eigenvalues of $P_{N,\gamma}^{Q}$ in $\widehat{\Omega}$. Fix
$\widetilde{\varepsilon}_{0}>0$ such that the Lebesgue measure of the set
$p^{-1}(\mathcal{B}_{p}^{\widetilde{\varepsilon}_{0}}\cap p(S^{1}))$ is less
than $\mu/(16\pi)$ (recall Definition 1.3). Then, by [8, Theorem 1.2] (see
also [65, Theorem 2.1]) it follows that
$\mathbf{P}(\widehat{\mathcal{N}}_{\widehat{\Omega}_{1}}\geqslant\mu N/4)\to
0$ as $N\to\infty$, where
$\widehat{\Omega}_{1}:=\mathcal{B}_{p}^{\widetilde{\varepsilon}_{0}}$. On the
other hand, by Theorems 5.4, 5.7, and Lemma 6.16(iii)-(v) it follows that
there exists some $0<\widehat{c}_{\gamma}<\infty$ such that
$\mathbf{P}(\widehat{\mathcal{N}}_{\widehat{\Omega}_{2}}\,>0)\to 0$ as
$N\to\infty$, where
$\widehat{\Omega}_{2}:=(p(S^{1}))^{\widehat{c}_{\gamma}\log
N/N}\setminus\widehat{\Omega}_{1}$. Finally, upon choosing
$\upalpha=4C_{\ref{thm-thintube}}/\mu$, and applying Theorem 7.1 with this
$\upalpha$ we deduce that
$\mathbf{P}(\widehat{\mathcal{N}}_{\widehat{\Omega}_{3}}\geqslant N\mu/4)\to
0$ as $N\to\infty$, where $\widehat{\Omega}_{3}:=\Omega_{\upalpha\log N/N}$
(recall (7.1)).

Setting $\varepsilon_{\ref{theo-location}}=\widetilde{\varepsilon}_{0}$ and
$C_{\ref{theo-location}}=\upalpha\vee\widehat{c}_{\gamma}^{-1}$ and taking a
union bound over the events that
$\widehat{\mathcal{N}}_{\widehat{\Omega}_{i}}\geqslant N\mu/4$ for $i=1,2,3$,
completes the proof. ∎

##  8\. Resolvent estimates close to the spectral curve

Given a symbol $p(\zeta)-z$, with $z\in\mathds{C}$, as in (4.16), we will see
that there is an important distinction between the regime where the spectral
parameter $z$ is inside a loop of $p(S^{1})$, i.e. where the winding number
$\mathrm{ind}_{p(S^{1})}(z)$ of the curve $p(S^{1})$ around $z$ is non-zero,
and the regime where $\mathrm{ind}_{p(S^{1})}(z)=0$. In the former case, we
will provide in Section 9 a construction of quasimodes, which are approximate
singular vectors, for finite Toeplitz matrices $P_{N}-z$. In regions where the
winding number is zero, we will be interested in obtaining resolvent
estimates. The key difficulty in both regimes will be in obtaining
sufficiently good estimates when $z$ is allowed to be at an $N$-dependent
distance from $p(S^{1})$.

In this section will prove the following estimate on the resolvent of a
parameter dependent Toeplitz matrix which, while being also of independent
interest, will be an essential ingredient in proving a spectral gap for the
small singular values, see Section 10.1.

######  Theorem 8.1.

Let $\Omega^{\prime}\Subset\mathds{C}$ be a non-empty relatively compact open
set. Let $N\in\mathds{N}$, let $\Omega_{N}\Subset\Omega^{\prime}$ be a family
of non-empty relatively compact open sets. Let $q_{z}$, $z\in\Omega^{\prime}$,
be a family of Laurent polynomials as in (4.21), with
$\mathfrak{N}_{z,\pm}=\mathfrak{N}_{\pm}\geqslant 0$ independent of $z$ in
$\Omega_{N}$ and satisfying (4.22), and with coefficients
$q_{z,n}\in\mathds{C}$ for $-\mathfrak{N}_{-}\leqslant
n\leqslant\mathfrak{N}_{+}$. Suppose that:

  * •

There exists a constant $0<C<\infty$ such that for all $z\in\Omega^{\prime};$

$q_{z,\mathfrak{N}_{+}}\neq 0,\quad|q_{z,-\mathfrak{N}_{-}}|\geqslant
1/C,\quad|q_{z,n}|\leqslant C,\text{ for }-\mathfrak{N}_{-}\leqslant
n\leqslant\mathfrak{N}_{+}.$ (8.1)

  * •

All roots $\zeta_{z}$ of $q_{z}(\zeta)$ are simple, for all
$z\in\Omega^{\prime}$.

  * •

There exists a constant $0<C<\infty$ such that for all $z\in\Omega^{\prime}$
and for any two distinct roots $\zeta_{z}\neq\omega_{z}$ we have

$|\zeta_{z}-\omega_{z}|\geqslant 1/C,$ (8.2)

and for any root $\zeta_{z}$

$0<|\zeta_{z}|\leqslant C.$ (8.3)

  * •

$0\notin q_{z}(S^{1})$ and there exists a constant $C_{0}>0$ such that for
$N>0$ large enough

$\mathrm{dist\,}(|\zeta_{z}|,1)\geqslant C_{0}\frac{\log N}{N},\quad\text{for
all }z\in\Omega_{N},\text{ and all roots }\zeta_{z}.$ (8.4)

  * •

There exists an $m_{0}\geqslant 0$ (unrelated to the constants $m_{0}$ in
(4.27) and (4.28)) and constants $C_{1}>1,C_{2}>0$ such that for all
$N\in\mathds{N}$ large enough and all $z\in\Omega_{N}$ we have that the roots
of $q_{z}(\zeta)$ inside $D(0,1)$, of total number $m_{z,+}\geqslant m_{0}$,
satisfy

$|\zeta_{1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{z,+}-m_{0}}^{+}|<1/C_{1}<1-C_{2}\frac{\log
N}{N}\leqslant|\zeta_{m_{z,+}-m_{0}+1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{z,+}}^{+}|.$
(8.5)

  * •

Let $\widehat{m}_{0}\in\\{0,m_{0}\\}$ and suppose that for all
$N\in\mathds{N}$ and all $z\in\Omega_{N}$ we have
$\mathrm{ind}_{q_{z}(S^{1})}(0)=\widehat{m}_{0}$.

Then there exists a constant $0<\widetilde{C}<+\infty$ such that for $N>0$
large enough and all $z\in\Omega_{N}$,

$\|P_{N}(q_{z})^{-1}\|\leqslant\widetilde{C}\frac{N^{1+C_{2}\Theta(\widehat{m}_{0})}}{\log
N}.$ (8.6)

######  Remark 8.2.

It will be clear from the proof that (8.6) holds when
$\mathrm{ind}_{q_{z}(S^{1})}(0)=0$ even without the structural assumption
(8.5). Furthermore, notice that in Theorem 8.1 we only consider the case of
non-negative winding numbers. This is all that we need in the sequel, however
one could prove the same result in the case of negative winding numbers
provided that one assume a similar assumption as (8.5) for the roots in
$\mathds{C}\backslash\overline{D(0,1)}$.

###### Proof of Theorem 8.1.

Throughout the proof, we will assume that $N>0$ is sufficiently large so that
the assumptions of the theorem hold. Moreover, all $O(1)$ terms and constants
are understood to be uniform in $N>0$ and $z\in\Omega_{N}$, without us
mentioning it explicitly at each occurrence. In fact the error terms will only
depend on the constants in the hypothesis of the theorem and global constants.

1\. Let $z\in\Omega_{N}$. Since $q_{z}$ satisfies the assumptions of Case 1 of
Lemma 4.1, we may order the roots of $q_{z}(\zeta)$ as in (4.26), and using
also (8.3) and (8.4),

$0<|\zeta_{1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{z,+}}^{+}|<1<|\zeta_{1}^{-}|\leqslant\cdots\leqslant|\zeta_{m_{z,-}}^{-}|\leqslant
C.$ (8.7)

Since $\mathrm{ind}_{q_{z}(S^{1})}(0)=\widehat{m}_{0}$ for all
$z\in\Omega_{N}$, it follows from a similar computation as in (4.34) that
$m_{z,\pm}=\mathfrak{N}_{\pm}\pm\widehat{m}_{0}$ are independent of $N$ and
$z$. Keeping in mind that $m_{z,\pm}$ are independent of $z$, but that the
coefficients $q_{z,n}$ do depend on $z$, we suppress from now on the $z$
subscript of $q_{z}$, $q_{z,n}$ and $m_{z,\pm}$. Hence, we are interested in
the symbol

$q(\zeta)=\sum_{-(m_{-}+\widehat{m}_{0})}^{m_{+}-\widehat{m}_{0}}q_{j}\zeta^{-j},$
(8.8)

where by (8.1)

$q_{m_{+}-\widehat{m}_{0}}\neq 0,\quad|q_{-(m_{-}+\widehat{m}_{0})}|\geqslant
1/C$ (8.9)

2\. We turn to estimating $\|P_{N}(q)^{-1}\|$. We begin by inverting
$\mathrm{Op}(q):\ell^{2}(\mathds{Z})\to\ell^{2}(\mathds{Z})$, see (4.17),
which is a convolution operator. We first search for a fundamental solution
$E:\mathds{Z}\to\mathds{C}$ to

$\mathrm{Op}(q)E=\delta_{0}\quad\text{on }\mathds{Z},$ (8.10)

where $\delta_{0}(\nu)=\delta_{0,\nu}$, see Section 1.6 for the Dirac
notation. Putting

$E(n)=\mathcal{F}^{-1}\\!\left(\frac{1}{q}\right)\\!(n)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{q(\mathrm{e}^{i\xi})}\mathrm{e}^{in\xi}d\xi.$
(8.11)

it follows from (4.5) that $E$ solves (8.10). Thus,

$\begin{split}&\mathrm{Op}(q)E*v=v,\quad
v\in\ell^{2}_{\mathrm{comp}}(\mathds{Z})\\\ &E*\mathrm{Op}(q)u=u,\quad
u\in\ell^{2}_{\mathrm{comp}}(\mathds{Z}),\end{split}$ (8.12)

where $*$ denotes the convolution on $\mathds{Z}$ and
$\ell^{2}_{\mathrm{comp}}$ denotes compactly supported functions in
$\ell^{2}$.

Notice that we may factor (8.8) as

$q(\zeta)=q_{-(m_{-}+\widehat{m}_{0})}\prod_{j=1}^{m_{+}-\widehat{m}_{0}}(1-\zeta^{+}_{j}/\zeta)\prod_{j=m_{+}-\widehat{m}_{0}+1}^{m_{+}}(\zeta-\zeta^{+}_{j})\prod_{k=1}^{m_{-}}(\zeta-\zeta_{k}^{-}).$
(8.13)

Performing the change of variables $\mathrm{e}^{i\xi}=\zeta\in S^{1}$ in
(8.11), we get by shrinking $S^{1}$ to $0$, the residue theorem and (8.13)
that for $n\geqslant 1$

$\begin{split}E(n)=\frac{1}{2\pi
i}\int_{S^{1}}\frac{\zeta^{n-1}}{q(\zeta)}d\zeta=\sum_{r=1}^{m_{+}}\frac{(\zeta^{+}_{r})^{n-1+m_{+}-\widehat{m}_{0}}}{q_{-(m_{-}+\widehat{m}_{0})}\prod_{j\neq
r}(\zeta^{+}_{r}-\zeta^{+}_{j})\prod_{k=1}^{m_{-}}(\zeta^{+}_{r}-\zeta_{k}^{-})},\end{split}$
(8.14)

Similarly, we get that

$\begin{split}E(0)=\sum_{r=1}^{m_{+}}\frac{(\zeta^{+}_{r})^{-1+m_{+}-\widehat{m}_{0}}}{q_{-(m_{-}+\widehat{m}_{0})}\prod_{j\neq
r}(\zeta^{+}_{r}-\zeta^{+}_{j})\prod_{k=1}^{m_{-}}(\zeta^{+}_{r}-\zeta_{k}^{-})}+\frac{\delta_{m_{+},\widehat{m}_{0}}}{q_{-(m_{-}+\widehat{m}_{0})}\prod_{j=1}^{m_{+}}(-\zeta^{+}_{j})\prod_{k=1}^{m_{-}}(-\zeta_{k}^{-})}.\end{split}$
(8.15)

By deforming $S^{1}$ to $|\zeta|=R$, $R\to\infty$, we obtain similarly that
for $n\leqslant-1$,

$E(n)=-\sum_{r=1}^{m_{-}}\frac{(\zeta^{-}_{r})^{n-1+m_{+}-\widehat{m}_{0}}}{q_{-(m_{-}+\widehat{m}_{0})}\prod_{j=1}^{m_{+}}(\zeta^{-}_{r}-\zeta^{+}_{j})\prod_{j\neq
r}(\zeta_{r}^{-}-\zeta_{j}^{-})}.$ (8.16)

Combining (8.14)-(8.16), (8.2), (8.7), (8.1) and (8.9), we have that

$|E(n)|=O(1)\left\\{\begin{array}[]{ll}|\zeta_{m_{+}}^{+}|^{n-1},&n\geqslant
1,\\\ 1,&n=0,\\\ |\zeta_{1}^{-}|^{n-1},&n\leqslant-1.\end{array}\right.$
(8.17)

Here, we replace $(\zeta_{1}^{-})^{-1}$ resp. $\zeta_{m_{+}}$ with $0$ when
$m_{-}=0$ resp. $m_{+}=0$. Thus, summing the geometric series and using (8.4)
we get that

$\|E\|_{\ell^{1}}=O(1)\frac{N}{\log N},$ (8.18)

and similarly that

$\|E\|_{\ell^{2}}=O(1)\sqrt{\frac{N}{\log N}}.$ (8.19)

Since $E$ is in $\ell^{1}$, see (8.18), the operator of convolution with $E$,
denoted $E*$, is a bounded operator $\ell^{2}\to\ell^{2}$. Therefore, (8.12)
can be extended to $u,v\in\ell^{2}(\mathds{Z})$, and $E*$ is the inverse of
$\mathrm{Op}(q)$ on $\ell^{2}(\mathds{Z})$.

3\. Next, for $v\in\ell^{2}(\mathds{Z})$ with support in $[0,N-1]$, we solve

$\mathrm{Op}(q)\psi=0\quad\text{ on }\mathds{Z}$ (8.20)

with “boundary” conditions

$\psi\\!\upharpoonright_{[-(m_{+}-\widehat{m}_{0}),-1]\cup[N,N+m_{-}+\widehat{m}_{0}-1]}=(E*v)\\!\upharpoonright_{[-(m_{+}-\widehat{m}_{0}),-1]\cup[N,N+m_{-}+\widehat{m}_{0}-1]}.$
(8.21)

We suppose here, and in the sequel, that $m_{\pm}\mp\widehat{m}_{0}>0$. The
cases when $m_{+}=\widehat{m}_{0}$ or $m_{-}=\widehat{m}_{0}=0$ can be treated
similarly, and we shall comment on these cases as we go along. In (8.21) it is
understood that when $m_{+}=\widehat{m}_{0}$, then we only have the boundary
condition on $[N,N+m_{-}+\widehat{m}_{0}-1]$, and similarly when
$m_{-}=\widehat{m}_{0}=0$.

Since $q(\zeta)$ satisfies the assumptions of Case 1 of Lemma 4.1, it follows
from Propositions 4.4 and 4.5, and the fact that all roots of $q(\zeta)$ are
simple, that the general solution to (8.20) is of the form

$\psi(\nu)=\sum_{j=1}^{m_{+}}a_{j}^{+}(\zeta_{j}^{+})^{\nu}+\sum_{j=1}^{m_{-}}a_{j}^{-}(\zeta_{j}^{-})^{\nu},\quad
a^{\pm}_{j}\in\mathds{C}.$ (8.22)

By (8.21), the coefficients are determined by

$\begin{pmatrix}V_{+}\Lambda_{+}^{\widehat{m}_{0}-m_{+}}&B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}}\\\
V_{-}\Lambda_{+}^{N}&B_{-}\Lambda_{-}^{N}\\\
\end{pmatrix}a=:Ma=c:=(E*v)\\!\upharpoonright_{[\widehat{m}_{0}-m_{+},-1]\cup[N,N+m_{-}+\widehat{m}_{0}-1]}.$
(8.23)

Here, the right hand side is seen as a vector in $\mathds{C}^{m_{+}+m_{-}}$,
$a=(a_{+},a_{-})^{\sf T}$,
$a_{\pm}=(a^{\pm}_{1},\dots,a^{\pm}_{m_{\pm}})\in\mathds{C}^{m_{\pm}}$,
$\Lambda_{+}:=\mathrm{diag}(\zeta_{1}^{+},\dots,\zeta_{m_{+}-\widehat{m}_{0}}^{+})$,
$\Lambda_{-}:=\mathrm{diag}(\zeta_{m_{+}-\widehat{m}_{0}+1}^{+},\dots,\zeta_{m_{+}}^{+},\zeta_{1}^{-},\dots,\zeta_{m_{-}}^{-})$,

$V_{\pm}:=\begin{pmatrix}1&\dots&1\\\
\zeta_{1}^{+}&\dots&\zeta_{m_{+}-\widehat{m}_{0}}^{+}\\\
\vdots&\vdots&\vdots\\\
(\zeta_{1}^{+})^{m_{\pm}\mp\widehat{m}_{0}-1}&\dots&(\zeta_{m_{+}-\widehat{m}_{0}}^{+})^{m_{\pm}\mp\widehat{m}_{0}-1}\end{pmatrix},$
(8.24)

and

$B_{\pm}:=\begin{pmatrix}1&\dots&1&1&\dots&1\\\
\zeta_{m_{+}-\widehat{m}_{0}+1}^{+}&\dots&\zeta_{m_{+}}^{+}&\zeta_{1}^{-}&\dots&\zeta_{m_{-}}^{-}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
(\zeta_{m_{+}-\widehat{m}_{0}+1}^{+})^{m_{\pm}\mp\widehat{m}_{0}-1}&\dots&(\zeta_{m_{+}}^{+})^{m_{\pm}\mp\widehat{m}_{0}-1}&(\zeta_{1}^{-})^{m_{\pm}\mp\widehat{m}_{0}-1}&\dots&(\zeta_{m_{-}}^{-})^{m_{\pm}\mp\widehat{m}_{0}-1}\end{pmatrix}.$
(8.25)

Here, we use the convention that when $\widehat{m}_{0}=0$ and $m_{+}>0$, then
the first column of $B_{\pm}$ is the one containing powers of $\zeta_{1}^{-}$.

######  Remark 8.3.

When $m_{+}=\widehat{m}_{0}$ then (8.23) is given by
$B_{-}\Lambda_{-}^{N}a=Ma=(E*v)\\!\upharpoonright_{[N,N+m_{-}-1]}$, and when
$m_{-}=\widehat{m}_{0}=0$ then only the first sum in (8.22) remains and (8.23)
is given by
$V_{+}\Lambda_{+}^{-m_{+}}a_{+}=Ma=(E*v)\\!\upharpoonright_{[-m_{+},-1]}$.

By the Cauchy-Schwarz inequality and (8.19), we have that for any
$k\in\mathds{Z}$

$|E*v(k)|=\left|\sum_{0}^{N-1}E(k-m)v(m)\right|\leqslant\|E\|_{\ell^{2}(\mathds{Z})}\|v\|_{\ell^{2}(\mathds{Z})}\leqslant
O(1)\sqrt{\frac{N}{\log N}}\|v\|_{\ell^{2}(\mathds{Z})}.$ (8.26)

We put $L:=L_{+}\oplus L_{-}$ with

$\begin{split}&L_{+}=\mathrm{diag}(G(\zeta_{1}^{+}),\dots,G(\zeta_{m_{+}-\widehat{m}_{0}}^{+})),\\\
&L_{-}=\mathrm{diag}(G(\zeta_{m_{+}-\widehat{m}_{0}+1}^{+}),\dots,G(\zeta_{m_{+}}^{+}),G(\zeta_{1}^{-}),\dots,G(\zeta_{m_{-}}^{-})),\quad
G(x)=\left(\frac{1-|x|^{2N}}{1-|x|^{2}}\right)^{1/2},\end{split}$ (8.27)

where we work with the same notational convention for $L_{-}$ as in (8.25).

Suppose that there exists a constant $0<C<\infty$ such that for $N>0$
sufficiently large $M$ is bijective and such that for all $z\in\Omega_{N}$

$\|LM^{-1}\|\leqslant C\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})},$ (8.28)

with $C_{2}$ as in (8.5). We will prove this fact in Step 4 below. It then
follows from the bijectivity of $M$ that (8.20) with boundary conditions
(8.21) has a unique solutions $\psi$ of the form (8.22) with coefficients
determined by (8.23). Hence, putting

$\widetilde{u}=(E*v-\psi)\\!\upharpoonright_{[0,N-1]},$

we have that $P_{N}(q)\widetilde{u}=v,$ so $P_{N}(q)$ is surjective, and thus,
being a square matrix, bijective. We get that for $N>0$ sufficiently large,

$\begin{split}\|\psi\|_{\ell^{2}([0,N-1])}&\stackrel{{\scriptstyle\eqref{re8}}}{{\leqslant}}\sum_{j=1}^{m_{+}}|a_{j}^{+}|G(\zeta_{j}^{+})+\sum_{j=1}^{m_{-}}|a_{j}^{-}|G(\zeta_{j}^{-})\\\
&\stackrel{{\scriptstyle\mbox{\tiny\rm
H\"{o}lder}}}{{\leqslant}}\sqrt{m_{+}+m_{-}}\|La\|_{\ell^{2}([1,m_{+}+m_{-}])}\\\
&\stackrel{{\scriptstyle\eqref{re9}}}{{\leqslant}}\sqrt{m_{+}+m_{-}}\|LM^{-1}c\|_{\ell^{2}([1,m_{+}+m_{-}])}\\\
&\stackrel{{\scriptstyle\eqref{nnre1}}}{{\leqslant}}O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})}\,\|c\|_{\ell^{2}([1,m_{+}+m_{-}])}\stackrel{{\scriptstyle\eqref{nnre2}}}{{\leqslant}}O(1)\frac{N^{1+C_{2}\Theta(\widehat{m}_{0})}}{\log
N}\|v\|_{\ell^{2}}.\end{split}$ (8.29)

Using Young’s convolution inequality and (8.18), we find that for $N>0$
sufficiently large

$\begin{split}\|\widetilde{u}\|_{\ell^{2}([0,N-1])}&\leqslant\|E*v\|_{\ell^{2}([0,N-1])}+\|\psi\|_{\ell^{2}([0,N-1])}\\\
&\leqslant\|E\|_{\ell^{1}}\|v\|_{\ell^{2}}+\|\psi\|_{\ell^{2}([0,N-1])}\stackrel{{\scriptstyle\eqref{re6.1},\eqref{eq-
sout}}}{{\leqslant}}O(1)\frac{N^{1+C_{2}\Theta(\widehat{m}_{0})}}{\log
N}\|v\|_{\ell^{2}},\end{split}$

uniformly for $z\in\Omega_{N}$. This proves (8.6), provided that (8.28) holds.

4\. It remains to prove (8.28). By (8.7), we have that

$\|V_{\pm}\|,\|B_{\pm}\|=O(1).$ (8.30)

Since $V_{+},B_{-}$ are Vandermonde matrices, we have that $\det
V_{+}=\prod_{1\leqslant i<j\leqslant
m_{+}-\widehat{m}_{0}}(\zeta_{i}^{+}-\zeta_{j}^{+})$ and similar expression
for $\det B_{-}$ given by a finite product of differences of the roots
$\zeta_{m_{+}-\widehat{m}_{0}+1}^{+},\dots,\zeta_{m_{+}}^{+}$ and
$\zeta_{1}^{-},\dots,\zeta_{m_{-}}^{-}$ (resp. just
$\zeta_{1}^{-},\dots,\zeta_{m_{-}}^{-}$ when $\widehat{m}_{0}=0$). Therefore,
(8.2) implies that both matrices are bijective and that their determinants are
uniformly bounded from below in modulus.

Together with (8.7), we deduce that

$\|(V_{+})^{-1}\|,\|(B_{-})^{-1}\|=O(1).$ (8.31)

To construct the inverse of $M$, we proceed via the Schur complement formula
and set

$\Gamma=B_{-}\Lambda_{-}^{N}-V_{-}\Lambda_{+}^{N+m_{+}-\widehat{m}_{0}}V_{+}^{-1}B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}}.$

Using (8.31), (8.30), (8.4), (8.7) and (8.5), we have that

$\begin{split}\|V_{-}\Lambda_{+}^{N+m_{+}-\widehat{m}_{0}}V_{+}^{-1}&B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}-N}B_{-}^{-1}\|\\\
&=O(1)\left\\{\begin{array}[]{ll}|\zeta_{m_{+}}|^{N+m_{+}}|\zeta_{1}^{-}|^{-(N+m_{+})},&\widehat{m}_{0}=0,\\\
|\zeta_{m_{+}-m_{0}}^{+}|^{N+m_{+}-m_{0}}|\zeta_{m_{+}-m_{0}+1}^{+}|^{m_{0}-(N+m_{+})},&\widehat{m}_{0}=m_{0},\end{array}\right.\\\
&=O(1)\left\\{\begin{array}[]{ll}N^{-2C_{0}},&\widehat{m}_{0}=0,\\\
\mathrm{e}^{-N\log
C_{1}}N^{C_{2}},&\widehat{m}_{0}=m_{0}\end{array}\right.\quad=:\varepsilon_{N}(\widehat{m}_{0}).\end{split}$
(8.32)

Hence, for $N>0$ large enough (depending only on the constants in the
assumptions of the theorem) we have that
$|\varepsilon_{N}(\widehat{m}_{0})|<1$, and we assume from now on that this is
the case. By a Neumann series argument we see that

$\Gamma^{-1}=\Lambda_{-}^{-N}B_{-}^{-1}(1+O(1)\varepsilon_{N}(\widehat{m}_{0})).$
(8.33)

Here the $O(1)$ term means an $m_{-}\times m_{-}$ matrix with operator norm
bounded by a constant uniformly in $N$ and $z\in\Omega_{N}$.

A straightforward computation yields that

$M^{-1}=\begin{pmatrix}\Lambda_{+}^{m_{+}-\widehat{m}_{0}}V_{+}^{-1}&-\Lambda_{+}^{m_{+}-\widehat{m}_{0}}V_{+}^{-1}B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}}\\\
0&1\\\ \end{pmatrix}\begin{pmatrix}1&0\\\
-\Gamma^{-1}V_{-}\Lambda_{+}^{N_{+}+m_{+}-\widehat{m}_{0}}V_{+}^{-1}&\Gamma^{-1}\\\
\end{pmatrix}.$ (8.34)

Using (8.27) and (8.34), we obtain that

$\displaystyle LM^{-1}=$ (8.35)
$\displaystyle\begin{pmatrix}L_{+}\Lambda_{+}^{m_{+}-\widehat{m}_{0}}V_{+}^{-1}(1+B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}}\Gamma^{-1}V_{-}\Lambda_{+}^{N_{+}+m_{+}-\widehat{m}_{0}}V_{+}^{-1}),&-L_{+}\Lambda_{+}^{m_{+}-\widehat{m}_{0}}V_{+}^{-1}B_{+}\Lambda_{-}^{\widehat{m}_{0}-m_{+}}\Gamma^{-1}\\\
-L_{-}\Gamma^{-1}V_{-}\Lambda_{+}^{N_{+}+m_{+}-\widehat{m}_{0}}V_{+}^{-1},&L_{-}\Gamma^{-1}\end{pmatrix}.$

To prove (8.28) note that it is sufficient to estimate the norm of each block
of the matrix (8.35) separately. Since $|\zeta_{m_{+}}^{+}|\leqslant
1\leqslant|\zeta_{1}^{-}|$ and (8.5) holds, it follows that

$\|\Lambda_{-}^{-n}\|,\|\Lambda_{+}^{n}\|\leqslant O(1),$ (8.36)

for any fixed $n\in\mathds{N}$. Using that $|\zeta_{m_{+}}^{+}|\leqslant 1$ we
find also that

$\|\Lambda_{+}^{N}\|\leqslant O(1).$ (8.37)

However, in view of (8.5) and since $1\leqslant|\zeta_{1}^{-}|$ we have that

$\|\Lambda_{-}^{-N}\|\leqslant
O(1)\left\\{\begin{array}[]{ll}|\zeta_{1}^{-}|^{-N},&\widehat{m}_{0}=0,\\\
|\zeta^{+}_{m_{+}-m_{0}+1}|^{-N},&\widehat{m}_{0}=m_{0}\end{array}\right.\leqslant
O(1)N^{C_{2}\Theta(\widehat{m}_{0})}$ (8.38)

Continuing, we deduce from (8.31), (8.33) and (8.38) that

$\|\Gamma^{-1}\|\leqslant O(1)\|\Lambda_{-}^{-N}B_{-}^{-1}\|\leqslant
O(1)N^{C_{2}\Theta(\widehat{m}_{0})}$ (8.39)

Recall (8.27). Then, by (8.4),

$\|L_{+}\|\leqslant\max_{1\leqslant j\leqslant
m_{+}-\widehat{m}_{0}}G(\zeta_{j}^{+})=O(1)\sqrt{\frac{N}{\log N}}.$

Combining this with (8.31) and (8.36), we see that

$\|L_{+}\Lambda_{+}^{m_{+}}V_{+}^{-1}\|=O(1)\sqrt{\frac{N}{\log N}}.$ (8.40)

Hence, using (8.40), (8.31), (8.36), (8.30), (8.39) and (8.37), we get that

$\|L_{+}\Lambda_{+}^{m_{+}}V_{+}^{-1}V_{-}\Lambda_{-}^{-m_{+}}\Gamma^{-1}B_{+}\Lambda_{+}^{N+m_{+}}V_{+}^{-1}\|=O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})}$ (8.41)

and

$\|L_{+}\Lambda_{+}^{m_{+}}V_{+}^{-1}V_{-}\Lambda_{-}^{-m_{+}}\Gamma^{-1}\|=O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})}.$ (8.42)

Using (8.4) and (8.5), we find that

$\|L_{-}\Lambda_{-}^{-N}\|=\max\\{\max_{j}|\zeta_{j}^{-}|^{-N}G(\zeta_{j}^{-}),\max_{m_{+}-\widehat{m}_{0}+1\leqslant
j\leqslant
m_{+}}|\zeta_{j}^{+}|^{-N}G(\zeta_{j}^{+})\\}=O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})}.$ (8.43)

Hence, similarly to (8.41), but using as well (8.43) and (8.33), we get that

$\|L_{-}\Gamma^{-1}B_{+}\Lambda_{+}^{N+m_{+}}V_{+}^{-1}\|=O(1)\|L_{-}\Lambda_{-}^{-N}\|=O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})},$ (8.44)

and

$\|L_{-}\Gamma^{-1}\|=O(1)\sqrt{\frac{N}{\log
N}}N^{C_{2}\Theta(\widehat{m}_{0})}.$ (8.45)

Hence, (8.35), (8.40)-(8.42), (8.44), and (8.45) yield (8.28) for $N>0$ large
enough.

######  Remark 8.4.

Recall Remark 8.3. When $m_{+}=\widehat{m}_{0}$ we have that
$M^{-1}=\Lambda_{-}^{-N}B_{-}^{-1}$ and we obtain (8.28) by a similar estimate
as in (8.44). When $m_{-}=\widehat{m}_{0}=0$ then
$M^{-1}=\Lambda_{+}^{m_{+}}V_{+}^{-1}$ and we obtain (8.28) by (8.40).

This completes the proof of Theorem 8.1. ∎

##  9\. Quasimodes for banded Toeplitz matrices

In this section we will construct exponentially decaying quasimodes for the
operator $P_{N}-z$, for $z$ _close_ to - however not on - the curve $p(S^{1})$
while avoiding a small neighborhood of the set of bad points, see Definition
1.3. We will see that the sign of the winding number $d(z)$ will decide
whether these quasimodes decay to the left ($d<0$) or to the right ($d>0$).
The construction will be based on exponential states $\mathfrak{z}^{\pm}$, see
Proposition 9.3, determined by the roots of the Laurent polynomial
$p(\zeta)-z$, with a particular focus on the decay rates of these exponential
states. We will separate between those states $\mathfrak{z}^{\pm}$ which decay
at a fixed rate $r>0$ and those which decay at an $N$-dependent rate
$r\asymp\log N/N$, where the former are determined by the roots of
$p(\zeta)-z$ “far” away from $p(S^{1})$ and the latter are determined by the
roots “close” to $p(S^{1})$.

Throughout this section we fix $z_{0}\in p(S^{1})$ such that

$dp\neq 0\text{ on }p^{-1}(z_{0}).$ (9.1)

Let $\Omega\Subset\mathds{C}$ be an open relatively compact simply connected
neighbourhood of $z_{0}$ such that

$|dp|\geqslant 1/C\text{ on }p^{-1}(z)\text{ for all }z\in\Omega.$ (9.2)

Note that by the implicit function theorem all roots $\zeta=\zeta(z)$ of
$p(\zeta)-z$ are simple and depend smoothly on $z\in\Omega$. Hence, after
possibly shrinking $\Omega$, there exists $0<C<\infty$ such that for any two
distinct roots $\zeta(z)\neq\omega(z)$ we have that

$|\zeta-\omega|\geqslant\frac{1}{C},$ (9.3)

uniformly for $z\in\Omega$. Recall the notation of the roots from Lemma 4.1.

Until further notice we will denote the winding number of the curve $p(S^{1})$
around a point $z\in\mathds{C}$, see also (4.34), by

$d\stackrel{{\scriptstyle\mathrm{def}}}{{=}}d(z)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{ind}_{p(S^{1})}(z).$
(9.4)

######  Assumption 9.1.

Let $z_{0}\in p(S^{1})$ be as in (9.1), and let $\Omega$ be as in (9.2), so
that (9.3) holds. Let $\Omega_{N}\Subset\Omega\backslash p(S^{1})$ be an open
relatively compact simply connected non-empty $N$-dependent set such that the
winding number $d(z)\neq 0$ is constant for all $z\in\Omega_{N}$, and such
that there exist constants $C_{\alpha},C_{\beta},C_{\gamma}>0$ and
$m^{0}_{\pm}\geqslant 0$, independent of $N$, such that for $N>0$ large enough
(depending only on these constants) and for all $z\in\Omega_{N}$ the following
holds:

  * •

we have that

$p^{-1}(z)\cap(S^{1}+D(0,C_{\gamma}N^{-1}\log N))=\emptyset;$ (9.5)

  * •

when $d>0$, then $0\leqslant m_{+}^{0}\leqslant d\leqslant m_{+}$ and

$0<|\zeta_{1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{+}-m_{+}^{0}}^{+}|<\frac{1}{C_{\beta}}<1-C_{\alpha}\frac{\log
N}{N}\leqslant|\zeta_{m_{+}-m_{+}^{0}+1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{+}}^{+}|<1;$
(9.6)

  * •

when $d<0$, then $0\leqslant m_{-}^{0}\leqslant-d\leqslant m_{-}$ and

$1<|\zeta_{1}^{-}|\leqslant\cdots\leqslant|\zeta_{m_{-}^{0}}^{-}|<1+C_{\alpha}\frac{\log
N}{N}<C_{\beta}\leqslant|\zeta_{m_{-}^{0}+1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{-}}^{-}|<+\infty.$
(9.7)

In (9.6) we work with the notation convention that when $m_{+}=m_{+}^{0}$ then
the inequalities before $1/C_{\beta}$ are void, and when $m_{+}^{0}=0$, then
the inequalities on the right hand side of $1-C_{\alpha}\log N/N$ are void. A
similar convention is to be understood for (9.7).

######  Remark 9.2.

1\. Unless otherwise stated, all future estimates will only depend on $\Omega$
and the constants in Assumption 9.1, and therefore will be uniform in
$\Omega_{N}$. This will be understood implicitly when we state that an
estimate is uniform in $\Omega_{N}$.

2\. We recall that we will work with $z\in\Omega(\varepsilon_{\ref{theo-
location}},C_{\ref{theo-location}},N)$, see (1.7) and the assumptions of
Theorem 1.6. In particular, ${\rm dist}(z,p(S^{1}))<C_{\ref{theo-
location}}\log N/N$. Let $z_{0}$ be the point on $p(S^{1})$ closest to $z$.
Then, since $dp(z_{0})\neq 0$ (as we work in $\mathcal{G}_{p,\varepsilon}$),
as well as the existence of $C_{\alpha}$ as in (9.6) or (9.7), e.g. that for
$d>0$, $\zeta_{m_{+}}^{+}\geqslant 1-C_{\alpha}\log N/N$. To see the
multiplicity count, i.e. that e.g. $m_{+}^{0}\leqslant d$ when $d>0$, note
that away from the set $\mathcal{B}_{2}$, if ${\sf g}(p)=1$ then there is
precisely one $\theta_{0}$ with $p(e^{i\theta_{0}})=z_{0}$, and since
$dp|_{\theta_{0}}\neq 0$, there is exactly one root $\zeta_{m_{+}}$ near $1$.
When ${\sf g}(p)>1$, there are ${\sf g}(p)$ such roots, while
$|d|\geqslant{\sf g}(p)$ if $d\neq 0$, so again the same conclusion holds. On
the other hand, by Remark 6.17 we have that for any eigenvalue $z$ in the good
set, with probability approaching one, the assumption (9.5) holds with
$C_{\gamma}$ as defined in Remark 6.17.

Our aim is to construct quasimodes for $P_{N}-z$, $z\in\Omega_{N}$, these are
approximate $\ell^{2}$-normalized eigenvectors $\psi\in\mathds{C}^{N}$ in the
sense that

$\|(P_{N}-z)\psi\|\to 0$

when $N\to\infty$. Our focus will be on obtaining quantitative estimates for
this decay. The eigenvectors of $P_{[0,\infty[}$ corresponding to the
eigenvalue $z$, when $d>0$, and shifted versions of the eigenvectors of
$P_{]-\infty,0]}$ corresponding to the eigenvalue $z$, when $d<0$, are
excellent candidates for quasimodes (after truncation).

We will construct these quasimodes as follows: Recall from Proposition 4.6
that these eigenvectors are exponential states which are created from powers
of the roots of $p(\zeta)-z$. Using assumption (9.6) and (9.7) we will take
care (whenever possible) to distinguish between eigenvectors which are built
up only by roots strictly away from the unit circle, and those eigenvectors
which contain also a contribution from roots close to the unit circle. After
truncation we orthonormalize the states within the two respective groups. As
we shall see below, we will need to slightly modify the behaviour at the
boundary of the second group to obtain a sufficiently fast decay. This way we
will obtain an almost orthogonal $|d|$-dimensional system of quasimodes, see
Proposition 9.6 below.

Recall Proposition 4.6. To construct quasimodes out of (4.46) respectively
(4.47), we begin with constructing a suitable system of solutions to (4.44)
when $N_{+}>0$ and its analog when $N_{-}>0$ with $m_{+},N_{+},\zeta_{j}^{+}$
replaced by $m_{-},N_{-},\zeta_{j}^{-}$ and the corresponding vectors
$\mathfrak{A}_{j}^{-}$ indexed over $1\leqslant\nu\leqslant N_{-}$.

We begin with considering the case $N_{+}>0$. Notice that this implies that
$0\notin p^{-1}(\Omega)$. As discussed after (9.2), the roots $\zeta_{z}$ of
$p(\zeta)-z$ depend smoothly on $z\in\Omega$, so, after potentially slightly
shrinking $\Omega$, there exists a constant $0<C<\infty$ such that any root
$\zeta_{z}$ satisfies

$1/C\leqslant|\zeta_{z}|,\text{ for all }z\in\Omega.$ (9.8)

Let $z\in\Omega_{N}$, suppose that $d>0$ and recall from (4.34) that
$d=m_{+}-N_{+}$. In view of (9.6), we find that $m_{+}-m_{+}^{0}\geqslant
N_{+}>0$. For the sake of the presentation we first suppose that
$d>m_{+}^{0}\geqslant 1$. Recall the matrix $\mathfrak{A}$ from (4.44) and
write

$\mathfrak{B}_{+}=(\mathfrak{A}_{1}^{+},\dots,\mathfrak{A}_{N_{+}}^{+})\in\mathds{C}^{N_{+}\times
N_{+}},\quad\mathfrak{C}_{+}=(\mathfrak{A}_{1}^{+},\dots,\mathfrak{A}_{m_{+}-m_{+}^{0}}^{+})\in\mathds{C}^{N_{+}\times(m_{+}-m_{+}^{0})}$

By (9.8) we have that for all $j=1,\dots,m_{+}$

$\|\mathfrak{A}_{j}^{+}\|\leqslant C^{N_{+}}\sqrt{N_{+}}.$ (9.9)

Notice that $\mathfrak{B}_{+}$ is a Vandermonde type matrix, so
$|\det\mathfrak{B}_{+}|=\prod_{k=1}^{N^{+}}|\zeta_{k}^{+}|^{-N^{+}}\prod_{1\leqslant
i<j\leqslant N_{+}}|\zeta_{i}^{+}-\zeta_{j}^{+}|$. Thus, by (9.8), (9.3), we
see that $\mathfrak{B}_{+}$ is invertible and, using (9.9), we find

$\|\mathfrak{B}_{+}\|,\|\mathfrak{B}_{+}^{-1}\|=O(1),$ (9.10)

where the constant depends only on $\Omega$. Hence, by (9.6), we see that
$\mathfrak{C}_{+}$ has rank $N_{+}$ and has a kernel of dimension
$\dim\mathcal{N}(\mathfrak{C}_{+})=m_{+}-m_{+}^{0}-N_{+}=d-m_{+}^{0}>0$. Let
$x_{1},\dots,x_{d-m_{+}^{0}}\in\mathds{C}^{m_{+}-m_{+}^{0}}$ be an orthonormal
system spanning $\mathcal{N}(\mathfrak{C}_{+})$. Clearly,

$\mathds{C}^{m_{+}}\ni a^{+}_{j}:=x_{j}\oplus
0_{m_{+}^{0}}\in\mathcal{N}(\mathfrak{A})\text{ for }j=1,\dots,d-m_{+}^{0}.$
(9.11)

For $j=d-m_{+}^{0}+1,\dots,d$, we put

$a^{+}_{j}:=(-\mathfrak{B}_{+}^{-1}\mathfrak{A}^{+}_{j+N_{+}},\delta_{j})\in\mathds{C}^{m_{+}},$
(9.12)

where $\delta_{k}\in\mathds{C}^{m_{+}-N_{+}}$ is so that
$\delta_{k}(n)=\delta_{k,n}$ for $n\in[m_{+}-N_{+}]$. By construction we have
that the $a^{+}_{j}$ are linearly independent and that they span
$\mathcal{N}(\mathfrak{A})$. Writing
$X_{+}=(x_{1},\dots,x_{d-m_{+}^{0}})\in\mathds{C}^{(m_{+}-m_{+}^{0})\times(d-m_{+}^{0})}$
and
$\Delta_{+}=(\delta_{d-m_{+}^{0}+1},\dots,\delta_{d})\in\mathds{C}^{d\times
m_{+}^{0}}$, we put

$A^{+}_{1}=\begin{pmatrix}X_{+}\\\ 0\\\ \end{pmatrix},\quad
A^{+}_{2}=\begin{pmatrix}-\mathfrak{B}_{+}^{-1}(\mathfrak{A}^{+}_{m_{+}-m_{+}^{0}+1},\dots,\mathfrak{A}^{+}_{m_{+}})\\\
\Delta_{+}\\\ \end{pmatrix},$

where on the left hand side $0\in\mathds{C}^{m_{+}^{0}\times(d-m_{+}^{0})}$.
Then,

$A^{+}=(a^{+}_{1},\dots,a^{+}_{d})=(A^{+}_{1},A^{+}_{2}).$ (9.13)

When $m_{+}^{0}=0$, then we have no solutions (9.12) and
$A^{+}=A^{+}_{1}=X_{+}$. When $d=m_{+}^{0}$, then we have no solutions (9.11)
and all vectors $a_{j}^{+}$ are of the form (9.12), so

$A^{+}=A_{2}^{+}=\begin{pmatrix}-\mathfrak{B}_{+}^{-1}(\mathfrak{A}^{+}_{N_{+}+1},\dots,\mathfrak{A}^{+}_{m_{+}})\\\
1_{d}\\\ \end{pmatrix}.$

Keeping in mind these special cases, we shall keep writing (9.13) for all
cases of $d\geqslant m_{+}^{0}\geqslant 0$, with the convention that
$A_{1}^{+}$ is void when $d=m_{+}^{0}$ and that $A^{+}_{2}$ is void when
$m_{+}^{0}=0$.

When $d<0$ (keeping in mind (4.34)) and $N_{-}>0$, we work under the
assumption (9.7) and we consider the analogue of $\mathfrak{A}$ from (4.44)
with $m_{+},N_{+},\zeta_{j}^{+}$ replaced by $m_{-},N_{-},\zeta_{j}^{-}$ and
the corresponding vectors $\mathfrak{A}_{j}^{-}$ indexed over
$1\leqslant\nu\leqslant N_{-}$. Notice that since $N_{-}>0$, it follows that
$\infty\notin p^{-1}(\Omega)$, so, after potentially slightly shrinking
$\Omega$, we have that any root $\zeta_{z}$ of $p(\zeta)-z$ satisfies
$|\zeta_{z}|\leqslant C$ for all $z\in\Omega$. Thus, the analogue of (9.9)
holds for $\mathfrak{A}_{j}^{-}$. Putting
$\mathfrak{B}_{-}=(\mathfrak{A}_{m_{-}-N_{-}+1}^{-},\dots,\mathfrak{A}_{m_{-}}^{-})$,
we find by a Van der Monde argument that the analogue of (9.10) holds for
$\mathfrak{B}_{-}$. Then, by the exact same steps as above, with the obvious
modifications, we can construct $|d|$ linearly independent solutions
$a^{-}_{j}$, to $\mathfrak{A}a^{-}=0$, such that

$A^{-}=(a^{-}_{1},\dots,a^{-}_{d})=\begin{pmatrix}-\mathfrak{B}_{-}^{-1}(\mathfrak{A}^{-}_{1},\dots,\mathfrak{A}^{-}_{m_{-}^{0}})&X_{-}\\\
\Delta_{-}&0\\\ \end{pmatrix}=(A^{-}_{1},A^{-}_{2}),$ (9.14)

where $\delta_{k}\in\mathds{C}^{m_{-}-N_{-}}$ is so that
$\delta_{k}(n)=\delta_{k,n}$ for $n\in[m_{-}-N_{-}]$,
$\Delta_{-}=(\delta_{1},\dots,\delta_{m_{-}^{0}})\in\mathds{C}^{|d|\times
m_{-}^{0}}$, and $X_{-}\in\mathds{C}^{(|d|-m_{-}^{0})\times(|d|-m_{-}^{0})}$
is a matrix with orthonormal columns spanning
$\mathcal{N}((\mathfrak{A}_{m_{-}^{0}+1}^{-},\dots,\mathfrak{A}_{m_{-}}^{-}))$.

In what follows we identify $\ell^{2}([0,N-1])\simeq\mathds{C}^{N}$, so for
the sake of consistency we index vectors and the columns of matrices starting
from $0$.

######  Proposition 9.3.

Let $N\geqslant N_{0}>1$ be integers, $z_{0}\in p(S^{1})$ be as in (9.1),
$\Omega\Subset\mathds{C}$ be a sufficiently small open simply connected
relatively compact neighbourhood of $z_{0}$, satisfying (9.2), and
$\Omega_{N}\Subset\Omega\backslash p(S^{1})$ be as in Assumption 9.1. Let
$N>0$ be sufficiently large, depending only on $\Omega$ and the constants in
Assumption 9.1 and the roots of $p(\zeta)-z$ be denoted as in Lemma 4.1. Write

$\mathfrak{z}_{j}^{+}=(1,\zeta_{j}^{+},\dots,(\zeta_{j}^{+})^{N-1})^{t}\in\mathds{C}^{N},\quad
j=1,\dots,m_{+},$ (9.15)

and

$\mathfrak{z}_{j}^{-}=((\zeta_{j}^{-})^{1-N},(\zeta_{j}^{-})^{2-N},\dots,1)^{t}\in\mathds{C}^{N},\quad
j=1,\dots,m_{-},$ (9.16)

with the convention that ${1}/{\infty}=0$. Moreover, let
$\mathfrak{Z}^{\pm}=(\mathfrak{z}_{1}^{\pm},\dots,\mathfrak{z}_{m_{\pm}}^{\pm})$.

  * •

Suppose that $d>0$. When $N_{+}>0$, let $A^{+}=(a_{1}^{+},\dots,a_{d}^{+})$ be
as in (9.13), and when $N_{+}\leqslant 0$ put
$A^{+}=I_{m_{+}}\in\mathds{C}^{m_{+}\times m_{+}}$. Define
$I_{+,1}:=[d-m_{+}^{0}]$ and $I_{+,2}:=[d]\backslash[d-m_{+}^{0}]$. Define

$(\widetilde{u}_{1}^{+},\dots,\widetilde{u}^{+}_{d})=\mathfrak{Z}^{+}A^{+}(G_{+,1}^{-1/2}\oplus\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2})$
(9.17)

with
$\mathfrak{L}_{+}=\mathrm{diag}(\|\mathfrak{z}_{m_{+}-m_{+}^{0}+1}^{+}\|,\dots,\|\mathfrak{z}_{m_{+}}^{+}\|)$,
$G_{+,2}=(\mathfrak{Z}^{+}A^{+}_{2}\mathfrak{L}_{+}^{-1})^{*}(\mathfrak{Z}^{+}A^{+}_{2}\mathfrak{L}_{+}^{-1})$,
and $G_{+,1}=(\mathfrak{Z}^{+}A^{+}_{1})^{*}(\mathfrak{Z}^{+}A^{+}_{1})$,
using the convention that
$(\widetilde{u}_{1}^{+},\dots,\widetilde{u}^{+}_{d})=\mathfrak{Z}^{+}A^{+}G_{+,1}^{-1/2}$
when $I_{+,2}=\emptyset$, and that
$(\widetilde{u}_{1}^{+},\dots,\widetilde{u}^{+}_{d})=\mathfrak{Z}^{+}A^{+}\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}$
when $I_{+,1}=\emptyset$. Then, uniformly for $z\in\Omega_{N}$,

$\langle\widetilde{u}_{n}^{+}|\widetilde{u}_{m}^{+}\rangle=\left\\{\begin{array}[]{ll}\delta_{n,m},&(n,m)\in
I_{+,1}\times I_{+,1}\cup I_{+,2}\times I_{+,2}\\\ \delta_{n,m}+O((\log
N/N)^{1/2}),&(n,m)\in I_{+,1}\times I_{+,2}\cup I_{+,2}\times
I_{+,1},\end{array}\right.$ (9.18)

and

$\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|\leqslant
O(1)\left\\{\begin{array}[]{ll}\mathrm{e}^{-N_{0}\log C_{\beta}},&j\in
I_{+,1},\\\ \mathrm{e}^{-\frac{N_{0}}{N}\log N^{C_{\gamma}}},&j\in
I_{+,2}.\end{array}\right.$ (9.19)

  * •

Suppose that $d<0$. When $N_{-}>0$, let $A^{-}=(a_{1}^{-},\dots,a_{|d|}^{-})$
be as in (9.14), and when $N_{-}\leqslant 0$ put
$A^{-}=I_{m_{-}}\in\mathds{C}^{m_{-}\times m_{-}}$. Define
$I_{-,2}:=[m_{-}^{0}]$ and $I_{-,1}:=[-d]\backslash[m_{-}^{0}]$. Define

$(\widetilde{u}_{1}^{-},\dots,\widetilde{u}^{-}_{|d|})=\mathfrak{Z}^{-}A^{-}(\mathfrak{L}_{-}^{-1}G_{-,1}^{-1/2}\oplus
G_{-,2}^{-1/2})$ (9.20)

with
$\mathfrak{L}_{-}=\mathrm{diag}(\|\mathfrak{z}_{m_{+}-m_{+}^{0}+1}^{+}\|,\dots,\|\mathfrak{z}_{m_{+}}^{+}\|)$,
$G_{-,1}=(\mathfrak{Z}^{-}A^{-}_{1}\mathfrak{L}_{-}^{-1})^{*}(\mathfrak{Z}^{-}A^{-}_{1}\mathfrak{L}_{-}^{-1})$
and $G_{-,2}=(\mathfrak{Z}^{-}A^{-}_{2})^{*}(\mathfrak{Z}^{-}A^{-}_{2})$,
using a similar notation convention as above. Then, uniformly for
$z\in\Omega_{N}$,

$\langle\widetilde{u}_{n}^{-}|\widetilde{u}_{m}^{-}\rangle=\left\\{\begin{array}[]{ll}\delta_{n,m},&(n,m)\in
I_{+,1}\times I_{+,1}\cup I_{+,2}\times I_{+,2}\\\ \delta_{n,m}+O((\log
N/N)^{1/2}),&(n,m)\in I_{-,1}\times I_{-,2}\cup I_{-,2}\times
I_{-,1},\end{array}\right.$ (9.21)

and

$\|\mathbf{1}_{[0,N-N_{0}-1]}\widetilde{u}_{j}^{-}\|\leqslant
O(1)\left\\{\begin{array}[]{ll}\mathrm{e}^{-\frac{N_{0}}{N}\log
N^{C_{\gamma}}},&j\in I_{-,2},\\\ \mathrm{e}^{-N_{0}\log C_{\beta}},&j\in
I_{-,1}.\end{array}\right.$ (9.22)

######  Remark 9.4.

When $d>0$, the normalized versions of the vectors defined in (9.15) will be
termed as the pure states associated with $z$. Similarly for $d<0$ the
normalized version vectors (9.16) will be the pure states corresponding to
$z$.

Thus, for $z$ satisfying (9.6) we note that the pure states
$\\{\mathfrak{z}_{j}^{+}/\|\mathfrak{z}_{j}^{+}\|\\}_{j=1}^{m_{+}-m_{+}^{0}}$
are completely localized and its entries decay exponentially in $N$, while the
remaining pure states decay and localize at a scale $N/\log N$. Similar
conclusion holds for $z$’s satisfying (9.7).

######  Remark 9.5.

For later use, let us record that for the Jordan block if $z$ is inside the
the unit disc then it has one pure state $\mathfrak{z}_{1}^{+}$, and
$\widetilde{u}_{1}^{+}=\mathfrak{z}_{1}^{+}/\|\mathfrak{z}_{1}^{+}\|$.

We postpone the proof of Proposition 9.3 to after that of Proposition 9.6.
Continuing, we notice that the $\widetilde{u}_{j}^{\pm}$ defined in (9.17)
resp. (9.20), are the truncated kernel elements of $P_{[0,+\infty[}-z$ resp.
$P_{]-\infty,N-1]}-z$, separated into two groups according to their decay
behaviour and orthonormalized within their respective groups. We could use
them already as quasimodes for $P_{N}-z$. In fact, it follows from (9.19) with
$N_{0}=N-(N_{+}\lor 0+N_{-}\lor 0)$, that when $d>0$

$\|(P_{N}-z)\widetilde{u}^{+}_{j}\|\leqslant
O(1)\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|\leqslant
O(1)\left\\{\begin{array}[]{ll}\mathrm{e}^{-N\log C_{\beta}},&j\in I_{+,1},\\\
N^{-C_{\gamma}},&j\in I_{+,2},\end{array}\right.$ (9.23)

and similarly from (9.22) when $d<0$. Ultimately we want to use these
quasimodes to obtain spectral gaps between the
$(|d|-m_{\mathrm{sign}(d)}^{0})$-th and the
$(|d|-m_{\mathrm{sign}(d)}^{0}+1)$-th, and between the $|d|$-th and
$(|d|+1)$-th singular value of $P_{N}-z$. We will see, that these quasimodes
will be sufficiently good for our purposes when $\gamma>2$. However, to obtain
sufficiently fast decay when $\gamma>1$ we need to modify them.

######  Proposition 9.6.

Under the assumptions of Proposition 9.3, we put $N_{0}=(1-1/\log N)(N-1)$,
and we have the following:

1\. If $d>0$, let $\widetilde{u}^{+}_{j}$, $j=1,\dots,d$, be as in (9.17) and
define $\psi^{+}_{j}\in\mathds{C}^{N}$, by

$\psi^{+}_{j}(\nu)=\left\\{\begin{array}[]{ll}\widetilde{u}^{+}_{j}(\nu),&j\in
I_{+,1},\\\
\mathbf{1}_{[0,N_{0}]}(\nu)\widetilde{u}^{+}_{j}(\nu)+\mathbf{1}_{[N_{0}+1,N-1]}(\nu)\frac{\cos(\frac{\pi}{2}\frac{\nu}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\widetilde{u}^{+}_{j}(\nu),&j\in
I_{+,2}.\end{array}\right.$ (9.24)

2\. If $d<0$, let $\widetilde{u}^{-}_{j}$, $j=1,\dots,-d$, be as in (9.20) and
define $\psi^{-}_{j}\in\mathds{C}^{N}$, by

$\psi^{-}_{j}(\nu)=\left\\{\begin{array}[]{ll}\mathbf{1}_{[0,N-N_{0}]}(\nu)\frac{\cos(\frac{\pi}{2}\frac{N-1-\nu}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\widetilde{u}^{-}_{j}(\nu)+\mathbf{1}_{[N-N_{0}+1,N-1]}(\nu)\widetilde{u}^{-}_{j}(\nu),&j\in
I_{-,2},\\\ \widetilde{u}^{-}_{j}(\nu),&j\in I_{-,1}.\end{array}\right.$
(9.25)

Then, for $i,j=1,\dots,|d|$, uniformly in $z\in\Omega_{N}$,

$\langle\psi^{\mathrm{sign}(d)}_{i}|\psi^{\mathrm{sign}(d)}_{j}\rangle=\delta_{i,j}+\left\\{\begin{array}[]{ll}0,&\text{
when }i,j\in I_{\mathrm{sign}(d),1}\\\ O(\log N)N^{-C_{\gamma}},&\text{ when
}i,j\in I_{\mathrm{sign}(d),2}\\\ O(\log N)N^{-C_{\gamma}}+O((\log
N/N)^{1/2}),&\text{else},\end{array}\right.$ (9.26)

and

$\|(P_{N}-z)\psi_{i}^{\mathrm{sign}(d)}\|=O(1)\left\\{\begin{array}[]{ll}\mathrm{e}^{-N\log
C_{\beta}},&\text{ when }i\in I_{\mathrm{sign}(d),1}\\\ \frac{\log
N}{N}N^{-C_{\gamma}},&\text{ when }i\in I_{\mathrm{sign}(d),2}.\\\
\end{array}\right.$ (9.27)

We refer to the vectors $\widetilde{\psi}^{\pm}_{j}$ as quasimodes.

###### Proof.

We will consider only the case when $d>0$, the case when $d<0$ is similar. Let
$N>0$ be large enough so that the conclusions of Proposition 9.3 hold.

1\. First notice that

$\cos\left(\frac{\pi}{2}\frac{N_{0}}{N-1}\right)=\sin\frac{\pi}{2\log
N}=\frac{\pi}{2\log N}(1+O((\log N)^{-2}).$ (9.28)

Let $j=d-m_{+}^{0}+1,\dots,d$, then

$\|\widetilde{u}_{j}^{+}-\psi_{j}^{+}\|^{2}=\sum_{\nu=N_{0}+1}^{N-1}\left|1-\frac{\cos(\frac{\pi}{2}\frac{\nu}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\right|^{2}|\widetilde{u}^{+}_{j}(\nu)|^{2}\leqslant
O((\log N)^{2})\|\mathbf{1}_{[N_{0}+1,N-1]}\widetilde{u}_{j}^{+}\|^{2}.$

Applying (9.19), we get that

$\|\widetilde{u}_{j}^{+}-\psi_{j}^{+}\|\leqslant O(\log N)N^{-C_{\gamma}}.$
(9.29)

######  Remark 9.7.

For later use we note here that the analogue of (9.29) in the case when $d<0$
holds, i.e.

$\|\widetilde{u}_{j}^{-}-\psi_{j}^{-}\|\leqslant O(\log
N)N^{-C_{\gamma}},\quad j\in I_{-,2}.$

This, together with the $\widetilde{u}_{j}^{+}$, $j=1,\dots,d$, being
normalized, see (9.18), yields that

$\|\psi_{j}^{+}\|=1+\left\\{\begin{array}[]{ll}0,&\text{ when }j\in I_{+,1}\\\
O(\log N)N^{-C_{\gamma}},&\text{ when }j\in I_{+,2}.\end{array}\right.$

Using furthermore the fact that the $\widetilde{u}_{j}^{+}$ are almost
orthogonal, see (9.18), we find that

$\begin{split}\langle\psi_{i}^{+}|\psi_{j}^{+}\rangle&=\langle\psi_{i}^{+}-\widetilde{u}_{i}^{+}|\psi_{j}\rangle+\langle\widetilde{u}_{i}^{+}|\psi_{j}^{+}-\widetilde{u}_{j}^{+}\rangle+\langle\widetilde{u}_{i}^{+}|\widetilde{u}_{j}^{+}\rangle\\\
&=\delta_{i,j}+\left\\{\begin{array}[]{ll}0,&\text{ when }i,j\in I_{+,1},\\\
O(\log N)N^{-C_{\gamma}},&\text{ when }i,j\in I_{+,2}\\\ O(\log
N)N^{-C_{\gamma}}+O((\log
N/N)^{1/2}),&\text{else}.\end{array}\right.\end{split}$

2\. It remains to prove (9.27). Recall from (9.17) and Proposition 4.6 that
the
$\widetilde{u}_{j}^{+}=(\mathfrak{Z}^{+}A^{+}(G_{+,1}^{-1/2}\oplus\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}))_{-,j}$
are the truncated and normalized (in fact almost orthonormalized) kernel
elements of $P_{[0,+\infty[}-z$. Here, and in the sequel we denote by
$A_{-,j}\in\mathds{C}^{n}$ the $j$-th column of some $n\times m$ matrix $A$ as
a vector in $\mathds{C}^{n}$. Let $u_{k}\in\ell^{2}([0,+\infty[)$,
$k=1,\dots,d$, be given by

$u_{k}(\nu)=\left\\{\begin{array}[]{ll}(\mathfrak{Z}^{+}A^{+})_{\nu,k},&0\leqslant\nu\leqslant
N-1,\\\ 0,&\nu\geqslant N.\end{array}\right.$ (9.30)

Then,

$((P_{[0,+\infty[}-z)u_{k})(\nu)=0,\quad 0\leqslant\nu\leqslant N-1-N_{-}\lor
0.$

By (4.34) we see that $N_{-}>0$ since we work here with $d>0$. To ease the
notation we write $\widetilde{N}_{\pm}=N_{\pm}\lor 0$, and
$\widetilde{a}_{k}=a_{k}-z\delta_{k,0}$ when $a_{k}$ exists, otherwise
$\widetilde{a}_{k}=-z\delta_{k,0}$, so that

$\sum_{k=-N_{-}}^{N_{+}}a_{k}\tau^{k}-z=\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\tau^{k}.$

Thus, for $0\leqslant\nu\leqslant N-1-\widetilde{N}_{-}$,

$\begin{split}(P_{N}-z)\widetilde{u}_{j}^{+}(\nu)&=\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\mathbf{1}_{[0,N-1]}(\nu-k)\widetilde{u}_{j}^{+}(\nu-k)\\\
&\stackrel{{\scriptstyle\eqref{qm5}}}{{=}}\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\mathbf{1}_{[0,N-1]}(\nu-k)(\mathfrak{Z}^{+}A^{+}(G_{+,1}^{-1/2}\oplus\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}))_{(\nu-k),j}\\\
&=\sum_{n=1}^{d}\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\mathbf{1}_{[0,N-1]}(\nu-k)(\mathfrak{Z}^{+}A^{+})_{(\nu-k),n}(G_{+,1}^{-1/2}\oplus\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2})_{n,j}\\\
&\stackrel{{\scriptstyle\eqref{eq-
pkak1}}}{{=}}\sum_{n=1}^{d}(G_{+,1}^{-1/2}\oplus\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2})_{n,j}((P_{[0,+\infty[}-z)u_{n})(\nu)=0.\end{split}$
(9.31)

Thus, for $0\leqslant\nu\leqslant N-1-\widetilde{N}_{-}$ and $j\in I_{+,1}$

$(P_{N}-z)\psi_{j}^{+}(\nu)=(P_{N}-z)\widetilde{u}_{j}(\nu)=0.$

Recall that $z\in\Omega\Subset\mathds{C}$ is relatively compact. Hence,
applying (9.19) with $N_{0}=N-1-\widetilde{N}_{-}-\widetilde{N}_{+}$, we see
that, uniformly in $z\in\Omega_{N}$,

$\|(P_{N}-z)\psi_{j}^{+}\|=O(1)\|\mathbf{1}_{[N-1-\widetilde{N}_{-}-\widetilde{N}_{+},N-1]}\psi_{i}^{+}\|=O(1)\mathrm{e}^{-N\log
C_{\beta}},\leavevmode\nobreak\ j\in I_{+,1},$ (9.32)

Until further notice let $j\in I_{+,2}$. Equation (9.31) implies that for
$0\leqslant\nu\leqslant N_{0}-\widetilde{N}_{-}$

$(P_{N}-z)\psi_{j}^{+}(\nu)=(P_{N}-z)\widetilde{u}_{j}^{+}(\nu)=0.$

Next, by (9.28), we have for
$N_{0}-(\widetilde{N}_{+}+\widetilde{N}_{-})<\nu\leqslant
N_{0}+(\widetilde{N}_{+}+\widetilde{N}_{-})$ that

$\left|1-\frac{\cos(\frac{\pi}{2}\frac{\nu}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\right|\leqslant
O(N^{-1}\log N)|N_{0}-\nu|=O(N^{-1}\log N).$ (9.33)

Hence, for $N_{0}-\widetilde{N}_{-}<\nu\leqslant N_{0}+\widetilde{N}_{+}$, we
get by (9.31), (9.33) that

$\begin{split}(P_{N}-z)\psi_{j}^{+}(\nu)&=\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\psi_{j}^{+}(\nu-k)\\\
&=\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\widetilde{u}_{j}^{+}(\nu-k)-\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\left(1-\frac{\cos(\frac{\pi}{2}\frac{\nu-k}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\right)\widetilde{u}_{j}^{+}(\nu-k)\mathbf{1}_{[N_{0}+1,N-1]}(\nu-k)\\\
&=O(N^{-1}\log N)\|\mathbf{1}_{I(\nu)}\widetilde{u}_{j}^{+}\|,\end{split}$
(9.34)

where $I(\nu):=[\nu-\widetilde{N}_{+},\nu+\widetilde{N}_{-}]$. For
$N_{0}+\widetilde{N}_{+}<\nu\leqslant N-1-\widetilde{N}_{-}$ we get by (9.31)
and a similar estimate as in (9.33), (9.34), that

$\begin{split}(P_{N}-z)\psi_{j}^{+}(\nu)&=\frac{\cos(\frac{\pi}{2}\frac{\nu}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\widetilde{u}_{j}^{+}(\nu-k)-\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\frac{\cos(\frac{\pi}{2}\frac{\nu}{N-1})-\cos(\frac{\pi}{2}\frac{\nu-k}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\widetilde{u}_{j}^{+}(\nu-k)\\\
&=O(N^{-1}\log N)\|\mathbf{1}_{I(\nu)}\widetilde{u}_{j}^{+}\|.\end{split}$
(9.35)

Now let $N-1-\widetilde{N}_{-}<\nu\leqslant N-1$. For
$-\widetilde{N}_{-}\leqslant k\leqslant\widetilde{N}_{+}$ we have that
$\cos(\frac{\pi}{2}\frac{\nu-k}{N-1})\asymp N^{-1}$, so by (9.28) we get that

$\begin{split}(P_{N}-z)\psi_{j}^{+}(\nu)&=\sum_{k=-\widetilde{N}_{-}}^{\widetilde{N}_{+}}\widetilde{a}_{k}\frac{\cos(\frac{\pi}{2}\frac{\nu-k}{N-1})}{\cos(\frac{\pi}{2}\frac{N_{0}}{N-1})}\widetilde{u}_{j}^{+}(\nu-k)\mathbf{1}_{[N_{0}+1,N-1]}(\nu-k)\\\
&=O(N^{-1}\log
N)\|\mathbf{1}_{I(\nu)}\mathbf{1}_{[N_{0}+1,N-1]}\widetilde{u}_{j}^{+}\|.\end{split}$
(9.36)

Combining (9.31), (9.34), (9.35) and (9.36) we obtain that

$\begin{split}\|(P_{N}-z)\psi_{j}^{+}\|^{2}&\leqslant O((N^{-1}\log
N)^{2})\sum_{\nu=N_{0}-\widetilde{N}_{-}}^{N-1}\|\mathbf{1}_{I(\nu)}\mathbf{1}_{[N_{0}-(\widetilde{N}_{+}+\widetilde{N}_{-}),N-1]}\widetilde{u}_{j}^{+}\|^{2}\\\
&\leqslant O((N^{-1}\log
N)^{2})\|\mathbf{1}_{[N_{0}-(\widetilde{N}_{+}+\widetilde{N}_{-}),N-1]}\widetilde{u}_{j}^{+}\|^{2}.\end{split}$
(9.37)

Applying now (9.19) with $N_{0}=(1-1/\log
N)N-(\widetilde{N}_{+}+\widetilde{N}_{-})$, we get that

$\|(P_{N}-z)\psi_{j}^{+}\|=O(N^{-1}\log N)N^{-C_{\gamma}}$ (9.38)

which concludes the proof of the proposition. ∎

###### Proof of Proposition 9.3.

We will only consider the case when $d>0$. The case $d<0$ can be proven
similarly with the obvious modifications. In what follows, we will assume that
$N>0$ is sufficiently large, depending only on $\Omega$ and the constants in
Assumption 9.1, without us mentioning this at every occurrence. Similarly, we
note that all constants $C>0$ and error estimates will be uniform in
$z\in\Omega_{N}$, and in fact they depend only on the above mentioned
parameters, so we shall only mention this at a few important points.

1\. Since the roots of $p(\zeta)-z$ depend smoothly on $z\in\Omega$, see the
discussion above (9.3), we have that either $0\in p^{-1}(z_{0})$ or we may
take $\Omega$ sufficiently small so that $0\notin p^{-1}(\Omega)$. In both
cases $0\notin p^{-1}(\Omega_{N})$. Hence, we have that the symbol
$p(\zeta)-z$ satisfies for all $z\in\Omega_{N}$ the assumptions of Case 1 of
Lemma 4.1, see also (4.40). We then know that $m_{+}>0$ since $d>0$, see
(4.34).

2\. Recall from (8.24) the $m_{+}\times m_{+}$ matrices $V=V_{+}$ and
$\Lambda=\Lambda_{+}$ (with $\widehat{m}_{0}=0$). Then, for $r=1,2$ and
$\psi_{1}\in\mathds{C}^{d-m_{+}^{0}}$, $\psi_{2}\in\mathds{C}^{m_{+}^{0}}$,

$\|\mathfrak{Z}^{+}A_{r}^{+}\psi_{r}\|^{2}\geqslant\sum_{j=0}^{\lfloor(N-2)/m_{+}\rfloor-1}\|V\Lambda^{jm_{+}}A_{r}^{+}\psi_{r}\|^{2}.$
(9.39)

Let $s_{1}(V)\geqslant\cdots\geqslant s_{m_{+}}(V)\geqslant 0$ denote the
singular values of $V$. Since $|\det V|=\prod_{1}^{m_{+}}s_{j}(V)$, and
$s_{1}(V)=\|V\|\leqslant m_{+}$, it follows that

$s_{m_{+}}(V)\geqslant\frac{|\det V|}{m_{+}^{m_{+}}}=\frac{\prod_{1\leqslant
i<j\leqslant
m_{+}}|\zeta_{i}^{+}-\zeta_{j}^{+}|}{m_{+}^{m_{+}}}\geqslant\frac{1}{C}.$
(9.40)

Here, in the equality we used the formula for the determinant of the Van der
Monde matrix $V$. Moreover, the constant in the last inequality is uniform $N$
and in $z\in\Omega_{N}$ by (9.3). Since $\Lambda$ is a diagonal matrix, we get
from (9.39) that

$\|\mathfrak{Z}^{+}A_{r}^{+}\psi_{r}\|^{2}\geqslant
s^{2}_{m_{+}}(V)\sum_{k=1}^{m_{+}}|(A_{r}^{+}\psi_{r})(k)|^{2}\sum_{j=0}^{\lfloor(N-2)/m_{+}\rfloor-1}|\zeta^{+}_{k}|^{2jm_{+}}.$
(9.41)

Putting

$d_{k}^{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j=0}^{\lfloor(N-2)/m_{+}\rfloor-1}|\zeta^{+}_{k}|^{2jm_{+}}=\frac{1-|\zeta^{+}_{k}|^{2\lfloor(N-2)/m_{+}\rfloor
m_{+}}}{1-|\zeta^{+}_{k}|^{2m_{+}}},$ (9.42)

and $D=\mathrm{diag}(d_{1},\dots,d_{m_{+}})$, we get from (9.41) that

$\|\mathfrak{Z}^{+}A_{r}^{+}\psi_{r}\|\geqslant
s_{m_{+}}(V)\|DA_{r}^{+}\psi_{r}\|.$ (9.43)

Observe that the smallest singular value of $D$ is $\min_{k}d_{k}\geqslant 1$.
Recall (9.13). Provided that $d>m_{+}^{0}$, we notice that $A^{+}_{1}$ is an
isometry since the columns of $X_{+}$ are orthonormal. Thus, it follows from
(9.43), (9.40) that

$\|\mathfrak{Z}^{+}A_{1}^{+}\psi_{1}\|\geqslant\frac{1}{C}\|\psi_{1}\|,$
(9.44)

so the smallest singular value of $G_{+,1}$ is at least $1/C$, which,
$G_{+,1}$ being self-adjoint, implies that it is bijective with inverse
bounded in norm by $C$. In particular, we have that

$\|A^{+}_{1}G_{+,1}^{-1/2}\|=\sqrt{C}.$ (9.45)

3\. Working with $m_{+}^{0}\geqslant 1$, we turn to $G_{+,2}$ and we will show
first that for $n\geqslant d-m_{+}^{0}+1$ the $u^{+}_{n}$ are, up to a small
error, given by $\mathfrak{z}_{n}/\|\mathfrak{z}_{n}\|$. First, we compute

$F_{n,m}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}|\langle\mathfrak{z}^{+}_{n}\mid\mathfrak{z}^{+}_{m}\rangle|=\left|\sum_{0}^{N-1}(\zeta^{+}_{n}\overline{\zeta}^{+}_{m})^{\nu}\right|=\left(\frac{(1-|\zeta_{n}^{+}|^{2N})(1-|\zeta_{n}^{+}|^{2N})+|(\zeta_{n}^{+})^{N}-(\zeta_{m}^{+})^{N}|^{2}}{(1-|\zeta_{n}^{+}|^{2})(1-|\zeta_{m}^{+}|^{2})+|\zeta_{n}^{+}-\zeta_{m}^{+}|^{2}}\right)^{1/2}.$
(9.46)

By (9.6), (9.5) and (9.3) we find that

$F_{n,m}=\left\\{\begin{array}[]{ll}\asymp 1,&\text{ when }1\leqslant
n,m\leqslant m_{+}-m_{+}^{0},\\\ \asymp 1,&\text{ when }n\neq m,\\\
\asymp\frac{N}{\log N},&\text{ when }m_{+}-m_{+}^{0}+1\leqslant n=m\leqslant
m_{+}.\end{array}\right.$ (9.47)

Second, notice that by (9.9), (9.10), we have that
$T:=-\mathfrak{B}_{+}^{-1}(\mathfrak{A}^{+}_{m_{+}-m_{+}^{0}+1},\dots,\mathfrak{A}^{+}_{m_{+}})$
has norm $\|T\|=O(1)$, depending only on $\Omega$. Recalling the definition of
$\Delta_{+}$ from the discussion before (9.13), we see that

$\mathfrak{Z}^{+}A_{2}^{+}\mathfrak{L}_{+}^{-1}=\widetilde{\mathfrak{Z}}^{+}T\mathfrak{L}_{+}^{-1}+\widehat{\mathfrak{Z}}^{+}\mathfrak{L}_{+}^{-1},$
(9.48)

where
$\widetilde{\mathfrak{Z}}^{+}=(\mathfrak{z}_{1}^{+},\dots,\mathfrak{z}_{N_{+}}^{+})$
and
$\widehat{\mathfrak{Z}}^{+}=(\mathfrak{z}_{m_{+}-m_{+}^{0}+1}^{+},\dots,\mathfrak{z}_{m_{+}}^{+})$.
Using (9.47), we get that

$\begin{split}G_{+,2}&=\mathfrak{L}_{+}^{-1}(\widehat{\mathfrak{Z}}^{+})^{*}\widehat{\mathfrak{Z}}^{+}\mathfrak{L}_{+}^{-1}+\mathfrak{L}_{+}^{-1}\left((\widehat{\mathfrak{Z}}^{+})^{*}\widetilde{\mathfrak{Z}}^{+}T+(\widetilde{\mathfrak{Z}}^{+}T)^{*}\widehat{\mathfrak{Z}}^{+}+(\widetilde{\mathfrak{Z}}^{+}T)^{*}\widetilde{\mathfrak{Z}}^{+}T\right)\mathfrak{L}_{+}^{-1}\\\
&=1+O(1)\frac{\log N}{N},\end{split}$ (9.49)

where the $O(1)$ term denotes an $m_{+}^{0}\times m_{+}^{0}$ matrix with norm
bounded by $O(1)$, depending only on $\Omega$ and the constants in Assumption
9.1. Hence, for $N>0$ sufficiently large (depending only on $\Omega$ and the
constants in Assumption 9.1), we have that $G_{+,2}$ is invertible and

$G_{+,2}^{-1/2}=1+O(1)\frac{\log N}{N}.$ (9.50)

Hence, using again (9.47) and denoting by $M_{-,n}$ the $n$-th column of a
matrix, we get that for $n\in I_{+,2}$

$\begin{split}\widetilde{u}^{+}_{n}&=\left(\mathfrak{Z}^{+}A_{2}^{+}\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}\right)_{-,n}\\\
&=\left(\widetilde{\mathfrak{Z}}^{+}T\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}\right)_{-,n}+\left(\widehat{\mathfrak{Z}}^{+}\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2}\right)_{-,n}=\frac{\mathfrak{z}_{n}^{+}}{\|\mathfrak{z}_{n}^{+}\|}+O(1)\sqrt{\frac{\log
N}{N}},\end{split}$ (9.51)

where the $O(1)$ term denotes a vector in $\mathds{C}^{N}$ whose $\ell^{2}$
norm is bounded by $O(1)$. In particular, (9.44) and (9.50) imply that (9.17)
is well-defined.

######  Remark 9.8.

For a later use we note that the analogue of (9.51) in the case when $d<0$
holds, i.e.

$\widetilde{u}^{-}_{n}=\frac{\mathfrak{z}_{n}^{-}}{\|\mathfrak{z}_{n}^{-}\|}+O(1)\sqrt{\frac{\log
N}{N}},\quad n\in I_{-,2}.$

4\. Since
$(\mathfrak{Z}^{+}A_{1}^{+}G_{+,1}^{-1/2})^{*}(\mathfrak{Z}^{+}A_{1}^{+}G_{+,1}^{-1/2})=1$,
and
$(\mathfrak{Z}^{+}A_{2}^{+}\mathfrak{L}_{+}G_{+,2}^{-1/2})^{*}(\mathfrak{Z}^{+}A_{2}^{+}\mathfrak{L}_{+}G_{+,2}^{-1/2})=1$,
we confirm the first line of (9.18). To prove the second line, let $(n,m)\in
I_{+,1}\times I_{+,2}$. Equations (9.51) and (9.17) then yield that

$\begin{split}\langle\widetilde{u}^{+}_{n}|\widetilde{u}^{+}_{m}\rangle&=\left\langle\mathfrak{Z}^{+}(A^{+}_{1}G_{+,1}^{-1/2})_{-,n}\Big{|}\frac{\mathfrak{z}_{m}^{+}}{\|\mathfrak{z}_{m}^{+}\|}\right\rangle+O(1)\sqrt{\frac{\log
N}{N}}\\\
&=\sum_{k=1}^{m_{+}-m_{+}^{0}}\langle\mathfrak{z}_{k}^{+}|\mathfrak{z}_{m}^{+}\rangle\|\mathfrak{z}_{m}^{+}\|^{-1}(A^{+}_{1}G_{+,1}^{-1/2})_{k,n}+O(1)\sqrt{\frac{\log
N}{N}}\stackrel{{\scriptstyle\eqref{qm12aa},\eqref{qm12c}}}{{\leqslant}}O(1)\sqrt{\frac{\log
N}{N}}.\end{split}$ (9.52)

The other case of the second line of (9.18) follows from symmetry.

5\. At last, we turn to proving (9.19). Let $j\in I_{+,1}$. Using Hölder’s
inequality, we compute

$\begin{split}\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|^{2}&=\sum_{\nu=N_{0}}^{N-1}|(\mathfrak{Z}^{+}A^{+}G_{+,1}^{-1/2})_{\nu,j}|^{2}\leqslant\sum_{\nu=N_{0}}^{N-1}\left|\sum_{\mu=1}^{m_{+}-m_{0}^{+}}(\mathfrak{Z}^{+})_{\nu,\mu}(A^{+}G_{+,1}^{-1/2})_{\mu,j}\right|^{2}\\\
&\leqslant\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{m_{+}-m_{0}^{+}}|(\mathfrak{Z}^{+})_{\nu,\mu}|^{2}\right)\left(\sum_{\mu=1}^{m_{+}-m_{0}^{+}}|(A^{+}G_{+,1}^{-1/2})_{\mu,j}|^{2}\right).\end{split}$
(9.53)

Since the right most term in the last line is controlled by the Hillbert-
Schmidt norm of the corresponding matrices, we get by using (9.45) and summing
the geometric series that

$\begin{split}\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|^{2}&\leqslant\|A^{+}G_{+,1}^{-1/2}\|^{2}_{\mathrm{HS}}\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{m_{+}-m_{0}^{+}}|(\mathfrak{Z}^{+})_{\nu,\mu}|^{2}\right)\\\
&\stackrel{{\scriptstyle\eqref{qm12aa}}}{{\leqslant}}O(1)\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{m_{+}-m_{0}^{+}}|\zeta_{\mu}^{+}|^{2\nu}\right)=O(1)\sum_{\mu=1}^{m_{+}-m_{0}^{+}}|\zeta_{\mu}^{+}|^{2N_{0}}\frac{1-|\zeta_{\mu}^{+}|^{2(N-N_{0})}}{1-|\zeta_{\mu}^{+}|^{2}}.\end{split}$
(9.54)

Since $|\zeta_{m_{+}-m_{+}^{0}}|\leqslant 1/C_{\beta}<1$, we have that for
$\mu=1,\dots,m_{+}-m_{+}^{0}$,

$\frac{1-|\zeta^{+}_{\mu}|^{2(N-N_{0})}}{1-|\zeta^{+}_{\mu}|^{2}}\leqslant\frac{C_{\beta}}{C_{\beta}-1}.$
(9.55)

Similarly we see that

$|\zeta^{+}_{\mu}|^{2N_{0}}\leqslant\mathrm{e}^{-2N_{0}\log C_{\beta}}.$
(9.56)

Combining this with (9.54) and (9.55) yields that

$\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|\leqslant
O(1)\mathrm{e}^{-N_{0}\log C_{\beta}},$

uniformly in $z\in\Omega_{N}$, and we conclude the first line in (9.19).

It remains to prove the second line of (9.19). Recall the definition of
$A_{2}^{+}$ (9.13), and notice that

$\begin{split}A_{2}^{+}\mathfrak{L}_{+}^{-1}&=\begin{pmatrix}-\mathfrak{B}^{-1}(\mathfrak{A}^{+}_{m_{+}-m_{+}^{0}+1},\dots,\mathfrak{A}^{+}_{m_{+}})\mathfrak{L}_{+}^{-1}\\\
\Delta_{+}\mathfrak{L}_{+}^{-1}\\\ \end{pmatrix}\\\
&=\widetilde{\mathfrak{L}}_{+}^{-1}\begin{pmatrix}-\mathfrak{B}^{-1}(\mathfrak{A}^{+}_{m_{+}-m_{+}^{0}+1},\dots,\mathfrak{A}^{+}_{m_{+}})\mathfrak{L}_{+}^{-1}\\\
\Delta_{+}\\\
\end{pmatrix}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\widetilde{\mathfrak{L}}_{+}^{-1}\widetilde{A}_{2}^{+}\end{split}$
(9.57)

where
$\widetilde{\mathfrak{L}}_{+}:=1_{N_{+}}\oplus\mathrm{diag}(\|\mathfrak{z}_{N_{+}+1}^{+}\|,\dots,\|\mathfrak{z}_{m_{+}}^{+}\|)$.
Furthermore, it follows from (9.9), (9.10) and (9.47) that

$\|\mathfrak{B}^{-1}(\mathfrak{A}^{+}_{m_{+}-m_{+}^{0}+1},\dots,\mathfrak{A}^{+}_{m_{+}})\mathfrak{L}_{+}^{-1}\|=O(1)\sqrt{\frac{\log
N}{N}},$ (9.58)

which implies that

$\|\widetilde{A}_{2}^{+}\|=O(1).$ (9.59)

Using the commutation relation (9.57), we compute for $j\in I_{+,2}$ that

$\begin{split}\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|^{2}&=\sum_{\nu=N_{0}}^{N-1}|(\mathfrak{Z}^{+}A_{2}^{+}\mathfrak{L}_{+}^{-1}G_{+,2}^{-1/2})_{\nu,j}|^{2}\leqslant\sum_{\nu=N_{0}}^{N-1}\left|\sum_{\mu=1}^{m_{+}}(\mathfrak{Z}^{+}\widetilde{\mathfrak{L}}_{+}^{-1})_{\nu,\mu}(\widetilde{A}_{2}^{+}G_{+,2}^{-1/2})_{\mu,j}\right|^{2}\\\
&\leqslant\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{m_{+}}|(\mathfrak{Z}^{+}\widetilde{\mathfrak{L}}_{+}^{-1})_{\nu,\mu}|^{2}\right)\left(\sum_{\mu=1}^{m_{+}}|(\widetilde{A}^{+}G_{+}^{-1/2})_{\mu,j}|^{2}\right).\end{split}$
(9.60)

Thus, by (9.59), (9.50)

$\begin{split}\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|^{2}&\leqslant\|\widetilde{A}^{+}G_{+}^{-1/2}\|_{\mathrm{HS}}^{2}\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{m_{+}}|(\mathfrak{Z}^{+}\widetilde{\mathfrak{L}}_{+}^{-1})_{\nu,\mu}|^{2}\right)\\\
&\leqslant
O(1)\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{N_{+}}|\zeta_{\mu}^{+}|^{2\nu}+\sum_{\mu=N_{+}+1}^{m_{+}}\frac{|\zeta_{\mu}^{+}|^{2\nu}}{\|\mathfrak{z}_{\mu}^{+}\|^{2}}\right).\end{split}$
(9.61)

Using (9.47), we get by a computation similar to (9.55) and (9.56) that

$\sum_{\nu=N_{0}}^{N-1}\left(\sum_{\mu=1}^{N_{+}}|\zeta_{\mu}^{+}|^{2\nu}+\sum_{\mu=N_{+}+1}^{m_{+}-m_{+}^{0}}\frac{|\zeta_{\mu}^{+}|^{2\nu}}{\|\mathfrak{z}_{\mu}^{+}\|^{2}}\right)=O(1)\mathrm{e}^{-N_{0}\log
C_{\beta}}.$ (9.62)

On the other hand, using (9.46) and the fact that $|\zeta_{m_{+}}|<1$, we find
$\mu=m_{+}-m_{+}^{0}+1,\dots,m_{+}$ that

$\sum_{\nu=N_{0}}^{N-1}\frac{|\zeta_{\mu}^{+}|^{2\nu}}{\|\mathfrak{z}_{\mu}^{+}\|^{2}}=|\zeta_{\mu}^{+}|^{2N_{0}}\frac{1-|\zeta_{\mu}^{+}|^{2(N-N_{0})}}{1-|\zeta_{\mu}^{+}|^{2N}}\leqslant|\zeta_{\mu}^{+}|^{2N_{0}}.$
(9.63)

By (9.5), we see that

$|\zeta_{\mu}^{+}|^{2N_{0}}\leqslant\exp\left(2N_{0}\log\left(1-C_{\gamma}\frac{\log
N}{N}\right)\right)\leqslant\exp\left(-2N_{0}C_{\gamma}\frac{\log
N}{N}\right)\leqslant\mathrm{e}^{-\frac{2N_{0}}{N}\log N^{C_{\gamma}}},$
(9.64)

which, together with (9.63), (9.62) and (9.61) yields that for $N>0$
sufficiently large

$\|\mathbf{1}_{[N_{0},N-1]}\widetilde{u}_{j}^{+}\|^{2}\leqslant
O(1)\mathrm{e}^{-\frac{2N_{0}}{N}\log N^{C_{\gamma}}},$ (9.65)

uniformly in $z\in\Omega_{N}$, and we conclude the first second line in
(9.19). ∎

##  10\. Singular values and vectors

We provide in this section various estimates on singular values and vectors of
$P_{N}-z$. The section is divided to subsections, dealing respectively with
small singular values, singular vectors, large singular values, and the
smallest singular value and Hilbert-Schmidt norm of the matrices appearing in
the resolvent expansion of the matrices discussed in Section 3.

###  10.1. Small singular values

We will work under the same assumptions as discussed in the beginning of
Section 9. We will use the quasimodes constructed in Proposition 9.6 to obtain
an upper bound on the $|d|$-th smallest singular values of $P_{N}-z$
corresponding to the decay speed of the quasimodes. We will also show with the
help of Theorem 8.1 that the $(|d|+1)$-th singular value of $P_{N}-z$ is
significantly larger than the $|d|$-th one, which leads to a spectral gap.
This is known as splitting phenomenon, see for instance [15, Section 9.2].
This phenomenon has typically been investigated for $z$ being at a fixed
distance from $p(S^{1})$ or for fixed symbols. In that case it is known that
the smallest $|d|$ singular values are of order $O(\mathrm{e}^{-N/C})$ whereas
the $(|d|+1)$-th singular value is bounded from below by some small $N$
independent constant. What is new here is that we allow the spectral parameter
$z$ to be at an $N$-dependent distance from $p(S^{1})$, which leads to a
significantly smaller spectral gap.

Moreover, we will show that due to the assumptions (9.6), (9.7), there exists
a second spectral gap between the first $(|d|-m_{\mathrm{sign}(d)}^{0})$
singular values, which are exponentially small in $N$, and the
$(|d|-m_{\mathrm{sign}(d)}^{0}+1)$-th singular value, which is at most
polynomially small in $N$.

Both spectral gaps will be crucial in describing the localization of the
eigenvectors of $P^{\delta}_{N}$.

######  Proposition 10.1.

Let $z_{0}\in p(S^{1})$ be as in (9.1) and let $\Omega\Subset\mathds{C}$ be a
sufficiently small open simply connected relatively compact neighborhood of a
point $z_{0}$ satisfying (9.2). Let $\Omega_{N}\Subset\Omega\backslash
p(S^{1})$ be as in Assumption 9.1. For $z\in\Omega_{N}$ let
$t_{1}\leqslant\cdots\leqslant t_{N}$ denote the eigenvalues of
$\sqrt{(P_{N}-z)^{*}(P_{N}-z)}$. Then there exists a constant $0<C<\infty$
such that, for all $N$ large enough (depending only on $\Omega$ and the
constants in Assumption 9.1) and for all $z\in\Omega_{N}$

$0\leqslant t_{1}\leqslant\cdots\leqslant t_{|d|}\leqslant C\frac{\log
N}{N}N^{-C_{\gamma}},$ (10.1)

and

$t_{|d|+1}\geqslant\frac{1}{C}\frac{\log N}{N}.$ (10.2)

Additionally, if $|d|-m_{\mathrm{sign}(d)}^{0}>0$, then

$0\leqslant t_{1}\leqslant\cdots\leqslant
t_{|d|-m_{\mathrm{sign}(d)}^{0}}\leqslant C\mathrm{e}^{-N\log C_{\beta}},$
(10.3)

and if $m_{\mathrm{sign}(d)}^{0}>0$

$\frac{1}{C}\frac{\log N}{N}N^{-C_{\alpha}}\leqslant
t_{|d|-m_{\mathrm{sign}(d)}^{0}+1}\leqslant\cdots\leqslant t_{|d|}\leqslant
C\frac{\log N}{N}N^{-C_{\gamma}}.$ (10.4)

######  Remark 10.2.

If in Assumption 9.1 we have that
$|\zeta_{m_{+}-m_{+}^{0}+1}^{+}|=|\zeta_{m_{+}}^{+}|$ (in case $d>0$) or
$|\zeta_{m_{-}^{0}+1}^{-}|=|\zeta_{m_{-}}^{-}|$ (in case $d<0$) then clearly
one can take $C_{\alpha}=C_{\gamma}+\epsilon$ for any $\epsilon>0$, improving
upon (10.4). By (5.85), this will always be the case for us.

###### Proof.

Recall that for a square $N\times N$ matrix $A$, $t_{j}(A)$ denote the
eigenvalues of $\sqrt{A^{*}A}$ (ordered non-decreasingly) and $s_{j}(A)$
denote the singular values (ordered non-increasingly), see (1.13). By the
singular value decomposition of the square matrix $A$, we have that
$t_{j}(A)=t_{j}(A^{*})$ and $s_{j}(A)=s_{j}(A^{*})$. Recall the Ky Fan
inequalities, see for instance [32]: for two square matrices $A,B$ we have

$\begin{split}&s_{n+m-1}(A+B)\leqslant s_{n}(A)+s_{m}(B),\\\
&s_{n+m-1}(AB)\leqslant s_{n}(A)s_{m}(B).\end{split}$ (10.5)

Further, if $A$ is an invertible $N\times N$ matrix, then

$s_{n}(A^{-1})=\frac{1}{s_{N-n+1}(A)},\quad n=1,\dots,N.$

From (4.6) we know that the symbol of the adjoint $\mathrm{Op}(p)^{*}$ is
given by $\overline{\widetilde{p}}(\omega)=\overline{p}(1/\omega)$. So in view
of Lemma 4.1, the roots of $\overline{p}(1/\omega)-\overline{z}$, or
equivalently the roots of $p(1/\overline{\omega})-{z}$ are given by
$\omega^{\mp}_{j}=1/\overline{\zeta}_{j}^{\,\pm}$, where $\zeta_{j}^{\pm}$ are
the roots of $p(\zeta)-z$. Hence, the discussion and results of Section 4
valid for $p(\zeta)-z$ remain valid for $\overline{p}(1/\omega)-\overline{z}$
with the roles of $m_{+},N_{+}$, and $m_{-},N_{-}$ exchanged. Furthermore, the
roles of the assumption (9.6) and (9.7) are then interchanged. This argument
will allow us to reduce the number cases of symbols $p-z$ we need to consider,
since we may always pass to studying the singular values of the adjoint
$(P_{N}-z)^{*}$. In particular, in the rest of the proof we may and will
assume that $d>0$ for $z\in\Omega_{N}$.

In what follows, we will work with $N>0$ sufficiently large, depending only on
$\Omega$ and the constants in Assumption 9.1, without us mentioning this
dependence at every occurrence. Similar, we note that all error estimates will
be uniform in $z\in\Omega_{N}$, and in fact they depend only on the above
mentioned parameters, so we shall only mention this at a few important points.

1\. Let $z\in\Omega_{N}$. We begin with the upper bound on $t_{d}$. Let
$\psi^{+}_{j}$, $j=1,\dots,d$ be as in (9.24), and note that for $N>0$
sufficiently large, they are linearly independent. Indeed if they were not,
then there would exist an $0\neq a=(a_{1},\dots,a_{d})\in\mathds{C}^{d|}$ such
that $a_{1}\psi^{+}_{1}+\dots+a_{d}\psi^{+}_{d}=0$. Let $j_{0}$ be such that
$|a_{j_{0}}|\geqslant|a_{j}|$ for $j\neq j_{0}$, with $|a_{j_{0}}|>0$. Since
the $\psi^{+}_{j}$ are almost-orthonormal by (9.26), we find that

$1+O(\log N)N^{-C_{\gamma}}=-\sum_{j\in[d],j\neq
j_{0}}\frac{a_{j}}{a_{j_{0}}}\langle\psi^{+}_{j}|\psi^{+}_{j_{0}}\rangle=O(\log
N)N^{-C_{\gamma}}+O((\log N/N)^{1/2}),$ (10.6)

a contradiction.

Let $K_{d}\subset\mathds{C}^{N}$ denote the linear subspace of
$\mathds{C}^{N}$ spanned by the $\psi_{j}^{+}$ and notice that it has
dimension $d$, for $N>0$ sufficiently large. For any $\psi\in K_{d}$, we write
$\psi=\sum_{j=1}^{d}a_{j}\psi^{+}_{j}$. If $\|\psi\|=1$, then we get by (9.26)
and the Cauchy-Schwarz inequality that for $N>0$ sufficiently large,

$1=\sum_{j=1}^{d}|a_{j}|^{2}\|\psi^{+}_{j}\|^{2}+\ \sum_{i\neq
j}a_{i}\overline{a}_{j}\langle\psi^{+}_{i}|\psi^{+}_{j}\rangle=\|a\|^{2}(1+\hat{\epsilon}_{N}),$

where $\hat{\epsilon}_{N}=O(1)(N^{-C_{\gamma}}\log N+(\log N/N)^{1/2})$ and
the constant is uniform in $z\in\Omega_{N}$ and independent of $\psi$. Thus,
for $N>0$ large enough, $\|a\|^{2}=(1+\hat{\epsilon}_{N})$, uniformly in the
choice of $z\in\Omega_{N}$ and $\psi$. The min-max principle and (9.27) yield
that

$\begin{split}t_{d}^{2}&=\min\limits_{L:\dim
L=d}\max\limits_{\begin{subarray}{c}\psi\in L\\\
\|\psi\|=1\end{subarray}}\|(P_{N}-z)\psi\|^{2}\leqslant\max\limits_{\begin{subarray}{c}\psi\in
K_{d}\\\ \|\psi\|=1\end{subarray}}\|(P_{N}-z)\psi\|^{2}\\\
&\leqslant\max\limits_{\begin{subarray}{c}\psi\in K_{d}\\\
\|\psi\|=1\end{subarray}}\left(\sum_{1}^{d}|a_{j}|\cdot\|(P_{N}-z)\psi^{+}_{j}\|\right)^{2}\stackrel{{\scriptstyle\eqref{qn4}}}{{\leqslant}}O(1)\left(\frac{\log
N}{N}N^{-C_{\gamma}}\right)^{2},\end{split}$

and we conclude the upper bound in (10.1) and (10.4).

Suppose that $d>m_{+}^{0}$ and recall (9.6) and (9.7). Letting
$K_{d-m_{+}^{0}}$ denote the linear subspace of $\mathds{C}^{N}$ spanned by
$\psi^{+}_{1},\dots,\psi^{+}_{d-m_{+}^{0}}$, and recalling from (9.26) that
these vectors are orthonormal, we deduce similarly as above from the min-max
principle and (9.27), that

$t_{d-m_{+}^{0}}^{2}\leqslant\max\limits_{\begin{subarray}{c}\psi\in
K_{d-m_{+}^{0}}\\\
\|\psi\|=1\end{subarray}}\|(P_{N}-z)\psi\|^{2}\leqslant\sum_{1}^{d-m_{+}^{0}}\|(P_{N}-z)\psi^{+}_{j}\|^{2}\stackrel{{\scriptstyle\eqref{qn4}}}{{\leqslant}}O(\mathrm{e}^{-2N\log
C_{\beta}}),$

and we conclude (10.3).

2\. We may write $\Omega$ as the disjoint union

$\Omega=\Omega^{\prime}\,\dot{\cup}\,(p(S^{1})\cap\Omega)\,\dot{\cup}\,\Omega^{\prime\prime}$

so that $\Omega_{N}\subset\Omega^{\prime}$. In particular
$\mathrm{ind}_{p(S^{1})}(z)=d$ for all $z\in\Omega^{\prime}$.

The rest of the proof will be concerned with the lower bounds on $t_{d+1}$ and
$t_{d-m_{+}^{0}+1}$ for $z\in\Omega_{N}$, and we will have to consider the two
cases of $N_{\pm}>0$ or $N_{+}\leqslant 0$. Note that by (4.15), these are the
only cases we need to consider, since $d>0$ while by Remark 4.2 and (4.34), we
have that $d\leqslant 0$ if $N_{-}<0$. We note here that in both cases
$\infty\notin p^{-1}(\Omega)$, see (4.15) and Lemma 4.1.

Recall from the discussion after (9.2) that all roots $\zeta_{z}$ of
$p(\zeta)-z$ depend smoothly on $z\in\Omega$. So, when either $N_{\pm}>0$, or
$N_{+}\leqslant 0$ and $0\notin p^{-1}(z_{0})$, we see that
$\\{0,\infty\\}\notin p^{-1}(\Omega)$ for $\Omega$ sufficiently small. On the
other hand, when $N_{+}\leqslant 0$ and $0\in p^{-1}(z_{0})$, then by the
smooth dependence of the roots on $z$ that $0\notin p^{-1}(\Omega^{\prime})$,
after possibly shrinking $\Omega$. Hence, for $\Omega$ sufficiently small,
there exists a constant $C>0$ such that any root $\zeta_{z}$ satisfies

$0<|\zeta_{z}|\leqslant C,\text{ for all }z\in\Omega^{\prime}.$ (10.7)

Since the symbol $p(\zeta)-z$ for $z\in\Omega^{\prime}$ satisfies the
assumptions of Case 1 of Lemma 4.1, we may order its roots as in (4.26). So,
in combination with (10.7) we get that

$0<|\zeta_{1}^{+}|\leqslant\cdots\leqslant|\zeta_{m_{+}}^{+}|<1<|\zeta_{1}^{-}|\leqslant\cdots\leqslant|\zeta_{m_{-}}^{-}|\leqslant
C,\text{ for all }z\in\Omega^{\prime}.$ (10.8)

By (4.34) we know that $m_{+}\geqslant d>0$, so we work in (10.8) with the
convention that when $m_{-}=0$ then only the estimates on the existing roots
in (10.8) apply.

Recall Assumption 9.1, and as in Theorem 8.1, let $\Theta$ denote the
Heaviside function, and let

$\widehat{m}_{0}\in\left\\{\begin{array}[]{ll}\\{0,m_{+}^{0}\\},&m_{+}^{0}>0\\\
\\{0\\},&m_{+}^{0}=0.\end{array}\right.$ (10.9)

To shorten the notation we will write $\Theta:=\Theta(\widehat{m}_{0})$
whenever convenient. The case $\widehat{m}_{0}=0$ will be used to prove the
lower bound in (10.2), and the case $\widehat{m}_{0}=m_{+}^{0}$ for the lower
bound in (10.4). Write $\widetilde{N}_{+}=N_{+}\lor 0$. For $\zeta\in S^{1}$
and $z\in\Omega^{\prime}$, we can write the symbol as

$p(\zeta)-z=\zeta^{-\widetilde{N}_{+}}\left(\sum_{0}^{\widetilde{N}_{+}+N_{-}}a_{N_{+}-j}\,\zeta^{j}-z\zeta^{\widetilde{N}_{+}}\right)=\zeta^{m_{+}-\widetilde{N}_{+}-m^{0}_{+}\Theta}q(\zeta)\stackrel{{\scriptstyle\eqref{at9.0}}}{{=}}\zeta^{d-m^{0}_{+}\Theta}q(\zeta),$
(10.10)

where

$q(\zeta)=\sum_{-m_{-}-m^{0}_{+}\Theta}^{m_{+}-m^{0}_{+}\Theta}q_{j}\,\zeta^{-j}:=a_{-N_{-}}\prod_{1}^{m_{+}-m^{0}_{+}\Theta}(1-\zeta^{+}_{j}/\zeta)\prod_{m_{+}-m^{0}_{+}\Theta+1}^{m_{+}}(\zeta-\zeta^{+}_{j})\prod_{1}^{m_{-}}(\zeta-\zeta^{-}_{j})$
(10.11)

Here
$q_{m_{+}-m^{0}_{+}\Theta}=a_{-N_{-}}\prod_{1}^{m_{+}}(-\zeta_{j}^{+})\prod_{1}^{m_{-}}(-\zeta_{j}^{-})$,
and $q_{-m_{-}-m^{0}_{+}\Theta}=a_{-N_{-}}$. So, by (10.8) and (4.15), we
conclude that there exists some constant $C>0$ such that for all
$z\in\Omega^{\prime}$

$q_{m_{+}-m^{0}_{+}\Theta}\neq 0,\quad|q_{-m_{-}-m^{0}_{+}\Theta}|\geqslant
1/C,\quad|q_{j}|\leqslant C.$

Equation (10.11) shows that there are no roots of $q$ on $S^{1}$ and, by
performing a computation as in (4.34), that
$\mathrm{ind}_{q(S^{1})}(0)=\widehat{m}_{0}$ for all $z\in\Omega_{N}$. This,
together with (10.7), (9.3), (9.5) and (9.6) shows that $q$, $\Omega_{N}$ and
$\Omega^{\prime}$ satisfy the assumptions of Theorem 8.1 with
$\widehat{m}_{0}$ as in (10.9), $C_{0}=C_{\alpha}$, $C_{1}=C_{\beta}$,
$C_{2}=C_{\gamma}$, and therefore there exists a constant $C>0$ such that for
$N>0$ sufficiently large and all $z\in\Omega_{N}$,

$s_{N}(P_{N}(q))\geqslant\frac{1}{C}\frac{\log
N}{N}N^{-C_{\alpha}\Theta(\widehat{m}_{0})}.$ (10.12)

When $d=m_{+}^{0}>0$ then $s_{N-j+1}=t_{j}$, and we immediately conclude the
lower bound in (10.4) using that $\widehat{m}_{0}=m_{+}^{0}$. Hence, from now
on we may and will assume that $d>m_{+}^{0}\Theta(\widehat{m}_{0})$, see also
Assumption 9.1.

Recall the definition of $\chi_{n}$ from (4.10). As noted after (4.12) we have
that $H(\chi_{n})=0$ when $n\leqslant 0$ and that $H(\chi_{n})$ has rank $n$,
when $n>0$. So by (4.14) and (10.10) we get that

$P_{N}(p-z)P_{N}(\chi_{m^{0}_{+}\Theta-d})=P_{N}(q)-\Pi_{N}H(p-z)H(\chi_{d-m^{0}_{+}\Theta})\Pi_{N},$
(10.13)

where the second term on the right hand side has rank $d-m^{0}_{+}\Theta$.
Since
$s_{1}(P_{N}(\chi_{m^{0}_{+}\Theta-d}))=\|P_{N}(\chi_{m^{0}_{+}\Theta-d})\|=1$,
we get by (10.13) and the second Ky Fan inequality in (10.5) that for any
$n=1,\dots,N$,

$s_{n}(P_{N}(p-z))=s_{n}(P_{N}(p-z))\,s_{1}(P_{N}(\chi_{m^{0}_{+}\Theta-d}))\geqslant
s_{n}(P_{N}(q)-\Pi_{N}H(p-z)H(\chi_{d-m^{0}_{+}\Theta})\Pi_{N}).$ (10.14)

By interleaving inequalities for singular values, see e.g. [12, Problem
III.6.4], for any matrices $A,K\in\mathds{C}^{N\times N}$ with $K$ of rank
$m\in\\{0,\dots,N-1\\}$,

$s_{N-m}(A)\geqslant s_{N}(A+K).$ (10.15)

Applying this estimate with

$A=P_{N}(q)-\Pi_{N}H(p-z)H(\chi_{d-m^{0}_{+}\Theta})\Pi_{N},\leavevmode\nobreak\
K=\Pi_{N}H(p-z)H(\chi_{d-m^{0}_{+}\Theta})\Pi_{N},$

and $m=d-m^{0}_{+}\Theta$, yields together with (10.14) and (10.12) that for
$N>0$ large enough (so that (10.12) holds)

$s_{N-d+m^{0}_{+}\Theta}(P_{N}(p-z))\geqslant
s_{N}(P_{N}(q))\geqslant\frac{1}{C}\frac{\log
N}{N}N^{-C_{\alpha}\Theta(\widehat{m}_{0})},$

for all $z\in\Omega_{N}$. Since $s_{N-j+1}=t_{j}$, we get (10.2) by taking
$\widehat{m}_{0}=0$, and we obtain the lower bound in (10.4), when
$d>m_{+}^{0}>0$, by taking $\widehat{m}_{0}=m_{+}^{0}$. This ends the proof of
Proposition 10.1. ∎

###  10.2. Singular vectors

Throughout this section we will continue to work with $z\in\Omega_{N}$
satisfying Assumption 9.1 and we will denote the orthonormal eigenvectors of
$(P_{N}-z)^{*}(P_{N}-z)$ corresponding to the eigenvalues $0\leqslant
t_{1}^{2}\leqslant\cdots\leqslant t_{N}^{2}$ by $e_{1},\dots,e_{N}$. Using the
spectral gap from Proposition 10.1 we will show that the quasimodes from
Proposition 9.6 span, up to a small error, the same space as the eigenvectors
$e_{1},\dots,e_{|d|}$. More precisely we will show that the eigenvectors
$e_{1},\dots,e_{|d|-m_{\mathrm{sign}(d)}^{0}}$ are well approximated by a
linear combination of the quasimodes
$\psi_{1},\dots,\psi_{|d|-m_{\mathrm{sign}(d)}^{0}}$, whereas the eigenvectors
$e_{|d|-m_{\mathrm{sign}(d)}^{0}+1},\dots,e_{|d|}$ are well approximated by a
linear combination of
$\psi_{|d|-m_{\mathrm{sign}(d)}^{0}+1},\dots,\psi_{|d|}$.

######  Proposition 10.3.

Consider the setup as in Proposition 10.1. Let
$\Pi_{\kappa}=\mathbf{1}_{[0,\kappa^{2}]}((P_{N}-z)^{*}(P_{N}-z))$,
$\kappa\geqslant 0$, $z\in\Omega_{N}$, be the spectral projector onto the
eigenspace of $(P_{N}-z)^{*}(P_{N}-z)$ corresponding to the eigenvalues
$0\leqslant t_{1}^{2}\leqslant\dots\leqslant t_{j}^{2}\leqslant\kappa^{2}$.
Let $\psi_{j}^{\pm}$ and $I_{\pm,1},I_{\pm,2}$ be as in Proposition 9.6, let
$N>0$ be sufficiently large (depending only on the $\Omega$ and the constants
in Assumption 9.1), and put

$\widetilde{e}_{j}:=\frac{\Pi_{t_{|d|-m_{\mathrm{sign}(d)}^{0}}}\psi_{j}^{\,\mathrm{sign}(d)}}{\|\Pi_{t_{|d|-m_{\mathrm{sign}(d)}^{0}}}\psi_{j}^{\,\mathrm{sign}(d)}\|},\quad
j\in I_{\mathrm{sign}(d),1},$

and

$\widetilde{e}_{j}:=\frac{\Pi_{t_{|d|}}\psi_{j}^{\,\mathrm{sign}(d)}}{\|\Pi_{t_{|d|}}\psi_{j}^{\,\mathrm{sign}(d)}\|},\quad
j\in I_{\mathrm{sign}(d),2}.$

Then, uniformly in $z\in\Omega_{N}$,

$\|\widetilde{e}_{j}-\psi_{j}^{\,\mathrm{sign}(d)}\|=O(1)\left\\{\begin{array}[]{ll}\frac{N^{(1+C_{\alpha})}}{\log
N}\mathrm{e}^{-N\log C_{\beta}},&j\in I_{\mathrm{sign}(d),1},\\\
N^{-C_{\gamma}}\log N,&j\in I_{\mathrm{sign}(d),2},\end{array}\right.$

and

$\langle\widetilde{e}_{i}|\widetilde{e}_{j}\rangle=\delta_{i,j}+\left\\{\begin{array}[]{ll}\frac{N^{(1+C_{\alpha})}}{\log
N}\mathrm{e}^{-N\log C_{\beta}},&\text{ when }i,j\in
I_{\mathrm{sign}(d),1},\\\ O(\log N)N^{-C_{\gamma}},&\text{ when }i,j\in
I_{\mathrm{sign}(d),2},\\\ O(\log N)N^{-C_{\gamma}}+O((\log
N/N)^{1/2}),&\text{else}.\end{array}\right.$ (10.16)

###### Proof.

To ease the notation we drop the $\pm$ superscripts and let $\psi_{j}$ be
either $\psi_{j}^{+}$ or $\psi_{j}^{-}$ depending on $\mathrm{sign}(d)$.
Furthermore, we write $B_{z}:=(P_{N}-z)^{*}(P_{N}-z)$ and we let $N\gg 1$. In
what follows all constants will be uniform in $N\gg 1$ and $z\in\Omega_{N}$
even when we do not state this fact explicitly.

By (9.27), we have that

$\langle
B_{z}\psi_{j}|\psi_{j}\rangle=\|(P_{N}-z)\psi_{j}\|^{2}=O(1)\left\\{\begin{array}[]{ll}\mathrm{e}^{-2N\log
C_{\beta}},&j\in I_{\mathrm{sign}(d),1}\\\ \left(\frac{\log
N}{N}N^{-C_{\gamma}}\right)^{2},&j\in
I_{\mathrm{sign}(d),2}.\end{array}\right.$ (10.17)

Let $\kappa=t_{|d|-m_{\mathrm{sign}(d)}^{0}}$ for $j\in
I_{\mathrm{sign}(d),1}$, and $\kappa=t_{|d|}$ for $j\in
I_{\mathrm{sign}(d),2}$. Let $\Pi=\Pi_{\kappa}$ and
$\Pi^{\perp}=\mathbf{1}_{]\kappa^{2},\infty[}(B_{z})$, and notice that both
are selfadjoint projectors onto disjoint eigenspaces and that
$\Pi+\Pi^{\perp}=1$. By the spectral theorem and the fact that $B_{z}\geqslant
0$,

$\langle B_{z}\psi_{j}|\psi_{j}\rangle=\langle
B_{z}\Pi\psi_{j}|\Pi\psi_{j}\rangle+\langle
B_{z}\Pi^{\perp}\psi_{j}|\Pi^{\perp}\psi_{j}\rangle\geqslant\|\Pi^{\perp}\psi_{j}\|^{2}\left\\{\begin{array}[]{ll}t_{|d|-m_{\mathrm{sign}(d)}^{0}+1}^{2},&j\in
I_{\mathrm{sign}(d),1},\\\ t_{|d|+1}^{2},&j\in
I_{\mathrm{sign}(d),2}.\end{array}\right.$ (10.18)

Combining (10.17), (10.18) and (10.2), (10.4) we get that

$\|\Pi^{\perp}\psi_{j}\|\stackrel{{\scriptstyle\eqref{p1}}}{{\leqslant}}\left\\{\begin{array}[]{ll}O(1)\left(\mathrm{e}^{-N\log
C_{\beta}}t_{|d|-m_{\mathrm{sign}(d)}^{0}+1}^{-1}\right)\stackrel{{\scriptstyle\eqref{sgv2a}}}{{\leqslant}}O(1)\frac{N^{(1+C_{\alpha})}}{\log
N}\mathrm{e}^{-N\log C_{\beta}},&j\in I_{\mathrm{sign}(d),1},\\\
O(1)\left(t_{|d|+1}^{-1}\frac{\log
N}{N}N^{-C_{\gamma}}\right)\stackrel{{\scriptstyle\eqref{sgv2}}}{{\leqslant}}O(1)N^{-C_{\gamma}},&j\in
I_{\mathrm{sign}(d),2}.\end{array}\right.$

Hence, by (9.26),

$|1-\|\Pi\psi_{j}\||\leqslant\left\\{\begin{array}[]{ll}\|\psi_{j}-\Pi\psi_{j}\|=\|\Pi^{\perp}\psi_{j}\|=O(1)\frac{N^{(1+C_{\alpha})}}{\log
N}\mathrm{e}^{-N\log C_{\beta}},&j\in I_{\mathrm{sign}(d),1}\\\
\|\psi_{j}-\Pi\psi_{j}\|+O(1)N^{-C_{\gamma}}\log N=O(1)N^{-C_{\gamma}}\log
N,&j\in I_{\mathrm{sign}(d),2}\end{array}\right.=:\varepsilon_{j}.$

Thus, for $N>0$ sufficiently large, so that $\varepsilon_{j}\ll 1$,

$\|\widetilde{e}_{j}-\psi_{j}\|=\left\|\frac{\Pi\psi_{j}}{1+O(\varepsilon_{j})}-\psi_{j}\right\|\leqslant
O(\varepsilon_{j}),$

which, together with (9.26) and

$\langle\widetilde{e}_{i}|\widetilde{e}_{j}\rangle=\langle\psi_{i}|\psi_{j}\rangle+\langle\widetilde{e}_{i}-\psi_{i}|\psi_{j}\rangle+\langle\psi_{i}|\widetilde{e}_{j}-\psi_{j}\rangle+\langle\widetilde{e}_{i}-\psi_{i}|\widetilde{e}_{j}-\psi_{j}\rangle,$

lets us conclude the proofs of the proposition. ∎

Notice that (10.16) implies by a similar argument as before (10.6), that for
$N>0$ large enough, the vectors $\widetilde{e}_{1},\dots,\widetilde{e}_{|d|}$
are linearly independent. Hence, we conclude the following.

######  Corollary 10.4.

Under the assumptions of Proposition 10.3, we have that the
$\widetilde{e}_{1},\dots,\widetilde{e}_{|d|-m_{\mathrm{sign}(d)}^{0}}$ span
the range $\mathcal{R}(\Pi_{t_{|d|-m_{\mathrm{sign}(d)}^{0}}})$ and that
$\widetilde{e}_{1},\dots,\widetilde{e}_{|d|}$ span
$\mathcal{R}(\Pi_{t_{|d|}})$.

######  Remark 10.5.

By Remark 9.5, (9.29), and Proposition 10.3 it follows that for the Jordan
block with $z$ inside the unit disc such that (9.5) holds we have that

$\left\|\frac{\mathfrak{z}_{1}^{+}}{\|\mathfrak{z}_{1}^{+}\|}-e_{1}\right\|=O(N^{-C_{\gamma}}\log
N).$

In what follows we suppose that $N>0$ is sufficiently large, so that the
conclusion of Proposition 10.3 holds. Furthermore, all error terms will be
uniform in $z\in\Omega_{N}$ even if we don’t state this explicitly.

In view of Proposition 10.3 and Corollary 10.4, we know that there exist
$a_{1},\dots,a_{|d|-m_{\mathrm{sign}(d)}^{0}}\in\ell^{2}(I_{\mathrm{sign}(d),1})$
and $b_{|d|-m_{\mathrm{sign}(d)}^{0}+1},\dots,b_{|d|}\in\ell^{2}([|d|])$ such
that

$e_{j}=\sum_{\nu\in
I_{\mathrm{sign}(d),1}}a_{j}(\nu)\widetilde{e}_{\nu},\leavevmode\nobreak\
\leavevmode\nobreak\ j\in I_{\mathrm{sign}(d),1},\quad\text{and}\quad
e_{j}=\sum_{1}^{|d|}b_{j}(\nu)\widetilde{e}_{\nu},\leavevmode\nobreak\
\leavevmode\nobreak\ j\in I_{\mathrm{sign}(d),2}.$

Let $n,m\in I_{\mathrm{sign}(d),1}$. Using (10.16) and the fact that the
$e_{j}$ are orthonormal, we get that

$\begin{split}\delta_{n,m}&=\langle
e_{n}|e_{m}\rangle=\sum_{\nu,\mu}a_{n}(\nu)\overline{a}_{m}(\mu)\langle\widetilde{e}_{\nu}|\widetilde{e}_{\mu}\rangle=\langle
a_{n}|a_{m}\rangle+O\\!\left(\|a\|^{2}N^{(1+C_{\alpha})}\mathrm{e}^{-N\log
C_{\beta}}/\log N\right),\end{split}$

where the constant in the error estimate is independent of $a$. We deduce that
$\|a_{n}\|=1+O(N^{(1+C_{\alpha})}\mathrm{e}^{-N/C}/\log N)$, and

$\langle
a_{n}|a_{m}\rangle=\delta_{n,m}+O\left(N^{(1+C_{\alpha})}\mathrm{e}^{-N\log
C_{\beta}}/\log N\right).$ (10.19)

Similarly, we deduce from (10.16) that for $n,m\in I_{\mathrm{sign}(d),2}$

$\langle b_{n}|b_{m}\rangle=\delta_{n,m}+O(N^{-C_{\gamma}}\log N)+O((\log
N/N)^{1/2}),$ (10.20)

where the constants in the error estimate are independent of $b$. Decompose

$b_{j}=\left\\{\begin{array}[]{ll}\widetilde{b}_{j}\oplus\widehat{b}_{j}\in\ell^{2}(I_{+,1})\oplus\ell^{2}(I_{+,2}),&\mathrm{sign}(d)>0,\\\
\widehat{b}_{j}\oplus\widetilde{b}_{j}\in\ell^{2}(I_{-,2})\oplus\ell^{2}(I_{-,1}),&\mathrm{sign}(d)<0.\end{array}\right.$

Exploiting the orthogonality of the $e_{j}$ and using (10.19), (10.20), we
deduce similarly as above that for $n\in I_{\mathrm{sign}(d),1}$ and $m\in
I_{\mathrm{sign}(d),2}$,

$\langle a_{n}|\widetilde{b}_{m}\rangle=O(N^{-C_{\gamma}}\log N)+O((\log
N/N)^{1/2}).$ (10.21)

By a similar argument as before (10.6), we deduce from (10.19) that for $N>0$
sufficiently large (depending only on the $\Omega$ and the constants in
Assumption 9.1), the $a_{j}$ are linearly independent and span
$\ell^{2}(I_{\pm,1})$. Thus, for each $\widetilde{b}_{j}$ there exists a
$d_{j}\in\mathds{C}^{|d|-m_{\mathrm{sign}(d)}^{0}}$ such that
$\widetilde{b}_{j}=\sum_{\nu}d_{j}(\nu)a_{\nu}$. Hence, by (10.21), (10.19),

$\|\widetilde{b}_{m}\|^{2}=\sum_{\nu,\mu}d_{m}(\nu)\overline{d_{m}(\mu)}\langle
a_{\nu}|a_{\mu}\rangle=\|d_{m}\|^{2}\left(1+O\\!\left(N^{(1+C_{\alpha})}\mathrm{e}^{-N\log
C_{\beta}}/\log N\right)\right).$ (10.22)

Combining this with (10.21), (10.22), (10.19), we deduce that

$\|d_{k}\|=O(N^{-C_{\gamma}}\log N)+O((\log
N/N)^{1/2})+O\\!\left(N^{(1+C_{\alpha})}\mathrm{e}^{-N\log C_{\beta}}/\log
N\right),$

where the error terms are independent of $a$, $b$, and $d$. Notice that the
error term which is exponentially small in $N$ may be absorbed into the other
two terms. Hence,

$\|\widetilde{b}_{m}\|=O(N^{-C_{\gamma}}\log N)+O((\log N/N)^{1/2}).$

Thus, for $j\in I_{\mathrm{sign}(d),2}$,

$e_{j}=\sum_{\nu\in
I_{\mathrm{sign}(d),2}}\widehat{b}_{j}(\nu)\psi_{\nu}^{\,\mathrm{sign}(d)}+O(N^{-C_{\gamma}}\log
N)+O((\log N/N)^{1/2}).$

In view of (9.29), (9.51) and Remarks 9.8 and 9.7, we may replace
$\psi_{\nu}^{\,\mathrm{sign}(d)}$ in the above expression by
${\mathfrak{z}_{\nu}^{\mathrm{sign}(d)}}/{\|\mathfrak{z}_{\nu}^{\mathrm{sign}(d)}\|}$,
see also Proposition 9.3.

Summing up what we have proven so far, we obtain in view of Propositions 10.3
and 9.6 the following. Recall that for $z\in\Omega_{N}$, $e_{1},\dots,e_{N}$
are an orthonormal set of eigenvectors of $(P_{N}-z)^{*}(P_{N}-z)$
corresponding to the eigenvalues $0\leqslant t_{1}^{2}\leqslant\cdots\leqslant
t_{N}^{2}$.

######  Proposition 10.6.

Under the assumptions of Proposition 10.3, we have that for $j\in
I_{\mathrm{sign}(d),1}$,

$e_{j}=\sum_{\nu\in
I_{\mathrm{sign}(d),1}}a_{j}(\nu)\psi_{\nu}^{\,\mathrm{sign}(d)}+O(1)N^{(1+C_{\alpha})}\mathrm{e}^{-N\log
C_{\beta}}/\log N,$

where $a_{j}\in\ell^{2}(I_{\mathrm{sign}(d),1})$ satisfy (10.19), uniformly in
$z\in\Omega_{N}$. Further, for $j\in I_{\mathrm{sign}(d),2}$,

$e_{j}=\sum_{\nu\in
I_{\mathrm{sign}(d),2}}b_{j}(\nu)\frac{\mathfrak{z}_{\nu}^{\mathrm{sign}(d)}}{\|\mathfrak{z}_{\nu}^{\mathrm{sign}(d)}\|}+O(N^{-C_{\gamma}}\log
N)+O((\log N/N)^{1/2}).$ (10.23)

where $b_{j}\in\ell^{2}(I_{\mathrm{sign}(d),2})$ satisfy

$\langle b_{n}|b_{m}\rangle=\delta_{n,m}+O(\log NN^{-C_{\gamma}})+O((\log
N/N)^{1/2}),$ (10.24)

uniformly in $z\in\Omega_{N}$.

######  Remark 10.7.

We recall, see points (i), (ii) of Lemma 6.16 and (5.85), that for
$\nu\neq\nu^{\prime}\in I_{\mathrm{sign}(d),2}$ we have that
$|\zeta^{\mathrm{sign}(d)}_{\nu}\cdot\overline{\zeta^{\mathrm{sign}(d)}_{\nu^{\prime}}}-1|$
is bounded away from $0$ by a constant $c>0$ independent of $N$, while for
$\nu=\nu^{\prime}\in I_{\mathrm{sign}(d),2}$ it is of order $\log N/N$ by
(9.5)-(9.7).

Next, we turn to proving localization estimates of the singular vectors.

######  Theorem 10.8.

Consider the setup and notation as in Proposition 10.1. Then, there exist
constants $0<C,C^{\prime}<\infty$ such that for $N>0$ sufficiently large
(depending only on $\Omega$ and the constants in Assumption 9.1) we have that
for any $z\in\Omega_{N}$, for any discrete interval
$I=[\ell_{0},\ell_{1}]\subset[0,N-1]$, with $|I|\geqslant 2(1+m_{+}^{0})$, and
for $j\in I_{\mathrm{sign}(d),2}$

$\|e_{j}\|^{2}_{\ell^{2}(I)}\leqslant C\min\left\\{|I|\frac{\log
N}{N},\frac{1}{C^{\prime}}\right\\}\mathrm{e}^{-2C_{\gamma}\mathfrak{d}\frac{\log
N}{N}}+CN^{-2C_{\gamma}}(\log N)^{2}+C\frac{\log N}{N},$ (10.25)

and

$\|e_{j}\|^{2}_{\ell^{2}(I)}\geqslant\frac{1}{C}\min\left\\{|I|\frac{\log
N}{N},\frac{1}{C^{\prime}}\right\\}\mathrm{e}^{-2C_{\alpha}\mathfrak{d}\frac{\log
N}{N}}-CN^{-2C_{\gamma}}(\log N)^{2}-C\frac{\log N}{N},$ (10.26)

and for $j\in I_{\mathrm{sign}(d),1}$,

$\|e_{j}\|^{2}_{\ell^{2}(I)}\leqslant C\mathrm{e}^{-2\mathfrak{d}\log
C_{\beta}}+CN^{2(1+C_{\alpha})}\mathrm{e}^{-2N\log C_{\beta}}(\log N)^{-2}.$
(10.27)

Here,

$\mathfrak{d}:=\left\\{\begin{array}[]{ll}\ell_{0}&\text{when }d>0,\\\
N-\ell_{1}&\text{when }d<0.\end{array}\right.$

We remark that we do not use (10.26) in the rest of the paper (instead, we use
(10.23)). Its inclusion here is to contrast with (10.25).

###### Proof.

We shall only consider the case when the winding number $d>0$, since the case
when $d<0$ is similar. Throughout we will assume that $N>0$ is sufficiently
large (depending only on $\Omega$ and the constants in Assumption 9.1) so that
the conclusions of Propositions 10.6, 10.3, 9.6, and 9.3 hold, and we have
that all error terms are uniform in $z\in\Omega_{N}$ without us mentioning
this explicitly.

1\. We begin by considering the case where $d-m_{+}^{0}>0$ and $j\in I_{+,1}$.
By Propositions 10.6 and 9.6, we find that

$\|e_{j}\|_{\ell^{2}(I)}^{2}\leqslant 2\left\|\sum_{\nu\in
I_{+,1}}a_{j}(\nu)\psi_{\nu}^{+}\right\|^{2}_{\ell^{2}(I)}+O(1)\frac{N^{2(1+C_{\alpha})}}{(\log
N)^{2}}\mathrm{e}^{-2N\log C_{\beta}}.$ (10.28)

Recall from Proposition 9.3 that $\psi_{\nu}^{+}$ is given by the $\nu$-th
column of $\mathfrak{Z}^{+}A^{+}_{1}G_{+,1}^{-1/2}$. Using the Cauchy-Schwarz
inequality we get that

$\begin{split}\left\|\sum_{\nu\in
I_{+,1}}a_{j}(\nu)\psi_{\nu}\right\|^{2}_{\ell^{2}(I)}&=\sum_{\mu\in
I}\left|\sum_{\nu\in
I_{+,1}}\sum_{\eta=1}^{m_{+}-m_{+}^{0}}\mathfrak{Z}^{+}_{\mu,\eta}(A^{+}_{1}G_{+,1}^{-1/2})_{\eta,\nu}a_{j}(\nu)\right|^{2}\\\
&=\sum_{\mu\in
I}\left|\sum_{\eta=1}^{m_{+}-m_{+}^{0}}\mathfrak{Z}^{\pm}_{\mu,\eta}(A^{+}_{1}G_{+,1}^{-1/2}a_{j})(\eta)\right|^{2}\leqslant\|A^{+}_{1}G_{+,1}^{-1/2}a_{j}\|^{2}\sum_{\eta=1}^{m_{+}-m_{+}^{0}}\|\mathfrak{z}^{+}_{\eta}\|^{2}_{\ell^{2}(I)}.\end{split}$
(10.29)

where $\mathfrak{z}^{+}_{\eta}$ is defined as in Proposition 9.3. For
$\eta=1,\dots,m_{+}$, we have that

$\|\mathfrak{z}^{+}_{\eta}\|^{2}_{\ell^{2}(I)}=\sum_{\nu=\ell_{0}}^{\ell_{1}}|\zeta^{+}_{\eta}|^{2\nu}=|\zeta_{\eta}^{+}|^{2\ell_{0}}\frac{1-|\zeta_{\eta}^{+}|^{2|I|}}{1-|\zeta_{\eta}^{+}|^{2}}.$
(10.30)

By (9.6) we have that $|\zeta_{\eta}^{+}|\leqslant 1/C_{\beta}$ for
$\eta=1,\dots,{m_{+}-m_{+}^{0}}$. Thus,

$|\zeta_{\eta}^{+}|^{2\ell_{0}}\leqslant\|\mathfrak{z}^{+}_{\eta}\|^{2}_{\ell^{2}(I)}\leqslant\frac{C_{\beta}^{2}}{C_{\beta}^{2}-1}|\zeta_{\eta}^{+}|^{2\ell_{0}}.$
(10.31)

Combining (9.45), (10.19), with (10.28) and (10.29), we conclude (10.27).

2\. Next, we let $j\in I_{+,2}$, so we may assume that $m_{+}^{0}>0$. By
Propositions 10.6 and 9.6, we find that

$\|e_{j}\|_{\ell^{2}(I)}^{2}\leqslant 2\left\|\sum_{\nu\in
I_{+,2}}b_{j}(\nu)\frac{\mathfrak{z}_{\nu}^{+}}{\|\mathfrak{z}_{\nu}^{+}\|}\right\|^{2}_{\ell^{2}(I)}+\varepsilon_{N},$
(10.32)

where, to ease the notation, we put $\varepsilon_{N}=O((\log
N)^{2})N^{-2C_{\gamma}}+O(\log N/N)$. For the sake of the presentation we will
keep this notation for the error term and we allow for the constant in the
estimate to change while remaining uniform in $z\in\Omega_{N}$.
From Proposition 10.6, we see that the coefficients $b_{j}$ are uniformly
bounded in $\ell^{2}$ by $O(1)$. Thus by a similar computation as in (10.29),
(10.31), we get together with (9.50), (9.59) that

$\|e_{j}\|_{\ell^{2}(I)}^{2}\leqslant O(1)\sum_{\nu\in
I_{+,2}}\frac{\|\mathfrak{z}^{+}_{\nu}\|^{2}_{\ell^{2}(I)}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}+\varepsilon_{N}.$
(10.33)

Recall (9.6) and (9.5). Let $\nu\in I_{+,2}$, and suppose first that $|I|\ll
N/\log N$, we find by Taylor expansion that

$2|I|C_{\alpha}\frac{\log N}{N}+O\\!\left(|I|\frac{\log
N}{N}\right)^{2}\leqslant 1-|\zeta_{\nu}^{+}|^{2|I|}\leqslant
2|I|C_{\gamma}\frac{\log N}{N}+O\\!\left(|I|\frac{\log N}{N}\right)^{2}.$
(10.34)

Thus,

$\frac{C_{\alpha}}{C_{\gamma}}|I|\left(1+O\\!\left(|I|\frac{\log
N}{N}\right)\right)\leqslant\frac{1-|\zeta_{\nu}^{+}|^{2|I|}}{1-|\zeta_{\nu}^{+}|^{2}}\leqslant\frac{C_{\gamma}}{C_{\alpha}}|I|\left(1+O\\!\left(|I|\frac{\log
N}{N}\right)\right).$ (10.35)

Plugging the estimates (10.34), (10.35) into (10.30), and using (9.47) as
well, we get that there exists a constant $C>0$ such that, for $N>0$ large
enough,

$\frac{1}{C}|I|\frac{\log N}{N}\mathrm{e}^{-2C_{\alpha}\ell_{0}(1+O(\frac{\log
N}{N}))\frac{\log
N}{N}}\leqslant\frac{\|\mathfrak{z}^{+}_{\nu}\|^{2}_{\ell^{2}(I)}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}\asymp|I|\frac{\log
N}{N}|\zeta_{\nu}|^{2\ell_{0}}\leqslant C|I|\frac{\log
N}{N}\mathrm{e}^{-2C_{\gamma}\ell_{0}\frac{\log N}{N}}.$ (10.36)

When $N/(C\log N)\leqslant|I|\leqslant N$ for some $C\gg 1$, then we see by
Taylor expansion that

$\frac{1-|\zeta_{\nu}^{+}|^{2|I|}}{1-|\zeta_{\nu}^{+}|^{2}}\asymp\frac{N}{\log
N}.$

Hence, by (9.47) and (10.30), we get that there exists a constant $C>0$ such
that, for $N>0$ large enough,

$\frac{1}{C}\mathrm{e}^{-2C_{\alpha}\ell_{0}(1+O(\frac{\log N}{N}))\frac{\log
N}{N}}\leqslant\frac{\|\mathfrak{z}^{+}_{\nu}\|^{2}_{\ell^{2}(I)}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}\asymp|\zeta_{\nu}|^{2\ell_{0}}\leqslant
C\mathrm{e}^{-2C_{\gamma}\ell_{0}\frac{\log N}{N}}.$ (10.37)

By (10.33), (10.36) and (10.37) we conclude (10.25).

3\. Next we prove the lower bound (10.26), so we work still with $j\in
I_{+,2}$ and $m_{+}^{0}>0$. Similar to (10.32), we find by Propositions 10.6
and 9.6 that

$\|e_{j}\|_{\ell^{2}(I)}^{2}\geqslant\frac{1}{2}\left\|\sum_{\nu\in
I_{+,2}}b_{j}(\nu)\frac{\mathfrak{z}_{\nu}^{+}}{\|\mathfrak{z}_{\nu}^{+}\|}\right\|^{2}_{\ell^{2}(I)}-\varepsilon_{N}.$

Let

$V=\begin{pmatrix}1&\dots&1\\\
\zeta_{m_{+}-m^{0}_{+}+1}^{+}&\dots&\zeta_{m_{+}}^{+}\\\
\vdots&\vdots&\vdots\\\
(\zeta_{m_{+}-m^{0}_{+}+1}^{+})^{m_{+}-m^{0}_{+}-1}&\dots&(\zeta_{m_{+}}^{+})^{m_{+}-m^{0}_{+}-1},\end{pmatrix}$

$\Lambda=\mathrm{diag}(\zeta_{m_{+}-m_{+}^{0}+1}^{+},\dots,\zeta_{m_{+}}^{+})$
and let $\mathfrak{L}_{+}$ be as in Proposition 9.3. By a similar argument as
for (9.40), we see that $s_{m_{+}-m_{+}^{0}}(V)\geqslant 1/C$ for some $C>0$,
depending only on $\Omega$. Let $|I|\geqslant 2(1+m_{+}^{0})$. Then,

$\begin{split}\left\|\sum_{\nu\in
I_{+,2}}b_{j}(\nu)\frac{\mathfrak{z}^{+}_{\nu}}{\|\mathfrak{z}^{+}_{\nu}\|}\right\|^{2}_{\ell^{2}(I)}&\geqslant\sum_{k=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}\|V\Lambda^{\ell_{0}+km_{+}^{0}}\mathfrak{L}_{+}^{-1}b\|^{2}\geqslant\frac{1}{C^{2}}\sum_{k=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}\|\Lambda^{\ell_{0}+km_{+}^{0}}\mathfrak{L}_{+}^{-1}b\|^{2}\\\
&\geqslant\frac{1}{C^{2}}\sum_{\nu\in
I_{+,2}}|b_{j}(\nu)|^{2}\sum_{k=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}\frac{|\zeta_{\nu}^{+}|^{2(\ell_{0}+km_{+}^{0})}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}\end{split}$
(10.38)

Let $\nu\in I_{+,2}$. A direct computation yields that

$\sum_{k=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}|\zeta_{\nu}^{+}|^{2(\ell_{0}+km_{+}^{0})}=|\zeta_{\nu}^{+}|^{2\ell_{0}}\frac{1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}(\lfloor(|I|-1)/m_{+}^{0}\rfloor-1)}}{1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}}}\geqslant|\zeta_{\nu}^{+}|^{2\ell_{0}}\frac{1-|\zeta_{\nu}^{+}|^{2(|I|-1-m_{+}^{0})}}{1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}}}$
(10.39)

Recall (9.5) and (9.6). Suppose first that $2(1+m_{+}^{0})\leqslant|I|\ll
N/\log N$, then $|I|/2\leqslant|I|-1-m_{+}^{0}\leqslant|I|$, and by Taylor
expansion we find that

$1-|\zeta_{\nu}^{+}|^{2(|I|-1-m_{+}^{0})}\geqslant|I|C_{\alpha}\frac{\log
N}{N}+O\\!\left(|I|\frac{\log N}{N}\right)^{2}.$

and similarly

$1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}}\leqslant 2m_{+}^{0}C_{\gamma}\frac{\log
N}{N}+O\\!\left(\frac{\log N}{N}\right)^{2}.$

Thus,

$\frac{1-|\zeta_{\nu}^{+}|^{2(|I|-1-m_{+}^{0})}}{1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}}}\geqslant\frac{C_{\alpha}}{2m_{+}^{0}C_{\gamma}}|I|\left(1+O\\!\left(|I|\frac{\log
N}{N}\right)\right).$ (10.40)

Combining (10.40), (10.39) and (10.38) with (9.47), yields that there exists a
constant $C>0$ such that for $N>0$ large enough and all $z\in\Omega_{N}$,

$\sum_{k=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}\frac{|\zeta_{\nu}^{+}|^{2(\ell_{0}+km_{+}^{0})}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}\geqslant\frac{1}{C}|I|\frac{\log
N}{N}|\zeta_{\nu}|^{2\ell_{0}}\geqslant\frac{1}{C}|I|\frac{\log
N}{N}\mathrm{e}^{-2C_{\alpha}\ell_{0}(1+O(\frac{\log N}{N}))\frac{\log
N}{N}}.$ (10.41)

When $N/(C\log N)\leqslant|I|\leqslant N$ for some $C\gg 1$, then we see by
Taylor expansion that

$\frac{1-|\zeta_{\nu}^{+}|^{2(|I|-1-m_{+}^{0})}}{1-|\zeta_{\nu}^{+}|^{2m_{+}^{0}}}\asymp\frac{1}{C}\frac{N}{\log
N}.$

We then deduce similarly to (10.41) that there exists a constant $C>0$ such
that for $N>0$ large enough and all $z\in\Omega_{N}$,

$\sum_{j=0}^{\lfloor(|I|-1)/m_{+}^{0}\rfloor-1}\frac{|\zeta_{\nu}^{+}|^{2(\ell_{0}+jm_{+}^{0})}}{\|\mathfrak{z}^{+}_{\nu}\|^{2}}\geqslant\frac{1}{C}\mathrm{e}^{-2C_{\alpha}\ell_{0}(1+O(\frac{\log
N}{N}))\frac{\log N}{N}}.$ (10.42)

From (10.24) we know that for $N>0$ sufficiently large $\|b_{j}\|\geqslant
1/2$. Hence, (10.41), (10.42) together with (10.38) imply (10.26), which
concludes the proof of the theorem. ∎

###  10.3. Large singular values

The next result gives estimates on the Hilbert-Schmidt and trace norm of $E$
from (2.9).

######  Proposition 10.9.

Let $z_{0}\in p(S^{1})$ be as in (9.1) and let $\Omega\Subset\mathds{C}$ be a
sufficiently small open relatively compact convex neighbourhood of $z_{0}$
satisfying (9.2). Let $N>1$ be sufficiently large, and let
$\Omega_{N}\Subset\Omega\backslash p(S^{1})$ satisfy (9.5). Let $d$ be as in
(9.4) and let $t_{j}$ be as in Proposition 10.1, then, uniformly in $N$ and
$z\in\Omega_{N}$,

$\sum_{j=|d|+1}^{N}t_{j}^{-2}=O\\!\left(\frac{N^{2}}{\log N}\right),$ (10.43)

and

$\sum_{j=|d|+1}^{N}t_{j}^{-1}=O(N{\log N}).$ (10.44)

###### Proof.

1\. Let $z_{0}$ and $\Omega$ as in the assumptions of the proposition and let
$z\in\Omega_{N}$. Let $\widetilde{N}=N_{+}\lor 0+N_{-}\lor 0$ and notice that
$\widetilde{N}\geqslant|d|$, see Lemma 4.1 and (4.34). By (4.20), (10.15) and
(1.13), we see that

$t_{j}(P_{N}-z)\geqslant
t_{j-\widetilde{N}}(P_{\mathds{Z}/N\mathds{Z}}-z),\quad
j=1+\widetilde{N},\dots,N.$ (10.45)

From (4.20) we deduce that the set of eigenvalues of
$(P_{\mathds{Z}/N\mathds{Z}}-z)^{*}(P_{\mathds{Z}/N\mathds{Z}}-z)$ is given by

$\\{t_{j}^{2}(P_{\mathds{Z}/N\mathds{Z}}-z);j=1,\dots,N\\}=\\{|p(\widehat{\zeta})-z|^{2};\widehat{\zeta}\in
S_{N}\\},$ (10.46)

where $S_{N}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{\mathrm{e}^{2\pi
ij/N};j=0,\dots,N-1\\}$. Note that
$t_{1}(P_{\mathds{Z}/N\mathds{Z}}-z)\geqslant\mathrm{dist\,}(z,p(S^{1}))>0$,
since $z\in\Omega_{N}$. Hence, combining (10.45) and (10.2), we get for
$\nu=1,2$,

$\sum_{j=|d|+1}^{N}\frac{1}{t_{j}^{\nu}(P_{N}-z)}\leqslant\sum_{j=1+\widetilde{N}}^{N}\frac{1}{t_{j}^{\nu}(P_{N}-z)}+\widetilde{N}\left(\frac{CN}{\log
N}\right)^{\nu}\leqslant\sum_{j=1}^{N}\frac{1}{t_{j}^{\nu}(P_{\mathds{Z}/N\mathds{Z}}-z)}+\widetilde{N}\left(\frac{CN}{\log
N}\right)^{\nu}.$ (10.47)

It remains to deal with the traces
$\mathrm{tr\,}((P_{\mathds{Z}/N\mathds{Z}}-z)^{*}(P_{\mathds{Z}/N\mathds{Z}}-z))^{-\nu/2}$
for $\nu=1,2$.

2\. We know by (4.16) that there are finitely many, say $N_{0}<+\infty$,
points in $p^{-1}(z_{0})$. We will enumerate them in the following way: for
$j=1,\dots,N_{0}^{\prime}$, with $N_{0}^{\prime}\geqslant 1$, let
$\zeta_{j}\in p^{-1}(z_{0})$ be so that $\zeta_{j}\in S^{1}$, and for
$j=N_{0}^{\prime}+1,\dots,N_{0}$, let $\zeta_{j}\in p^{-1}(z_{0})$ be so that
$\zeta_{j}\notin S^{1}$. If $N_{0}^{\prime}=N_{0}$ then the latter set of
points is empty.

By (9.2) and the Implicit function theorem, it follows that, after potentially
shrinking $\Omega$ (while keeping it convex), there exist complex open
neighbourhoods $Z_{j}$ of $\zeta_{j}$, such that $p:Z_{j}\to\Omega$ is a
diffeomorphism and such that $Z_{j}\cap S^{1}=\emptyset$ for
$j=N_{0}^{\prime}+1,\dots,N_{0}$. Furthermore, after potentially further
shrinking $\Omega$ and the sets $Z_{j}$, we can arrange so that $p(S^{1}\cap
Z_{j})=\Omega\cap p(S^{1})=\Gamma$ for all $j=1,\dots,N_{0}^{\prime}$. Let
$\Omega^{\prime}\subset\Omega$ be a smaller open convex neighbourhood of
$z_{0}$ such that $\overline{\Omega^{\prime}}\subset\Omega$, and put
$Z_{j}^{\prime}:=(p\\!\upharpoonright_{Z_{j}\to\Omega})^{-1}(\Omega^{\prime})\subset
Z_{j}$. A convenient choice is

$\Omega^{\prime}=\\{z\in\mathds{C};\mathrm{dist\,}(z,\Gamma^{\prime})\leqslant
1/C_{\Omega^{\prime}},\pi(z)\in\Gamma^{\prime}\\},$

where $0<C_{\Omega^{\prime}}<\infty$ is some sufficiently large constant,
$\Gamma^{\prime}\subset\Gamma$ is a connected non-empty subsegment, containing
$z_{0}$, so that the endpoints of the segments $\Gamma$ and $\Gamma^{\prime}$
are separated by some fixed positive distance, i.e. for some large constant
$C^{\prime}>0$ we have that
$\mathrm{dist\,}(\partial\Gamma,\partial\Gamma^{\prime})\geqslant
1/C^{\prime}$. Furthermore, $\pi(z)$ denotes the point in $p(S^{1})$ with
$|\Pi(z)-z|=\mathrm{dist\,}(z,p(S^{1}))$. Notice that this point is unique if
$C_{\Omega^{\prime}}$ is large enough.

We note that there exists a constant $C>0$ (depending only on $\Omega$ and
$\Omega^{\prime}$) such that for all $z\in\Omega^{\prime}$ and all $\zeta\in
S^{1}\backslash\bigcup_{j=1}^{N_{0}^{\prime}}(Z_{j}\cap S^{1})$

$|p(\zeta)-z|\geqslant\mathrm{dist\,}(\partial\Omega,\partial\Omega^{\prime})\geqslant
1/C.$ (10.48)

Furthermore, observe that since two consecutive points in $S_{N}$ differ by an
angle of $2\pi/N$, we have that

$\\#\\{S_{N}\cap
Z_{j}\\}=N\int_{(p\upharpoonright_{Z_{j}\to\Omega})^{-1}(\Omega)}L_{S^{1}}(d\theta)+O(1),\quad
j=1,\dots,N_{0}^{\prime},$ (10.49)

where $L_{S^{1}}$ denotes the normalized Lebesgue measure on $S^{1}$.

Let $j\in\\{1,\dots,N_{0}^{\prime}\\}$,
$z\in\Omega_{N}^{\prime}:=\Omega_{N}\cap\Omega^{\prime}$, and
$\zeta_{0}^{j}\in Z_{j}^{\prime}$ be the unique point in $Z_{j}^{\prime}$ such
that $p(\zeta_{0}^{j})=\Pi(z)$. For $N>1$ sufficiently large (depending only
on $\Omega$ and $\Omega^{\prime}$) there exists a point
$\widehat{\zeta}^{j}_{*}\in Z_{j}\cap S_{N}$ which is the closest point in
$Z_{j}\cap S_{N}$ to $\zeta_{0}^{j}$. There may be two such points, in which
case we pick one of them. Then,

$|\zeta_{0}^{j}-\widehat{\zeta}_{*}^{j}|\leqslant|1-\mathrm{e}^{2\pi
i/N}|\leqslant O(N^{-1}).$ (10.50)

By Taylor expansion around $\zeta^{j}_{0}$, we find that

$|p(\widehat{\zeta}^{j}_{*})-z|\leqslant|p(\zeta^{j}_{0})-z|+O(N^{-1}),$

where the constant is independent of $z$. In view of (10.46), we have that

$t_{1}(P_{\mathds{Z}/N\mathds{Z}}-z)=\min\limits_{\widehat{\zeta}\in
S_{N}}|p(\widehat{\zeta})-z|.$

Setting $\widetilde{t}:=|p(\zeta^{j}_{0})-z|$, as $p(\zeta_{0}^{j})=\Pi(z)$
and hence $|p(\zeta_{0}^{j})-z|={\rm dist}(z,p(S^{1}))$, we see that

$\widetilde{t}\leqslant
t_{1}(P_{\mathds{Z}/N\mathds{Z}}-z)\leqslant\min\limits_{j=1,\dots,N_{0}^{\prime}}|p(\widehat{\zeta}^{j}_{*})-z|\leqslant\max\limits_{j=1,\dots,N_{0}^{\prime}}|p(\widehat{\zeta}^{j}_{*})-z|\leqslant\widetilde{t}+O(N^{-1}),$
(10.51)

where the constant is independent of $z$. By Taylor expansion and (9.2) we get
that for $\widehat{\zeta}^{j}\in Z_{j}$

$\begin{split}|p(\widehat{\zeta}^{j})-z|&=|p(\zeta^{j}_{0})+(\partial_{\zeta}p)(\zeta^{j}_{0})(\widehat{\zeta}^{j}-\zeta^{j}_{0})+O(|\widehat{\zeta}^{j}-\zeta^{j}_{0}|^{2})-z|\\\
&\geqslant\frac{1}{C}|\widehat{\zeta}^{j}-\zeta^{j}_{0}|(1-C_{1}|\widehat{\zeta}^{j}-\zeta^{j}_{0}|)-\widetilde{t}.\end{split}$
(10.52)

Here the constant $0<C<\infty$ only depends on $\Omega$, see (9.2), and the
constant $0<C_{1}<\infty$ depends only on $C$ and the second derivative of $p$
in the $Z_{j}$’s, which is uniformly bounded. Suppose that

$|\widehat{\zeta}^{j}-\zeta^{j}_{0}|\leqslant\frac{1}{2C_{1}},\quad|\widehat{\zeta}^{j}-\zeta^{j}_{0}|\geqslant
8C\widetilde{t},$ (10.53)

where $C,C_{1}$ are as in (10.52). Hence, we find by (10.52), (10.53) that

$|p(\widehat{\zeta}^{j})-z|\geqslant\frac{1}{4C}|\widehat{\zeta}^{j}-\zeta^{j}_{0}|.$
(10.54)

3\. We turn to proving the two estimates claimed in the proposition. We let
$\Omega$ and $\Omega^{\prime}$ be as in Step 2. Recall (10.51) and note that
if $\widetilde{t}\geqslant 1/C$ for some arbitrarily large but fixed constant
$C$, then

$\sum_{j=1}^{N}\frac{1}{t_{j}(P_{\mathds{Z}/N\mathds{Z}}-z)},\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{j=1}^{N}\frac{1}{t_{j}^{2}(P_{\mathds{Z}/N\mathds{Z}}-z)}=O(N),$

and we conclude (10.43) and (10.44) in view of (10.47).

Continuing, we may assume from now on that $\widetilde{t}\leqslant 1/C_{0}$,
for some arbitrarily large but fixed constant $C_{0}$. Since the roots of
$p(\zeta)-z$ depend smoothly on $z\in\Omega$, see the discussion after (9.2),
it follows from (9.5) that there exists a constant $C>0$ such that for all
$N>0$,

$\mathrm{dist\,}(\Omega_{N},p(S^{1}))\geqslant\frac{1}{C}\frac{\log N}{N},$

Hence, we may from now on assume that

$\frac{1}{C}\frac{\log N}{N}\leqslant\widetilde{t}\leqslant\frac{1}{C_{0}}.$
(10.55)

for some large $C>0$, independent of $N>0$ and $z\in\Omega_{N}$. Let
$\nu\in\\{1,2\\}$. Then, in view of (10.46), (10.48) we get

$\sum_{j=1}^{N}\frac{1}{t_{j}^{\nu}(P_{\mathds{Z}/n\mathds{Z}}-z)}=\sum_{j=1}^{N_{0}^{\prime}}\sum_{\widehat{\zeta}\in
Z_{j}\cap S_{N}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}+O(N),$ (10.56)

where the constant in the error term depends only on $\Omega$ and
$\Omega^{\prime}$. Using (10.49), we see that by potentially shrinking
$\Omega$, and the sets $Z_{j}$ and $\Omega^{\prime}$ accordingly, we may
suppose that the first estimate in (10.53) holds for all $\widehat{\zeta}\in
Z_{j}\cap S_{N}$, $j=1,\dots,N_{0}^{\prime}$. Next, pick a
$j\in\\{1,\dots,N_{0}^{\prime}\\}$, while keeping in mind that
$N_{0}^{\prime}<\infty$, and split the corresponding inner sum on the right
hand side of (10.56) into two parts: one where the second inequality in
(10.53) holds and one where it does not, so

$\sum_{\widehat{\zeta}\in Z_{j}\cap
S_{N}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}=\sum_{\begin{subarray}{c}\widehat{\zeta}\in
Z_{j}\cap S_{N},\\\ |\widehat{\zeta}-\zeta_{0}^{j}|\geqslant
8C\widetilde{t}\end{subarray}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}+\sum_{\begin{subarray}{c}\widehat{\zeta}\in
Z_{j}\cap S_{N},\\\
|\widehat{\zeta}-\zeta_{0}^{j}|<8C\widetilde{t}\end{subarray}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}.$
(10.57)

We begin with treating the second sum on the right hand side. Since the points
in $S_{N}$ differ by an angle of $2\pi/N$, we see that for $N>1$

$\\#\\{\widehat{\zeta}\in Z_{j}\cap
S_{N};|\widehat{\zeta}-\zeta_{0}^{j}|<8C\widetilde{t}\\}=O(N\widetilde{t}\,),$

where the constant depends only on the symbol $p$, $\Omega$ and the constant
$C$ from (10.52). Hence, by (10.46) and (10.51),

$\sum_{\begin{subarray}{c}\widehat{\zeta}\in Z_{j}\cap S_{N},\\\
|\widehat{\zeta}-\zeta_{0}^{j}|<8C\widetilde{t}\end{subarray}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}=O(N\widetilde{t}^{\,1-\nu}).$
(10.58)

Next, we deal with the first sum on the right hand side of (10.57). Since for
this term (10.53) holds, we get by (10.54) that

$\sum_{\begin{subarray}{c}\widehat{\zeta}\in Z_{j}\cap S_{N},\\\
|\widehat{\zeta}-\zeta_{0}^{j}|\geqslant
8C\widetilde{t}\end{subarray}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}\leqslant\sum_{\begin{subarray}{c}\widehat{\zeta}\in
Z_{j}\cap S_{N},\\\ |\widehat{\zeta}-\zeta_{0}^{j}|\geqslant
8C\widetilde{t}\end{subarray}}\frac{(4C)^{\nu}}{|\widehat{\zeta}-\zeta^{j}_{0}|^{\nu}}$
(10.59)

By (10.50) and (10.55) we find that there exist constants
$0<C_{2},C_{2}^{\prime}<\infty$ such that for $N>0$ large enough (depending
only on the constants in the referenced inequalities) we have that for all
$\widehat{\zeta}\in Z_{j}\cap S_{N}$ with
$|\widehat{\zeta}-\zeta_{0}^{j}|\geqslant 8C\widetilde{t}$ and for all
$z\in\Omega_{N}$, such that (10.55) holds,

$|\widehat{\zeta}-\zeta^{j}_{0}|\geqslant\frac{1}{C_{2}}|\widehat{\zeta}-\widehat{\zeta}^{j}_{*}|\geqslant\frac{1}{C_{2}^{\prime}}\widetilde{t}.$
(10.60)

By rotational invariance, we may assume that ${\widehat{\zeta}}^{j}_{*}=1$.
Then, we have that $\widehat{\zeta}_{k}=\mathrm{e}^{2\pi ik/N}$,
$k=1,\dots,M_{1}$ are the points in $Z_{k}$ to the left of $1$ and
$\widehat{\zeta}_{k}=\mathrm{e}^{2\pi ik/N}$, $k=-1,\dots,-M_{2}$ are the
points in $Z_{k}$ to the right of $1$. By (10.49) it is clear that after
potentially further shrinking $\Omega$ and taking $N>0$ large enough, we have
that $M_{1},M_{2}\leqslant N/C_{3}$, for $C_{3}\gg 1$. Hence, we find that
there exists a constant $0<C_{4}<\infty$ such that

$|\widehat{\zeta}_{k}-\zeta^{j}_{*}|=|\mathrm{e}^{\pm 2\pi
ik/N}-1|\geqslant\frac{1}{C_{4}N}|k|,\quad-M_{2}\leqslant k\leqslant
M_{1},\leavevmode\nobreak\ k\neq 0.$ (10.61)

On the other hand, we see by (10.55) that if $|k|\leqslant B^{-1}\log N$,
$B\gg 1$, then

$|\widehat{\zeta}_{k}-{\widehat{\zeta}}^{j}_{*}|=|\mathrm{e}^{\pm 2\pi
ik/N}-1|\ll\widetilde{t},$ (10.62)

for $N>1$. Combining (10.61), (10.62) with (10.59) and (10.60), we conclude
that

$\sum_{\begin{subarray}{c}\widehat{\zeta}\in Z_{j}\cap S_{N},\\\
|\widehat{\zeta}-\zeta_{0}^{j}|\geqslant
8C\widetilde{t}\end{subarray}}\frac{1}{|p(\widehat{\zeta})-z|^{\nu}}\leqslant\sum_{\begin{subarray}{c}k=-M_{2}\\\
|k|\geqslant B^{-1}\log
N\end{subarray}}^{M_{1}}\frac{(4CC_{2}C_{4}N)^{\nu}}{|k|^{\nu}}\leqslant\left\\{\begin{array}[]{ll}O(N\log
N),&\nu=1\\\ O(N^{2}(\log N)^{-1}),&\nu=2.\end{array}\right.$

Together with (10.47), (10.55), (10.56), (10.57) and (10.58), we conclude
(10.43) and (10.44). ∎

###  10.4. Smallest singular value and Hilbert-Schmidt norms

In this short section we provide bounds on minimal singular values and
Hilbert-Schmidt norms of matrices appearing in the Grushin problem. Recall the
definition of $E(\cdot)$ from (2.9).

######  Lemma 10.10.

Consider the setup and notation as in Proposition 10.1. Fix $M\in\mathds{N}$
and $\gamma>1$. Assume that the entries of $Q$ are i.i.d. with zero mean, unit
variance, and finite fourth moment. If $\gamma>3/2$ then

$\mathbf{P}\left(\inf_{z\in\Omega_{N}}s_{\min}(I_{N}+N^{-\gamma}QE(z))\leqslant
1/2\right)\leqslant N^{-(\gamma-3/2)/2},$

for all large $N$. Otherwise, if $\gamma\in(1,3/2]$ then given any
$\alpha_{0}>0$ there exists $\beta=\beta(\alpha_{0},\gamma)<\infty$, such that

$\mathbf{P}\left(s_{\min}(I_{N}+N^{-\gamma}QE(z))\leqslant
N^{-\beta}\right)\leqslant N^{-\alpha_{0}},$

uniformly for all $z\in\Omega_{N}$ and all large $N$.

###### Proof.

1\. We first consider the easier case $\gamma>3/2$. By Proposition 10.1,
applied with $M=|d|$, we have that $\|E(z)\|=O(N)$ uniformly for
$z\in\Omega_{N}$. As the entries of $Q$ have zero mean and finite fourth
moments, applying [42, Theorem 2] and Markov’s inequality we obtain that

$\mathbf{P}\left(N^{-\gamma}\sup_{z\in\Omega_{N}}\|QE(z)\|\geqslant
1/2\right)\leqslant\mathbf{P}\left(\|Q\|\geqslant
cN^{\gamma-1}\right)\leqslant
c^{-1}N^{-(\gamma-1)}\cdot\mathds{E}\|Q\|\leqslant N^{-(\gamma-3/2)/2},$

for all large $N$, and some $c>0$. Thus, by the triangle inequality we derive
that

$\mathbf{P}\left(\inf_{z\in\Omega_{N}}s_{\min}(I_{N}+N^{-\gamma}QE(z))\leqslant
1/2\right)\leqslant\mathbf{P}\left(N^{-\gamma}\sup_{z\in\Omega_{N}}\|QE(z)\|\geqslant
1/2\right)\leqslant N^{-(\gamma-3/2)/2}.$

This yields the result for $\gamma>3/2$.

2\. We turn to the case of $\gamma\in(1,3/2]$. The proof uses a change of
basis, a lower bound on the smallest singular value of matrices with i.i.d.
entries shifted by a deterministic matrix, and an upper bound on the maximal
singular value of a matrix with i.i.d. entries.

To carry out these steps we introduce additional notation. Let $\mathsf{E}(z)$
and $\mathsf{F}(z)$ be the two unitary matrices whose columns are
$\\{e_{1}(z),e_{2}(z),\ldots,e_{N}(z)\\}$ and
$\\{f_{1}(z),f_{2}(z),\ldots,f_{N}(z)\\}$, respectively (Recall the
definitions of $e_{i}$’s and $f_{j}$’s from Section 2.1 applied to $P=P_{N}$).
Further define $\mathsf{E}_{1}(z)$ and $\mathsf{E}_{2}(z)$ to be the $N\times
M$ and $N\times(N-M)$ matrices consisting of the first $M$ columns and the
remaining $(N-M)$ columns of $\mathsf{E}(z)$, respectively. Similarly define
$\mathsf{F}_{1}(z)$ and $\mathsf{F}_{2}(z)$. Finally, let $\mathsf{T}(z)$ be
the diagonal matrices with entries
$\\{t_{M+1}(z)^{-1},t_{M+2}(z)^{-1},\ldots,t_{N}(z)^{-1}\\}$, where
$t_{j}(z)=t_{j}$, $j\in[N]$ are as in Proposition 10.1.

Equipped with this notation, upon recalling the definition of $E(z)$, we see
that

$I_{N}+N^{-\gamma}QE(z)=I_{N}+N^{-\gamma}Q\mathsf{E}_{2}(z)\mathsf{T}(z)\mathsf{F}_{2}(z)^{*}.$

Fix $x\in S^{N-1}$. As the columns of $\mathsf{F}(z)$ form a basis of
$\mathds{C}^{N}$, there exist $c_{1}\in\mathds{C}^{M}$ and
$c_{2}\in\mathds{C}^{N-M}$ such that

$x=\mathsf{F}_{1}(z)c_{1}+\mathsf{F}_{2}(z)c_{2}\qquad\text{ and
}\qquad\|c_{1}\|^{2}+\|c_{2}\|^{2}=1.$

Therefore, using the orthonormality property of the columns of $\mathsf{F}(z)$
we derive that

$\|(I_{N}+N^{-\gamma}QE(z))x\|^{2}=\|\mathsf{F}_{1}(z)c_{1}+\mathsf{F}_{2}(z)c_{2}+N^{-\gamma}Q\mathsf{E}_{2}(z)\mathsf{T}(z)c_{2}\|^{2}\\\
=\left\|\mathsf{F}_{1}(z)(c_{1}+N^{-\gamma}\mathsf{F}_{1}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z)c_{2})+\mathsf{F}_{2}(z)(I_{N-M}+N^{-\gamma}\mathsf{F}_{2}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z))c_{2}\right\|^{2}\\\
=\left\|c_{1}+N^{-\gamma}\mathsf{F}_{1}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z)c_{2}\right\|^{2}+\left\|(I_{N-M}+N^{-\gamma}\mathsf{F}_{2}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z))c_{2}\right\|^{2}\\\
=\left\|\begin{bmatrix}I_{M}&N^{-\gamma}\mathsf{F}_{1}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z)\\\
0&I_{N-M}+N^{-\gamma}\mathsf{F}_{2}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z)\end{bmatrix}\cdot\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}\right\|^{2},$ (10.63)

where in the second step we have used that
$\mathsf{F}_{1}(z)\mathsf{F}_{1}(z)^{*}+\mathsf{F}_{2}(z)\mathsf{F}_{2}(z)^{*}=I_{N}$
and in the third step we have again used the fact that the columns of
$\mathsf{F}(z)=\begin{bmatrix}\mathsf{F}_{1}(z)&\mathsf{F}_{2}(z)\end{bmatrix}$
form an orthonormal basis of $\mathds{C}^{N}$. So (10.63) shows that it is
enough to find a lower bound on the smallest singular value of the $2\times 2$
block matrix appearing in its rhs.

To this end, we note that any block matrix $A$ of the form
$A=\begin{bmatrix}I&A_{1}\\\ 0&A_{2}\end{bmatrix}$ is invertible iff $A_{2}$
is, and in that case

$A^{-1}=\begin{bmatrix}I&-A_{1}A_{2}^{-1}\\\ 0&A_{2}^{-1}\end{bmatrix}.$

Hence, using the triangle inequality we obtain that

$\|A^{-1}\|\leqslant 1+(\|A_{1}\|+1)\cdot\|A_{2}^{-1}\|.$

Recalling the fact that $\|c_{1}\|^{2}+\|c_{2}\|^{2}=1$, using the above and
(10.63) we therefore deduce that

$\displaystyle(s_{\min}(I_{N}+N^{-\gamma}QE(z)))^{-1}$
$\displaystyle=\|(I_{N}+N^{-\gamma}QE(z))^{-1}\|$ (10.64)
$\displaystyle\leqslant
1+(N^{-\gamma}\|\mathsf{A}_{1}(z)\|+1)\cdot\|(I_{N-M}+N^{-\gamma}\mathsf{A}_{2}(z))^{-1}\|,$

where for brevity we write

$\mathsf{A}_{1}(z):=\mathsf{F}_{1}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z)\qquad\text{
and
}\qquad\mathsf{A}_{2}(z):=\mathsf{F}_{2}(z)^{*}Q\mathsf{E}_{2}(z)\mathsf{T}(z).$

Using [42], Proposition 10.1, and Markov’s inequality again we further have
that

$\mathbf{P}\left(N^{-\gamma}\|\mathsf{A}_{1}(z)\|+1\geqslant
2N^{2\alpha_{0}+1/2}\right)\leqslant t_{M+1}(z)^{-1}\cdot
N^{-(\gamma+2\alpha_{0}+1/2)}\cdot\mathds{E}\|Q\|=O(N^{-2\alpha_{0}}),$
(10.65)

where we have also used the facts that $t_{M+1}(z)^{-1}=O(N)$ and $\gamma>1$.
Hence, by (10.64), we observe that in order to find a lower bound on the
smallest singular value of $I_{N}+N^{-\gamma}QE(z)$ it remains to find the
same for $I_{N-M}+N^{-\gamma}\mathsf{A}_{2}(z)$. Upon using Weyl’s inequality
for singular values it is straightforward to notice that

$s_{\min}(I_{N-M}+N^{-\gamma}\mathsf{A}_{2}(z))\geqslant
s_{\min}(I_{N}+N^{-\gamma}\mathsf{A}(z)),$ (10.66)

where

$\mathsf{A}(z):=\mathsf{F}(z)^{*}Q\mathsf{E}(z)\widehat{\mathsf{T}}(z)\qquad\text{
and }\qquad\widehat{\mathsf{T}}(z):=\begin{bmatrix}I_{M}&0\\\
0&\mathsf{T}(z)\end{bmatrix}.$

On the other hand

$\displaystyle s_{\min}(I_{N}+N^{-\gamma}\mathsf{A}(z))$
$\displaystyle\geqslant s_{\min}(\widehat{\mathsf{T}}(z))\cdot
s_{\min}(\widehat{\mathsf{T}}(z)^{-1}+N^{-\gamma}\mathsf{F}(z)^{*}Q\mathsf{E}(z))$
$\displaystyle=t_{N}(z)^{-1}\cdot N^{-\gamma}\cdot
s_{\min}(N^{\gamma}\mathsf{F}(z)\widehat{\mathsf{T}}(z)^{-1}\mathsf{E}(z)^{*}+Q)$
(10.67)

Since $P_{N}$ has a bounded norm and $\Omega_{N}$ is a bounded set we have
that $\|\widehat{\mathsf{T}}(z)^{-1}\|=t_{N}(z)=O(1)$. As $\mathsf{E}(z)$ and
$\mathsf{F}(z)$ are unitary matrices, an application of [69, Theorem 2.1]
together with (10.67) now yield that

$\mathbf{P}\left(s_{\min}(I_{N}+N^{-\gamma}\mathsf{A}(z))\leqslant
N^{-\beta_{0}}\right)\leqslant N^{-2\alpha_{0}},$ (10.68)

for all large $N$, where $\beta_{0}<\infty$ is some constant depending only on
$\alpha_{0}$ and $\gamma$. Finally, using (10.64) and (10.66), the probability
bounds (10.65) and (10.68), and a union bound we deduce that

$\mathbf{P}\left(s_{\min}(I_{N}+N^{-\gamma}QE(z))\leqslant\frac{1}{3}N^{-(2\alpha_{0}+\beta_{0}+1/2)}\right)=O(N^{-2\alpha_{0}}),$

for all large $N$. This completes the proof of the lemma. ∎

Next we derive bounds on the Hilbert-Schmidt norms of some matrices. This will
allow us to control the error terms in the resolvent expansion of various
terms appearing in the Grushin problem.

######  Proposition 10.11.

Consider the setup and notation as in Proposition 10.1. Additionally assume
that $\Omega$ is convex. Fix $\gamma>1$, $\kappa\in\mathds{N}$ and
$d\in[-\widetilde{m},\widetilde{m}]\cap\mathds{Z}\setminus\\{0\\}$. Let
Assumption 1.1 holds. Then for any $\upalpha_{0}>0$, we have, uniformly in
$z\in\Omega_{N}$ and all $N$ large, that

$\mathbf{P}\left(N^{-\gamma\kappa}\|(E(z)Q)^{\kappa}E_{+}(z)\|_{{\rm
HS}}\geqslant N^{-\frac{(\gamma-1)\kappa}{2}}\right)\leqslant
N^{-\upalpha_{0}},$ (10.69)
$\mathbf{P}\left(N^{-\gamma\kappa}\|(E(z)Q)^{\kappa}E(z)\|_{{\rm HS}}\geqslant
N^{1-\frac{(\gamma-1)\kappa}{2}}\right)\leqslant N^{-\upalpha_{0}},$ (10.70)
$\mathbf{P}\left(N^{-\gamma\kappa}\|E_{-}(z)(QE(z))^{\kappa}\|_{{\rm
HS}}\geqslant N^{-\frac{(\gamma-1)\kappa}{2}}\right)\leqslant
N^{-\upalpha_{0}},$ (10.71)

and

$\mathbf{P}\left(N^{-\gamma(\kappa+1)}\|E_{-}(z)(QE(z))^{\kappa}QE_{+}(z)\|_{{\rm
HS}}\geqslant N^{-\gamma-\frac{(\gamma-1)\kappa}{2}}\right)\leqslant
N^{-\upalpha_{0}}.$ (10.72)

To prove Proposition 10.11 we will rely on bounds on moments of Hilbert-
Schmidt norm of matrices appearing in (10.69)-(10.72), as well as bounds on
$\|E_{\pm}(\cdot)\|_{{\rm HS}}$ and on the sum of the inverse of the non-zero
singular values of $E(\cdot)$.

######  Proposition 10.12.

Fix $\kappa\in\mathds{N}$. Let $\\{\mathsf{E}^{(\ell)}\\}_{\ell=1}^{\kappa}$
and $\\{\mathsf{F}^{(\ell)}\\}_{\ell=1}^{\kappa}$ be two collections of
$N\times N$ deterministic matrices, possibly with complex-valued entries, such
that

$\max\left\\{\max_{\ell=1}^{\kappa}\|\mathsf{E}^{(\ell)}\|_{{\rm
HS}},\max_{\ell=1}^{\kappa}\|\mathsf{F}^{(\ell)}\|_{{\rm
HS}}\right\\}\leqslant 1.$ (10.73)

Define the product

$\widehat{\mathsf{E}}(\kappa):=\mathsf{E}^{(1)}Q\mathsf{F}^{(1)}\cdot\mathsf{E}^{(2)}Q\mathsf{F}^{(2)}\cdots\mathsf{E}^{(\kappa)}Q\mathsf{F}^{(\kappa)}.$
(10.74)

If the entries of $Q$ satisfies Assumption 1.1 then for any
$h_{0}\in\mathds{N}$

$\mathbf{E}\|\widehat{\mathsf{E}}(\kappa)\|_{{\rm HS}}^{2h_{0}}\leqslant
C_{\ref{prop:mom-HS}}\cdot\mathfrak{C}_{\kappa h_{0}},$

where $C_{\ref{prop:mom-HS}}<\infty$ is some constant depending only on
$\kappa$ and $h_{0}$.

The proof of Proposition 10.12 is straightforward. It relies on bounds on
moments of quadratic forms of independent random variables, derived in [75].

###### Proof of Proposition 10.12.

We begin with an auxiliary computation. Let $A,B$ be deterministic matrices.
Then

$W:=\|AQB\|_{{\rm
HS}}^{2}=\sum_{i,j,k,k^{\prime},\ell,\ell^{\prime}}A_{i,k}\overline{A_{i,k^{\prime}}}Q_{k,\ell}\overline{Q_{k^{\prime},\ell^{\prime}}}B_{\ell,j}\overline{B_{\ell^{\prime},j}}=\sum_{k,\ell,k^{\prime},\ell^{\prime}}Q_{kl}\overline{Q_{k^{\prime},\ell^{\prime}}}C_{k,\ell,k^{\prime},\ell^{\prime}},$

where

$C_{k,\ell,k^{\prime},\ell^{\prime}}:=\sum_{i,j}A_{ik}\overline{A_{ik^{\prime}}}B_{\ell,j}\overline{B_{\ell^{\prime},j}}.$

This, in particular, shows that $W$ is a quadratic form in the entries of $Q$.
Hence, using [75, Theorem 2]111Whittle in [75] works with real linear and
quadratic forms of real valued random variables. Upon splitting the sums into
real and imaginary parts, analogous bounds can be derived in the complex case.
Hence, without loss of generality we continue to use [75] for complex
quadratic forms of complex valued random variables. and Assumption 1.1, for
any $s\geqslant 2$, we find that

$\mathbf{E}|W-\mathbf{E}W|^{s}\leqslant
C(s)\left(\sum_{k,k^{\prime},\ell,\ell^{\prime}}|C_{k,\ell,k^{\prime},\ell^{\prime}}|^{2}\right)^{s/2}\cdot\mathfrak{C}_{s}\leqslant
C(s)\|A\|_{{\rm HS}}^{2s}\|B\|_{{\rm HS}}^{2s}\cdot\mathfrak{C}_{s},$

for some absolute constant $C(s)<\infty$, depending only on $s$, where in the
last step we used that

$|C_{k,\ell,k^{\prime},\ell^{\prime}}|\leqslant\|a_{k}\|\cdot\|a_{k^{\prime}}\|\cdot\|b_{\ell}\|\cdot\|b_{\ell^{\prime}}\|,$

and $a_{k}$ and $b_{k}$ are the $k$-th column and row of $A$ and $B$,
respectively.

On the other hand we note that

$|\mathbf{E}W|\leqslant\mathfrak{C}_{1}\sum_{k,\ell}\|a_{k}\|^{2}\|b_{\ell}\|^{2}=\|A\|_{{\rm
HS}}^{2}\|B\|_{{\rm HS}}^{2}\cdot\mathfrak{C}_{1}.$

Thus

$\mathbf{E}|W|^{s}\leqslant
2^{s-1}(\mathbf{E}|W-\mathbf{E}W|^{s}+|\mathbf{E}W|^{s})\leqslant
2^{s}C(s)\mathfrak{C}_{s}\cdot\|A\|_{{\rm HS}}^{2s}\|B\|_{{\rm HS}}^{2s},$
(10.75)

where we also used that by Jensen’s inequality it follows
$\mathfrak{C}_{1}^{s}\leqslant\mathfrak{C}_{s}$. Next by Hölder’s inequality
we obtain that

$\mathbf{E}\|\widehat{\mathsf{E}}(\kappa)\|_{{\rm
HS}}^{2h_{0}}\leqslant\mathbf{E}\left(\prod_{j=1}^{\kappa}\|\mathsf{E}^{(j)}Q\mathsf{F}^{(j)}\|_{{\rm
HS}}^{2h_{0}}\right)\leqslant\left(\prod_{j=1}^{\kappa}\mathbf{E}\|\mathsf{E}^{(j)}Q\mathsf{F}^{(j)}\|_{{\rm
HS}}^{2\kappa
h_{0}}\right)^{1/\kappa}\leqslant\max_{j=1}^{\kappa}\mathbf{E}\|\mathsf{E}^{(j)}Q\mathsf{F}^{(j)}\|_{{\rm
HS}}^{2\kappa h_{0}}.$

Now the desired bound follows from (10.75), upon setting $s=\kappa h_{0}$,
$A=\mathsf{E}^{(j)}$, and $B=\mathsf{F}^{(j)}$, for $j\in[\kappa]$. This
concludes the proof. ∎

Equipped with Proposition 10.12, we now prove Proposition 10.11.

###### Proof of Proposition 10.11.

Recall that by Proposition 10.9, there exists a constant $C_{\ref{prop:HS-
bd}}$ so that

$\sum_{j=|d|+1}^{N}t_{j}(z)^{-2}\leqslant C_{\ref{prop:HS-
bd}}\frac{N^{2}}{\log N}\qquad\mbox{ and
}\qquad\sum_{j=|d|+1}^{N}t_{j}(z)^{-1}\leqslant C_{\ref{prop:HS-bd}}N{\log
N}.$ (10.76)

For $j\in[\kappa]$ we let

$\mathsf{F}^{(j)}:=\frac{1}{\sqrt{C_{\ref{prop:HS-bd}}N\log
N}}\sum_{i=|d|+1}^{N}\frac{1}{\sqrt{t_{i}(z)}}e_{i}(z)\circ e_{i}^{*}(z)$

and

$\mathsf{E}^{(j+1)}:=\frac{1}{\sqrt{C_{\ref{prop:HS-bd}}N\log
N}}\sum_{i=|d|+1}^{N}\frac{1}{\sqrt{t_{i}(z)}}e_{i}(z)\circ f_{i}^{*}(z).$

Further let

$\mathsf{E}^{(1)}:=|d|^{-1/2}E_{-}(z)\qquad\mbox{ and
}\qquad\mathsf{F}^{(\kappa+1)}:=|d|^{-1/2}E_{+}(z).$

By Proposition 10.9 and (2.9)-(2.10) it follows that (10.73) holds with these
choices of $\\{\mathsf{E}^{(j)}\\}_{j=1}^{\kappa+1}$ and
$\\{\mathsf{F}^{(j)}\\}_{j=1}^{\kappa+1}$. On the other hand note that
$E(z)=C_{\ref{prop:HS-bd}}N\log N\cdot\mathsf{F}^{(j)}\mathsf{E}^{(j+1)}$ for
$j\in[\kappa]$. Therefore, applying Proposition 10.12 we now derive that

$\mathbf{E}\left[\|\widetilde{\mathsf{E}}\|_{{\rm
HS}}^{2h_{0}}\right]\leqslant d^{2h_{0}}\cdot(C_{\ref{prop:HS-bd}}N\log
N)^{2h_{0}\kappa}\cdot C_{\ref{prop:mom-HS}}\cdot\mathfrak{C}_{\kappa h_{0}}.$

where $\widetilde{\mathsf{E}}:=E_{-}(z)(QE(z))^{\kappa}QE_{+}(z)$. Hence
applying Markov’s inequality with
$h_{0}=\lceil\frac{2\upalpha_{0}}{\kappa(\gamma-1)}\rceil$ we obtain that

$\mathbf{P}\left(N^{-\gamma(\kappa+1)}\|E_{-}(z)(QE(z))^{\kappa}QE_{+}(z)\|_{{\rm
HS}}\geqslant N^{-\gamma-\frac{(\gamma-1)\kappa}{2}}\right)\leqslant
N^{-(\gamma+1)\kappa h_{0}}\mathbf{E}\left[\|\widetilde{\mathsf{E}}\|_{{\rm
HS}}^{2h_{0}}\right]\leqslant N^{-\upalpha_{0}},$

for all large $N$. This concludes the proof of (10.72). To obtain (10.71) we
let

$\mathsf{E}_{\star}:=\mathsf{E}^{(1)}Q\mathsf{F}^{(1)}\cdot\mathsf{E}^{(2)}Q\mathsf{F}^{(2)}\cdots\mathsf{E}^{(\kappa-1)}Q\mathsf{F}^{(\kappa-1)}\cdot\mathsf{E}^{(\kappa)}Q\mathsf{F}_{\star}^{(\kappa)},$

where $\\{\mathsf{E}^{(i)}\\}_{i=1}^{\kappa}$ and
$\\{\mathsf{F}^{(i)}\\}_{i=1}^{\kappa-1}$ are as above, and
$\mathsf{F}^{(\kappa)}:=C_{\ref{prop:HS-bd}}^{-1/2}N^{-1}({\log N})^{1/2}\cdot
E(z)$. Notice that, by (10.76), we have
$\|\mathsf{F}_{\star}^{(\kappa)}\|_{\rm HS}\leqslant 1$. Therefore, applying
applying Proposition 10.12 again we obtain that

$\mathbf{E}\left[\|\mathsf{E}_{\star}\|_{{\rm HS}}^{2h_{0}}\right]\leqslant
d^{h_{0}}\cdot(C_{\ref{prop:HS-bd}}N\log N)^{2h_{0}\kappa}\cdot
C_{\ref{prop:mom-HS}}\cdot\mathfrak{C}_{\kappa h_{0}}.$

Now (10.71) follows by choosing $h_{0}$ as above and applying Markov’s
inequality. The proofs of (10.69)-(10.70) are similar. For (10.69) we take
$\mathsf{E}^{(1)}=E(z)/(C_{\ref{prop:HS-bd}}N\log N)$ and
$\mathsf{F}^{(\kappa+1)}=E_{+}(z)/\sqrt{|}d|$, while for (10.70) we take
$\mathsf{F}^{(\kappa+1)}=E(z)/(C_{\ref{prop:HS-bd}}N\log N)$. Further details
are omitted. ∎

The next corollary will come handy in the proof of Theorem 1.6.

######  Corollary 10.13.

In the setup of Proposition 10.11, we have that for any $\upalpha_{0}>0$, for
all $N$ large,

$\mathbf{P}\big{(}N^{-\gamma}\|E(z)(I+N^{-\gamma}QE(z))^{-1}QE_{+}(z)\|_{\rm
HS}\geqslant N^{-(\gamma-1)/4}\big{)}\leqslant N^{-\upalpha_{0}},$ (10.77)
$\mathbf{P}\big{(}N^{-2\gamma}\|E_{-}(z)(I+N^{-\gamma}QE(z))^{-1}QE(z)QE_{+}(z)\|_{\rm
HS}\geqslant N^{-\gamma-(\gamma-1)/4}\big{)}\leqslant N^{-\upalpha_{0}},$
(10.78) $\mathbf{P}\big{(}\|E_{-}(z)(I+N^{-\gamma}QE(z))^{-1}\|_{\rm
HS}>2\sqrt{M}\big{)}\leqslant N^{-\upalpha_{0}},$ (10.79)

and

$\mathbf{P}\big{(}\|E(z)(I+N^{-\gamma}QE(z))^{-1}\|_{\rm HS}>2C_{\ref{prop:HS-
bd}}N/\sqrt{\log N}\big{)}\leqslant N^{-\upalpha_{0}}.$ (10.80)

###### Proof.

The proof uses the resolvent expansion and Proposition 10.11. First let us
prove (10.77).

Using the resolvent expansion, we write, with $\hat{Q}=N^{-\gamma}Q$ and $k$
positive integer and omitting throughout the argument $z$,

$(I+\hat{Q}E)^{-1}=\sum_{i=0}^{k-1}(-\hat{Q}E)^{i}+(-\hat{Q}E)^{k}(I+\hat{Q}E)^{-1}.$
(10.81)

Considering the first term in the right hand side of (10.81), we have by
(10.69) that, for $i\geqslant 0$,

$\|E(-\hat{Q}E)^{i}\hat{Q}E_{+}\|_{\rm HS}\leqslant N^{-(\gamma-1)(i+1)}/2,$
(10.82)

with probability $1-N^{-2\upalpha_{0}}$, for all $N$ large. On the other hand,
by (10.70) and the fact that for any two matrices $A,B$ one has $\|BA\|_{\rm
HS}\leqslant\|A\|\cdot\|B\|_{\rm HS}$, we obtain that

$\|E(-\hat{Q}E)^{k}(I+\hat{Q}E)^{-1}QE_{+}\|_{\rm HS}\leqslant
N^{1-\frac{(\gamma-1)k}{2}}\cdot\|(I+\hat{Q}E)^{-1}\|\cdot\|Q\|\cdot\|E_{+}\|,$
(10.83)

on a set of event with probability at least $1-N^{-2\upalpha_{0}}$. Now by the
second part of Lemma 10.10, [42] (which gives $\mathbf{E}\|Q\|=O(N^{1/2})$),
and (2.10), we further obtain that
$\|(I+\hat{Q}E)^{-1}\|\cdot\|Q\|\cdot\|E_{+}\|\leqslant N^{\beta}$ for some
finite $\beta$, with probability at least $1-N^{-2\upalpha_{0}}$. By choosing
$k$ large enough, we conclude that on these events, the right hand side of
(10.83) can be made arbitrarily small. Combining this with (10.81) and
(10.82), we obtain (10.77). The proof of (10.78), being similar, is omitted.

To see (10.79), we use the resolvent expansion (10.81) and write

$E_{-}(I+N^{-\gamma}QE)^{-1}=E_{-}+\sum_{i=1}^{k-1}E_{-}(\hat{Q}E)^{i}+E_{-}(\hat{Q}E)^{k}(I+\hat{Q}E)^{-1}.$

Recall that $\|E_{-}\|_{\rm HS}=\sqrt{M}$. As above, the other terms are
controlled by (10.71) and Lemma 10.10. This completes the proof of (10.79). To
prove (10.80) we observe that by (10.76) one has that $\|E\|_{{\rm
HS}}\leqslant C_{\ref{prop:HS-bd}}N/\sqrt{\log N}$ (without loss of generality
assume $C_{\ref{prop:HS-bd}}\geqslant 1$). Therefore, arguing similarly as
above one derives (10.80). Further details are omitted. ∎

######  Remark 10.14.

The reader may inspect the proofs of Proposition 10.11 and Corollary 10.13 to
observe that (10.77) continues to hold even if we can replace the term
$N^{-(\gamma-1)/4}$ by $N^{-(1-\upepsilon)\cdot(\gamma-1)}$ for any
$\upepsilon>0$. This improvement will be necessary later in Section 11 to
argue that the eigenvectors of a randomly perturbed Jordan block is completely
delocalized in sup-norm.

######  Remark 10.15.

In Sections 9 and 10.1-10.4 we worked with Assumption 9.1. Since $p(S^{1})$ is
compact, it follows that Assumption 9.1 holds uniformly in
$z\in\Omega_{\varepsilon_{\ref{theo-location}},C_{\ref{theo-location}},N}$
(recall (1.7)). This observation will be used in Section 11, in the proofs of
Theorem 1.6 and Corollary 1.7.

##  11\. Proofs of Theorem 1.6 and Corollary 1.7

We start with the proof of Theorem 1.6.

###  11.1. Proof of Theorem 1.6

We begin by creating a net $\mathcal{N}_{\gamma}$ in
$\Omega(\varepsilon_{\ref{theo-location}},C_{\ref{theo-location}},N)$, of
spacing $N^{-\theta}$ and cardinality at most $O(N^{2\theta-1}\log N)$, with
$\theta>\gamma\vee 4$. Note that for any
$\hat{z}\in\Omega(\varepsilon_{\ref{theo-location}},C_{\ref{theo-
location}},N)$, there exists a $z^{*}=z^{*}(\hat{z})\in\mathcal{N}_{\gamma}$
with $|z^{*}-\hat{z}|\leqslant N^{-\theta}$.

1\. We begin with the proof of the first point in Theorem 1.6, and proceed as