function decay for energies $E_{*}$ near the codomain of $\mathbf{E}_{s}$.
Under these assumptions, we apply Section 4 to $\mathbf{E}_{s}$ to classify
energy regions $J_{s}^{(j_{s+1})}\subset\mathbf{J}_{s}$ as being simple- or
double-resonant (indicated by $j_{s+1}\in\\{1,2\\}$) and prevent recurrence to
those energy regions for long times. The resonant sites for $\mathbf{E}_{s}$
will be so well-separated that we can find an even integer $L_{s+1}$ for each
energy region well-separated from resonances of all the ancestors of
$\mathbf{E}_{s}$. Thus, the intervals $\Lambda_{s+1}$ of length $L_{s+1}$ will
have long shoulder intervals with Green’s function decay by our inductive
assumption.

Having found the integers $L_{s+1}$, we can then apply Sections 2 and 3 to
construct Rellich children $\mathbf{E}_{s+1}$ which are likewise cosine-like.
Finally, for energies which are not resonant with any child $\mathbf{E}_{s+1}$
for long intervals $\Lambda$, we can use these constructed intervals to build
coverings of $\Lambda$ like those described in Appendix A, proving the
inductive nonresonance hypothesis. Here we crucially use the uniform local
separation between double-resonant Rellich children established in Proposition
3.6; since the separation is much larger than the scale-$s+1$ resonance
parameter, energies near the codomain of one of a pair of double-resonant
Rellich functions do not allow resonance with the other.

The perturbative upper bound $\varepsilon_{0}(v,\alpha)$ on the interaction
$\varepsilon$ will be polynomially small, depending on $v$ and $\alpha$, in
the initial length scale $L$; the size of $L$ effectively will be dictated by
avoiding substantial cumulative loss in the Green’s function decay parameters
$\gamma_{k}$, cf. Lemma 5.2 below.

$\mathbf{E}_{s}$$\mathcal{E}^{(1)}(\mathbf{E}_{s})$$\mathcal{E}^{(2)}(\mathbf{E}_{s})$$\mathbf{E}_{s+1,i}^{(1)}$$\mathbf{E}_{s+1,n_{s},\vee}^{(2)}$$\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}$$n_{s}\in\mathcal{N}_{s}$$\cdots$$\cdots$$\cdots$$\cdots$$L_{s+1}\sim
M_{s+1}^{(1)}$$L_{s+1}\sim M_{s+1}^{(2)}$ Figure 7: The inductive expansion of
our Rellich tree at a particular node $\mathbf{E}_{s}$.

###  5.1 Preparing the induction

####  5.1.1 The initial scale

Suppose $\mathbf{E}_{0}:=v$, $v:\mathbb{T}\to[-1,1]$, is cosine-like in the
sense of the main theorem, with Morse constants

$d_{0}\leq|\partial_{\theta}v(\theta)|+|\partial_{\theta}^{2}v(\theta)|\leq
D_{0},\quad\theta\in\mathbb{T},$

and that $\alpha\in DC_{C,\tau}$ is Diophantine. Letting $L$ be a large even
integer, we consider the initial length scales

$L_{1}^{(1)}:=L,\quad L_{1}^{(2)}:=L^{2},$

and initialize the following parameters:

$\nu_{0}=\frac{C_{\alpha}d_{0}}{14(8L)^{\tau}},\quad\rho_{-1}=\frac{d_{0}}{24D_{0}^{2}}\frac{1}{(8L)^{2\tau}},\quad\bar{\rho}_{0}=\frac{d_{0}}{15000D_{0}^{2}}\rho_{-1}^{2},$
$\rho_{0}=\bar{\rho}_{0}^{3},\quad\delta_{0}:=\varepsilon<\varepsilon_{0}:=\rho_{0}^{4},\quad\gamma_{-1}=\gamma_{0}=\frac{1}{4}|\log\varepsilon|,\quad\ell_{-1}=\ell_{0}=1.$

With these parameters, we recall a classical Neumann series argument ensuring
off-diagonal Green’s function decay on nonresonant intervals:

######  Lemma 5.1.

Let $\Lambda\subset\mathbb{Z}$ and suppose
$|v(\theta_{*}+m\alpha)-E_{*}|\geq\rho_{0}$ for $m\in\Lambda$. Then for
$m,n\in\Lambda$, $|\theta-\theta_{*}|<\rho_{0}/8D_{0}$, and
$|E-E_{*}|<\rho_{0}/2$,

$\displaystyle|R^{\Lambda}_{\theta,E}(m,n)|\leq\frac{8}{\rho_{0}}\left(\frac{8\varepsilon}{\rho_{0}}\right)^{|m-n|}.$

In particular, for $\gamma_{0}=\frac{1}{4}|\log\varepsilon|$ and $|m-n|\geq
1$,

$\displaystyle\log|R^{\Lambda}_{\theta,E}(m,n)|\leq-\gamma_{0}|m-n|,$ (5.1)

i.e. the Green’s function decay property for $(\ell_{0},\gamma_{0})$ holds.

###### Proof.

For $m\in\Lambda$, $|\theta-\theta_{*}|<\rho_{0}/8D_{0}$, and
$|E-E_{*}|<\rho_{0}/2$, one has

$\displaystyle|v(\theta+m\alpha)-E|$
$\displaystyle\geq|v(\theta_{*}+m\alpha)-E_{*}|-|E-E_{*}|-D_{0}|\theta-\theta_{*}|>\rho_{0}/4.$

Thus $|v(\theta+m\alpha)-E|\geq\rho_{0}/4>4\varepsilon$ for all $m\in\Lambda$,
and $\|(V(\theta)-E)^{-1}\varepsilon\Delta\|_{\mathbb{C}^{\Lambda}}<1/2$;
hence, by the Neumann series, one has for $m,n\in\Lambda$

$\displaystyle|R^{\Lambda}_{\theta,E}(m,n)|$
$\displaystyle=|\langle\delta_{m},(H^{\Lambda}(\theta)-E)^{-1}\delta_{n}\rangle|$
$\displaystyle\leq\frac{4}{\rho_{0}}\sum_{k\geq|m-n|}\left(\frac{8\varepsilon}{\rho_{0}}\right)^{k}$
$\displaystyle\leq\frac{8}{\rho_{0}}\left(\frac{8\varepsilon}{\rho_{0}}\right)^{|m-n|}.$

Provided $|m-n|\geq 1$, we have

$\displaystyle|R^{\Lambda}_{\theta,E}(m,n)|\leq\frac{8}{\rho_{0}}\left(\frac{8\varepsilon}{\rho_{0}}\right)^{|m-n|}$
$\displaystyle\leq\rho_{0}^{2|m-n|-1}\leq\rho_{0}^{|m-n|}.\qed$

This lemma, combined with our assumptions on $v$, will be the foundation of
our induction.

####  5.1.2 Inductive definitions

Fix a scale $s\geq 0$. If $s\geq 1$, we suppose that, for $0\leq k\leq s-1$,
we have collections

$\displaystyle\mathcal{E}_{k}^{(j)}$
$\displaystyle=\bigcup_{\mathbf{E}\in\mathcal{E}_{k-1}}\mathcal{E}^{(j)}(\mathbf{E}),\quad
j\in\\{1,2\\},$ $\displaystyle\mathcal{E}_{k}$
$\displaystyle=\mathcal{E}_{k}^{(1)}\cup\mathcal{E}_{k}^{(2)}$

of Rellich functions $\mathbf{E}_{k}:\mathbf{I}_{k}\to\mathbf{J}_{k}$ of
certain Dirichlet restrictions $H^{\Lambda_{k}}$,
$\Lambda_{k}=[-L_{k}/2,L_{k}/2]$, where $L_{k}=L_{k}(\mathbf{E}_{k})$ are even
integers on scale $k$

$\displaystyle\frac{1}{2}M_{k}^{(j)}\leq L_{k}\leq
M_{k}^{(j)},\quad\mathbf{E}_{k}\in\mathcal{E}_{k}^{(j)},$

where we define the length scales

$M_{k}^{(j)}:=(L^{4^{k-1}})^{j},\quad k\geq 1,\;j\in\\{1,2\\}.$

We take as convention that $M_{0}^{(1)}=M_{0}^{(2)}=1$. For each
$\mathbf{E}_{k}\in\mathcal{E}_{k}$, $k\geq 1$, we define the corresponding
parameters:

  1. $d_{k}=\frac{\nu_{k-1}}{12}$, $D_{k}=2D_{0}(1+\frac{D_{0}}{\sigma_{k}})$, (Morse)

  2. $\nu_{k}=\bar{\nu}_{k-1}/2$, $\bar{\nu}_{k}:=\frac{3d_{k}}{8D_{0}}\bar{\rho}_{k}$ (Derivative control)

  3. $\rho_{k}=\varepsilon^{L_{k}^{2/3}}$, $\bar{\rho}_{k}=\frac{d_{k}}{15000D_{0}^{2}}\rho_{k-1}^{2}$, $\sigma_{k}=\varepsilon^{72M_{k}^{(1)}}$ (Resonance)

  4. $\delta_{k}=\varepsilon^{L_{k}/8}$ (Eigenfunction interaction)

  5. $\ell_{k}=L_{k}^{5/6}$, $\gamma_{k}=\gamma_{0}(1-64\sum_{i=1}^{k}\frac{|\log(\rho_{i}/8)|}{\ell_{i}})$ (Green’s function decay)

We likewise define $\widetilde{\mathbf{J}}(\mathbf{E}_{k})$ and
$\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$ as in equation (4.1) with
$\rho=\rho_{k}$ and $\widecheck{\rho}=\rho_{k-1}$, cf. Figure 6. As matters of
convention, we define
$\widetilde{\mathbf{J}}(\mathbf{E}_{-1})=\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{-1})=\mathbb{R}$
and $\mathcal{E}_{0}=\mathcal{E}(\mathbf{E}_{-1}):=\\{v\\}$.

We have the following relations among these parameters:

######  Lemma 5.2.

Let $\mathbf{E}_{k}\in\mathcal{E}_{k}$ with $L_{k}$ and constants defined as
above. For $L$ sufficiently large (depending only on $C_{\alpha}$, $\tau$,
$d_{0}$, and $D_{0}$), we have the following for all $k\geq 0$:

  1. 1.

$\gamma_{k}\geq\frac{1}{2}\gamma_{0}\geq\log 7$.

  2. 2.

$10M_{k}^{(2)}<M_{k+1}^{(1)}$, $24L_{k+1}^{2/3}<8L_{k+1}^{5/6}<L_{k+1}$,
$\ell_{k+1}\gg 16|\log\varepsilon||\log\rho_{k+1}|$.

  3. 3.

$\frac{C_{\alpha}}{2(8M_{k+1}^{(1)})}\geq\frac{7\nu_{k}}{d_{k}}$,
$\frac{3d_{k}}{8D_{0}}\bar{\rho}_{k}<\nu_{k}$, and
$\frac{9\rho_{k}}{8\nu_{k}}\ll\bar{\rho}_{k}\ll\frac{\nu_{k}^{2}\rho_{k-1}}{100D_{0}^{3}}$.

  4. 4.

$\delta_{k}<\rho_{k}^{3}/2\ll\rho_{k-1}^{2}\nu_{k}^{5/2}$.

  5. 5.

If $j_{k}=2$, then $2\rho_{k}<\frac{\nu_{k-1}\sigma_{k}}{2D_{0}+\nu_{k-1}}$.

###### Proof.

We proceed with each item in turn:

  1. 1.

For $1\leq i\leq k$, we have by definition that

$\displaystyle\frac{|\log(\rho_{i}/8)|}{\ell_{i}}$ $\displaystyle\leq
L_{i}^{-1/6}|\log\varepsilon|\lesssim\tau L^{-\frac{1}{6}4^{i-1}}\log
L\lesssim\tau L^{-\frac{1}{7}(4^{i-1})}$

Since this sequence is subgeometric, we may find $L$ sufficiently large such
that

$\displaystyle
64\sum_{i=1}^{k}\frac{|\log(\rho_{i}/8)|}{\ell_{i}}\leq\frac{1}{2}$

independently of $k$. By possibly making $L$ larger, one can insist that

$\frac{1}{2}\gamma_{0}=\frac{1}{8}\log|\varepsilon|\sim\tau\log|L|>\log 7.$

  2. 2.

This is immediate from the fact that, for $k\geq 1$,

$M_{k+1}^{(1)}\geq LM_{k}^{(2)},$

alongside the observation that $|\log\varepsilon|$ is comparable to $\log L$.

  3. 3.

One can check by definitions that, for $L$ sufficiently large (depending on
$\tau$, $d_{0}$, and $D_{0}$), one has for $k\geq 0$ that:

$\displaystyle\nu_{k}^{5/4}<d_{k+1}<\nu_{k}<\frac{d_{k}}{2}$
$\displaystyle\rho_{k-1}^{9/4}<\bar{\rho}_{k}<\rho_{k-1}^{2}$
$\displaystyle\rho_{k-1}^{9/4}<\nu_{k+1},\bar{\nu}_{k}<\rho_{k-1}^{2}$
$\displaystyle\rho_{k}\ll\rho_{k-1}^{6}.$

The inequalities follow for sufficiently large $L$.

  4. 4.

Given the above inequalities, one has

$\rho_{k-1}^{2}\nu_{k}^{5/2}\gg\rho_{k-1}^{2}\rho_{k-2}^{45/8}\gg\rho_{k-1}^{2}\rho_{k-2}^{6}\gg\rho_{k-1}^{3}\gg\rho_{k}^{3}/2,$

and $L_{k}>24L_{k}^{2/3}$.

  5. 5.

Since $j_{k}=2$, we have
$L_{k}^{2/3}\geq(M_{k}^{(2)}/2)^{2/3}\geq\frac{1}{2}(M_{k}^{(1)})^{4/3}\gg
144M_{k}^{(1)}$ provided $L$ is sufficiently large, and thus $\log\rho_{k}\ll
2\log\sigma_{k}$. The inequality follows because
$\log\nu_{k-1}\gg\log\rho_{k-1}\gg\log\sigma_{k}$. ∎

It will be convenient to fix language describing “nonresonant” regions at each
scale. Fix $(\theta_{*},E_{*})\in\mathbb{T}\times\mathbb{R}$ and a Rellich
function $\mathbf{E}_{k}\in\mathcal{E}_{k}$. We say that a set
$S\subset\mathbb{Z}$ is _$k$ -nonresonant_ (relative to $(\theta_{*},E_{*})$
and $\mathbf{E}_{k}$) if

$\theta_{*}+m\alpha\notin\mathbf{I}_{k}\,\text{ or
}\,|\mathbf{E}_{k}(\theta_{*}+m\alpha)-E_{*}|\geq\rho_{k}\quad\forall m\in S.$

Note that, with this language, Lemma 5.1 shows that $0$-nonresonant intervals
have $(\ell_{0},\gamma_{0})$ Green’s function decay.

To show Green’s function decay at future scales $k\geq 1$, we must avoid
resonant sites for all ancestors of $\mathbf{E}_{k}$; we codify this condition
as $k$-regularity. Namely, we say that a point $m\in\mathbb{Z}$ is _$k$ -left-
regular_ if, for each ancestor $\mathbf{E}_{i}$ ($0\leq i\leq k-1$) of
$\mathbf{E}_{k}$, $[m,m+\frac{3}{4}L_{i+1}]$ is $i$-nonresonant. Analogously,
we say that $m$ is _$k$ -right-regular_ if, for each ancestor $\mathbf{E}_{i}$
($0\leq i\leq k-1$) of $\mathbf{E}_{k}$, $[m-\frac{3}{4}L_{i+1},m]$ is
$i$-nonresonant, and we say that $m$ is _$k$ -regular_ if it is both $k$-left-
regular and $k$-right-regular.

Finally, we define the slight enlargement of the set of $k$-resonant points by

$\mathcal{S}_{k}(\theta_{*},E_{*}):=\left\\{m\in\mathbb{Z}:\theta_{*}+m\alpha\in\mathbf{I}_{k},\;|\mathbf{E}_{k}(\theta_{*}+m\alpha)-E_{*}|\leq\frac{25}{24}\rho_{k}\right\\}.$

###  5.2 The inductive proposition

Subject to these definitions, we suppose that the following inductive
proposition holds for $0\leq k\leq s-1$:

######  Proposition 5.3 (Induction, scale $k$).

Suppose the hypotheses below hold:

######  H 5.3.1 (Cosine-like Rellich functions, scale $k$).

Each $\mathbf{E}_{k}\in\mathcal{E}_{k}$ satisfies Assumption 3 with parameters
$\nu=\nu_{k},d=d_{k},$ and $D=D_{k}$.

######  H 5.3.2 (Resonant orbits, scale $k$).

If $k\geq 1$, let $\mathbf{E}_{k}\in\mathcal{E}^{(j_{k})}(\mathbf{E}_{k-1})$,
$\mathbf{E}_{k}:\mathbf{I}_{k}\to\mathbf{J}_{k}$, denote

$\mathcal{N}_{k}:=\mathcal{N}(8M_{k+1}^{(1)},\alpha,\mathbf{E}_{k}),$

and let $\theta_{*}\in\mathbb{T}$. We can characterize aspects of the resonant
orbits depending on the resonance type $j_{k}$:

  1. 1.

$j_{k}=1$:

We have

$\min\\{|n|:n\in\mathcal{N}_{k}\\}>8M_{k}^{(1)}>L_{k}.$

Furthermore, suppose either $E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{k})$
or the closest point in $\mathbf{J}_{k}$ to $E_{*}$ is a critical value of
$\mathbf{E}_{k}$. Then, for any $m\in\mathcal{S}_{k}(\theta_{*},E_{*})$,

$B_{\rho_{k-1}/24D_{0}}(\theta_{*}+m\alpha)\subset\mathbf{I}_{k}.$

  2. 2.

$j_{k}=2$ (and thus $\mathbf{E}_{k}=\mathbf{E}_{k,n_{k-1},\bullet}^{(2)}$):

We have

$\mathcal{N}_{k}=\emptyset.$

Furthermore, suppose
$E_{*}\in[\inf\widetilde{\mathbf{J}}(\mathbf{E}_{k,n_{k-1},\wedge}^{(2)}),\sup\widetilde{\mathbf{J}}(\mathbf{E}_{k,n_{k-1},\vee}^{(2)})].$
Then, for any $m\in\mathcal{S}_{k}(\theta_{*},E_{*})$, for some
$p\in\\{m,m-n_{k-1}\\}$,

$B_{\rho_{k-1}/24D_{0}}(\theta_{*}+p\alpha)\subset\mathbf{I}_{k,n_{k-1},\vee}\cup\mathbf{I}_{k,n_{k-1},\wedge}.$

If
$\theta_{*}+p\alpha\in\mathbf{I}_{k,n_{k-1},\wedge}^{(2)}\setminus\mathbf{I}_{k,n_{k-1},\vee}^{(2)}$,
then
$|\mathbf{E}_{k,n_{k-1},\wedge}^{(2)}(\theta_{*}+p\alpha)-E_{*}|<\frac{9}{8}\rho_{k-1}$;
similarly, if
$\theta_{*}+p\alpha\in\mathbf{I}_{k,n_{k-1},\vee}^{(2)}\setminus\mathbf{I}_{k,n_{k-1},\wedge}^{(2)}$,
then
$|\mathbf{E}_{k,n_{k-1},\vee}^{(2)}(\theta_{*}+p\alpha)-E_{*}|<\frac{9}{8}\rho_{k-1}$.

######  H 5.3.3 (Nonresonance, scale $k$).

Let $\mathbf{E}_{k-1}\in\mathcal{E}_{k-1}$, let $\theta_{*}\in\mathbb{T}$, and
suppose $E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{k-1})$ is such that

  * •

If $\mathbf{E}_{k-1}$ does not attain its maximum at a critical point, then
$E_{*}\leq\sup\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k-1})}\widetilde{\mathbf{J}}(\mathbf{E})$.

  * •

If $\mathbf{E}_{k-1}$ does not attain its minimum at a critical point, then
$E_{*}\geq\inf\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k-1})}\widetilde{\mathbf{J}}(\mathbf{E})$.

Let $\mathbf{E}_{k}\in\mathcal{E}(\mathbf{E}_{k-1})$ be a Rellich curve
minimizing $\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}))$
among $\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k-1})$. Suppose
$\Lambda=[a,b]\subset\mathbb{Z}$ is $k$-nonresonant and that $a$ and $b$ are
$k$-left- and $k$-right-regular, respectively. Then for $|E-E_{*}|<\rho_{k}/2$
and $\theta\in B_{\rho_{k}/8D_{0}}(\theta_{*})$,

$\displaystyle\log|R^{\Lambda}_{\theta,E}(m,n)|\leq-\gamma_{k}|m-n|,\quad|m-n|\geq\ell_{k},$

and

$\displaystyle\|R^{\Lambda}_{\theta,E}\|\leq 4\rho^{-1}_{k}.$

Furthermore, if $\widetilde{\Lambda}$ is $k$-nonresonant with
$|\widetilde{\Lambda}|\geq 2L_{k}+3M_{k}^{(1)}$, then there exists a
$k$-nonresonant subinterval $\Lambda=[a,b]\subset\widetilde{\Lambda}$ with
$|\Lambda|\geq|\widetilde{\Lambda}|-(2L_{k}+3M_{k}^{(1)})$ such that $a$ and
$b$ are $k$-left- and $k$-right-regular, respectively.

Subject to the above hypotheses, we have the following: Let
$\mathbf{E}_{k}\in\mathcal{E}_{k}$ with
$\mathbf{E}_{k}:\mathbf{I}_{k}\to\mathbf{J}_{k}$. There exists a collection
$\mathcal{I}_{k+1}(\mathbf{E}_{k})=\mathcal{I}_{k+1}^{(1)}\cup\mathcal{I}_{k+1}^{(2)}$
of intervals $I_{k+1}$ and corresponding even integers $L_{k+1}$ so that,
denoting $\Lambda_{k+1}=[-L_{k+1}/2,L_{k+1}/2]$, the following hold:

  1. (Simple resonance): If $I_{k+1}\in\mathcal{I}_{k+1}^{(1)}$, $\theta_{*}\in I_{k+1}$, and $E_{*}=\mathbf{E}_{k}(\theta_{*})$, then $H^{\Lambda_{k+1}}(\theta_{*})$ satisfies Assumption 1 with

$\displaystyle\widehat{\Lambda}=\Lambda_{k+1},\;{\Lambda}=\Lambda_{k},$
$\displaystyle\rho=\rho_{k},\;\delta=\delta_{k},\;\gamma=\gamma_{k},\;\ell=\ell_{k}.$

  2. (Double resonance): If $I_{k+1}\in\mathcal{I}_{k+1}^{(2)}$, there exists a unique $n$ with $L_{k}<|n|\leq 8M_{k+1}^{(1)}$ and a unique point $\theta_{**}=\theta_{k,n,-}\in I_{k+1}$ such that $E_{**}=\mathbf{E}_{k}(\theta_{**})=\mathbf{E}_{k}(\theta_{**}+n\alpha)$. Furthermore, $H^{\Lambda_{k+1}}(\theta_{**})$ satisfies Assumption 2 with the following assignments:

$\displaystyle\widehat{\Lambda}=\Lambda_{k+1},\;{\Lambda}_{-}=\Lambda_{k},\;{\Lambda}_{+}=\Lambda_{k}+n,$
$\displaystyle\widecheck{\rho}=\frac{2}{3}\rho_{k-1},\;\delta=\delta_{k},\;\gamma=\gamma_{k},\;\ell=\ell_{k},\;\nu=\nu_{k}.$

Moreover, we can choose the additional parameters introduced in Section 3 as
follows:

$\displaystyle\widehat{\sigma}=\sigma_{k+1},\;\eta=\frac{5\rho_{k}}{4\nu_{k}},\;\rho=\rho_{k}.$

In particular, there exists a family
$\mathcal{E}(\mathbf{E}_{k})=\mathcal{E}^{(1)}(\mathbf{E}_{k})\cup\mathcal{E}^{(2)}(\mathbf{E}_{k})$
of Rellich children $\mathbf{E}_{k+1}:\mathbf{I}_{k+1}\to\mathbf{J}_{k+1}$
such that each $\mathbf{E}_{k+1}$ is a Rellich function of $H^{\Lambda_{k+1}}$
satisfying Assumption 3, such that Hypotheses 5.3.1, 5.3.2, and 5.3.3 hold at
scale $k+1$.

Additionally, we have the following:

  1. 1.

The even integers $L_{k+1}$ can be chosen such that $\pm L_{k+1}/2$ and
$\pm(L_{k+1}/2+1)$ are $k$-regular for any $\theta_{*}\in\mathbf{I}_{k+1}$ for
any $E_{*}\in B_{\rho_{k}}(\mathbf{E}_{k}(\theta_{*}))$, and such that
$L_{k+1}/2+1$ (respectively, $-(L_{k+1}/2+1)$) is $k+1$-left-regular
(respectively, $k+1$-right-regular) for any $E_{*}\in
B_{\rho_{k}}(\mathbf{E}_{k+1}(\theta_{*}))$

  2. 2.

The intervals $\mathbf{I}_{k+1}$ and their relevant translates cover
$\mathbf{I}_{k}$, in the sense that if $\theta\in\mathbf{I}_{k}$ and
$\theta\notin\mathbf{I}_{k+1}$ for all
$\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$, then there exists a unique
$n$ with $8M_{k}^{(1)}<|n|\leq 8M_{k+1}^{(1)}$ and a Rellich curve
$\mathbf{E}_{k+1}=\mathbf{E}_{k+1,n,\bullet}\in\mathcal{E}^{(2)}(\mathbf{E}_{k})$
such that $\theta\in\mathbf{I}_{k+1}+n\alpha$.

  3. 3.

If $E_{*}=E_{n_{k+1}}(\mathbf{E}_{k+1})$ for $n_{k+1}\in\mathcal{N}_{k+1}$,
then
$\overline{B}_{\rho_{k}}(E_{*})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{k})$.
Furthermore, we have

$\displaystyle\bar{\rho}_{k}/2\leq|\mathbf{J}_{k+1}|\leq 2\bar{\rho}_{k}.$

Subject to these hypotheses at scale $s-1$, we verify that the statement holds
at scale $s$ with appropriately defined constants:

######  Theorem 5.4.

There exists $\varepsilon_{0}=\varepsilon_{0}(\alpha,v)$ such that, for
$\varepsilon\leq\varepsilon_{0}$, Proposition 5.3 holds for all $k\geq 0$.

For the remainder of this section, we suppose that the proposition holds for
$0\leq k\leq s-1$, that Hypotheses 5.3.1, 5.3.2, and 5.3.3 hold for $0\leq
k\leq s$, and prove the proposition for $k=s$. We note that, in light of Lemma
5.1, the proof below also verifies the initial case $s=0$.

###  5.3 The family of descendants of $\mathbf{E}_{s}$

If $s=0$, let $\mathbf{E}_{s}=v$; otherwise, let
$\mathbf{E}_{s}\in\mathcal{E}^{(j_{s})}(\mathbf{E}_{s-1})$,
$\mathbf{E}_{s}:\mathbf{I}_{s}\to\mathbf{J}_{s}$, $j_{s}\in\\{1,2\\}$, be a
Rellich function of $H^{\Lambda_{s}}$, $\Lambda_{s}=[-L_{s}/2,L_{s}/2]$
constructed as in Proposition 5.3 at scale $s-1$. In the case $j_{s}=2$ (and
thus $\mathbf{E}_{s}=\mathbf{E}_{s,n_{s-1},\bullet}^{(2)}$), we denote

$\mathbf{I}_{s,n_{s-1},\cup}:=\mathbf{I}_{s,n_{s-1},\wedge}^{(2)}\cup\mathbf{I}_{s,n_{s-1},\vee}^{(2)}.$

####  5.3.1 Construction of the next length scale

By our inductive hypotheses, $\mathbf{E}_{s}$ satisfies Assumption 3, with
Morse constants $d_{s},D_{s}$ and boundary derivative lower bound $\nu_{s}$.
By Lemma 5.2, Section 4 applies to $\mathbf{E}_{s}$ with these values, and,
recalling that

$\bar{\rho}_{s}=\frac{d_{s}}{15000D_{0}^{2}}\rho_{s-1}^{2},$

we have the following:

######  Lemma 5.5.

$\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{s})\cap\mathbf{J}_{s}$ can be
covered by closed intervals $J_{s,i}^{(j)}$, $j\in\\{1,2\\}$, such that

  1. 1.

$J_{s,n_{s}}^{(2)}=\overline{B}_{\bar{\rho}_{s}}(E_{n_{s}})$, where
$\theta_{*}=\theta_{s,n_{s},-}$ is the unique point in $\mathbf{I}_{s}$ such
that

$E_{n_{s}}:=\mathbf{E}_{s}(\theta_{*})=\mathbf{E}_{s}(\theta_{*}+n_{s}\alpha)$

for any
$n_{s}\in\mathcal{N}_{s}:=\mathcal{N}(8M_{s+1}^{(1)},\alpha,\mathbf{E}_{s})$
(cf. Section 4).

  2. 2.

$\bar{\rho}_{s}\leq|J_{s,i}^{(j)}|\leq 2\bar{\rho}_{s}$

  3. 3.

$J_{s,i}\cap J_{s,i^{\prime}}\neq\emptyset\implies|J_{s,i}\cap
J_{s,i^{\prime}}|=3\rho_{s}$

  4. 4.

For any $E\in J_{s,i}^{(j)}$,
$\overline{B}_{\rho_{s}}(E)\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$.
Furthermore, if $j=2$ and $E_{*}=E_{n_{s}}$, then
$\overline{B}_{\rho_{s-1}}(E_{*})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s-1})$.

  5. 5.

The total number of such intervals $J_{s,i}^{(j)}$ does not exceed
$5\lceil|\mathbf{J}_{s}|\bar{\rho}_{s}^{-1}\rceil$.

These intervals are defined such that, on the corresponding preimages

$\displaystyle
I_{s,i,\pm}^{(j)}:=(\mathbf{E}_{s})_{\pm}^{-1}(J_{s,i}^{(j)}),\quad
I_{s,i}^{(j)}:=I_{s,i,-}^{(j)}\cup I_{s,i,+}^{(j)},$

one has either

  1. (Simple resonance): If $\theta_{*}\in I_{s,i}^{(1)}$, then for any $n$ with $0\neq|n|\leq 8M_{s+1}^{(1)}$ such that $\theta_{*}+n\alpha\in\mathbf{I}_{s}$,

$\displaystyle|\mathbf{E}_{s}(\theta_{*})-\mathbf{E}_{s}(\theta_{*}+n\alpha)|>3\rho_{s}.$

  2. (Double resonance): If $\theta_{*}=\theta_{s,n_{s},-}\in I_{s,n_{s},-}^{(2)}$, then for $\theta\in B_{\rho_{s-1}/8D_{0}}(\theta_{*})$ and any $n$ with $n\notin\\{0,n_{s}\\},|n|\leq 8M_{s+1}^{(2)}$ such that $\theta+n\alpha\in\mathbf{I}_{s}$,

$\displaystyle|\mathbf{E}_{s}(\theta)-\mathbf{E}_{s}(\theta+n\alpha)|>2\rho_{s-1}$

and

$\displaystyle\min\\{-\partial_{\theta}\mathbf{E}_{s}(\theta),\partial_{\theta}\mathbf{E}_{s}(\theta+n_{s}\alpha)\\}\geq\nu_{s}.$

Finally, recalling that

$\bar{\nu}_{s}=\frac{3d_{s}}{8D_{0}}\bar{\rho}_{s}<\nu_{s},$

each function $\mathbf{E}_{s}|_{I_{s,i}^{(j)}}:I_{s,i}^{(j)}\to J_{s,i}^{(j)}$
satisfies Assumption 3 with Morse constants $d_{s},D_{s}$ and boundary
derivative constant $\bar{\nu}_{s}$.

###### Proof.

This follows immediately from Propositions 4.9 and 4.10, except the second
half of item 4, which follows from the inductive proposition. ∎

$\mathbf{J}_{s}$$E_{n_{s}}(\mathbf{E}_{s})$$\mathbf{E}_{s}$$J_{s,n_{s}}^{(2)}$$\widetilde{\mathbf{J}}(\mathbf{E}_{s})\cap\mathbf{J}_{s}$$\overline{\rho}_{s}$$n_{s}\alpha$$\theta_{s,n_{s},-}$$\theta_{s,n_{s},+}$$I_{s,n_{s},+}$$I_{s,n_{s},-}$$J_{s,i}^{(1)}$$3\rho_{s}$
Figure 8: A cartoon illustration of the covering constructed in Lemma 5.5.

Fix an interval $J_{s,i}^{(j)}$ ($j\in\\{1,2\\}$), and note that
$|I_{s,i,\pm}^{(j)}|\leq\rho_{s-1}/24D_{0}$ by Lemma 4.2, the definition of
$\bar{\rho}_{s}$, and item 2 of Lemma 5.5. Consider all ‘ancestors’
$\mathbf{E}_{k}:\mathbf{I}_{k}\to\mathbf{J}_{k}$ ($0\leq k\leq s-1$) of
$\mathbf{E}_{s}$. For each such $k$, we define two sets $\mathcal{S}_{k,\pm}$
as follows:

$\mathcal{S}_{k,\pm}:=\bigcup_{\theta\in
I_{s,i,\pm}^{(j)}}\mathcal{S}_{k}(\theta,\mathbf{E}_{s}(\theta)).$

The integers in $\mathcal{S}_{k,\pm}$ are well-separated uniformly in $\theta$
at each scale $0\leq k\leq s-1$ in the following sense:

######  Lemma 5.6.

Let $0\leq k\leq s-1$, and suppose $m\in\mathcal{S}_{k,\pm}$.

  1. 1.

If $j_{k+1}=1$, then for any $n\neq m$ with $|n-m|\leq 8M_{k+1}^{(1)}$,
$n\notin\mathcal{S}_{k,\pm}$.

  2. 2.

If $j_{k+1}=2$ (and thus $\mathbf{E}_{k+1}=\mathbf{E}_{k+1,n_{k},\bullet}$ for
some $M_{k}^{(2)}<|n_{k}|\leq 8M_{k+1}^{(1)}$), then for some
$p\in\\{m,m-n_{k}\\}$ and for any $n\notin\\{p,p+n_{k}\\}$ with $|n-p|\leq
8M_{k+1}^{(2)}$, $n\notin\mathcal{S}_{k,\pm}$.

###### Proof.

Since $m\in\mathcal{S}_{k,\pm}$, there is some $\theta_{*}\in
I_{s,i,\pm}^{(j)}$ such that
$m\in\mathcal{S}_{k}(\theta_{*},\mathbf{E}_{s}(\theta_{*}))$; note that
$J_{s,i}^{(j)}\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{k})$,
so $E_{*}=\mathbf{E}_{s}(\theta_{*})$ satisfies the assumptions of the
inductive hypothesis 5.3.2. As noted above, we also have that
$|I_{s,i,\pm}^{(j)}|\leq\rho_{s-1}/24D_{0}\leq\rho_{k}/24D_{0}$.

We have two cases:

  1. 1.

$j_{k+1}=1$:

Take $p=m$. By the length bound on $I_{s,i,\pm}^{(j)}$, we have

$I_{s,i,\pm}^{(j)}+p\alpha\subset
B_{\rho_{k}/24D_{0}}(\theta_{*}+p\alpha)\subset\mathbf{I}_{s}.$

  2. 2.

$j_{k+1}=2$: By Hypothesis 5.3.2 and the length bound on $I_{s,i,\pm}^{(j)}$,
for some $p\in\\{m,m-n_{k+1}\\}$, we have

$I_{s,i,\pm}^{(j)}+p\alpha\subset
B_{\rho_{k}/24D_{0}}(\theta_{*}+p\alpha)\subset\mathbf{I}_{k+1,\,\cup}\subset
B_{\rho_{k}/8D_{0}}(\theta_{k+1,n_{k+1},-}).$

In both cases, by Lemma 5.5, for any $\theta\in I_{s,i,\pm}^{(j)}$ and $n\neq
p$ with $|n-p|\leq 8M_{k+1}^{(j_{k+1})}$ such that
$\theta+n\alpha\in\mathbf{I}_{k}$, we have

$\displaystyle|\mathbf{E}_{k}(\theta+n\alpha)-\mathbf{E}_{s}(\theta)|\geq$
$\displaystyle|\mathbf{E}_{k}(\theta+n\alpha)-\mathbf{E}_{k}(\theta+p\alpha)|-|\mathbf{E}_{k}(\theta+p\alpha)-\mathbf{E}_{k}(\theta_{*}+p\alpha)|$
$\displaystyle-|\mathbf{E}_{k}(\theta_{*}+p\alpha)-\mathbf{E}_{s}(\theta_{*})|-|\mathbf{E}_{s}(\theta_{*})-\mathbf{E}_{s}(\theta)|$
$\displaystyle\geq
3\rho_{k}-\frac{1}{24}\rho_{s-1}-\frac{25}{24}\rho_{k}-\frac{1}{24}\rho_{s-1}\geq\frac{25}{24}\rho_{k},$

and so $n\notin\mathcal{S}_{k,\pm}$, as claimed. ∎

This separation will, in turn, allow us to construct even integers $L_{s+1}$
so that $\pm L_{s+1}/2$ are $s$-regular:

######  Lemma 5.7.

There is an even integer $L_{s+1}$ with $M_{s+1}^{(j)}/2\leq L_{s+1}\leq
M_{s+1}^{(j)}$ so that, for all $\theta_{*}\in I_{s,i,\pm}^{(j)}$ for all
$E_{*}\in B_{\rho_{s}}(\mathbf{E}_{s}(\theta_{*}))$, $\pm L_{s+1}/2$ and
$\pm(L_{s+1}/2+1)$ are $s$-regular relative to $(\theta_{*},E_{*})$.

###### Proof.

Let $a_{s}:=M_{s+1}^{(j)}/2$. We define a nonincreasing sequence of integers
$a_{s},a_{s-1},\dots,a_{0}$ as follows. For each $1\leq k\leq s$, we have by
the previous lemma that

$\\#\left\\{a\in[a_{k}-8L_{k},a_{k}]:\operatorname{dist}(a,\mathcal{S}_{k-1,\pm})<\frac{4}{5}L_{k}\right\\}<\frac{9}{5}L_{k},$

where we have crucially used here that $5M_{k-1}^{(2)}<M_{k}^{(1)}/2$. The
same counting argument holds replacing $[a_{k}-8L_{k},a_{k}]$ by
$[-a_{k},-a_{k}+8L_{k}]$; since $4(9/5)<8$, we can find some
$a_{k-1}\in[a_{k}-8L_{k},a_{k}]$ such that

$\operatorname{dist}(\pm a_{k-1},\mathcal{S}_{k-1,\pm})\geq\frac{4}{5}L_{k}$

for all four choices of signs.

We define $L_{s+1}=2a_{0}$. By construction, we have

$M^{(j)}_{s+1}\geq L_{s+1}\geq M^{(j)}_{s+1}-2\sum_{k=1}^{s}8L_{k}\geq
M^{(j)}_{s+1}/2,$

and

$\operatorname{dist}(\pm
L_{s+1}/2,\mathcal{S}_{k-1,\pm})\geq\frac{4}{5}L_{k}-8\sum_{i=1}^{k-1}L_{i}>\frac{3}{4}L_{k}+1$

for all $1\leq k\leq s$ and all four choices of signs. ∎

####  5.3.2 Decay of resonant eigenpairs

For the fixed interval $J_{s,i}^{(j)}$ we denote by

$\Lambda_{s+1}:=[-L_{s+1}/2,L_{s+1}/2]$

and recall that

$\ell_{s}=L_{s}^{5/6}\geq
16L_{s}^{2/3}|\log\varepsilon|^{2}=16|\log\varepsilon||\log\rho_{s}|.$

From our inductive assumptions, we immediately find Green’s function decay
away from $\\{0\\}$ (and the double-resonant site $\\{n_{s}\\}$, if it
exists):

######  Lemma 5.8.

Fix $\theta_{*}\in\mathbf{I}_{s}$ and find $\Lambda_{s+1}$ as above. Then we
have the following:

  1. 1.

If $\theta_{*}\in I_{s,i,\pm}^{(1)}$, then for
$|E-\mathbf{E}_{s}(\theta_{*})|<\frac{3}{2}\rho_{s}$ and $\theta\in
B_{\rho_{s}/8D_{0}}(\theta_{*})$,

$\log|R_{\theta,E}^{\Lambda_{s+1}\setminus\Lambda_{s}}(m,n)|\leq-\gamma_{s}|m-n|,\quad|m-n|\geq\ell_{s}$

and

$\|R_{\theta,E}^{\Lambda_{s+1}\setminus\Lambda_{s}}\|\leq 4\rho_{s}^{-1}.$

Furthermore, there are intervals
$\Lambda_{l}\supset[-\frac{7}{8}L_{s},-\frac{1}{8}L_{s}]$ and
$\Lambda_{r}\supset[\frac{1}{8}L_{s},\frac{7}{8}L_{s}]$ such that for
$|E-\mathbf{E}_{s}(\theta_{*})|<\frac{3}{2}\rho_{s}$ and $\theta\in
B_{\rho_{s-1}/8D_{0}}(\theta_{*})$,

$\displaystyle\log|R^{\Lambda_{l/r}}_{\theta,E}(m,n)|\leq-\gamma_{s-1}|m-n|,\quad|m-n|\geq\ell_{s-1}.$

  2. 2.

If $\theta_{*}=\theta_{s,n_{s},-}$, then for
$|E-\mathbf{E}_{s}(\theta_{*})|<\frac{3}{2}\rho_{s-1}$ and $\theta\in
B_{\rho_{s-1}/8D_{0}}(\theta_{*})$,

$\log|R_{\theta,E}^{\Lambda_{s+1}\setminus(\Lambda_{s}\cup\Lambda_{s}+n_{s})}(m,n)|\leq-\gamma_{s-1}|m-n|,\quad|m-n|\geq\ell_{s-1}$

and

$\|R_{\theta,E}^{\Lambda_{s+1}\setminus(\Lambda_{s}\cup\Lambda_{s}+n_{s})}\|\leq
4\rho_{s-1}^{-1}.$

Furthermore, letting $c_{l}=\min\\{0,n_{s}\\}$ and $c_{r}=\max\\{0,n_{s}\\}$,
there are intervals
$\Lambda_{l}\supset[c_{l}-\frac{7}{8}L_{s},c_{l}-\frac{1}{8}L_{s}]$,
$\Lambda_{c}\supset[c_{l}+\frac{1}{8}L_{s},c_{r}-\frac{1}{8}L_{s}]$, and
$\Lambda_{r}\supset[c_{r}+\frac{1}{8}L_{s},c_{r}+\frac{7}{8}L_{s}]$ such that,
for $|E-\mathbf{E}_{s}(\theta_{*})|<\frac{3}{2}\rho_{s-1}$ and $\theta\in
B_{\rho_{s-1}/8D_{0}}(\theta_{*})$,

$\displaystyle\log|R^{\Lambda_{l/c/r}}_{\theta,E}(m,n)|\leq-\gamma_{s-1}|m-n|,\quad|m-n|\geq\ell_{s-1}.$

In either case, the intervals $\Lambda_{s}\cap\Lambda_{l/c/r}$ also satisfy
the Green’s function decay property for $(\ell_{s-1},\gamma_{s-1})$, as do the
intervals $(\Lambda_{s}+n_{s})\cap\Lambda_{l/c/r}$.

###### Proof.

Noting the $s$-regularity relative to $(\theta_{*},E_{*})$ for any $E_{*}\in
B_{\rho_{s}}(\mathbf{E}_{s}(\theta_{*}))$, these statements are precisely the
nonresonance hypothesis 5.3.3 applied to various intervals. In particular, the
maximal connected components of $\Lambda_{s+1}\setminus\Lambda_{s}$ are
inductively $s$-nonresonant with $s$-directionally regular endpoints.
Similarly, the maximal connected components of
$\Lambda_{s+1}\setminus(\Lambda_{s}\cup\Lambda_{s}+n_{s})$ are
$s-1$-nonresonant, as are the maximal components of
$[-L_{s},L_{s}]\setminus\Lambda_{s-1}$ (in case
$\mathbf{E}_{s}\in\mathcal{E}^{(1)}(\mathbf{E}_{s-1})$) or
$[-L_{s},L_{s}]\setminus(\Lambda_{s-1}\cup\Lambda_{s-1}+n_{s-1})$ (in case
$\mathbf{E}_{s}\in\mathcal{E}^{(2)}(\mathbf{E}_{s-1})$).

The existence of the subintervals $\Lambda_{l/c/r}$ follows from the
nonresonance hypothesis after noting that

$\frac{1}{8}L_{s}-M_{s-1}^{(2)}-1\geq\frac{1}{16}(M_{s-1}^{(2)})^{2}-M_{s-1}^{(2)}-1\gg
5L_{s-1}$

for the left and right intervals, and

$c_{r}-c_{l}-2(1+M_{s-1}^{(2)})\geq L_{s}-2(1+M_{s-1}^{(2)})\gg 5L_{s-1}$

for the center interval in the double-resonant case.

Finally, the decay estimates on the intervals $\Lambda_{s}\cap\Lambda_{l/c/r}$
likewise follow from the inductive nonresonance hypothesis and the
$s-1$-directional regularity of $L_{s}/2$. The decay estimates on
$(\Lambda_{s}+n_{s})\cap\Lambda_{l/c/r}$ follow from the corresponding
regularities of $n_{s}\pm L_{s}/2$, which follow from the observation that
$H^{\Lambda_{s}}(\theta_{*}+n_{s}\alpha)=H^{\Lambda_{s}+n_{s}}(\theta_{*})$.

∎

Noting that

$\frac{3}{8}L_{s}>L_{s}^{5/6}=\ell_{s}\gg\ell_{s-1},$

we apply the above decay to conclude stable eigenvector decay for resonant
eigenvectors of $H^{\Lambda_{s+1}}$:

######  Lemma 5.9.

Suppose $\theta_{*}\in\mathbf{I}_{s}$, and let
$E_{*}=\mathbf{E}_{s}(\theta_{*})$.

  1. 1.

If $\theta_{*}\in I_{s,i,\pm}^{(1)}$, let $\rho=\rho_{s}$ and let
$\mathcal{P}$ be the partition of $\Lambda_{s+1}$ into $\Lambda_{s}$ and its
complement.

  2. 2.

If $\theta_{*}=\theta_{s,n_{s},-}$, let $\rho=\rho_{s-1}$ and let
$\mathcal{P}$ be the partition of $\Lambda_{s+1}$ into
$\Lambda_{s}\cup\Lambda_{s}+n_{s}$ and its complement.

Then, for $\theta\in B_{\rho/8D_{0}}(\theta_{*})$, for any eigenpair
$(E,\psi)$ of $H^{\Lambda_{s+1}}(\theta)$ with $|E-E_{*}|<\frac{3}{2}\rho$,
the unit eigenvector $\psi$ has $\mathcal{P}$-boundary values no larger than
$\delta_{s}/\varepsilon$:

$\|\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}\psi\|\leq 4\delta_{s}.$

Furthermore, the resonant unit eigenvector $\psi_{s}$ of $H^{\Lambda_{s}}$
corresponding to $\mathbf{E}_{s}$ (as well as the associated unit eigenvector
of $H^{\Lambda_{s}+n_{s}}$ in case 2) has stably-small $\mathcal{P}$-boundary
values:

$\displaystyle\|\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}\psi_{s}\|$
$\displaystyle\leq 2\delta_{s}$
$\displaystyle\|\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}(\partial_{\theta}\psi_{s})\|$
$\displaystyle\leq 24D_{0}\frac{\delta_{s}}{\rho}$

###### Proof.

The estimate on $\|\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}\psi\|$ follows in all
cases from Lemma 5.8 and the Poisson formula (1.11); for example, in the case
$\theta_{*}\in I_{s}^{(1)}$, denoting $\Lambda_{l}=[a_{l},b_{l}]$, one has

$\displaystyle|\psi(-L_{s}/2)|$
$\displaystyle\leq\varepsilon\left(|R_{\theta,E}^{\Lambda_{l}}(-L_{s}/2,a_{l})|+|R_{\theta,E}^{\Lambda_{l}}(-L_{s}/2,b_{l})|\right)$
$\displaystyle\leq 2\varepsilon e^{-3\gamma_{s-1}L_{s}/8}\leq\delta_{s}.$

The other cases follow analogously.

The estimate on $\|\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}\psi_{s}\|$ is made
similarly, instead applying the decay on the intervals
$\Lambda_{s}\cap\Lambda_{l/c/r}$.

Let $\chi_{s}$ denote the linear projection onto the coordinates in the
interval $\Lambda_{s}$, and $\widetilde{\chi}_{s}$ denote the linear
projection onto the interval

$\\{m:|m|\leq L_{s}/4\\}\subset\Lambda_{s}.$

Let $\Gamma_{s}:=\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}\chi_{s}$, and note that

$\Gamma_{s}(\partial_{\theta}\psi_{s})=\Gamma_{\mathcal{P}}^{\Lambda_{s+1}}(\partial_{\theta}\psi_{s}).$

Recall that

$\partial_{\theta}\psi_{s}=-R_{\perp}(H^{\Lambda_{s}},\mathbf{E}_{s})V^{\prime}\psi_{s};$

we break our estimate of $\|\Gamma_{s}(\partial_{\theta}\psi_{s})\|$ into
pieces corresponding to $\widetilde{\chi}_{s}$ and its relative complement
$(\chi_{s}-\widetilde{\chi}_{s})$.

By again applying the Poisson formula on the intervals
$\Lambda_{s}\cap\Lambda_{l/c/r}$ from Lemma 5.8, we have

$\displaystyle\|(\chi_{s}-\widetilde{\chi}_{s})V^{\prime}\psi_{s}\|^{2}$
$\displaystyle\leq\sum_{L_{s}/4\leq|m|\leq L_{s}/2}D_{0}^{2}|\psi_{s}(m)|^{2}$
$\displaystyle\leq
8D_{0}^{2}\sum_{j>L_{s}/8}\varepsilon^{2}e^{-2\gamma_{s-1}j}$
$\displaystyle\leq(3D_{0}\delta_{s})^{2}.$

Now let $\widecheck{\mathcal{P}}$ be the partition of $\Lambda_{s}$ into
$\Lambda_{s}\cap(\Lambda_{l}\cup\Lambda_{c}\cup\Lambda_{r})$ and its
complement, and recall that $E_{*}=\mathbf{E}_{s}(\theta_{*})$. Letting
$R_{\perp}=R_{\perp}(H^{\Lambda_{s}},\mathbf{E}_{s}),\widecheck{R}=R_{\widecheck{\mathcal{P}}}^{\Lambda_{s}}(E_{*}),$
and $P_{s}=\psi_{s}\psi_{s}^{\top}$, we expand

$\Gamma_{s}R_{\perp}\widetilde{\chi}_{s}=\Gamma_{s}\widecheck{R}(I-P_{s}-\Gamma_{\widecheck{\mathcal{P}}}^{\Lambda_{s}}R_{\perp})\widetilde{\chi}_{s}$

by the resolvent identity. Since the intervals in
$\Lambda_{s}\cap(\Lambda_{l}\cup\Lambda_{c}\cup\Lambda_{r})$ are
$s-1$-nonresonant, we get

$\max\\{\|\Gamma_{s}\widecheck{R}\widetilde{\chi}_{s}\|,\|\Gamma_{s}\widecheck{R}\Gamma_{\widecheck{\mathcal{P}}}^{\Lambda_{s}}\|\\}\leq
2\varepsilon e^{-\gamma_{s-1}L_{s}/4}.$

By expanding
$P_{s}=\widetilde{\chi}_{s}P_{s}+(\chi_{s}-\widetilde{\chi}_{s})P_{s}$, we
likewise estimate

$\displaystyle\|\Gamma_{s}\widecheck{R}P_{s}\widetilde{\chi}_{s}\|$
$\displaystyle\leq\|\Gamma_{s}\widecheck{R}\widetilde{\chi}_{s}\|\|P_{s}\widetilde{\chi}_{s}\|+\|\Gamma_{s}\widecheck{R}\|\|(\chi_{s}-\widetilde{\chi}_{s})P_{s}\|\|\widetilde{\chi}_{s}\|$
$\displaystyle\leq 2\varepsilon
e^{-\gamma_{s-1}L_{s}/4}+\frac{8\varepsilon}{\rho_{s-1}}\cdot 3\varepsilon
e^{-\gamma_{s-1}L_{s}/8}$ $\displaystyle\leq 6\frac{\delta_{s}}{\rho_{s-1}}.$

It remains to estimate $\|R_{\perp}\|$, but this follows inductively.
Specifically, by Hypothesis 5.3.2, if $\theta_{*}=\theta_{s,n_{s},-}$, it must
be that $j_{s}=1$ and so $\|R_{\perp}\|\leq 4\rho_{s-1}^{-1}$ by (2.11). If
$\theta_{*}\in I_{s}^{(1)}$, we have the estimate $\|R_{\perp}\|\leq
4\rho_{s}^{-1}$, either by (2.11) if $j_{s}=1$, or by (3.13) and Proposition
3.6 if $j_{s}=2$, since the separation guaranteed by Proposition 3.6 is bigger
than $2\rho_{s}$. In either case, $\|R_{\perp}\|\leq 4\rho^{-1}$ by our
definition of $\rho$. Thus, we compute

$\displaystyle\|\Gamma_{s}(\partial_{\theta}\psi_{s})\|$
$\displaystyle=\|\Gamma_{s}R_{\perp}V^{\prime}\psi_{s}\|$
$\displaystyle\leq\|\Gamma_{s}R_{\perp}\|\|(\chi_{s}-\widetilde{\chi}_{s})V^{\prime}\psi_{s}\|+\|\Gamma_{s}R_{\perp}\widetilde{\chi}_{s}\|\|V^{\prime}\psi_{s}\|$
$\displaystyle\leq\|\Gamma_{s}R_{\perp}\|\|(\chi_{s}-\widetilde{\chi}_{s})V^{\prime}\psi_{s}\|+D_{0}\left(\|\Gamma_{s}\widecheck{R}\widetilde{\chi}_{s}\|+\|\Gamma_{s}\widecheck{R}P_{s}\widetilde{\chi}_{s}\|+\|\Gamma_{s}\widecheck{R}\Gamma_{\widecheck{\mathcal{P}}}^{\Lambda_{s}}\|\|R_{\perp}\|\|\widetilde{\chi}_{s}\|\right)$
$\displaystyle\leq 8\varepsilon\rho^{-1}\cdot
3D_{0}\delta_{s}+D_{0}\left(2\delta_{s}+6\delta_{s}\rho_{s-1}^{-1}+8\delta_{s}\rho^{-1}\right)$
$\displaystyle\leq 24D_{0}\frac{\delta_{s}}{\rho},$

as claimed. The estimate for the unit eigenvector of $H^{\Lambda_{s}+n_{s}}$
follows analogously using the intervals
$(\Lambda_{s}+n_{s})\cap\Lambda_{l/c/r}$ instead of the intervals
$\Lambda_{s}\cap\Lambda_{l/c/r}$. ∎

####  5.3.3 Constructing the descendants $\mathbf{E}_{s+1}$

With these estimates, we are ready to invoke the content of Sections 2 and 3:

######  Lemma 5.10.

We have the following:

  1. 1.

(Simple resonance): If $\theta_{*}\in I_{s}^{(1)}$ and
$E_{*}=\mathbf{E}_{s}(\theta_{*})$, then $H^{\Lambda_{s+1}}(\theta_{*})$
satisfies Assumption 1 with the following assignments:

$\displaystyle\widehat{\Lambda}=\Lambda_{s+1}$
$\displaystyle,\;\Lambda=\Lambda_{s},$
$\displaystyle\rho=\rho_{s},\;\delta=\delta_{s}$
$\displaystyle,\;\gamma=\gamma_{s},\;\ell=\ell_{s}.$

  2. 2.

(Double resonance): If $\theta_{**}=\theta_{s,n_{s},-}$ and
$E_{**}=\mathbf{E}_{s}(\theta_{*})$, then $H^{\Lambda_{s+1}}(\theta_{**})$
satisfies Assumption 2 with the following assignments:

$\displaystyle\widehat{\Lambda}=\Lambda_{s+1},\;{\Lambda}_{-}$
$\displaystyle=\Lambda_{s},\;{\Lambda}_{+}=\Lambda_{s}+n_{s},$
$\displaystyle\widecheck{\rho}=\frac{2}{3}\rho_{s-1},\;\delta=\delta_{s}$
$\displaystyle,\;\gamma=\gamma_{s},\;\ell=\ell_{s},\;\nu={\nu}_{s}.$

###### Proof.

We verify the relevant assumptions in each case:

  1. 1.

The values $\delta$, $\rho$, and $\gamma$ satisfy (2.1) and (2.2) by
definition. Taking ${\psi}=\psi_{s}$, item 1 of Assumption 1 is immediate by
definition of $E_{*}$. Items 2 and 5 follow immediately from Lemma 5.9, and
item 3 follows immediately from Lemma 5.8. Finally, item 4 follows from the
inductive bound $\|R^{\Lambda_{s}}_{\perp,\theta,E}\|\leq 4\rho_{s}^{-1}$
(following from either Proposition 2.1 or 3.1, depending on the value of
$j_{s}$) and the bound
$\|R^{\Lambda_{s+1}\setminus\Lambda_{s}}_{\theta,E}\|\leq 4\rho_{s}^{-1}$ from
Lemma 5.8.

  2. 2.

Again, $\delta$, $\widecheck{\rho}$, and $\gamma$ satisfy (3.1), (3.2), and
(3.3) by definition. As in case 1, item 1 of Assumption 2 is immediate by
definition of $E_{**}$, items 2 and 5 follow immediately from Lemma 5.9, and
item 3 follows immediately from Lemma 5.8. Since
$n_{s}\in\mathcal{N}_{s}\neq\emptyset$, we must have $j_{s}=1$, and so
inductively $\|R_{\perp,\theta,E}^{\Lambda_{s}}\|\leq
4\rho_{s-1}^{-1}<4\widecheck{\rho}^{-1}$ for
$|E-E_{**}|\leq\rho_{s-1}=\frac{3}{2}\widecheck{\rho}$ (and similarly for
$\|R_{\perp,\theta,E}^{\Lambda_{s}+n_{s}}\|$) by Proposition 2.1. Item 4
follows from this observation and the bound
$\|R^{\Lambda_{s+1}\setminus(\Lambda_{s}\cup\Lambda_{s}+n)}_{\theta,E}\|\leq
4\rho_{s-1}^{-1}<4\widecheck{\rho}^{-1}$ from Lemma 5.8. Finally, item 6
follows from Lemma 5.5. ∎

As a consequence of the above, for each phase interval $I_{s}\in\mathcal{I}$,
we have found an interval $\Lambda_{s+1}=\Lambda_{s+1}(I_{s})$ such that any
resonant Rellich function is Morse with at most one critical point and whose
eigenvector is localized. Specifically, if $I_{s}=I_{s,i}^{(1)}$ we have found
a unique Rellich function

$\mathbf{E}_{s+1,i}^{(1)}:I_{s,i}^{(1)}\to J_{s+1,i}^{(1)}$

of $H^{\Lambda_{s+1}}$, where we define $J_{s+1,i}^{(1)}$ to be the image of
$I_{s,i}^{(1)}$ under $\mathbf{E}_{s+1,i}^{(1)}$. Similarly, for each
$I_{s,n,-}^{(2)}$, $n\in\mathcal{N}_{s}$, we have found two Rellich functions

$\mathbf{E}_{s+1,n,\bullet}^{(2)}:I_{s,n,-}^{(2)}\to
J_{s+1,n}^{(2)},\quad\bullet\in\\{\vee,\wedge\\}$

of $H^{\Lambda_{s+1}}$, where we define $J_{s+1,n}^{(2)}$ to be the smallest
interval containing the images of $I_{s,n,-}^{(2)}$ under both Rellich
functions. We note that, since $\varepsilon<\frac{1}{3}$ and
$\ell_{s},L_{s}<M_{s+1}^{(2)}$, we may choose $\widehat{\sigma}=\sigma_{s+1}$
in Theorem 3.2, as this choice satisfies (3.14). To finish constructing the
children of $\mathbf{E}_{s}$, we must modify the domains of these new Rellich
functions so as to again satisfy Assumption 3.

In the simple-resonant case, by Lemma 5.5 and Proposition 2.2, the function
$\mathbf{E}_{s+1,i}^{(1)}$ satisfies the assumptions of Lemma 4.12 relative to
the function $\mathbf{E}_{s,i}^{(1)}$ with $\delta=\delta_{s}$ and
$\rho=\rho_{s}$; thus, we may find an interval
$\mathbf{I}_{s+1,i}^{(1)}\subset I^{(1)}_{s,i}$ and its image
$\mathbf{J}_{s+1,i}^{(1)}:=\mathbf{E}_{s+1,i}^{(1)}(\mathbf{I}_{s+1,i}^{(1)})$
with

$|I_{s,i}^{(1)}\setminus\mathbf{I}_{s+1,i}^{(1)}|\leq\frac{32\delta_{s}}{\bar{\nu}_{s}}\ll\frac{\rho_{s}}{24D_{0}}.$

We abuse notation and henceforth denote by $\mathbf{E}_{s+1,i}^{(1)}$ the
restriction of the Rellich function of $H^{\Lambda_{s+1}}$ to the interval
$\mathbf{I}_{s+1,i}^{(1)}$.

The double-resonant case requires a bit more care. If
$I_{s}=I_{s,n_{s},-}^{(2)}$, $n_{s}\in\mathcal{N}_{s}$, we recall the regions

$J^{(2)}_{s,n_{s}}:=\overline{B}_{\bar{\rho}_{s}}(E_{n_{s}}),\quad
I_{s,n_{s},\pm}^{(2)}:=\mathbf{E}_{s,\pm}^{-1}(J^{(2)}_{s,n_{s}}).$

Note that one has

$B_{\frac{\bar{\rho}_{s}}{D_{0}}}(\theta_{s,n_{s},\pm})\subset
I_{s,n_{s},\pm}^{(2)}\subset
B_{\frac{\bar{\rho}_{s}}{\nu_{s}}}(\theta_{s,n_{s},\pm})$

since $\nu_{s}\leq|\partial_{\theta}\mathbf{E}_{s}|\leq D_{0}$ on
$I_{s,n_{s},\pm}^{(2)}$.

Denote by

$\displaystyle\tilde{\mathbf{E}}_{s,n_{s},\vee}^{(2)}(\theta)$
$\displaystyle:=\max\\{\mathbf{E}_{s,-}(\theta),\mathbf{E}_{s,+}(\theta+n_{s}\alpha)\\},\quad\theta\in\tilde{I}_{s,n_{s},\vee}^{(2)}:=[\inf
I_{s,n_{s},-}^{(2)},\sup I_{s,n_{s},+}^{(2)}-n_{s}\alpha],$
$\displaystyle\tilde{\mathbf{E}}_{s,n_{s},\wedge}^{(2)}(\theta)$
$\displaystyle:=\min\\{\mathbf{E}_{s,-}(\theta),\mathbf{E}_{s,+}(\theta+n_{s}\alpha)\\},\quad\theta\in\tilde{I}_{s,n_{s},\wedge}^{(2)}:=[\inf
I_{s,n_{s},+}^{(2)}-n_{s}\alpha,\sup I_{s,n_{s},-}^{(2)}].$

We denote also

$I_{s,n_{s},\cap}^{(2)}:=I_{s,n_{s},-}^{(2)}\cap(I_{s,n_{s},+}^{(2)}-n_{s}\alpha)\subset\tilde{I}_{s,n_{s},\bullet}^{(2)},\quad\bullet\in\\{\vee,\wedge\\}.$

Since $\theta_{s,n_{s},+}=\theta_{s,n_{s},-}+n_{s}\alpha$, we have that

$\overline{B}_{\frac{\bar{\rho}_{s}}{D_{0}}}(\theta_{s,n_{s},-})\subset
I_{s,n_{s},\cap}^{(2)}.$

By construction,

$|\tilde{\mathbf{E}}_{s,n_{s},\bullet}^{(2)}(\tilde{I}_{s,n_{s},\bullet}^{(2)})|=\bar{\rho}_{s},\quad\bullet\in\\{\vee,\wedge\\},$

and thus, by Lemma 4.2,

$\displaystyle|\\{\theta\in\tilde{I}_{s,n_{s},\bullet}^{(2)}:\tilde{\mathbf{E}}_{s,n_{s},\bullet}^{(2)}(\theta)=\mathbf{E}_{s}(\theta+p\alpha)\\}|\geq\frac{\bar{\rho}_{s}}{D_{0}},\quad\bullet\in\\{\vee,\wedge\\},\;p\in\\{0,n_{s}\\}.$

The functions $\tilde{\mathbf{E}}_{s,n_{s},\bullet}^{(2)}$ satisfy Assumption
3; by Lemma 5.5 and Theorem 3.2, the Rellich functions
$\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}$ satisfy the assumptions of Lemma 4.11
relative to the functions $\tilde{\mathbf{E}}_{s,n_{s},\bullet}^{(2)}$ with
$\delta=C\delta_{s}$; thus, we may find an interval
$\mathbf{I}_{s+1,n_{s},\bullet}^{(2)}\subset\tilde{I}_{s,n_{s},\bullet}^{(2)}$
and its image
$\mathbf{J}_{s+1,n_{s},\bullet}^{(2)}:=\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}(\mathbf{I}_{s+1,n_{s},\bullet}^{(2)})$
with

$|\tilde{I}_{s,n_{s},\bullet}^{(2)}\setminus\mathbf{I}_{s+1,n_{s},\bullet}^{(2)}|\leq\frac{32C\delta_{s}}{\bar{\nu}_{s}}\ll\frac{\rho_{s}}{24D_{0}}.$

In particular, we note that
$|\mathbf{I}^{(2)}_{s+1,n_{s},\cap}|\geq\frac{\overline{\rho}_{s}}{D_{0}}-\frac{\rho_{s}}{24D_{0}}\gg\frac{5\rho_{s}}{4\nu_{s}}$,
where
$\mathbf{I}^{(2)}_{s+1,n_{s},\cap}:=\mathbf{I}^{(2)}_{s+1,n_{s},\vee}\cap\mathbf{I}^{(2)}_{s+1,n_{s},\wedge}$,
so we may choose $\eta=\frac{5\rho_{s}}{4\nu_{s}}$ and $\rho=\rho_{s}$ in
Theorem 3.2, with
$B_{\eta}(\theta_{s,n_{s},-})\subset\mathbf{I}^{(2)}_{s+1,n_{s},\cap}$. We
abuse notation and henceforth denote by $\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}$
the restriction of the corresponding Rellich function of $H^{\Lambda_{s+1}}$
to the interval $\mathbf{I}_{s+1,n_{s},\bullet}^{(2)}$,
$\bullet\in\\{\vee,\wedge\\}$.

Denote the collection of all descendants of $\mathbf{E}_{s}$ constructed above
by $\mathcal{E}(\mathbf{E}_{s})$.

######  Lemma 5.11.

Let $\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$ be constructed as above.

  1. 1.

$\mathbf{E}_{s+1}:\mathbf{I}_{s+1}\to\mathbf{J}_{s+1}$ satisfies Assumption 3
with Morse constants

$\displaystyle d_{s+1}$ $\displaystyle=\nu_{s}/12,$ $\displaystyle D_{s+1}$
$\displaystyle=2D_{0}(1+D_{0}\sigma_{s+1}^{-1})$

and boundary derivative constant

$\displaystyle\nu_{s+1}$
$\displaystyle=\min\left\\{\frac{d_{s+1}}{14(8M_{s+2}^{(1)})^{\tau}},\frac{\bar{\nu}_{s}}{2}\right\\}=\frac{\bar{\nu}_{s}}{2}$

  2. 2.

$\mathbf{J}_{s+1}\cup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$

  3. 3.

If $\mathbf{E}_{s}$ does not attain its maximum at a critical point, then

$\sup\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{s})\leq\sup\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}).$

Similarly, if $\mathbf{E}_{s}$ does not attain its minimum at a critical
point, then

$\inf\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{s})\geq\inf\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}).$

  4. 4.

If $\mathbf{E}_{s+1}$ does not attain its supremum $\sup\mathbf{J}_{s+1}$ at a
critical point and
$\sup\mathbf{J}_{s+1}\neq\sup_{\mathbf{E}^{\prime}\in\mathcal{E}(\mathbf{E}_{s})}\sup\mathbf{J}^{\prime}$,
then there is some $\mathbf{E}_{s+1}^{\prime}\in\mathcal{E}(\mathbf{E}_{s+1})$
such that

$B_{\frac{11\rho_{s}}{4}}(\sup\mathbf{J}_{s+1})\subset\mathbf{J}_{s+1}^{\prime}.$

The same statement holds with $\inf$ in place of $\sup$.

###### Proof.

Item 1 follows immediately from Lemma 5.10; from the same result, we likewise
have

$|\sup J_{s+1,i}^{(j)}-\sup J_{s,i}^{(j)}|,|\inf J_{s+1,i}^{(j)}-\inf
J_{s,i}^{(j)}|\leq C\delta_{s}\ll\rho_{s}/2$

for $j\in\\{1,2\\}$ and all $i$ for which these intervals are defined. The
process of “trimming” the function by Lemma 4.11 or 4.12 does not change the
supremum or infimum; thus, we still have

$\displaystyle|\sup\mathbf{J}_{s+1,i}^{(1)}-\sup
J_{s,i}^{(1)}|,|\inf\mathbf{J}_{s+1,i}^{(1)}-\inf J_{s,i}^{(1)}|$
$\displaystyle\leq C\delta_{s}\ll\rho_{s}/2,$
$\displaystyle|\sup\mathbf{J}_{s+1,n_{s},\vee}^{(2)}-\sup
J_{s,n_{s}}^{(2)}|,|\inf\mathbf{J}_{s+1,n_{s},\wedge}^{(2)}-\inf
J_{s,n_{s}}^{(2)}|$ $\displaystyle\leq C\delta_{s}\ll\rho_{s}/2.$

In particular, since each $\mathbf{E}_{s+1,n,\vee}^{(2)}$ attains its infimum
at a critical point, and each $\mathbf{E}_{s+1,n,\wedge}^{(2)}$ attains it
supremum at a critical point, this estimate applies at every non-critical
value extremum of any $\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$.

The next three items follow. Specifically, by construction,

$\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s})-\sup J_{s+1,i}^{(j)}\geq\rho_{s};$

and by definition,

$\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})-\sup\mathbf{J}_{s+1}\leq\rho_{s+1}.$

These estimates, combined with the analogous estimates with $\inf$ and with
the estimates established above, give item 2. For item 3, we use the fact that
by construction,

$\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s})-\sup_{i,j}\sup J_{s,i}^{(j)}\leq
2\bar{\rho}_{s}+\rho_{s}\ll\rho_{s-1}/24,$

and the fact that, by definition, for each
$\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})$,
$\mathbf{E}:\mathbf{I}\to\mathbf{J}$, we have

$\sup\mathbf{J}-\sup\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\leq\frac{5}{4}\rho_{s}\ll\rho_{s-1}/24.$

Since, by definition,
$\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s})-\sup\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{s})=\rho_{s-1}/8$,
and the analogous estimates with $\inf$ hold, item 3 follows from the above
estimates. Item 4 follows from the above estimates and the fact that, by
construction, each pair of adjacent intervals $J_{s,i}^{(j)}$ overlap by
$3\rho_{s}$. ∎

Finally, we verify the directional regularity of our new endpoints:

######  Lemma 5.12.

For all $\theta_{*}\in\mathbf{I}_{s+1}$ for all $E_{*}\in
B_{\rho_{s}}(\mathbf{E}_{s+1}(\theta_{*}))$, $L_{s+1}/2+1$ is $s+1$-left-
regular and $-(L_{s+1}/2+1)$ is $s+1$-right-regular relative to
$(\theta_{*},E_{*})$.

###### Proof.

We will show the point $L_{s+1}/2+1$ is $s+1$-left-regular; the other case is
analogous. Since $L_{s+1}/2+1$ is $s$-regular by Lemma 5.7, it suffices to
check that $[L_{s+1}/2+1,5L_{s+1}/4+1]$ is $s$-nonresonant.

In the case $j_{s+1}=2$, we have
$\theta_{*}\in\mathbf{I}^{(2)}_{s+1,n_{s},\cup}$, an interval of length at
most $\rho_{s-1}/12D_{0}$; thus
$\operatorname{dist}(\theta_{*},\mathbf{I}_{s,-})\leq\rho_{s-1}/12D_{0}$, and
$\operatorname{dist}(\theta_{*}+n_{s}\alpha,\mathbf{I}_{s,+})\leq\rho_{s-1}/12D_{0}$.
Then, by the Diophantine condition,
$\operatorname{dist}(\theta_{*}+m\alpha,\mathbf{I}_{s})>\rho_{s-1}/12D_{0}$
for all $m\in[L_{s+1}/2+1,5L_{s+1}/4+1]$, which implies that interval is
$s$-nonresonant.

Consider instead the case $j_{s+1}=1$. We have

$|\mathbf{E}_{s+1}(\theta_{*})-\mathbf{E}_{s}(\theta_{*})|\leq
C\delta_{s}\ll\rho_{s}/2,$

and for any $m\in[L_{s+1}/2+1,5L_{s+1}/4+1]$,

$0<|m|\leq(5L_{s+1}/4+1)<8M_{s+1}^{(1)};$

by Lemma 5.5, it follows that, for any such $m$, if
$\theta_{*}+m\alpha\in\mathbf{I}_{s}$,

$\displaystyle|\mathbf{E}_{s}(\theta_{*}+m\alpha)-E_{*}|$
$\displaystyle\geq|\mathbf{E}_{s}(\theta_{*})-\mathbf{E}_{s}(\theta_{*}+m\alpha)|-|E_{*}-\mathbf{E}_{s}(\theta_{*})|$
$\displaystyle\geq|\mathbf{E}_{s}(\theta_{*})-\mathbf{E}_{s}(\theta_{*}+m\alpha)|-2\rho_{s}>\rho_{s}.\qed$

###  5.4 Resonant orbits

We note that double resonance of any descendant $\mathbf{E}_{s+1}$ can happen
only at lengths greater than $L_{s+1}$:

######  Lemma 5.13.

If $j_{s+1}=1$, then

$\displaystyle\min\\{|n|:n\in\mathcal{N}_{s+1}\\}>8M_{s+1}^{(1)}\geq L_{s+1}.$

If $j_{s+1}=2$, then $\mathcal{N}_{s+1}=\emptyset$.

###### Proof.

In the case $j_{s+1}=2$, the interval $\mathbf{I}_{s+1}$ is a single interval
of length at most $\rho_{s-1}/24D_{0}$, and $\mathcal{N}_{s+1}=\emptyset$ by
the Diophantine condition.

Consider instead the case $j_{s+1}=1$, and suppose for the sake of
contradiction that there exists $n\in\mathcal{N}_{s+1}$ with $|n|\leq
8M_{s+1}^{(1)}$. Then by definition there exists
$\theta_{s+1,n,-}\in\mathbf{I}_{s+1,-}$ such that
$\theta_{s+1,n,-}+n\alpha\in\mathbf{I}_{s+1,+}$ and
$\mathbf{E}_{s+1}(\theta_{s+1,n,-})=\mathbf{E}_{s+1}(\theta_{s+1,n,-}+n\alpha)$.

Define the function
$g(\theta)=\mathbf{E}_{s}(\theta)-\mathbf{E}_{s}(\theta+n\alpha)$. Since
$j_{s+1}=1$, we have that $|g(\theta_{s+1,n,-})|\leq 2C\delta_{s}$;
furthermore, we have

$|\partial_{\theta}g(\theta)|\geq
2\left(\nu_{s}-C\frac{\delta_{s}}{\rho_{s}}\right)\geq\nu_{s}$

for $\theta\in I_{s+1,n,-}^{(2)}$ by Lemma 5.5. Thus, there exists some point
$\theta_{s,n,-}$ with

$|\theta_{s,n,-}-\theta_{s+1,n,-}|<2C\frac{\delta_{s}}{\nu_{s}}\ll\frac{3}{4D_{0}}\bar{\rho}_{s}$

such that $|g(\theta_{s,n,-})|=0$; since $|n|\leq 8M_{s+1}^{(1)}$,
$n\in\mathcal{N}_{s}$. But $B_{3\bar{\rho}_{s-1}/4D_{0}}(\theta_{s-1,n,-})\cap
I_{s-1,i,-}^{(1)}=\emptyset$ for any $i$ and any $n\in\mathcal{N}_{s}$; thus,
$j_{s+1}=2$, a contradiction. ∎

This allows us to verify a remaining item of the inductive proposition:

######  Lemma 5.14.

If $E_{*}=E_{n_{s+1}}(\mathbf{E}_{s+1})$ for $n_{s+1}\in\mathcal{N}_{s+1}$,
then
$\overline{B}_{\rho_{s}}(E_{*})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$.

###### Proof.

Let $n_{s+1}\in\mathcal{N}_{s+1}$ and $E_{*}=E_{n_{s+1}}$. We have two cases:

  * •

If $\mathbf{E}_{s+1}$ does not attain its minimum at a critical point, since
$E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})$,
$E_{*}-\rho_{s}\in\mathbf{J}_{s+1}\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$,
by Lemma 5.11.

  * •

If $\mathbf{E}_{s+1}$ attains its minimum at a critical point, note that as
$\mathcal{N}_{s+1}$ is nonempty, $j_{s+1}=1$ by Lemma 5.13. Thus, by Lemma
5.10, $\mathbf{E}_{s+1}$ is $C^{2}$-close to $\mathbf{E}_{s}$, so
$\mathbf{E}_{s}$ also attains its minimum at a critical point, with
$|\inf\mathbf{J}_{s}-\inf\mathbf{J}_{s+1}|\leq 2\delta_{s}$. By Lemma 4.7,
$E_{*}-\inf\mathbf{J}_{s+1}\gg 2\delta_{s}$; thus

$\displaystyle E_{*}-\rho_{s}$
$\displaystyle\geq\inf\mathbf{J}_{s+1}+2\delta_{s}-\rho_{s}$
$\displaystyle\geq\inf\mathbf{J}_{s}-\rho_{s}$
$\displaystyle=\inf\widetilde{\mathbf{J}}(\mathbf{E}_{s}).$

In either case,
$E_{*}-\rho_{s}\geq\inf\widetilde{\mathbf{J}}(\mathbf{E}_{s})$; analogously,
$E_{*}+\rho_{s}\leq\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s})$. ∎

We likewise demonstrate that $s$-resonant points $m$ correspond to points
$\theta_{*}+m\alpha$ in the domain of the relevant Rellich function (part of
Hypothesis 5.3.2):

######  Lemma 5.15.

Let $\theta_{*}\in\mathbb{T}$ and
$\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$, and suppose that $E_{*}$
satisfies the following:

  1. 1.

If $j_{s+1}=1$, either $E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})$, or
the closest point in $\mathbf{J}_{s+1}$ to $E_{*}$ is a critical value of
$\mathbf{E}_{s+1}$.

  2. 2.

If $j_{s+1}=2$ (and so
$\mathbf{E}_{s+1}=\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}$), then
$E_{*}\in[\inf\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}),\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\vee}^{(2)})]$

Let $m\in\mathcal{S}_{s}(\theta_{*},E_{*})$. Then:

  1. 1.

If $j_{s+1}=1$, then
$B_{\rho_{s}/24D_{0}}(\theta_{*}+m\alpha)\subset\mathbf{I}_{s+1}$.

  2. 2.

If $j_{s+1}=2$ (and so
$\mathbf{E}_{s+1}=\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}$), then for some
$p\in\\{m,m-n_{s}\\}$,
$B_{\rho_{s}/24D_{0}}(\theta_{*}+p\alpha)\subset\mathbf{I}_{s+1,n_{s},\cup}^{(2)}$.
Furthermore, if
$\theta_{*}+p\alpha\in\mathbf{I}_{s+1,n_{s},\bullet}\setminus\mathbf{I}_{s+1,n_{s},\cap}$,
then

$|\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}(\theta_{*}+p\alpha)-E_{*}|<\frac{9}{8}\rho_{s}.$

###### Proof.

The proof proceeds by case analysis:

  1. 1.

$j_{s+1}=1$:

    1. (a)

$\mathbf{I}_{s+1}$ is a union of two disjoint intervals: In this case,
$\mathbf{E}_{s+1}$ has no critical points, so
$E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})=[\inf\mathbf{J}_{s+1}+\frac{9}{8}\rho_{s},\sup\mathbf{J}_{s+1}-\frac{9}{8}\rho_{s}]$.
Thus,

$\mathbf{E}_{s}(\theta_{*}+m\alpha)\in[\inf\mathbf{J}_{s+1}+\frac{1}{12}\rho_{s},\sup\mathbf{J}_{s+1}-\frac{1}{12}\rho_{s}],$

and so

$\mathbf{E}_{s}\left(B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha)\right)\subset[\inf\mathbf{J}_{s+1}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}].$

On the other hand, one has

$\mathbf{E}_{s,\pm}(\mathbf{I}_{s+1,\pm})\supset[\inf\mathbf{J}_{s+1}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}]$

since $j_{s+1}=1$, and in particular

$\mathbf{I}_{s+1}\supset\mathbf{E}_{s}^{-1}\left([\inf\mathbf{J}_{s+1}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}]\right)\supset
B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha).$

    2. (b)

$\mathbf{I}_{s+1}$ is a single interval: In this case, $\mathbf{E}_{s+1}$ has
a unique critical point in $\mathbf{I}_{s+1}$. Suppose that this critical
point is a minimum (the other case is analogous). Since $j_{s+1}=1$,
$\mathbf{E}_{s}$ likewise achieves its minimum at a critical point, and by
definition of $\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})$ and our assumption on
$E_{*}$ we have

$E_{*}\leq\sup\mathbf{J}_{s+1}-\frac{9}{8}\rho_{s}.$

As above, we find

$\mathbf{E}_{s}\left(B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha)\right)\subset[\inf\mathbf{J}_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}].$

On the other hand, one has that

$\mathbf{E}_{s,\pm}(\mathbf{I}_{s+1,\pm})\supset[\inf\mathbf{J}_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}]$

since $j_{s+1}=1$, and in particular

$\mathbf{I}_{s+1}\supset\mathbf{E}_{s}^{-1}\left([\inf\mathbf{J}_{s},\sup\mathbf{J}_{s+1}-\frac{1}{24}\rho_{s}]\right)\supset
B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha).$

  2. 2.

$j_{s+1}=2$ and $\mathbf{E}_{s+1}=\mathbf{E}_{s+1,n_{s},\bullet}^{(2)}$: By
assumption, we have

$E_{*}\in[\inf\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}),\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\vee}^{(2)})],$

from which it follows that

$\mathbf{E}_{s}\left(B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha)\right)\supset[\inf\mathbf{J}_{s+1,\wedge}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1,\vee}-\frac{1}{24}\rho_{s}]$

in analog to the above. We have two subcases:

    1. (a)

$\theta_{*}+m\alpha\in\mathbf{I}_{s,-}$: Let
$I=[\inf\mathbf{I}_{s+1,\vee},\sup\mathbf{I}_{s+1,\wedge}]\subset\mathbf{I}_{s+1,n_{s},\cup}$.
Since $j_{s+1}=2$, we have

$\mathbf{E}_{s,-}(I)\supset[\inf\mathbf{J}_{s+1,\wedge}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1,\vee}-\frac{1}{24}\rho_{s}],$

from which it follows that

$I\supset\mathbf{E}_{s,-}^{-1}\left([\inf\mathbf{J}_{s+1,\wedge}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1,\vee}-\frac{1}{24}\rho_{s}]\right)\supset
B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha).$

Furthermore, for
$\theta\in(\mathbf{I}_{s+1,n_{s},\vee}\setminus\mathbf{I}_{s+1,n_{s},\wedge})\cap
I$, we have by Lemma 3.9 that

$|\mathbf{E}_{s+1,n_{s},\vee}^{(2)}(\theta)-\mathbf{E}_{s}(\theta)|<C\delta_{s}\ll\rho_{s}/12,$

and similarly for $\vee$ and $\wedge$ swapped. The lemma thus holds in this
case taking $p=m$.

    2. (b)

$\theta_{*}+m\alpha\in\mathbf{I}_{s,+}$: Let
$I=[\inf\mathbf{I}_{s+1,\wedge},\sup\mathbf{I}_{s+1,\vee}]\subset\mathbf{I}_{s+1,n_{s},\cup}$.
Since $j_{s+1}=2$, we have

$\mathbf{E}_{s,+}(I+n_{s}\alpha)\supset[\inf\mathbf{J}_{s+1,\wedge}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1,\vee}-\frac{1}{24}\rho_{s}],$

from which it follows that

$I+n_{s}\alpha\supset\mathbf{E}_{s,+}^{-1}\left([\inf\mathbf{J}_{s+1,\wedge}+\frac{1}{24}\rho_{s},\sup\mathbf{J}_{s+1,\vee}-\frac{1}{24}\rho_{s}]\right)\supset
B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+m\alpha),$

i.e. for $p=m-n_{s}$,

$\mathbf{I}_{s+1,n_{s},\cup}\supset I\supset
B_{\frac{\rho_{s}}{24D_{0}}}(\theta_{*}+p\alpha).$

Furthermore, for
$\theta\in(\mathbf{I}_{s+1,n_{s},\vee}\setminus\mathbf{I}_{s+1,n_{s},\wedge})\cap
I$, we have by Lemma 3.9 that

$|\mathbf{E}_{s+1,n_{s},\vee}^{(2)}(\theta)-\mathbf{E}_{s}(\theta+n_{s}\alpha)|<C\delta_{s}\ll\rho_{s}/12,$

and similarly for $\vee$ and $\wedge$ swapped. The lemma thus holds in this
case taking $p=m-n_{s}$. ∎

###  5.5 Nonresonance

We now prove that the nonresonance hypothesis 5.3.3 holds at scale $s+1$. We
begin by establishing Green’s function decay for $s+1$-nonresonant intervals
provided their left and right endpoints are $(s+1)$-left- and $(s+1)$-right-
regular, respectively.

######  Lemma 5.16.

Let $\theta_{*}\in\mathbb{T}$ and
$E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E}_{s})$ be such that

  * •

If $\mathbf{E}_{s}$ does not attain its maximum at a critical point, then
$E_{*}\leq\sup\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})}\widetilde{\mathbf{J}}(\mathbf{E})$.

  * •

If $\mathbf{E}_{s}$ does not attain its minimum at a critical point, then
$E_{*}\geq\inf\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})}\widetilde{\mathbf{J}}(\mathbf{E})$.

Let $\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$ be a Rellich curve
minimizing $\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}))$
among all $\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})$, and let
$\widehat{\Lambda}=[a,b]\subset\mathbb{Z}$ be an $(s+1)$-nonresonant interval
with $a$ being $(s+1)$-left-regular and $b$ being $(s+1)$-right-regular. Then
for $E\in B_{\rho_{s+1}/2}(E_{*})$ and $\theta\in
B_{\rho_{s+1}/8D_{0}}(\theta_{*})$,

$\log|R^{\widehat{\Lambda}}_{\theta,E}(m,n)|\leq-\gamma_{s+1}|m-n|,\quad|m-n|\geq\ell_{s+1},$

and

$\|R^{\widehat{\Lambda}}_{\theta,E}\|\leq 4\rho_{s+1}^{-1}.$

###### Proof.

Recall the set $\mathcal{S}_{s}(\theta_{*},E_{*})$ (defined relative to the
parent curve $\mathbf{E}_{s}$), and define
$\mathcal{S}_{s}^{\widehat{\Lambda}}:=\mathcal{S}_{s}\cap\widehat{\Lambda}$.
If $\mathcal{S}_{s}^{\widehat{\Lambda}}$ is empty, then $\widehat{\Lambda}$ is
$s$-nonresonant and the lemma follows from the inductive hypothesis. We thus
assume $\mathcal{S}_{s}^{\widehat{\Lambda}}$ is nonempty.

In order to apply Lemmas A.3 and A.1, we wish to find a finite family
$\\{\Lambda_{s+1}+p\\}_{p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}}$
of translates of $\Lambda_{s+1}$ such that each
$m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$ is near the center of
$\Lambda_{s+1}+p$ for some
$p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}$ and
$\|R^{\Lambda_{s+1}+p}_{\theta_{*},E_{*}}\|\leq\rho_{s+1}^{-1}$ for all
$p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}$. We do so by case
analysis:

  1. 1.

$j_{s+1}=1$:

For any $m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$,
$\theta_{*}+m\alpha\in\mathbf{I}_{s+1}$ by Lemma 5.15, and since
$|\mathbf{E}_{s}(\theta_{*}+m\alpha)-E_{*}|<25\rho_{s}/24$ and
$\widehat{\Lambda}$ is $(s+1)$-nonresonant, we have

$\frac{9}{8}\rho_{s}>|\mathbf{E}_{s+1}(\theta_{*}+m\alpha)-E_{*}|\geq\rho_{s+1}.$

By the eigenvalue separation estimates in Section 2 coming from Lemma 5.10, it
follows that $|\lambda-E_{*}|\geq\rho_{s+1}$ for any eigenvalue $\lambda$ of
$H^{\Lambda_{s+1}+m}(\theta_{*})$, and so
$\|R_{\theta_{*},E_{*}}^{\Lambda_{s+1}+m}\|\leq\rho_{s+1}^{-1}$. In this case,
we define
$\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}=\mathcal{S}_{s}^{\widehat{\Lambda}}$.

  2. 2.

$j_{s+1}=2$:

Let $\mathbf{E}_{s+1}=\mathbf{E}_{s+1,n_{s},\vee}^{(2)}$ (the $\wedge$ case is
analogous), and fix $m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$. By Lemma 5.15,
$\theta_{*}+p\alpha\in\mathbf{I}_{s+1,n_{s},\cup}^{(2)}$ for some
$p=p_{m}\in\\{m,m-n_{s}\\}$. By the regularity assumption on the endpoints of
$\widehat{\Lambda}$, we must have that $p\in\widehat{\Lambda}$. We have
subcases:

    1. (a)

$\theta_{*}+p\alpha\in\mathbf{I}_{s+1,n_{s},\cap}^{(2)}$: We have
$|\mathbf{E}_{s+1}(\theta_{*}+p\alpha)-E_{*}|\geq\rho_{s+1}$. To separate
$E_{*}$ from $\mathbf{E}^{(2)}_{s+1,n_{s},\wedge}(\theta_{*}+p\alpha)$, we
crucially use the uniform separation between the images of
$\mathbf{E}^{(2)}_{s+1,n_{s},\vee}$ and $\mathbf{E}^{(2)}_{s+1,n_{s},\wedge}$
guaranteed by Proposition 3.6. This separation is greater than $2\rho_{s+1}$,
which implies that $\widetilde{\mathbf{J}}(\mathbf{E}^{(2)}_{s+1,n_{s},\vee})$
and $\widetilde{\mathbf{J}}(\mathbf{E}^{(2)}_{s+1,n_{s},\wedge})$ are
disjoint; since, by assumption,

$\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}_{s+1}))\leq\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\wedge}^{(2)})),$

we have

$E_{*}\geq\sup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1,n_{s},\wedge}^{(2)})=\sup\mathbf{J}_{\wedge}+\rho_{s+1},$

and so

$|\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}(\theta_{*}+p\alpha)-E_{*}|\geq\rho_{s+1}.$

It remains to separate $E_{*}$ from other eigenvalues of
$H^{\Lambda_{s+1}+p}$. Because

$\operatorname{dist}\left(E_{*},\mathbf{E}_{s}(\mathbf{I}_{s+1,n_{s},\cap}^{(2)}\cup(\mathbf{I}_{s+1,n_{s},\cap}^{(2)}+n_{s}\alpha))\right)<\frac{25}{24}\rho_{s}\ll\frac{1}{12}\rho_{s}$

and by construction

$|\mathbf{E}_{s}(\mathbf{I}_{s+1,n_{s},\cap}^{(2)}\cup(\mathbf{I}_{s+1,n_{s},\cap}^{(2)}+n_{s}\alpha))|\ll\frac{1}{4}\rho_{s-1},$

it follows that

$|E_{*}-\mathbf{E}_{s}(\theta_{s,n_{s},-})|<\frac{1}{3}\rho_{s-1}=\frac{1}{2}\left(\frac{2}{3}\rho_{s-1}\right).$

Thus we may apply the estimate on
$\|R_{\perp}(E_{*};H^{\Lambda_{s+1}+p},\\{\mathbf{E}_{s+1,\vee}\\}\cup\\{\mathbf{E}_{s+1,\wedge}\\})\|$
coming from Section 3 and Lemma 5.10. It follows that
$\|R_{\theta_{*},E_{*}}^{\Lambda_{s+1}+p}\|\leq\rho_{s+1}^{-1}$.

    2. (b)

$\theta_{*}+p\alpha\in\mathbf{I}_{s+1,n_{s},\vee}^{(2)}\setminus\mathbf{I}_{s+1,n_{s},\wedge}^{(2)}$
(The case with $\vee$ and $\wedge$ swapped is analogous.): Lemma 5.15 and the
eigenvalue separation estimate from Lemma 3.9 ensure $|\lambda-
E_{*}|\geq\rho_{s+1}$ for any eigenvalue
$\lambda\neq\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}$ of
$H^{\Lambda_{s+1}+p}(\theta_{*})$. As in the $\mathbf{I}_{s+1,n_{s},\cap}$
case,

$|\mathbf{E}_{s+1,n_{s},\wedge}^{(2)}(\theta_{*}+p\alpha)-E_{*}|\geq\rho_{s+1}.$

It follows that
$\|R_{\theta_{*},E_{*}}^{\Lambda_{s+1}+p}\|\leq\rho_{s+1}^{-1}$.

We denote by $\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}$ the set of all
integers $p$ coming from $m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$ as above.

For each interval $\Lambda_{s+1}+p$,
$p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}$, it follows immediately
from $\|R_{\theta_{*},E_{*}}^{\Lambda_{s+1}+p}\|\leq\rho_{s+1}^{-1}$ that, for
$E\in B_{3\rho_{s+1}/4}(E_{*})$ and $\theta\in
B_{\rho_{s+1}/8D_{0}}(\theta_{*})$, that

$\|R_{\theta,E}^{\Lambda_{s+1}+p}\|\leq 8\rho_{s+1}^{-1}.$

By the $s+1$-regularity of the endpoints of $\widehat{\Lambda}=[a,b]$, for any
$m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$, we have

$\min\\{|m-a|,|m-b|\\}\geq 3L_{s+1}/4;$

since $|p-m|\leq 8M_{s+1}^{(1)}\ll L_{s+1}/4$, we have that
$\Lambda_{s+1}+p\subset\Lambda$ for all
$p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}$. Moreover, by Lemma 5.5,
the intervals
$\\{\Lambda_{s+1}+p\\}_{p\in\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}}$
are non-overlapping. Enumerating
$\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}=\\{p_{i}\\}_{i=1}^{|\overline{\mathcal{S}}_{s}^{\widehat{\Lambda}}|}$
in increasing order, we denote by
$\widetilde{\Lambda}_{i}:=[p_{i},p_{i}+n_{s}]$ (with the convention that
$n_{s}=0$ if $j_{s+1}=1$) and define the intervals
$\widetilde{\Lambda}_{i,l/r}$ to be the maximal intervals to the left/right of
$\widetilde{\Lambda}_{i}$, i.e.

$\displaystyle\widetilde{\Lambda}_{i,l}$
$\displaystyle=[\max\widetilde{\Lambda}_{i-1}+1,\min\widetilde{\Lambda}_{i}-1]=:[\widetilde{a}_{i,l},\widetilde{b}_{i,l}],$
$\displaystyle\widetilde{\Lambda}_{i,r}$
$\displaystyle=[\max\widetilde{\Lambda}_{i}+1,\min\widetilde{\Lambda}_{i+1}-1]=:[\widetilde{a}_{i,r},\widetilde{b}_{i,r}].$

We note that $\widetilde{\Lambda}_{i,r}=\widetilde{\Lambda}_{i+1,l}$, but we
fix this notation to emphasize the relationship of these intervals to
$\widetilde{\Lambda}_{i}$.

Again by Lemma 5.5, each interval $\widetilde{\Lambda}_{i,l/r}$ is
$s$-nonresonant and has length at least $L_{s+1}\gg 2L_{s}+3M_{s}^{(1)}$, and
thus each such interval satisfies the inductive hypothesis 5.3.3. In
particular, there exist intervals
${\Lambda}_{i,l/r}=[{a}_{i,l/r},{b}_{i,l/r}]\subset\widetilde{\Lambda}_{i,l/r}$
satisfying the Green’s function decay property for $(\ell_{s},\gamma_{s})$
with

$\max\\{|\widetilde{a}_{i,l/r}-{a}_{i,l/r}|,|\widetilde{b}_{i,l/r}-{b}_{i,l/r}|\\}\leq
2L_{s}+3M_{s}^{(1)}\ll\frac{1}{2}M_{s+1}^{(1)}.$

Moreover, by the regularity of the endpoints of $\widehat{\Lambda}$, we can
choose

${a}_{1,l}=\widetilde{a}_{1,l}=a,\quad{b}_{|\overline{\mathcal{S}}^{\widehat{\Lambda}}_{s}|,r}=\widetilde{b}_{|\overline{\mathcal{S}}^{\widehat{\Lambda}}_{s}|,r}=b.$

Define ${\Lambda}_{i}=[{b}_{i,l}+1,{a}_{i,r}-1]$. We have
${\Lambda}_{i}\subset\Lambda_{s+1}+p_{i}\subset\widehat{\Lambda}$ and
$\|R^{\Lambda_{s+1}+p_{i}}\|\leq 8\rho_{s+1}^{-1}$, so we may apply Lemma A.3
to conclude that $\widehat{\Lambda}$, partitioned by the $\Lambda_{i}$ and
$\Lambda_{i,l/r}$ subintervals, satisfies Assumption 4 with
$({\ell},{\gamma})=(\ell_{s},\gamma_{s})$ and $\widehat{\ell}=\ell_{s+1}$. By
Lemma A.1, $\widehat{\Lambda}$ satisfies the Green’s function decay property
for $(\ell_{s+1},\gamma_{s+1})$ where
$\gamma_{s+1}=\gamma_{s}-16|\log\varepsilon\log(\rho_{s+1}/8)|/\ell_{s+1}$.

Since we have shown the Green’s function decay property for
$|E-E_{*}|<\frac{3}{4}\rho_{s+1}$, it follows that the resolvent is well-
defined for $|E-E_{*}|<\frac{3}{4}\rho_{s+1}$, i.e.,
$E\notin\operatorname{sp}H^{\widehat{\Lambda}}(\theta)$. Thus, for
$|E-E_{*}|\leq\rho_{s+1}/2$, $\|R^{\widehat{\Lambda}}_{\theta,E}\|\leq
4\rho_{s+1}^{-1}$. ∎

Finally, the following lemma shows that if the endpoints of
$\widehat{\Lambda}$ are not regular, we can adjust them (provided
$\widehat{\Lambda}$ is sufficiently long) to find a subinterval which does
satisfy the conditions of Lemma 5.16.

######  Lemma 5.17.

Let
$\theta_{*},E_{*},\mathbf{E}_{s+1}\in\mathcal{E}^{(j_{s+1})}(\mathbf{E}_{s})$
satisfy the relations stated in Lemma 5.16, and let $\widehat{\Lambda}=[a,b]$
with $|\widehat{\Lambda}|\geq 2L_{s+1}+3M_{s+1}^{(1)}$. Then there exists a
subinterval $\tilde{\Lambda}=[\tilde{a},\tilde{b}]\subset\Lambda$ with
$|\tilde{\Lambda}|\geq|\widehat{\Lambda}|-(2L_{s+1}+3M_{s+1}^{(1)})$ such that
$\tilde{a}$ is $s+1$-left-regular and $\tilde{b}$ is $s+1$-right-regular.

###### Proof.

In the following, if $j_{s+1}=1$, we adopt the convention that $n_{s}=0$.

Let $m_{a}$ minimize $|m-a|$ among $m\in\mathcal{S}_{s}^{\widehat{\Lambda}}$.
If $\theta_{*}+m_{a}\alpha\in\mathbf{I}_{s}$ and $|m_{a}-a|<L_{s}$, we replace
$a$ with $a_{s-1}=\max\\{m_{a}+1,m_{a}+n_{0}+1\\}$; similarly, if
$\theta_{*}+m_{a}\alpha\in\mathbf{I}_{s+1}+n_{s}\alpha$ and
$|m_{a}-a|<L_{s+1}$, replace $a$ with $a_{s}=\max\\{m_{a}+1,m_{a}-n_{s}+1\\}$.
Otherwise $a_{s}=a$. We perform analogous adjustments to $b$. We repeat this
process for each $s-1\geq i\geq 1$, with $i$ replacing $s$, and set
$\tilde{a}=a_{0}$ and $\tilde{b}=b_{0}$. By the inductive proposition, these
endpoints satisfy the necessary regularity conditions. ∎

###  5.6 Proof of Proposition 5.3 and Theorem 5.4

###### Proof.

The lemmas above complete the proof of inductive Proposition 5.3 at scale $s$.
The integers $L_{s+1}$ satisfy the Proposition by Lemmas 5.7 and 5.12. Any
function in $\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$ is a Rellich
function of some $H^{\Lambda_{s+1}}$ satisfying Assumption 3 by Lemma 5.11;
such a function satisfies inductive hypothesis 5.3.1 by Lemma 5.10, hypothesis
5.3.2 by Lemmas 5.13 and 5.15, and hypothesis 5.3.3 by Lemmas 5.16 and 5.17.
For any $\mathbf{E}_{s+1}\in\mathcal{E}(\mathbf{E}_{s})$,
$\mathbf{J}_{s+1}\cup\widetilde{\mathbf{J}}(\mathbf{E}_{s+1})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$
by Lemma 5.11; and if $E_{*}=E_{n_{s+1}}(\mathbf{E}_{s+1})$ for
$n_{s+1}\in\mathcal{N}_{s+1}$, then
$\overline{B}_{\rho_{s}}(E_{*})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{s})$
by Lemma 5.14. Item 4 follows from Lemma 5.5.

Since our choice of $\varepsilon<\varepsilon_{0}=\rho_{0}^{4}$ was arbitrary
provided $L$ was sufficiently long as in Lemma 5.2, Theorem 5.4 holds. ∎

##  6 Localization and Cantor spectrum

With the induction argument of Section 5 complete, we have now constructed a
tree of Rellich functions $\mathbf{E}_{s}$, cf. Figure 4, with each child
classified as being either simple- or double-resonant relative to its parent.
In this section we relate this tree to spectral information about the limiting
operator $H$. In particular, we will show the following:

  1. 1.

The spectral points of $H$ are precisely the limit points of the modified
codomains $\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$ along any
infinite path $\\{\mathbf{E}_{k}\\}_{k=0}^{\infty}$ in our Rellich tree.

  2. 2.

Every Rellich function $\mathbf{E}$ has some double-resonant descendant; by
the uniform local separation estimate from Proposition 3.6, the gap that this
double resonance opens is large enough to remain open for all future scales,
guaranteeing Cantor spectrum.

  3. 3.

The set $\mathrm{B}$ of bad phases $\theta_{*}$ which encounter a scale-$k$
double resonance on an orbit of approximate size $M_{k+1}^{(2)}$ for
infinitely many scales $k$ has zero measure. The full-measure complement
$\Theta$ of this bad set will have the property that, for $\theta\in\Theta$,
any generalized eigenvalue $E(\theta)$ of $H(\theta)$ corresponds to a path
$\\{\mathbf{E}^{(j_{k})}_{k}\\}_{k=0}^{\infty}$ which eventually consists only
of simple resonances, i.e. there exists some $K$ such that $j_{k}=1$ for all
$k\geq K$. In fact, we will show that this path can be chosen so that the
corresponding center of localization $m_{k}$ eventually remains fixed; this
will yield exponential localization of the corresponding generalized
eigenfunction, hence Anderson localization.

To this end, it will be useful to introduce some notation to refer to
different parts of our Rellich tree. Recall that $\mathcal{E}(\mathbf{E})$
denotes the collection of all immediate children of the Rellich curve
$\mathbf{E}$, and that $\mathcal{E}_{s}$ denotes the collection of all
scale-$s$ Rellich functions, i.e. the $s^{th}$ “generation” of the tree. For a
Rellich function $\mathbf{E}\in\mathcal{E}_{k}$, $k\leq s$, we denote by
$\mathcal{E}_{s}(\mathbf{E})$ the collection of all the scale-$s$ descendants
of $\mathbf{E}$; i.e., for a scale-$k$ Rellich function $\mathbf{E}_{k}$,
$\mathcal{E}_{k}(\mathbf{E}_{k})=\\{\mathbf{E}_{k}\\}$, and for all $s\geq k$,
if $\mathbf{E}^{\prime}\in\mathcal{E}_{s}(\mathbf{E}_{k})$ and
$\mathbf{E}^{\prime\prime}\in\mathcal{E}(\mathbf{E}^{\prime})$, then
$\mathbf{E}^{\prime\prime}\in\mathcal{E}_{s+1}(\mathbf{E}_{k})$. With this
notation we have $\mathcal{E}_{k}=\mathcal{E}_{k}(\mathbf{E}_{0})$; we denote
by $\mathcal{E}:=\bigcup_{k\geq 0}\mathcal{E}_{k}$ the entire Rellich tree.

###  6.1 Characterization of the spectrum

In this first subsection, we use paths through the tree of Rellich curves to
characterize the spectrum of $H$.

First, note that there can be at most one energy common to the modified
codomains $\widetilde{\mathbf{J}}(\mathbf{E}_{k})$ of all Rellich functions in
an infinite path:

######  Lemma 6.1.

Let $\\{\mathbf{E}_{k}\\}_{k=0}^{\infty}$ be a sequence of Rellich functions
such that $\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$ for all $k\geq 0$.
Then
$\liminf_{k\to\infty}|{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})|=\liminf_{k\to\infty}|\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})|=0$.
In particular,
$\bigcap_{k=0}^{\infty}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$
contains at most a single point.

###### Proof.

By construction,
$|\widetilde{\mathbf{J}}(\mathbf{E}_{k})|,|\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})|\leq
4\bar{\rho}_{k-1}\to 0$. ∎

Thus, each infinite path characterizes a single point; we will show these are
spectral points. Define the sets

$\Sigma:=\left\\{E\in\mathbb{R}:E=\bigcap_{k=0}^{\infty}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k}),\;\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})\,\forall
k\geq 0\right\\}$

and

$G:=(\mathbb{R}\setminus\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{0}))\cup\bigcup_{\mathbf{E}\in\mathcal{E}}\left(\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\setminus\bigcup_{\mathbf{E}^{\prime}\in\mathcal{E}(\mathbf{E})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}^{\prime})\right).$

We will show that $\Sigma$ characterizes the spectrum of $H(\theta)$, while
$G$ characterizes the spectral gaps. We first show that points in $\Sigma$ are
indeed spectral:

######  Lemma 6.2.

$\Sigma\subset\operatorname{sp}H(\theta).$

###### Proof.

Let $E\in\Sigma$. Then for each $k\geq 0$, we can find a
$(\theta_{k},E_{k})\in\mathbb{T}\times\mathbb{R}$, with
$|E-E_{k}|\leq\rho_{k}$, and $\mathbf{E}_{k}\in\mathcal{E}$ such that
$H^{\Lambda_{k}}\psi_{k}=\mathbf{E}_{k}(\theta_{k})\psi_{k}=E_{k}\psi_{k}$. By
irrationality of $\alpha$, we can find some $m_{k}\in\mathbb{Z}$ such that
$\|(\theta+m_{k}\alpha)-\theta_{k}\|_{\mathbb{T}}<\rho_{k}/D_{0}$. Let
$\mathcal{P}$ be the partition of $\mathbb{Z}$ into $\Lambda_{k}$ and its
complement, and let $\psi_{k}^{m_{k}}$ denote $\psi_{k}$ shifted by $m_{k}$.
Then

$\displaystyle\|(H(\theta)-E)\psi_{k}^{m_{k}}\|$
$\displaystyle=\|(H(\theta+m_{k}\alpha)-E)\psi_{k}\|$
$\displaystyle\leq\|H(\theta+m_{k}\alpha)-H(\theta_{k})\|+\|(H(\theta_{k})-E_{k})\psi_{k}\|+|E_{k}-E|$
$\displaystyle\leq\|V(\theta+m_{k}\alpha)-V(\theta_{k})\|+\|\Gamma_{\mathcal{P}}\psi_{k}\|+|E_{k}-E|$
$\displaystyle<3\rho_{k}$

Thus $\\{\psi_{k}^{m_{k}}\\}_{k=0}^{\infty}$ forms a Weyl sequence for $E$,
and $E\in\operatorname{sp}H(\theta)$. ∎

We now establish an important technical lemma demonstrating that energies
outside of the modified codomains of all children of a Rellich curve are
uniformly nonresonant on sufficiently long intervals $\Lambda$:

######  Lemma 6.3.

Let
$E_{*}\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\setminus\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})$.
Then for any $\theta_{*}\in\mathbb{T}$, there are arbitrarily long intervals
$\tilde{\Lambda}$ with the $(\ell_{k+1},\gamma_{k+1})$ Green’s function decay
property.

###### Proof.

It suffices to show that the hypotheses of Lemma 5.17 are met by $E_{*}$, any
$\theta_{*}\in\mathbb{T}$, and any sufficiently long interval
$\Lambda\subset\mathbb{Z}$.

Suppose $\mathbf{E}_{k}$ does not attain its maximum at a critical point. We
have
$E_{*}\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\subset{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$,
and, by item 2 of Lemma 5.11,

$E_{*}\leq\sup\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\leq\sup\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\leq\sup\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\mathbf{J}}(\mathbf{E})$

Similarly
$E_{*}\geq\inf\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\mathbf{J}}(\mathbf{E})$
if $\mathbf{E}_{k}$ does not attain its minimum at a critical point.

Now suppose that
$E_{*}\in\widetilde{\mathbf{J}}(\mathbf{E})\setminus\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})$
for some $\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})$. By definition of
$\widetilde{\mathbf{J}}$ and $\widetilde{\widetilde{\mathbf{J}}}$, this means
that $E_{*}$ is near a non-critical-point extremum of $\mathbf{E}$, and by
item 3 of Lemma 5.11, it follows that
$E_{*}\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}^{\prime})$ for some
other $\mathbf{E}^{\prime}\in\mathcal{E}(\mathbf{E}_{k})$, a contradiction.
Thus we have $E_{*}\notin\widetilde{\mathbf{J}}(\mathbf{E})$ for any
$\mathbf{E}\in\mathcal{E}(\mathbf{E}_{s})$.

Now let $\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})$ minimize
$\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}))$ among all
$\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})$, let $\mathbf{I}$ be its domain,
and let $\mathbf{J}$ be its image. By the above comments, this distance is
nonzero; thus, $E_{c}:=\arg\min_{E\in\mathbf{J}}|E_{*}-E|$ is an extremum of
$\mathbf{E}$. But by item 3 of Lemma 5.11, if this extremum were not a
critical point, there would be some
$\mathbf{E}^{\prime}\in\mathcal{E}(\mathbf{E}_{k})$ with
$\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}^{\prime}))<\operatorname{dist}(E_{*},\widetilde{\mathbf{J}}(\mathbf{E}))$,
which is not the case; thus, $E_{c}$ is a critical point of $\mathbf{E}$.
Furthermore, since $E_{*}\notin\widetilde{\mathbf{J}}(\mathbf{E})$,
$\operatorname{dist}(E_{*},\mathbf{J})>\rho_{k+1}$. It follows that for any
$m\in\mathbb{Z}$, either $|\mathbf{E}(\theta_{*}+m\alpha)-E_{*}|>\rho_{k+1}$,
or $\theta_{*}+m\alpha\notin\mathbf{I}$. Thus, all the hypotheses of Lemma
5.17 are met. ∎

As a consequence of this result, we can separate the energies in $G$ from
generalized eigenvalues of $H$:

######  Lemma 6.4.

If $E_{*}\in G$, then there is some $\rho>0$ such that for $|E-E_{*}|<\rho$,
$E$ is not a generalized eigenvalue of $H(\theta)$ corresponding to a
generalized eigenvector $\psi$ growing at most quadratically
($|\psi(j)|<C(j^{2}+1)$).

###### Proof.

By Lemma 6.3, there is some scale $k\geq 0$ for which we have
$(\ell_{k},\gamma_{k})$ Green’s function decay on arbitrary long intervals;
i.e., for $|E-E_{*}|<\rho_{k}/2$,

$\log|R^{\tilde{\Lambda}}_{E,\theta}(m,n)|\leq-\gamma_{k}|m-n|,\quad|m-n|\geq\ell_{k},$

for arbitrarily long intervals $\tilde{\Lambda}$. Denote
$\tilde{\Lambda}=[\tilde{a},\tilde{b}]$.

Assume $E$ is generalized eigenvalue of $H(\theta)$ corresponding to a
generalized eigenvector $\psi$ growing at most quadratically
($|\psi(j)|<C(j^{2}+1)$). Fix $j\in\mathbb{Z}$, and choose $\tilde{\Lambda}\ni
j$ large enough so
$\min\\{|\tilde{a}-j|,|\tilde{b}-j|\\}\geq|\tilde{\Lambda}|/3\geq\ell_{k}$. By
the Poisson formula (1.11),

$\displaystyle|\psi(j)|$ $\displaystyle\leq
e^{-\gamma_{k}|\tilde{\Lambda}|/3}\left(|\psi(\tilde{a}-1)|+|\psi(\tilde{b}+1)|\right)$
$\displaystyle\leq
2Ce^{-\gamma_{k}|\tilde{\Lambda}|/3}((|j|+|\tilde{\Lambda}|)^{2}+1)$

As $|\tilde{\Lambda}|\to\infty$, the right-hand side approaches $0$, so we
must have $|\psi(j)|=0$. Since this is true for arbitrary $j$, $\psi$ must be
identically zero, i.e., $E$ must not be a generalized eigenvalue. ∎

Combining the above results with the sets’ definitions, we arrive at the
characterization of $\operatorname{sp}H(\theta)$:

######  Lemma 6.5.

$\operatorname{sp}H(\theta)=\mathbb{R}\setminus G=\Sigma.$

###### Proof.

By Lemma 6.2, $\Sigma\subset\operatorname{sp}H(\theta)$; and since generalized
eigenvalues are dense in $\operatorname{sp}H(\theta)$ by Schnol’s Lemma, Lemma
6.4 implies that $\operatorname{sp}H(\theta)\subset\mathbb{R}\setminus G$. It
remains to show that $\mathbb{R}\setminus G\subset\Sigma$. Let
$E\in\mathbb{R}\setminus G$. The definition of $G$ ensures that
$E\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{0})$, and if
$E\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$, then
$E\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k+1})$ for some
$\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$. But this means precisely
that $E\in\Sigma$; thus $\mathbb{R}\setminus G\subset\Sigma$. ∎

###  6.2 Cantor spectrum

Having characterized the spectrum, we now demonstrate that $\Sigma$ is a
Cantor set, i.e. it is a (closed) nowhere-dense set without isolated points.
To do so, we demonstrate that the spectral gaps $G$ always meet our modified
codomains:

######  Lemma 6.6.

For any $\mathbf{E}\in\mathcal{E}$, ${\widetilde{\mathbf{J}}}(\mathbf{E})\cap
G$ is nonempty.

###### Proof.

We proceed by contradiction: suppose there is some
$\mathbf{E}_{k}\in\mathcal{E}_{k}$ such that
${\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\cap G=\varnothing$.

We first note that $\mathbf{E}_{k}$ cannot contain a critical point. Suppose
it attained its minimum at a critical point; then we would have

$\displaystyle\inf\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$
$\displaystyle=\inf\mathbf{\operatorname{im}}\mathbf{E}_{k}-\rho_{k}$
$\displaystyle<\inf_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\inf\operatorname{im}\mathbf{E}-\rho_{k+1}$
$\displaystyle=\inf_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\inf\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}).$

Then
$\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\setminus\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\subset{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\cap
G$ is nonempty, contradicting our initial assumption. Analogously, if
$\mathbf{E}_{k}$ attains its maximum at a critical point, we reach the same
contradiction. Thus, $\mathbf{E}_{k}$ does not contain a critical point.

Let $n\in\mathcal{N}(\alpha,\mathbf{E}_{k})$ such that
$E_{n}\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$ and
$\operatorname{dist}(E_{n},\partial\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k}))\geq\rho_{k-1}/12$;
by irrationality of $\alpha$, such an $n$ exists. We will show that for some
$\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$,
$n\in\mathcal{N}(\alpha,\mathbf{E}_{k+1})$ and
$E_{n}(\mathbf{E}_{k+1})\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k+1})$
with
$\operatorname{dist}(E_{n}(\mathbf{E}_{k+1}),\partial\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k+1}))\geq\rho_{k}/12$.
First note that if $E_{n}\in J_{k,n_{0}}^{(2)}$ for some
$n_{0}\in\mathcal{N}_{k+1}$, then, by construction, there are a pair of
double-resonant Rellich functions
$\mathcal{E}(\mathbf{E}_{k})\ni\mathbf{E}_{k+1,\bullet}:\mathbf{I}^{(2)}_{k+1,n,\bullet}\to\mathbf{J}^{(2)}_{k+1,n,\bullet}\,(\bullet\in\\{\vee,\wedge\\})$
with
$\mathbf{J}^{(2)}_{k+1,n,\bullet}\subset\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$;
since the uniform local separation between double-resonant Rellich pairs
guaranteed by Proposition 3.6 is larger than $2\rho_{k+1}$, there is some
$E\in(\sup\mathbf{J}^{(2)}_{k+1,n,\wedge}+\rho_{k+1},\inf\mathbf{J}^{(2)}_{k+1,n,\vee}-\rho_{k+1})$,
which must then satisfy

$E\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})\setminus\bigcup_{\mathbf{E}\in\mathcal{E}(\mathbf{E}_{k})}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{k})\cap
G,$

contradicting our assumption. Thus $E_{n}$ is contained in the images of only
simple-resonant functions $\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$. By
Lemma 5.11 item 4, we can find some such function with
$B_{\rho_{k}/8}(E_{n})\subset\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$.
Let $E_{\pm}=E_{n}\pm\rho_{k}/24$. Then, since
$\partial_{E}T_{\mathbf{E}_{k}}\geq 2/D_{0}$ by the inverse function theorem,

$\displaystyle T_{\mathbf{E}_{k+1}}(E_{+})$ $\displaystyle\geq
T_{\mathbf{E}_{k}}(E_{+})-\frac{4\delta_{s}}{\nu_{s}}$ $\displaystyle\geq
T_{\mathbf{E}_{k}}(E_{n})+\frac{\rho_{k}}{12D_{0}}-\frac{4\delta_{s}}{\nu_{s}}$
$\displaystyle\geq T_{\mathbf{E}_{k}}(E_{n})$

where the first line follows from Lemma 4.13. Similarly,

$\displaystyle T_{\mathbf{E}_{k+1}}(E_{-})$ $\displaystyle\leq
T_{\mathbf{E}_{k}}(E_{n}).$

By continuity of $T_{\mathbf{E}_{k+1}}$, there is some $E_{*}\in[E_{-},E_{+}]$
with $T_{\mathbf{E}_{k+1}}(E_{*})=T_{\mathbf{E}_{k}}(E_{n})$; by definition,
this means $n\in\mathcal{N}(\alpha,\mathbf{E}_{k+1})$ with
$E_{n}(\mathbf{E}_{k+1})=E_{*}$. By construction,
$E_{*}\in\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k+1})$ with
$\operatorname{dist}(E_{*},\partial\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k+1}))\geq\rho_{k+1}/12$.

But
$\widetilde{\mathbf{J}}(\mathbf{E}_{k+1})\subset\widetilde{\mathbf{J}}(\mathbf{E}_{k})$
by Lemma 5.11, so we can repeat the above procedure to find
$\mathbf{E}_{k+2}\in\mathcal{E}(\mathbf{E}_{k+1})$ with
$n\in\mathcal{N}(\alpha,\mathbf{E}_{k+2})$ satisfying the above conditions. As
we continue to repeat this procedure, we will eventually reach some scale $s$
with $|n|\leq 8M_{s+1}^{(1)}$; since $n\in\mathcal{N}(\alpha,\mathbf{E}_{s})$,
we will have $n\in\mathcal{N}_{s}$, and we will construct a double-resonant
interval $J_{s,n}^{(2)}$ with $E_{n}(\mathbf{E}_{s})\in J_{s,n}^{(2)}$. As we
have shown above, this leads to a contradiction with our original assumption.
∎

###### Proof of Cantor spectrum.

It suffices to show that for each $E\in\operatorname{sp}H(\theta)$, there are
points arbitrarily close to $E$ which are not in $\operatorname{sp}H(\theta)$.
Fix $E\in\operatorname{sp}H(\theta)$. By Lemmas 6.1 and 6.5, we can find some
$\mathbf{E}\in\mathcal{E}$ with
$\widetilde{\mathbf{J}}(\mathbf{E})\supset\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\ni
E$ and $|{\widetilde{\mathbf{J}}}(\mathbf{E})|$ arbitrarily small. By Lemma
6.6, $\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E})\ni E^{\prime}$ for some
$E^{\prime}\in G$; by Lemma 6.5, $E^{\prime}\notin\operatorname{sp}H(\theta)$.
∎

###  6.3 Localization

We first construct the full-measure set on which localization occurs. Using
the definitions preceding Lemma 4.8, we define the double-resonant regions

$J^{DR}_{n}(\overline{\rho}_{k},12M_{k+1}^{(2)},\alpha,\mathbf{E}):=\overline{B}_{\overline{\rho}_{k}}(E_{n}(\mathbf{E})),\quad
n\in\mathcal{N}(12M_{k+1}^{(2)},\alpha,\mathbf{E})$

of each function $\mathbf{E}\in\mathcal{E}_{k}$, and we define

$J^{DR}=\bigcup_{n\in\mathcal{N}(12M_{k+1}^{(2)},\alpha,\mathbf{E})}J^{DR}_{n}.$

We now define and measure the sets

$\mathrm{B}_{k}:=\bigcup_{\mathbf{E}\in\mathcal{E}_{k}}\bigcup_{|n|\leq
12M_{k+1}^{(2)}}\left(\mathbf{E}^{-1}(J^{DR})+n\alpha\right)$

on which two resonances at scale $k$ occur at nearby shifts, and define

$\mathrm{B}:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}\mathrm{B}_{k}.$

######  Lemma 6.7.

$|\mathrm{B}|=0$.

###### Proof.

By construction, for any $\mathbf{E}\in\mathcal{E}_{k}$,
$|\mathcal{E}(\mathbf{E})|\leq
10\lceil\bar{\rho}_{k}^{-1}\rceil\ll\rho_{k-1}^{-3}$. Thus, for $k\geq 1$,

$|\mathcal{E}_{k}|\leq|\mathcal{E}_{0}|\prod_{i=0}^{k-1}\rho_{i-1}^{-3}\leq\rho_{k-2}^{-4}.$
(6.1)

Fix $\mathbf{E}\in\mathcal{E}_{k}$. By definition, each
$|J_{n}^{DR}|=2\overline{\rho}_{k}$; by Lemma 4.2,
$|\mathbf{E}^{-1}(J_{n}^{DR})|\leq
2\sqrt{\frac{12}{d_{k}}\overline{\rho}_{k}}\ll\rho_{k-1}^{1/2}$. Then
$|\mathbf{E}^{-1}(J^{DR})|=\left|\sum_{n\in\mathcal{N}(12M_{k+1}^{(2)},\alpha,\mathbf{E})}\mathbf{E}^{-1}(J^{DR}_{n})\right|\leq
25M_{k+1}^{(2)}\rho_{k-1}^{1/2}$, and $\left|\bigcup_{|n|\leq
12M_{k+1}^{(2)}}\left(\mathbf{E}^{-1}(J^{DR})+n\alpha\right)\right|\leq
625(M_{k+1}^{(2)})^{2}\rho_{k-1}^{1/2}$.

Combining this with the above estimate (6.1), we have $|\mathrm{B}_{k}|\leq
625(M_{k+1}^{(2)})^{2}\rho_{k-2}^{-4}\rho_{k-1}^{1/2}$, and
$\sum_{k=1}^{\infty}|\mathrm{B}_{k}|\leq
625\sum_{k=0}^{\infty}(M_{k+1}^{(2)})^{2}\rho_{k-2}^{-4}\rho_{k-1}^{1/2}<\infty$.
Thus, by the Borel-Cantelli Lemma, $|\mathrm{B}|=0$. ∎

Thus $\Theta:=\mathbb{T}\setminus\mathrm{B}$ is a set of full measure. For the
remainder of this section, we fix $\theta\in\Theta$.

To show that $H(\theta)$ is Anderson localized, it suffices by Schnol’s lemma
[22] to show that every generalized eigenvector $\psi$ of $H(\theta)$ that
grows at most quadratically ($|\psi(j)|<C(j^{2}+1)$) in fact decays
exponentially. For the remainder of this section, we fix a generalized
eigenvalue $E(\theta)$ and corresponding generalized eigenvector $\psi$. By
Lemma 6.5, we can also fix a sequence $\\{\mathbf{E}_{k}\\}_{k=0}^{\infty}$,
with $\mathbf{E}_{k+1}\in\mathcal{E}(\mathbf{E}_{k})$ for all $k\geq 0$, such
that
$E\in\bigcap_{k=0}^{\infty}\widetilde{\widetilde{\mathbf{J}}}(\mathbf{E}_{k})$.

To locate the center of localization for $\psi$, we first employ an argument
similar to Lemma 6.4:

######  Lemma 6.8.

There is a $K\in\mathbb{N}$ such that for all $k\geq K$, there is an
$m_{k}\in\mathbb{Z}$ satisfying $|m_{k}|\leq 3M_{k}^{(2)}$ such that
$\theta+m_{k}\alpha\in\mathbf{I}_{k}$ and
$|E-\mathbf{E}_{k}(\theta+m_{k}\alpha)|<\rho_{k}$.

###### Proof.

If not, there is an increasing integer sequence $\\{k_{i}\\}_{i=0}^{\infty}$
such that $[-3M_{k_{i}}^{(2)},3M_{k_{i}}^{(2)}]$ satisfies the hypotheses of
Lemma 5.17 at scale $k_{i}$. Thus, for each $i$, there is an interval
$\Lambda_{i}:=[a_{i},b_{i}]\supset[-\frac{2}{3}M_{k_{i}}^{(2)},\frac{2}{3}M_{k_{i}}^{(2)}]$
which satisfies the Green’s function decay property for
$(\ell_{k_{i}},\gamma_{k_{i}})$. For any $|n|\leq\frac{1}{3}M_{k_{i}}^{(2)}$,
$|n-a_{i}|,|n-b_{i}|\geq\frac{1}{3}M_{k_{i}}^{(2)}\geq\ell_{k_{i}}$; thus the
Poisson formula (1.11) gives

$\displaystyle|\psi(n)|$ $\displaystyle\leq
e^{-\gamma_{k_{i}}M_{k_{i}}^{(2)}/3}\left(|\psi(a_{i}-1)|+|\psi(b_{i}+1)|\right)$
$\displaystyle\leq
2Ce^{-\gamma_{0}M_{k_{i}}^{(2)}/24}((3M_{k_{i}}^{(2)}+1)^{2}+1)$

Fixing $n\in\mathbb{Z}$, we have $|n|\leq\frac{1}{3}M_{k_{i}}^{(2)}$ as
$i\to\infty$, and the right-hand side of the above inequality approaches $0$.
Thus we must have $|\psi(n)|=0$. But $\psi$ cannot be identically zero. ∎

The next lemma confirms that $m_{k}$ is unique at scale $12M_{k+1}^{(2)}$, and
is in fact (eventually) independent of $k$, for $\theta\in\Theta$.

######  Lemma 6.9.

There is a $K\in\mathbb{N}$ and $m_{\infty}\in\mathbb{Z}$, with
$|m_{\infty}|\leq 3M_{K}^{(2)}$, such that for all $k\geq K$,

$\theta+m_{\infty}\alpha\in\mathbf{I}_{k},\text{ and
}|E-\mathbf{E}_{k}(\theta+m_{\infty}\alpha)|<\rho_{k},$ (6.2)

and
$[m_{\infty}-12M_{k+1}^{(2)},m_{\infty}-1]\cup[m_{\infty}+1,m_{\infty}+12M_{k+1}^{(2)}]$
is $k$-nonresonant (relative to $(\theta,E)$ and $\mathbf{E}_{k}$).

###### Proof.

By Lemma 6.8, for sufficiently large $k$, we can choose $|m_{k}|\leq
3M_{k}^{(2)}$ satisfying (6.2) (with $m_{k}$ replacing $m_{\infty}$). If
$\theta\in\Theta$, then $\mathbf{E}_{k}(\theta+m\alpha)\notin J^{DR}$ for all
$|m|\leq 12M_{k+1}^{(2)}$, and thus $\theta+m_{k}\alpha$ is in the simple
resonance case of Proposition 4.10 (with $12M_{k+1}^{(2)}$ in the role of
$L^{(1)}$), for sufficiently large $k$; thus, for $0<|m-m_{k}|\leq
12M_{k+1}^{(2)}$, if $\theta+m\alpha\in\mathbf{I}_{k}$, then

$|E-\mathbf{E}(\theta+m\alpha)|\geq|\mathbf{E}(\theta+m_{k}\alpha)-\mathbf{E}(\theta+m\alpha)|-|E-\mathbf{E}(\theta+m_{k}\alpha)|>2\rho_{k}-\rho_{k}=\rho_{k};$

i.e., $[m_{k}-12M_{k+1}^{(2)},m_{k}-1]\cup[m_{k}+1,m_{k}+12M_{k+1}^{(2)}]$ is
$k$-nonresonant.

It remains only to show that $m_{k}$ must be independent of $k$. Suppose
$m_{k+1}\neq m_{k}$. Note that $|m_{k+1}|\leq 3M_{k+1}^{(2)}$, so
$|m_{k+1}-m_{k}|\leq 12M_{k+1}^{(2)}$. Since

$\theta+m_{k+1}\alpha\in\mathbf{I}_{k+1}\subset\mathbf{I}_{k}$

and $[m_{k}-12M_{k+1}^{(2)},m_{k}-1]\cup[m_{k}+1,m_{k}+12M_{k+1}^{(2)}]$ is
$k$-nonresonant, we have $|E-\mathbf{E}_{k}(\theta+m_{k+1}\alpha)|>\rho_{k}$.
Furthermore, since

$\theta+m_{k+1}\alpha\notin
J^{DR}(\overline{\rho}_{k},12M_{k+1}^{(2)},\alpha,\mathbf{E}_{k})\supset
J^{DR}(\overline{\rho}_{k},8M_{k+1}^{(1)},\alpha,\mathbf{E}_{k}),$

we must have $\mathbf{E}_{k+1}\in\mathcal{E}^{(1)}(\mathbf{E}_{k})$; thus, by
Proposition 2.2,

$|\mathbf{E}_{k+1}(\theta+m_{k+1}\alpha)-\mathbf{E}_{k}(\theta+m_{k+1}\alpha)|\leq
2\delta_{k}\ll\rho_{k}/2$

. Thus,

$\displaystyle|E-\mathbf{E}_{k+1}(\theta+m_{k+1}\alpha)|$
$\displaystyle\geq|E-\mathbf{E}_{k}(\theta+m_{k+1}\alpha)|-|\mathbf{E}_{k+1}(\theta+m_{k+1}\alpha)-\mathbf{E}_{k}(\theta+m_{k+1}\alpha)|$
$\displaystyle>\rho_{k}-\rho_{k}/2\gg\rho_{k+1}.$

In particular, this means $\\{m_{k+1}\\}$ is $k+1$-nonresonant, which
contradicts the construction of $m_{k+1}$; we must therefore have
$m_{k+1}=m_{k}$, and consequently $m_{k}$ is in fact independent of $k$, and
we may set $m_{\infty}=m_{k}$ for all sufficiently large $k$. ∎

###### Proof of Anderson localization.

As noted above, we need only show that $\psi$ decays exponentially. Let $K$
and $m_{\infty}$ be the integers defined in Lemma 6.9. For
$|n-m_{\infty}|>8M_{K+1}^{(2)}$, let $k\geq K+1$ be such that
$8M_{k}^{(2)}<|n-m_{\infty}|\leq 8M_{k+1}^{(2)}$. Consider the interval

$\Lambda=\left[n-\left\lceil\frac{|n-m_{\infty}|}{2}\right\rceil,n+\left\lceil\frac{|n-m_{\infty}|}{2}\right\rceil\right].$

All $m\in\Lambda$ satisfy $0<|m-m_{\infty}|\leq 12M_{k+1}^{(2)}$. Thus, by
Lemma 6.9, $\Lambda$ satisfies the hypotheses of Lemma 5.17 at scale $k$;
therefore, there is some subinterval

$\displaystyle\tilde{\Lambda}:=[\tilde{a},\tilde{b}]$
$\displaystyle\supset\left[n-\left\lceil\frac{|n-m_{\infty}|}{2}\right\rceil+3M_{k}^{(2)},n+\left\lceil\frac{|n-m_{\infty}|}{2}\right\rceil-3M_{k}^{(2)}\right]$
$\displaystyle\supset\left[n-\left\lceil\frac{|n-m_{\infty}|}{9}\right\rceil,n+\left\lceil\frac{|n-m_{\infty}|}{9}\right\rceil\right]$

which satisfies the Green’s function decay property for
$(\ell_{k},\gamma_{k})$. Note that
$\left\lceil\frac{|n-m_{\infty}|}{9}\right\rceil>\frac{8}{9}M_{k}^{(2)}\geq\ell_{k}$;
thus, by the Poisson formula (1.11),

$\displaystyle|\psi(n)|$ $\displaystyle\leq
e^{-|\log\varepsilon||n-m_{\infty}|/9}\left(|\psi(\tilde{a}-1)|+|\psi(\tilde{b}+1)|\right).$

We have

$\displaystyle\max\\{|\tilde{a}-1|,|\tilde{b}+1|\\}$
$\displaystyle\leq|n|+\left\lceil\frac{|n-m_{\infty}|}{2}\right\rceil+1$
$\displaystyle\leq|n-m_{\infty}|+\frac{|n-m_{\infty}|}{2}+2+3M_{K}^{(2)}$
$\displaystyle\leq\frac{3}{2}|n-m_{\infty}|+M_{k}^{(2)}$
$\displaystyle<2|n-m_{\infty}|.$

Therefore,

$\displaystyle|\psi(n)|$ $\displaystyle\leq
4Ce^{-|\log\varepsilon||n-m_{\infty}|/9}(|n-m_{\infty}|^{2}+1),$

which decays exponentially away from $m_{\infty}$. ∎

##  Appendix A Bootstrapped Green’s function decay

In this appendix, we will show that, if one can partition an interval
$\widehat{\Lambda}$ into subintervals alternating between long intervals with
Green’s function decay and short intervals with resolvent bounds, one can
iterate the resolvent identity to bootstrap the decay to all of
$\widehat{\Lambda}$. Results of this variety are well-established in the
literature (cf. e.g. [13]); we include the details relative to our specific
application for the reader’s convenience.

Let $v\in C^{2}(\mathbb{T},[-1,1])$ with
$\|\partial_{\theta}v\|_{\infty}+\|\partial_{\theta}^{2}v\|_{\infty}\leq
D_{0}$, $(\theta_{*},E_{*})\in\mathbb{T}\times\mathbb{R}$,
$0<\varepsilon<1/7$, and $0<{\widehat{\rho}}<1/2$. Additionally, let
$\ell\in\mathbb{N}$, $1<{\gamma}\leq|\log\varepsilon|$, and

$M:=\max\\{{\ell},|\log{\widehat{\rho}}|\\}.$

Consider an integer $\widehat{\ell}\in\mathbb{N}$ with
$\widehat{\ell}>16|\log\varepsilon|M$, and define

$\displaystyle\widehat{\gamma}$
$\displaystyle={\gamma}-16|\log\varepsilon|M/\widehat{\ell}.$

######  Assumption 4.

Suppose that $\widehat{\Lambda}\subset\mathbb{Z}$ admits a partition
$\mathcal{P}$ satisfying:

  1. 1.

Every other interval of $\mathcal{P}$ satisfies the Green’s function decay
property for $({\ell},{\gamma})$. We denote the intervening intervals by
$\Lambda_{i}$ and use the notation $\Lambda_{i,l/r}$ to refer to the interval
(on which the Green’s function decay property for $({\ell},{\gamma})$ is
satisfied) immediately to the left/right of $\Lambda_{i}$.

Note that, with this convention, one will have, e.g.,
${\Lambda}_{i,r}={\Lambda}_{i+1,l}$.

  2. 2.

For each ${\Lambda}_{i}=[a_{i},b_{i}]$, the interval
$\widehat{\Lambda}_{i}:=\Lambda_{i,l}\cup\Lambda_{i}\cup\Lambda_{i,r}$
satisfies

$\displaystyle\|{P}_{i}R^{\widehat{\Lambda}_{i}}\Gamma_{\mathcal{P}}\|\leq
2{\widehat{\rho}}^{-1},$ (A.1)

where ${P}_{i}$ denotes the projection onto coordinates
$[a_{i}-{\ell},b_{i}+{\ell}]\cap\widehat{\Lambda}_{i}$. Furthermore,
$R^{\widehat{\Lambda}_{i}}$ is well-defined.

  3. 3.

$|\Lambda_{i,l/r}|>2\widehat{\ell}$.

  4. 4.

$|\Lambda_{i}|\leq M$.

We illustrate one segment of Assumption 4 in Figure 9. Under these
assumptions, we get $(\widehat{\ell},\widehat{\gamma})$ decay on
$\widehat{\Lambda}$:

${\Lambda}_{i,l}$${\Lambda}_{i,r}$${\ell}$$\left({\ell},{\gamma}\right)$decay$\left({\ell},{\gamma}\right)$decay${\Lambda}_{i}$decoupled
resolvent bound${\ell}$ Figure 9: An illustration of one segment
$\widehat{\Lambda}_{i}$ of the partition $\mathcal{P}$ from Assumption 4; note
that the naming convention for the decay intervals is non-unique, i.e.
$\Lambda_{i,r}=\Lambda_{i+1,l}$.

######  Lemma A.1.

Suppose $\widehat{\Lambda}$ satisfies Assumption 4. Then $\widehat{\Lambda}$
satisfies the Green’s function decay property for
$(\widehat{\ell},\widehat{\gamma})$.

Before proving Lemma A.1, we will first handle the special case where there is
precisely one interval $\Lambda_{i}$:

######  Assumption 5.

Suppose that $\widehat{\Lambda}$ is partitioned by $\mathcal{P}$ into
${\Lambda}_{l}\cup{\Lambda}\cup{\Lambda}_{r}$, such that

  1. 1.

${\Lambda}_{l/r}$ satisfies the Green’s function decay property for
$({\ell},{\gamma})$.

  2. 2.

$R^{\widehat{\Lambda}}$ is well-defined, and, denoting $\Lambda=[a,b]$,

$\displaystyle\|{P}R^{\widehat{\Lambda}}\Gamma_{\mathcal{P}}\|\leq
2{\widehat{\rho}}^{-1},$ (A.2)

where ${P}$ denotes the projection onto coordinates
$[a-{\ell},b+{\ell}]\cap\widehat{\Lambda}$.

  3. 3.

$|\Lambda|\leq M$.

Letting

$\widetilde{\gamma}:={\gamma}-6|\log\varepsilon|M/\widehat{\ell},$

in this case we get slightly improved decay on $\widehat{\Lambda}$:

######  Lemma A.2.

Suppose $\widehat{\Lambda}$ satisfies Assumption 5. Then $\widehat{\Lambda}$
satisfies the Green’s function decay property for
$(\widehat{\ell},\widetilde{\gamma})$.

###### Proof.

Suppose $m,n\in\widehat{\Lambda}$ and $|m-n|\geq\widehat{\ell}\geq 3M\geq
2{\ell}+|\Lambda|$. Then at most one of $m,n$ satisfies
$\operatorname{dist}(\cdot,{\Lambda})\leq{\ell}$. In the case neither $m$ nor
$n$ is within ${\ell}$ of ${\Lambda}$, we use the resolvent expansion

$R^{\widehat{\Lambda}}=R^{\widehat{\Lambda}}_{\mathcal{P}}-R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}=R^{\widehat{\Lambda}}_{\mathcal{P}}-R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}_{\mathcal{P}}+R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}_{\mathcal{P}}.$

Since $m,n$ do not belong to adjacent intervals in $\mathcal{P}$, the middle
term of $R^{\widehat{\Lambda}}(m,n)$ in this expansion vanishes. By Green’s
function decay on ${\Lambda}_{l/r}$ and (A.2),

$\displaystyle\log|R^{\widehat{\Lambda}}(m,n)|$
$\displaystyle\leq\log\left(e^{-{\gamma}|m-n|}+2\varepsilon{\widehat{\rho}}^{-1}e^{-{\gamma}(|m-n|-|\Lambda|-1)}\right)$
$\displaystyle\leq\log\left(3{\widehat{\rho}}^{-1}e^{-{\gamma}(|m-n|-|\Lambda|-2{\ell}-1)}\right)$
$\displaystyle=-\left({\gamma}-\frac{\log
3+|\log{\widehat{\rho}}|+{\gamma}|\Lambda|+2{\gamma}{\ell}+{\gamma}}{|m-n|}\right)|m-n|$
$\displaystyle\leq-\left({\gamma}-\frac{6|\log\varepsilon|M}{\ell}\right)|m-n|.$

On the other hand, if $\operatorname{dist}(n,\Lambda)\leq{\ell}$, we have
$|m-n|\leq\operatorname{dist}(m,\Lambda)+|\Lambda|+2{\ell}$. We use the
resolvent expansion

$R^{\widehat{\Lambda}}=R^{\widehat{\Lambda}}_{\mathcal{P}}-R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}.$

By Green’s function decay on ${\Lambda}_{l/r}$ and (A.2),

$\log|R^{\widehat{\Lambda}}(m,n)|\leq\log\left(e^{-{\gamma}|m-n|}+2{\widehat{\rho}}^{-1}e^{-{\gamma}(\operatorname{dist}(m,\Lambda)-1)}\right)\leq\log\left(3{\widehat{\rho}}^{-1}e^{-{\gamma}(|m-n|-|\Lambda|-2{\ell}-1)}\right),$

and the proof concludes as in the first case. ∎

The following lemma is useful to ensure that the decoupled resolvent bounds
(A.1) or (A.2) are satisfied:

######  Lemma A.3.

Suppose $\widehat{\ell}\in\mathbb{N}$ is an integer with

$\widehat{\ell}\geq\max\left\\{{\ell},\frac{|\log{\widehat{\rho}}|+1}{{\gamma}}\right\\}.$

Suppose that $\widehat{\Lambda}$ is partitioned by $\mathcal{P}$ into
${\Lambda}_{l}\cup{\Lambda}\cup{\Lambda}_{r}$, such that

  1. 1.

${\Lambda}_{l/r}$ satisfies the Green’s function decay property for
$({\ell},{\gamma})$.

  2. 2.

There exists an interval
$\widehat{\Lambda}\supset\widetilde{\Lambda}=[\widetilde{a},\widetilde{b}]\supset{\Lambda}=[{a},{b}]$
such that

$\displaystyle\min\\{|\widetilde{a}-{a}|,|\widetilde{b}-{b}|\\}$
$\displaystyle\geq\ell,$
$\displaystyle\|R^{\widetilde{\Lambda}}\|\leq{\widehat{\rho}}^{-1}.$

Then $R^{\widehat{\Lambda}}$ is well-defined, (A.2) is satisfied, and
$\widehat{\Lambda}$ satisfies Assumption 5.

###### Proof.

Denote by $\widetilde{\mathcal{P}}$ the partition of $\widehat{\Lambda}$ into
$\widetilde{\Lambda}$ and its complement
$\widehat{\Lambda}\setminus\widetilde{\Lambda}$, and denote as shorthand
$\widetilde{R}=R^{\widehat{\Lambda}}_{\widetilde{\mathcal{P}}}$ and
$\widetilde{\Gamma}=\Gamma^{\Lambda}_{\widetilde{\mathcal{P}}}$.

We expand the resolvent $R^{\widehat{\Lambda}}$ by alternating the resolvent
formulas

$\displaystyle R^{\widehat{\Lambda}}$
$\displaystyle=\widetilde{R}-\widetilde{R}\widetilde{\Gamma}R^{\widehat{\Lambda}},$
$\displaystyle R^{\widehat{\Lambda}}$
$\displaystyle=R^{\widehat{\Lambda}}_{\mathcal{P}}-R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}},$

to get the expansion

$R^{\widehat{\Lambda}}=\widetilde{R}-\widetilde{R}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}+\widetilde{R}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}\widetilde{R}-\dots.$
(A.3)

Note that ${P}\widetilde{R}={P}R^{\widetilde{\Lambda}}$, where ${P}$ is
defined as in (A.2); similarly, because the boundary points of $\mathcal{P}$
are in $\widetilde{\Lambda}$,
$\Gamma_{\mathcal{P}}\widetilde{R}=\Gamma_{\mathcal{P}}R^{\widetilde{\Lambda}}$.
Thus multiplying on the left by ${P}$ gives

$\displaystyle{P}R^{\widehat{\Lambda}}$
$\displaystyle={P}R^{\widetilde{\Lambda}}-{P}R^{\widetilde{\Lambda}}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}+{P}R^{\widetilde{\Lambda}}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}R^{\widetilde{\Lambda}}-\dots.$

By the Green’s function decay on
${\Lambda}_{l/r}\supset\widetilde{\Lambda}\setminus{\Lambda}$,

$\displaystyle\|\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}}\|\leq\|\widetilde{\Gamma}R^{{\Lambda}_{l}}\Gamma_{\mathcal{P}}\|+\|\widetilde{\Gamma}R^{{\Lambda}_{r}}\Gamma_{\mathcal{P}}\|\leq\exp(-{\gamma}\ell),$
(A.4)

and so we get that

$\displaystyle\|{P}R^{\widehat{\Lambda}}\Gamma_{\mathcal{P}}\|$
$\displaystyle=\left\|{P}\sum_{k\geq
0}(R^{\widetilde{\Lambda}}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}}\Gamma_{\mathcal{P}})^{k}(R^{\widetilde{\Lambda}}-R^{\widetilde{\Lambda}}\widetilde{\Gamma}R^{\widehat{\Lambda}}_{\mathcal{P}})\Gamma_{\mathcal{P}}\right\|$
$\displaystyle\leq\sum_{k\geq
0}\left(\frac{e^{-{\gamma}\ell}}{{\widehat{\rho}}}\right)^{k}\frac{2\varepsilon+e^{-{\gamma}\ell}}{{\widehat{\rho}}}$
$\displaystyle\leq 2{\widehat{\rho}}^{-1}.$

Using the same method to estimate the expansion (A.3) gives an upper bound on
$\|R^{\widehat{\Lambda}}\|$, which ensures $R^{\widehat{\Lambda}}$ is well-
defined. ∎

###### Proof of Lemma A.1.

Each interval
$\widehat{\Lambda}_{i}=\Lambda_{i,l}\cup\Lambda_{i}\cup\Lambda_{i,r}$
satisfies the Green’s function decay property for
$(\widehat{\ell},\widetilde{\gamma})$ by Lemma A.2. Define the partitions
$\mathcal{P}_{1}:=\bigcup_{i\text{
odd}}\widehat{\Lambda}_{i}\cup\bigcup_{i\text{ even}}\Lambda_{i}$ and
$\mathcal{P}_{0}:=\bigcup_{i\text{
even}}\widehat{\Lambda}_{i}\cup\bigcup_{i\text{ odd}}\Lambda_{i}$. For ease of
notation, we define $\mathcal{P}_{i}=\mathcal{P}_{0}$ whenever $i$ is an even
integer, and $\mathcal{P}_{i}=\mathcal{P}_{1}$ whenever $i$ is an odd integer.

We first show that $R^{\widehat{\Lambda}}$ is well-defined. Partition
$\widehat{\Lambda}=\check{\Lambda}_{0}\cup\check{\Lambda}_{1}$ such that
$\bigcup_{i\text{ even}}\Lambda_{i}\subset\check{\Lambda}_{0}$ and
$\bigcup_{i\text{ odd}}\Lambda_{i}\subset\check{\Lambda}_{1}$, and let
$Q_{0},Q_{1}$ be projections onto the coordinates
$\check{\Lambda}_{0},\check{\Lambda}_{1}$ respectively. Then, by Assumption 4,
we have upper bound estimates on
$\|Q_{0}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\|$ and
$\|Q_{1}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\|$, and we have
$\|\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{1}}\|,\|\Gamma_{\mathcal{P}_{1}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\Gamma_{\mathcal{P}_{0}}\|\ll
1$. Thus, we can write
$R^{\widehat{\Lambda}}=Q_{0}R^{\widehat{\Lambda}}+Q_{1}R^{\widehat{\Lambda}}$
and expand each term using an infinite resolvent expansion alternating between
$\mathcal{P}_{0}$ and $\mathcal{P}_{1}$, the first term starting with
$\mathcal{P}_{0}$ and the second term starting with $\mathcal{P}_{1}$. This
will give an upper bound on $\|R^{\widehat{\Lambda}}\|$, ensuring it is well-
defined.

Suppose $m,n\in\widehat{\Lambda}$ and $|m-n|\geq\widehat{\ell}$. Because the
$\widehat{\Lambda}_{i}$ cover $\widehat{\Lambda}$ with overlaps of size at
least $2\widehat{\ell}$, there must some $\widehat{\Lambda}_{i}\ni m$ such
that $m$ is separated from its boundary by at least $\widehat{\ell}$. Without
loss of generality, let this be $\widehat{\Lambda}_{0}$.

Suppose first that $\operatorname{dist}(n,\Lambda_{i})>{\ell}$ for all
$\Lambda_{i}$. We expand the resolvent first from the left using
$\mathcal{P}_{0}$, then from the right using the partition $\mathcal{P}$
defined in Assumption 4, and then from the left alternating between
$\mathcal{P}_{1}$ and $\mathcal{P}_{0}$:

$R^{\widehat{\Lambda}}=R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}-R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}}+R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}_{\mathcal{P}}-R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\Gamma_{\mathcal{P}_{1}}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}}R^{\widehat{\Lambda}}_{\mathcal{P}}+\dots.$
(A.5)

We now expand $R^{\widehat{\Lambda}}(m,n)$ using (A.5). Since
$|m-n|\geq\widehat{\ell}$, the first term is bounded by
$e^{-\widetilde{\gamma}|m-n|}$ by Lemma A.2. The second term gives a sum of
terms of the form $\varepsilon R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}(m,p\pm
1)R^{\widehat{\Lambda}}_{\mathcal{P}}(p,n)$ where $p\in\Lambda_{i}$ for some
$i$; then $R^{\widehat{\Lambda}}_{\mathcal{P}}(p,n)=0$ because $n$ does not
belong to any $\Lambda_{i}$. Subsequent terms are of the form

$(-1)^{k+1}\varepsilon^{k+1}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}(p_{0},q_{0})R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}(p_{1},q_{1})\cdots
R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})R^{\widehat{\Lambda}}_{\mathcal{P}}(p_{k+1},n)$
(A.6)

for $k\geq 1$, where $p_{0}=m$, and for $0\leq i<k$, $q_{i}$ is a boundary
point of the interval in $\mathcal{P}_{i}$ containing $p_{i}$, and
$p_{i+1}=q_{i}\pm 1$; $p_{k+1}$ is a boundary point of the interval in
$\mathcal{P}$ containing $n$, and $q_{k}=p_{k+1}\pm 1$.

The structure of the intervals, our assumption on $m$, and Lemma A.2 guarantee
that
$\log|R^{\widehat{\Lambda}}_{\mathcal{P}_{i}}(p_{i},q_{i})|\leq-\widetilde{\gamma}|p_{i}-q_{i}|$
for all $0\leq i<k$. Similarly, Assumption 4 guarantees
$\log|R^{\widehat{\Lambda}}_{\mathcal{P}}(p_{k+1},n)|\leq-{\gamma}|p_{k+1}-n|$.
To estimate $|R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})|$, we note
that $q_{k}$ must be near some $\Lambda_{j}$ (i.e., either
$q_{k}\in\Lambda_{j}$ or $\operatorname{dist}(q_{k},\Lambda_{j})\leq{\ell}$),
and we make the following considerations:

  * •

If $k$ shares the same parity as $j$, then in order for
$R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})$ to be nonzero, we must
have $p_{k}\in\Lambda_{j}\in\mathcal{P}_{k}$. Then (A.2) applies, and
$|p_{k}-q_{k}|\leq 2{\ell}+|\log{\widehat{\rho}}|\leq 3M$, giving
$|R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})|=|R^{\widehat{\Lambda}_{j}}(p_{k},q_{k})|\leq
2\varepsilon^{-1}{\widehat{\rho}}^{-1}\leq
2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-\widetilde{\gamma}|p_{k}-q_{k}|}$.

  * •

If $k$ and $j$ have opposite parities, then $|p_{k}-q_{k}|\geq
2\widehat{\ell}-{\ell}\geq 5M\geq\widehat{\ell}$. Thus either
$R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})=0$, or Lemma A.2
applies, giving
$\log|R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})|\leq-\widetilde{\gamma}|p_{k}-q_{k}|$.

Finally, since $|m-n|\leq|p_{k+1}-n|+\sum_{i=0}^{k}|p_{i}-q_{i}|+k+1$, and
$\varepsilon^{k+1}\leq e^{-\widetilde{\gamma}(k+1)}$, each term of the form
(A.6) is bounded by
$2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-\widetilde{\gamma}|m-n|}$.
By the same considerations, but using the fact that
$|p_{i}-q_{i}|\geq\widehat{\ell}$ for all $0\leq i<k$, each term of the form
(A.6) is bounded by
$2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-k\widetilde{\gamma}\widehat{\ell}}$.
We consider the first term $R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}(m,n)$ from
(A.5) to be the sole term of the expansion with $k=0$, and note that the same
bounds apply.

In the case that $n\in\Lambda_{j}$ or
$\operatorname{dist}(n,\Lambda_{j})\leq{\ell}$, we follow the same procedure
but do not use $\mathcal{P}$, i.e., we use the expansion

$R^{\widehat{\Lambda}}=R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}-R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}+R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\Gamma_{\mathcal{P}_{1}}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}-R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}\Gamma_{\mathcal{P}_{1}}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}\Gamma_{\mathcal{P}_{0}}R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}+\dots.$
(A.7)

to obtain terms of the form

$(-1)^{k}\varepsilon^{k}R^{\widehat{\Lambda}}_{\mathcal{P}_{0}}(p_{0},q_{0})R^{\widehat{\Lambda}}_{\mathcal{P}_{1}}(p_{1},q_{1})\cdots
R^{\widehat{\Lambda}}_{\mathcal{P}_{k}}(p_{k},q_{k})$ (A.8)

where $p_{0}=m$, $q_{k}=n$, and for $0\leq i<k$, $q_{i}$ is a boundary point
of the interval in $\mathcal{P}_{i}$ containing $p_{i}$, and $p_{i+1}=q_{i}\pm
1$. By considerations analogous to those in the first case, we obtain the same
bounds
$2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-\widetilde{\gamma}|m-n|}$
and
$2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-k\widetilde{\gamma}\widehat{\ell}}$
for each term of the form (A.8).

Since each $p_{i}$ gives two choices for $q_{i}$, there are at most $2^{k+1}$
nonzero terms of the form (A.6) or (A.8) for each nonzero value of $k$. Let
$k_{0}=\lceil\frac{|m-n|}{\widehat{\ell}}\rceil$. For the at most
$2^{k_{0}+1}$ terms with $k<k_{0}$, we use the bound
$2{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-\widetilde{\gamma}|m-n|}$,
while we use the bound
$2{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-k\widetilde{\gamma}\widehat{\ell}}$
for $k\geq k_{0}$, giving

$\displaystyle|R^{\widehat{\Lambda}}(m,n)|$ $\displaystyle\leq
2\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}\left(2^{k_{0}+1}e^{-\widetilde{\gamma}|m-n|}+\sum_{k=k_{0}}^{\infty}2^{k+1}e^{-k\widetilde{\gamma}\widehat{\ell}}\right)$
$\displaystyle\leq
4\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}\left(2^{k_{0}}e^{-\widetilde{\gamma}|m-n|}+2^{k_{0}+1}e^{-\widetilde{\gamma}k_{0}\widehat{\ell}}\right)$
$\displaystyle\leq
4\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}2^{k_{0}+2}e^{-\widetilde{\gamma}|m-n|}$
$\displaystyle\leq
2^{|m-n|/\widehat{\ell}+5}\varepsilon^{-1}{\widehat{\rho}}^{-1}e^{3M\widetilde{\gamma}}e^{-\widetilde{\gamma}|m-n|}.$

It follows that

$\displaystyle\log|R^{\widehat{\Lambda}}(m,n)|$
$\displaystyle\leq-\left(\widetilde{\gamma}-\frac{\log
2}{\widehat{\ell}}-\frac{5\log
2+|\log\varepsilon|+|\log{\widehat{\rho}}|+3M\widetilde{\gamma}}{|m-n|}\right)|m-n|$
$\displaystyle\leq-\left(\widetilde{\gamma}-\frac{10|\log\varepsilon|M}{\widehat{\ell}}\right)|m-n|$
$\displaystyle\leq-\widehat{\gamma}|m-n|.\qed$

##  Appendix B Proofs of preliminaries

Below we collect proofs of the foundational results from the introduction.

###  B.1 Perturbation theory lemmas

###### Proof of Lemma 1.1.

We proceed by contraposition. Suppose
$\operatorname{rank}(\chi)<\operatorname{rank}(P)$. Then
$\dim(\ker(\chi))+\dim(\operatorname{im}(P))>\dim(V)$ by rank-nullity.
Consequently, $\ker(\chi)\cap\operatorname{im}(P)$ contains a unit vector
$Pv$. But then

$\displaystyle\|(I-\chi)Pv\|$ $\displaystyle=\|Pv\|=1,$

so $\|(I-\chi)P\|\geq 1$. ∎

###### Proof of Lemma 1.2.

There must be an eigenvalue in $\overline{B}_{\delta}(E_{*})$, since

$\displaystyle\delta^{2}\geq\|(A-E_{*})\phi\|^{2}=\|(A-E_{*})\chi_{\mathbb{R}}(A)\phi\|^{2}\geq\inf_{\lambda_{j}\in\,\operatorname{sp}A}|\lambda_{j}-E_{*}|^{2}.$

By assumption we have

$\displaystyle\delta^{2}\geq\|(A-E_{*})\phi\|^{2}$
$\displaystyle\geq\|(I-\chi_{B_{\rho}(E_{*})}(A))(A-E_{*})\phi\|^{2}$
$\displaystyle\geq\rho^{2}\|(I-\chi_{B_{\rho}(E_{*})}(A))\phi\|^{2}$

which is (1.4). To see (1.5), notice that, since

$\displaystyle\left\|(I-\chi_{B_{\rho}(E_{*})}(A))\phi\right\|^{2}$
$\displaystyle=1-\|\chi_{B_{\rho}(E_{*})}(A)\phi\|^{2}\leq(\delta/\rho)^{2},$

it follows from $\chi_{B_{\rho}(E_{*})}(A)^{2}=\chi_{B_{\rho}(E_{*})}(A)$ that

$\displaystyle\|\phi-\psi\|^{2}$
$\displaystyle=2(1-\|\chi_{B_{\rho}(E_{*})}(A)\phi\|)$ $\displaystyle\leq
2(1-\|\chi_{B_{\rho}(E_{*})}(A)\phi\|^{2})$ $\displaystyle\leq
2(\delta/\rho)^{2},$

which was the claim. ∎

###### Proof of Lemma 1.3.

The family of eigenpairs $(\mathbf{E},\psi)(\theta)$ in question are the
implicit function defined by the vanishing of

$\displaystyle F(E,\phi,\theta):=\begin{bmatrix}(A(\theta)-E)\phi\\\
\langle\phi,\phi\rangle-1\end{bmatrix}.$

Indeed, since $E_{*}$ is simple, one has that

$\displaystyle(D_{(E,\phi)}F)(E_{*},\psi_{*},\theta_{*})=\begin{bmatrix}A(\theta_{*})-E_{*}&-\psi_{*}\\\
2\psi_{*}^{\top}&0\end{bmatrix}$

is non-singular: denoting $P=\psi_{*}\psi_{*}^{\top}\oplus 1$ and $Q=I-P$, one
computes via the Schur complement that

$\displaystyle\det((D_{(E,\phi)}F)(E_{*},\psi_{*},\theta_{*}))$
$\displaystyle=\det\left(\begin{bmatrix}0&-1\\\
2&0\end{bmatrix}\right)\det\left(Q(A(\theta_{*})-E_{*})Q\right)$
$\displaystyle=2\prod_{\lambda_{j}\neq E_{*}}(\lambda_{j}-E_{*})\neq 0.$

Thus the implicit function $(\mathbf{E},\psi)(\theta)$ is defined in a
neighborhood of $\theta_{*}$ and twice-differentiable at $\theta_{*}$.

We differentiate the eigenvalue relation $(A-\mathbf{E})\psi=0$ at
$\theta_{*}$ to get that

$\displaystyle(A^{\prime}-\partial_{\theta}\mathbf{E})\psi=-(A-\mathbf{E})\partial_{\theta}\psi.$
(B.1)

Equation (1.7) follows by taking inner products of the relation (B.1) with
$\psi$; (1.8) follows from the uniqueness of the implicit function since (1.8)
satisfies (B.1); and (1.9) follows by differentiating (1.7). Finally, (1.10)
is immediate from (1.8). ∎

Finally, we have the simple proof of the Poisson formula:

###### Proof of Lemma 1.4.

This follows by applying the resolvent $R^{\Lambda}(E)$ to the observation

$\displaystyle(H^{\Lambda}-E)\psi$
$\displaystyle=\varepsilon\psi(a-1)\delta_{a}+\varepsilon\psi(b+1)\delta_{b}.\qed$

###  B.2 Cauchy Interlacing Theorem

The Cauchy Interlacing Theorem 3.7 follows from the Min-Max Principle:

###### Proof of Theorem 3.7.

If $w_{j}$ is an eigenvector for $B$ with eigenvalue $\beta_{j}$ and
$W_{j}^{m}:=\operatorname{span}\\{w_{j},\dots,w_{m}\\}$, then one has that
$P^{*}W_{1}^{k}$ is a $k$-dimensional subspace of ${\mathbb{R}}^{n}$ and

$\displaystyle\beta_{k}$ $\displaystyle=\max_{y\in W_{1}^{k}}\frac{\langle
By,y\rangle}{\|y\|^{2}}=\max_{y\in W_{1}^{k}}\frac{\langle
PAP^{*}y,y\rangle}{\|y\|^{2}}=\max_{y\in W_{1}^{k}}\frac{\langle
A(P^{*}y),P^{*}y\rangle}{\|P^{*}y\|^{2}}$
$\displaystyle\geq\min_{U}\left\\{\max_{x}\left\\{\frac{\langle
Ax,x\rangle}{\|x\|^{2}}\;:\;x\in
U\setminus\\{0\\}\right\\}:\dim(U)=k\right\\}=\alpha_{k}.$

On the other hand, one has that $P^{*}W_{k}^{m}$ is an
$m-k+1=n-(n-m+k)+1$-dimensional subspace, and one likewise has that

$\displaystyle\beta_{k}$ $\displaystyle=\min_{y\in W_{k}^{m}}\frac{\langle
A(P^{*}y),P^{*}y\rangle}{\|P^{*}y\|^{2}}$
$\displaystyle\leq\max_{U}\left\\{\min_{x}\left\\{\frac{\langle
Ax,x\rangle}{\|x\|^{2}}\;:\;x\in
U\setminus\\{0\\}\right\\}:\dim(U)=m-k+1\right\\}=\alpha_{n-m+k}.\qed$

###  B.3 Eigenvalue separation lemma

This lemma is classical (cf. [14, 20]), but we include a proof for the sake of
completeness. Consider the transfer matrix

$\displaystyle A(V,E)$
$\displaystyle=\begin{bmatrix}\frac{V_{0}-E}{-\varepsilon}&-1\\\
1&0\end{bmatrix}$

with associated Schrödinger cocycle

$\displaystyle M_{[a,b]}(V,E):=\prod_{k=b}^{a}A(S^{k}V,E)$

where $(S^{k}V)_{m}=V_{m+k}$, the product concatenates on the right, and
$b\geq a$.

By definition, one has that $\psi=\psi(n)$ solves the formal Schrödinger
difference equation if and only if

$\displaystyle M_{[a,b]}(V,E)\begin{bmatrix}\psi(a)\\\ \psi(a-1)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\psi(b)\\\ \psi(b-1)\end{bmatrix}$

for all $n\in{\mathbb{Z}}$. We also have the fundamental cocycle identity

$\displaystyle M_{[a,b]}(V,E)$
$\displaystyle=M_{[c+1,b]}(V,E)M_{[a,c]}(V,E),\quad a\leq c<b.$ (B.2)

The proof of Lemma 1.5 comes from the simplicity of the eigenvalues of
$H^{\widehat{\Lambda}}$ and the orthogonality of the corresponding
eigenvectors; namely, forcing a sufficient proportion of the masses of two
distinct eigenvectors into the same window pushes the corresponding
eigenvalues apart. We make this quantitative below:

###### Proof of Lemma 1.5.

If
$|\widehat{E}_{2}-\widehat{E}_{1}|\geq\varepsilon\left(\left(\frac{m_{\Lambda}}{\varepsilon}\right)^{2}+2\right)/|\Lambda|^{2}$,
the claim follows immediately; suppose for the remainder of the proof that

$\displaystyle|\widehat{E}_{2}-\widehat{E}_{1}|<\varepsilon\left(\left(\frac{m_{\Lambda}}{\varepsilon}\right)^{2}+2\right)/|\Lambda|^{2}.$

Denote $[a,b]:=\widehat{\Lambda}$, $[c,d]:={\Lambda}$, and
$\vec{\psi}_{k}:=[\widehat{\psi}_{k}(c),\widehat{\psi}_{k}(c-1)]^{\top}$.
Since the eigenvectors $\widehat{\psi}_{k}$ are real, we may suppose they are
normalized at the left endpoint $c$ of ${\Lambda}$ such that

$\displaystyle\|\vec{\psi}_{k}\|^{2}=1,\quad k=1,2,$

and

$\displaystyle\langle\vec{\psi}_{1},\vec{\psi}_{2}\rangle\geq 0.$

In particular, we note for later use that
$\|\widehat{\psi}_{k}\|^{2}\geq\|\vec{\psi}_{k}\|^{2}\geq 1$. With this
normalization, if $\theta\in(0,\pi/2]$ denotes the acute angle between
$\vec{\psi}_{1}$ and $\vec{\psi}_{2}$, one has

$\displaystyle 1-\langle\vec{\psi}_{1},\vec{\psi}_{2}\rangle$
$\displaystyle=1-\cos\theta\leq 1-\cos^{2}\theta$
$\displaystyle\leq\sin^{2}\theta\leq\sin\theta$
$\displaystyle=|\widehat{\psi}_{1}(c)\widehat{\psi}_{2}(c-1)-\widehat{\psi}_{1}(c-1)\widehat{\psi}_{2}(c)|=:|W(\widehat{\psi}_{1},\widehat{\psi}_{2})(c)|$

and consequently

$\displaystyle\frac{1}{2}\|\vec{\psi}_{1}-\vec{\psi}_{2}\|^{2}\leq|W(\widehat{\psi}_{1},\widehat{\psi}_{2})(c)|.$

By Green’s identity, one has

$\displaystyle\frac{\widehat{E}_{2}-\widehat{E}_{1}}{\varepsilon}\sum_{j=c}^{b}\widehat{\psi}_{1}(j)\widehat{\psi}_{2}(j)$
$\displaystyle={\frac{1}{\varepsilon}}\sum_{j=c}^{b}\widehat{\psi}_{1}(j)(H^{\widehat{\Lambda}}\widehat{\psi}_{2})(j)-(H^{\widehat{\Lambda}}\widehat{\psi}_{1})(j)\widehat{\psi}_{2}(j)$
$\displaystyle=W(\widehat{\psi}_{1},\widehat{\psi}_{2})(c)-W(\widehat{\psi}_{1},\widehat{\psi}_{2})({b}+1).$

Since $\widehat{\psi}_{k}$ are eigenfunctions for $H^{[a,b]}$, they satisfy
the Dirichlet boundary condition at $b$; in particular,
$W(\widehat{\psi}_{1},\widehat{\psi}_{2})(b+1)=0$, and we see

$\displaystyle|W(\widehat{\psi}_{1},\widehat{\psi}_{2})(c)|=\left|\frac{\widehat{E}_{2}-\widehat{E}_{1}}{\varepsilon}\sum_{j=c}^{b}\widehat{\psi}_{1}(j)\widehat{\psi}_{2}(j)\right|\leq|\widehat{E}_{2}-\widehat{E}_{1}|\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|$

by the Cauchy-Schwarz inequality. Combining these observations, we see that

$\displaystyle\frac{1}{\sqrt{2}}\|\vec{\psi}_{1}-\vec{\psi}_{2}\|\leq\left(\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|\right)^{1/2}.$
(B.3)

We will apply observation (B.3) with a telescoping estimate on transfer
matrices to get an upper estimate on
$\|\widehat{\psi}_{1}-\widehat{\psi}_{2})\|_{{\Lambda}}^{2}$ in terms of
$|\widehat{E}_{2}-\widehat{E}_{1}|$. Specifically, we have

$\displaystyle\|M_{[c,d]}(V,\widehat{E}_{1})-M_{[c,d]}(V,\widehat{E}_{2})\|$
$\displaystyle\leq\sum_{j\in{\Lambda}}\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}\|M_{[c,j-1]}(V,\widehat{E}_{1})\|\|M_{[j+1,d]}(V,\widehat{E}_{2})\|$
$\displaystyle\leq\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}|{\Lambda}|\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{(|{\Lambda}|-1)/2},$

and consequently

$\displaystyle\|\widehat{\psi}_{1}$
$\displaystyle-\widehat{\psi}_{2}\|_{{\Lambda}}\leq\sum_{j\in{\Lambda}}|\widehat{\psi}_{1}(j)-\widehat{\psi}_{2}(j)|$
$\displaystyle\leq\sum_{j\in{\Lambda}}\|M_{[c,j]}(V,\widehat{E}_{1})\vec{\psi}_{1}-M_{[c,j]}(V,\widehat{E}_{2})\vec{\psi}_{2}\|$
$\displaystyle\leq\sum_{j\in{\Lambda}}\|M_{[c,j]}(V,\widehat{E}_{1})-M_{[c,j]}(V,\widehat{E}_{2})\|+\|M{[c,j]}(V,\widehat{E}_{2})(\vec{\psi}_{1}-\vec{\psi}_{2})\|$
$\displaystyle\leq\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}|{\Lambda}|^{2}\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{(|{\Lambda}|-1)/2}+|{\Lambda}|\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{|{\Lambda}|/2}\left(\frac{2|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|\right)^{1/2}.$

Since
$|\widehat{E}_{2}-\widehat{E}_{1}|<\varepsilon\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)/|{\Lambda}|^{2}$
by our very first assumption, it follows that

$\displaystyle\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|_{{\Lambda}}$
$\displaystyle\leq\left(\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}\right)^{1/2}|{\Lambda}|\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{|{\Lambda}|/2}\left(1+\sqrt{2}(\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|)^{1/2}\right)$

and thus

$\displaystyle\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|_{{\Lambda}}^{2}$
$\displaystyle\leq\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}|{\Lambda}|^{2}\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{|{\Lambda}|}\left(1+\sqrt{2}(\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|)^{1/2}\right)^{2}$
$\displaystyle<6\frac{|\widehat{E}_{2}-\widehat{E}_{1}|}{\varepsilon}|{\Lambda}|^{2}\left(\left(\frac{m_{{\Lambda}}}{\varepsilon}\right)^{2}+2\right)^{|{\Lambda}|}\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|,$

where in the last line we used that
$1\leq(\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|)^{1/2}$ and
$(1+\sqrt{2})^{2}<6$.

On the other hand, since $\widehat{\psi}_{1}$ and $\widehat{\psi}_{2}$ are
eigenfunctions for different eigenvalues, they are orthogonal, and we have

$\displaystyle\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|_{{\Lambda}}^{2}$
$\displaystyle=\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|^{2}-\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|_{\widehat{\Lambda}\setminus{\Lambda}}^{2}$
$\displaystyle\geq\left(\|\widehat{\psi}_{1}\|^{2}+\|\widehat{\psi}_{2}\|^{2}\right)-\frac{1}{2}\left(\|\widehat{\psi}_{1}\|^{2}+\|\widehat{\psi}_{2}\|^{2}\right)$
$\displaystyle\geq\frac{1}{2}\left(\|\widehat{\psi}_{1}\|^{2}+\|\widehat{\psi}_{2}\|^{2}\right)$
$\displaystyle\geq\|\widehat{\psi}_{1}\|\|\widehat{\psi}_{2}\|$

where the first line above uses orthogonality of vectors with disjoint
supports, the second uses orthogonality of distinct eigenvectors and the
localization assumption
$\|\widehat{\psi}_{k}\|_{{\Lambda}}^{2}\geq\|\widehat{\psi}_{k}\|^{2}/2$, and
the fourth line uses the inequality of arithmetic and geometric means.

Combining the upper and lower bounds on
$\|\widehat{\psi}_{1}-\widehat{\psi}_{2}\|_{{\Lambda}}^{2}$ above yields the
claim. ∎

## References

  * [1] I. Binder, D. Kinzebulatov, and M. Voda.  Non-perturbative localization with quasiperiodic potential in continuous time.  Commun. Math. Phys. 351 (2017), 1149–1175.
  * [2] J. Bourgain.  Positivity and continuity of the Lyapounov exponent for shifts on $\mathbb{T}^{d}$ with arbitrary frequency vector and real analytic potential with arbitrary frequency vector and real analytic potential.  J. Anal. Math. 96 (2005), 313–355.
  * [3] J. Bourgain and M. Goldstein.  On nonperturbative localization with quasi-periodic potential.  Ann. of Math. 152 (2000), 835–879.
  * [4] J. Bourgain, M. Goldstein, and W. Schlag.  Anderson localization for Schrödinger operators on $\mathbb{Z}$ with potentials given by the skew-shift.  Commun. Math. Phys. 220 (2001), 583–621.
  * [5] J. Bourgain and W. Schlag. Anderson localization for Schrödinger operators on $\mathbb{Z}$ with strongly mixing potentials, Commun. Math. Phys. 215 (2000), 143–175.
  * [6] V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, F. Wang, and Z. Zhang.  Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent.  Trans. Amer. Math. Soc. 372 (2019), 3619–3667.
  * [7] V. Bucaj, D. Damanik, J. Fillman, V. Gerbuz, T. VandenBoom, F. Wang, and Z. Zhang.  Positive Lyapunov exponents and a large deviation theorem for continuum Anderson models, briefly.  J. Func. Anal. 277 (2019), 3179–3186.
  * [8] R. Carmona, A. Klein, and F. Martinelli. Anderson localization for Bernoulli and other singular potentials, Commun. Math. Phys. 108 (1987), 41–66.
  * [9] D. Damanik, J. Fillman, and S. Sukhtaiev.  Localization for Anderson models on metric and discrete tree graphs.  Math. Ann., 376 (2020), 1337–1393.
  * [10] H. von Dreifus and A. Klein. A new proof of localization in the Anderson tight binding model, Commun. Math. Phys. 124 (1989), 285–299.
  * [11] A. Elgart and A. Klein. An eigensystem approach to Anderson localization, J. Funct. Anal. 271 (2016), 3465–3512.