarrow \mathcal {H} \) be a bounded operator. The spectrum of A is denoted \( \sigma (A) \). Note \( \sigma (A) \) is a nonempty compact subset of \( \mathbb {C} \). Also note, if A is self-adjoint, then \( \sigma (A)\subseteq \mathbb {R} \).
Potentials and Schrödinger Operators
A (d-dimensional lattice) potential is a bounded real-valued function V on \( \mathbb {Z}^d \). Moreover, V also denotes the bounded self-adjoint multiplication operator on \( \ell ^2(\mathbb {Z}^d)\) defined by \( [V\psi ](n)=V(n)\psi (n) \) for every \( \psi \in \ell ^2(\mathbb {Z}^d),n\in \mathbb {Z}^d \). The (d-dimensional lattice) Schrödinger operator with respect to a potential \( V:\mathbb {Z}^d\rightarrow \mathbb {R} \) is the bounded self-adjoint operator H on \(\ell ^2(\mathbb {Z}^d) \) defined by \( H=\Delta +V \), where \( \Delta =\Delta ^{(d)} \) is the (d-dimensional lattice) Laplacian defined by \([\Delta \psi ](n)=\sum _{m\in \left\{ e_1,\ldots ,e_d\right\} }\psi (n+m)+\psi (n-m) \) for every \( \psi \in \ell ^2(\mathbb {Z}^d),n\in \mathbb {Z}^d \); \( \left\{ e_1,\ldots ,e_d\right\} \) is the standard basis.
Separable Potentials and the Laplacian
Let \( V:\mathbb {Z}^d\rightarrow \mathbb {R} \) be a potential. Say V is separable if there exist (sub)potentials \( V_1,\ldots ,V_d:\mathbb {Z}\rightarrow \mathbb {R} \) such that \( V(n)=V_1(n_1)+\cdots +V_d(n_d) \) for every \( n\in \mathbb {Z}^d \). The proof of the following theorem can be found within [12].
Theorem 2.1
Let \( V:\mathbb {Z}^d\rightarrow \mathbb {R}:n\mapsto V_1(n_1)+\cdots +V_d(n_d) \) be a separable potential. Define \( H{:}{=}\Delta ^{(d)}+V \). For each k, define \( H_k{:}{=}\Delta ^{(1)}+V_k \). Then,
$$\begin{aligned} \sigma (H)=\sigma (H_1)+\cdots +\sigma (H_d). \end{aligned}$$
By the spectral mapping theorem,
$$\begin{aligned} \textstyle \sigma (\Delta ^{(1)})\!=\!\sigma ( U_{1} \!+\!U_1)=\sigma (\Phi _{U_1}(z^*+z)) \! =\! \{z^*+z:z\in \sigma (U_1)\}=\{z^*+z:|z|=1\}\!=\![-2,2]. \end{aligned}$$
By Theorem 2.1,
$$\begin{aligned} \textstyle \sigma (\Delta ^{(d)})=\sigma (\Delta ^{(1)})+ \cdots +\sigma (\Delta ^{(1)})\text {(d terms)}=[-2d,2d]. \end{aligned}$$
Here, \( U_m \) is the unitary operator from \( \ell ^2(\mathbb {Z}^d) \) to \( \ell ^2(\mathbb {Z}^d) \) defined by \( [U_m\psi ](n)=\psi (n-m) \) for every \( \psi \in \ell ^2(\mathbb {Z}^d),n\in \mathbb {Z}^d \). Also here, \( \Phi _A \) is the Borel functional calculus with respect to a bounded operator A.
Quasiperiodic Potentials and the Almost Mathieu Operator
Let \( V:\mathbb {Z}\rightarrow \mathbb {R} \) be a potential. Let b be a positive integer. Say V is (b-frequency) quasiperiodic if there exists \( v\in C(\mathbb {T}^b,\mathbb {R}) \) and there exist \( \alpha ,\omega \in \mathbb {T}^b \) such that v is nonconstant and \( \left\{ 1,\alpha _1,\ldots ,\alpha _b\right\} \) is independent over the rationals and \( V(n)=v(n\alpha +\omega ) \) for every \( n\in \mathbb {Z} \). The proof of the following theorem can be found within [36].
Theorem 2.2
Let \( V_\omega :\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential with parameter \( \omega \). Let \( H_\omega \) be the Schrödinger operator. Then, \( \sigma (H_\omega )=\Sigma _{\omega }{=}{:}\Sigma \) is independent of \( \omega \). Furthermore, \( \Sigma \) is a nonempty compact subset of \( \mathbb {R} \) and \( \Sigma \) has no isolated points.
The potential of the almost Mathieu operator \( V_{\lambda ,\alpha ,\omega }:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto 2\lambda \cos (2\pi (n\alpha +\omega )) \) is 1-frequency quasiperiodic when \( \alpha \) is irrational. By the resolution of the Ten Martini Problem within [4], the spectrum \( \Sigma _{\lambda ,\alpha ,\omega } \) of the almost Mathieu operator is a Cantor set when \( \alpha \) is irrational.
Integrated Density of States
Let \( V_{\omega }:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential with parameter \( \omega \). Let \( H_{\omega } \) be the Schrödinger operator. The integrated density of states (IDS) is
$$\begin{aligned} \mathfrak {N}:\mathbb {R}\rightarrow [0,1]:x&\mapsto \textstyle \int \langle \delta _0, \mathbb {1}_{(-\infty ,x]}(H_{\theta })\delta _0\rangle d\mu (\theta ). \end{aligned}$$
Here, \( \mu \) is the normalized Haar measure on \( \mathbb {T}^b \). Note \( \mathfrak {N}\upharpoonright (-\infty ,\inf \Sigma ]\equiv 0 \) and \( \mathfrak {N}\upharpoonright [\sup \Sigma ,+\infty )\equiv 1 \). The proof of Theorem 2.3 can be found within [36] and of Theorem 2.4 can be found within [11].
Theorem 2.3
(Gap Labeling) Let \( V:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential. Let H be the Schrödinger operator. Let \( \mathfrak {N} \) be the IDS. Define \( \Sigma {:}{=}\sigma (H) \). Let \( \mathrm {Gap}_{\mathsf {b}}(\Sigma ) \) be the collection of all bounded gaps of \( \Sigma \).
-
(i)
\( \mathfrak {N} \) is monotone and continuous.
-
(ii)
\( \left\{ x\in \mathbb {R}:\left( \forall \varepsilon>0\right) [\mathfrak {N}(x+\varepsilon )-\mathfrak {N}(x-\varepsilon )>0]\right\} =\Sigma \).
-
(iii)
\( \left\{ \mathfrak {N}(x):x\in U\in \mathrm {Gap}_{\mathsf {b}}(\Sigma )\right\} \subseteq \{\mathbf {n}\alpha -\left\lfloor \mathbf {n}\alpha \right\rfloor :\mathbf {n}\in \mathbb {Z}^b\}{\setminus }\left\{ 0\right\} \).
-
(iv)
\( \left\{ \mathfrak {N}(x):x\in U\in \mathrm {Gap}_{\mathsf {b}}(\Sigma )\right\} \supseteq \{\mathbf {n}\alpha -\left\lfloor \mathbf {n}\alpha \right\rfloor :\mathbf {n}\in \mathbb {Z}^b\}{\setminus }\left\{ 0\right\} \) \( \implies \) \( \Sigma \) is a Cantor set.
Theorem 2.4
(Hölder Continuity) Let \( V_{\lambda ,v,\alpha ,\omega }:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto \lambda v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential with parameters \( \lambda ,v,\alpha ,\omega \). Let \( H_{\lambda ,v,\alpha ,\omega } \) be the Schrödinger operator. Fix \( v,\alpha \). Let \( \mathfrak {N}_{\lambda } \) be the IDS. Assume the following:
-
(i)
\(\alpha \) satisfies a Diophantine condition:
for some \( c>0,t>b \).
-
(ii)
v satisfies a regularity condition: v is r-differentiable on \( \mathbb {T}^b \) for some \( r\ge 550t \).
There exist \( \lambda _0=\lambda _0(v,b,c,t,r)>0, C_{\mathsf {H}}=C_{\mathsf {H}}(b,c,t)>0 \) such that
$$\begin{aligned} \smash {|\mathfrak {N}_\lambda (x)-\mathfrak {N}_\lambda (y)|\le C_{\mathsf {H}}|x-y|^{{1}/{2}}} \end{aligned}$$
for every \( \lambda _0\ge |\lambda |>0,x,y \).
Spectral Gaps
Let \( V:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential. Let H be the Schrödinger operator. Let \( \mathfrak {N} \) be the IDS. Define \( \Sigma {:}{=}\sigma (H) \). Let \( \mathrm {Gap}_{\mathsf {b}}(\Sigma ) \) be the collection of all bounded gaps of \( \Sigma \). Fix \( \mathbf {n}\in \mathbb {Z}^b{\setminus }\left\{ \mathbf {0}\right\} \). The \( \mathbf {n} \)-th spectral gap of \( \Sigma \) is either the element \( G^{(\mathbf {n})} \) in \( \mathrm {Gap}_{\mathsf {b}}(\Sigma ) \) such that \( \mathfrak {N}\upharpoonright \smash {\overline{G^{(\mathbf {n})}}}\equiv \mathbf {n}\alpha -\lfloor \mathbf {n} \alpha \rfloor \) or the empty set when such an element \( G^{(\mathbf {n})} \) does not exist. Moreover, \( (-\infty ,\inf \Sigma )\cup (\sup \Sigma ,+\infty ) \) is called the \( \mathbf {0} \)-th spectral gap of \( \Sigma \). The proof of the following theorem can be found within [29].
Theorem 2.5
(Gap Estimate) Let \( V_{v,\alpha ,\omega }:\mathbb {Z}\rightarrow \mathbb {R}:n\mapsto v\big |_{\mathbb {T}^b}(n\alpha +\omega ) \) be a quasiperiodic potential with parameters \( v,\alpha ,\omega \). Let \( H_{v,\alpha ,\omega } \) be the Schrödinger operator. Fix \( \alpha \). Fix \( r_0>r>0 \). Define \( \Sigma _{v}{:}{=}\sigma (H_{v,\alpha ,\omega }) \). For each \( \mathbf {n}\ne \mathbf {0} \), let \( G_{v} ^{(\mathbf {n})} =(E_{v}^{(\mathbf {n}){ -}}\), \( E_{v}^{(\mathbf {n}){+}}\)) be the \(\mathbf {n} \)-th spectral gap of \( \Sigma _{v} \). Assume the following:
-
(i)
\( \alpha \) satisfies a Diophantine condition:
for some \( c>0,t>b \).
-
(ii)
v satisfies a regularity condition: v is analytic on \( (\mathbb {T}+i(-r_0,r_0))^b \).
There exists \( \varepsilon =\varepsilon (b,c,t,r_0,r)>0 \) such that
$$\begin{aligned} {| E_{v} ^{(\mathbf {n}){+}} - E_{v}^{(\mathbf {n}) {\scriptscriptstyle -}} |\le (|v|_{r_0})^{2/3}e^{-2\pi r\Vert \mathbf {n}\Vert }} \end{aligned}$$
for every \( \varepsilon \ge |v|_{r_0}>0,\mathbf {n}\ne \mathbf {0} \).
Thickness and Gap Lemmas
Let K be a nonempty compact subset of \( \mathbb {R} \). Henceforth \( K^{\scriptscriptstyle -}{:}{=}\inf K \) and \( K^{\scriptscriptstyle +}{:}{=}\sup K \). Let \( \mathrm {Gap}_{\mathsf {b}}(K) \) be the collection of all bounded gaps of K. Let U be a bounded gap of K. The left-plank of U, denoted \( \pi _{\scriptscriptstyle -}(K,U) \), is the length-maximal interval [a, b] contained in \( [K^{\scriptscriptstyle -},K^{\scriptscriptstyle +}] \) such that \( b\in \partial U \) and for each \( V\in \mathrm {Gap}_{\mathsf {b}}(K) \), if \( V\cap [a,b]\ne \varnothing \), then \( \mathrm {length}(V)< \mathrm {length}(U) \). The right-plank of U, denoted \( \pi _{\scriptscriptstyle +}(K,U) \), is the length-maximal interval [a, b] contained in \( [K^{\scriptscriptstyle -},K^{\scriptscriptstyle +}] \) such that \( a\in \partial U \) and for each \( V\in \mathrm {Gap}_{\mathsf {b}}(K) \), if \( V\cap [a,b]\ne \varnothing \), then \( \mathrm {length}(V)< \mathrm {length}(U) \). The (local) thickness of U is
$$\begin{aligned} \inf \nolimits _{{\bullet }\in \left\{ -,+\right\} } \frac{\mathrm {length}(\pi _{\scriptscriptstyle \bullet }(K,U))}{\mathrm {length}(U)}{{}{=}{:}\tau (K,U).} \end{aligned}$$
The thickness of K is
$$\begin{aligned} \smash {\inf \nolimits _{U\in \mathrm {Gap}_{\mathsf {b}}(K)} \tau (K,U){{}{=}{:}\tau (K).}} \end{aligned}$$
Note \(\tau (K)=0 \) when K has an isolated point, \( \tau (K)=1 \) when K is the middle-thirds Cantor set, and \( \tau (K)=+\infty \) when K is an interval. The proof of Theorem 2.6 can be found within [12] and of Theorem 2.7 can be found within [1, 2]. The former theorem is the classical Gap Lemma, but we require the latter theorem which is a generalized version.
Theorem 2.6
(Gap Lemma) Let \( K_1 \) and \( K_2 \) be nonempty compact subsets of \( \mathbb {R} \). For each nonempty compact subset K of \( \mathbb {R} \), let \( \mathrm {Gap}_{\mathsf {b}}(K) \) be the collection of all bounded gaps of K. For each nonempty compact subset K of \( \mathbb {R} \), define \(\Gamma (K){:}{=}\sup \{\mathrm {length}(U):U\in \mathrm {Gap}_{\mathsf {b}}(K)\} \). Assume \( [K_{1}^{\scriptscriptstyle -},K_{1}^{\scriptscriptstyle +}]\cap [K_{2}^{\scriptscriptstyle -},K_{2}^{\scriptscriptstyle +}]\ne \varnothing \), \( \Gamma (K_2)\le \mathrm {diam}(K_1) \), \( \Gamma (K_1)\le \mathrm {diam}(K_2) \), and \( 1\le \tau (K_1)\cdot \tau (K_2) \). Then, \( K_1 \cap K_2\ne \varnothing \). Furthermore, \( K_1+K_2=[K_{1}^{\scriptscriptstyle -}+K_{2}^{\scriptscriptstyle -},K_{1}^{\scriptscriptstyle +}+K_{2}^{\scriptscriptstyle +}] \).
Theorem 2.7
(Gap Lemma) Let \( K_1,\ldots ,K_d \) (\( d\ge 2 \)) be nonempty compact subsets of \( \mathbb {R} \). For each nonempty compact subset K of \( \mathbb {R} \), let \( \mathrm {Gap}_{\mathsf {b}}(K) \) be the collection of all bounded gaps of K. For each nonempty compact subset K of \( \mathbb {R} \), define \(\Gamma (K){:}{=}\sup \{\mathrm {length}(U):U\in \mathrm {Gap}_{\mathsf {b}}(K)\} \). Assume
$$\begin{aligned} {\left\{ \begin{array}{ll} \left( \forall i:2\le i\le d\right) \left( \forall j:1 \le j\le i-1\right) [\Gamma (K_j)\le \mathrm {diam}(K_i)],\\ \left( \forall i:2\le i\le d\right) [\Gamma (K_i)\le \mathrm {diam}(K_1)+\cdots +\mathrm {diam}(K_{i-1})]. \end{array}\right. } \end{aligned}$$
If \( 1\le \frac{\tau (K_1)}{\tau (K_1)+1}+\cdots +\frac{\tau (K_d)}{\tau (K_d)+1} \), then \( \tau (K_1+\cdots +K_d)=+\infty \) and
$$\begin{aligned} K_1+\cdots +K_d=[K_{1}^{\scriptscriptstyle -}+\cdots +K_{d}^{\scriptscriptstyle -},K_{1}^{\scriptscriptstyle +}+\cdots +K_{d}^{\scriptscriptstyle +}]. \end{aligned}$$
If \(\frac{\tau (K_1)}{\tau (K_1)+1}+\cdots +\frac{\tau (K_d)}{\tau (K_d)+1}<1\), then \( \tau (K_1+\cdots +K_d)\ge \frac{\frac{\tau (K_1)}{\tau (K_1)+1}+\cdots + \frac{\tau (K_d)}{\tau (K_d)+1}}{1-(\frac{\tau (K_1)}{\tau (K_1)+1}+\cdots +\frac{\tau (K_d)}{\tau (K_d)+1})} \).