# Spectral Asymptotics at Thresholds for a Dirac-type Operator on
$\mathbb{Z}^{2}$

Pablo Miranda  Departamento de Matemática y Ciencia de la Computación,
Universidad de Santiago de Chile, Las Sophoras 173. Santiago, Chile.
<EMAIL_ADDRESS>,  Daniel Parra  Departamento de Matemática y Ciencia
de la Computación, Universidad de Santiago de Chile, Las Sophoras 173.
Santiago, Chile<EMAIL_ADDRESS>and  Georgi Raikov† Facultad de
Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna
4860, Santiago, Chile.

###### Abstract.

In this article, we provide the spectral analysis of a Dirac-type operator on
$\mathbb{Z}^{2}$ by describing the behavior of the spectral shift function
associated with a sign–definite trace–class perturbation by a multiplication
operator. We prove that it remains bounded outside a single threshold and
obtain its main asymptotic term in the unbounded case. Interestingly, we show
that the constant in the main asymptotic term encodes the interaction between
a flat band and whole non–constant bands. The strategy used is the reduction
of the spectral shift function to the eigenvalue counting function of some
compact operator which can be studied as a toroidal pseudo–differential
operator.

##  1\. Introduction and main results

We start by giving a streamlined presentation of our operators acting on
$\mathbb{Z}^{2}$. First we define a graph structure by setting the set of
vertices $\mathcal{V}=\mathbb{Z}^{2}$ and the set of oriented edges
$\mathcal{A}=\\{(x,y)\in\mathbb{Z}^{2}\times\mathbb{Z}^{2}:y=x\pm\delta_{i}\\}$,
where $\\{\delta_{1},\delta_{2}\\}$ form the canonical basis of
$\mathbb{Z}^{2}$. We note an edge in $\mathcal{A}$ as $\mathrm{e}=(x,y)$ and
its transpose by $\overline{\mathrm{e}}=(y,x)$. Given a vertex $x$, we set
$\mathcal{A}_{x}=\\{\mathrm{e}\in\mathcal{A}:\mathrm{e}=(x,y)\\}$.

Les us denote by $X=(\mathcal{V},\mathcal{A})$ this graph structure defined on
$\mathbb{Z}^{2}$, and consider the vector spaces of _$0-$ cochains_ $C^{0}(X)$
and _$1-$ cochains_ $C^{1}(X)$ given by:

$C^{0}(X):=\\{f:\mathcal{V}\to\mathbb{C}\\}\text{ ; }\qquad
C^{1}(X):=\\{f:\mathcal{A}\to\mathbb{C}\mid
f(\mathrm{e})=-f(\overline{\mathrm{e}})\\}\text{ .}$

We will denote by $C(X)$ the direct sum $C^{0}(X)\oplus C^{1}(X)$ and will
refer to an $f\in C{(X)}$ as a _cochain_. This suggests the notation
$C_{c}(X)$ for the space of cochains that are finitely supported.

For $f,g\in C(X)$ we take the inner product

(1) $\langle
f,g\rangle:=\sum_{x\in\mathcal{V}}f(x)\overline{g(x)}+\frac{1}{2}\sum_{\mathrm{e}\in\mathcal{A}}f(\mathrm{e})\overline{g(\mathrm{e})}\
.$

The Hilbert space $l^{2}(X)$ is defined as the closure of $C_{c}(X)$ in the
norm induced by Eq. 1. It coincides with $\\{f\in C(X)\mid\left\lVert
f\right\rVert:=\langle f,f\rangle^{\frac{1}{2}}<\infty\\}$.

The _coboundary operator_ (or _difference operator_) $d:C^{0}(X)\to C^{1}(X)$
is defined by:

$df(\mathrm{e}):=f(y)-f(x)\text{, for }\mathrm{e}=(x,y)\ .$

Its formal adjoint $d^{*}:C^{1}(X)\to C^{0}(X)$ is given by the finite sum

$d^{*}f(x)=-\sum_{\mathrm{e}\in A_{x}}f(\mathrm{e})\ .$

Note that since
$A_{x}=\\{(x,x+\delta_{1}),(x,x+\delta_{2}),(x,x-\delta_{1}),(x,x-\delta_{2})\\}$,
the sum has only four terms.

We can now define the operator on $l^{2}(X)$

$H_{0}:=\begin{pmatrix}m&d^{*}\\\ d&-m\end{pmatrix}\ ,$

where $m$ is just the multiplication operator by the constant $m\geq 0$. It is
not difficult to see that

$H_{0}^{2}=\begin{pmatrix}\Delta_{\mathcal{V}}+m^{2}&0\\\
0&\Delta_{\mathcal{A}}+m^{2}\end{pmatrix}\ ,$

where $\Delta_{\mathcal{V}}$ and $\Delta_{\mathcal{A}}$ are the discrete
Laplacians in edges and vertices, respectively. So it is natural to say that
$H_{0}$ is a Dirac-type operator [Eck45, AT15, Par17].

The operator $H_{0}$ is bounded self-adjoint, and is analytically fibered over
$\mathbb{T}^{2}$ (see Section 2.1 for details). The analysis of its band
functions shows that the spectrum of $H_{0}$ is

(2)
$\sigma(H_{0})=\sigma_{ac}(H_{0})=[-\sqrt{m^{2}+8},-m]\bigcup[m,\sqrt{m^{2}+8}]\
.$

Moreover, there exist a discrete set $\mathcal{T}$ of thresholds in the
spectrum of $H_{0}$, given by

(3) $\mathcal{T}=\left\\{\pm m,\pm\sqrt{m^{2}+4},\pm\sqrt{m^{2}+8}\right\\}\
.$

In this article we understand the set of thresholds as those point in the
spectrum of $H_{0}$ where the Mourre estimates does not hold and hence we do
not have _a priori_ a Limiting Absorption Principle (LAP) [Par17].

We will see that there are three types of thresholds and one could expect
qualitatively different properties of the spectrum near each type: for $m>0$
the points $\\{\pm\sqrt{m^{2}+8},m\\}$ are thresholds of _elliptic_ type,
$\\{\pm\sqrt{m^{2}+4}\\}$ are of _hyperbolic_ type, and $-m$ is elliptic and
an eigenvalue of infinite multiplicity. The case $m=0$ is particular: there is
no finite gap in the spectrum and the threshold at zero is not of elliptic
type. It is called a _Dirac point_ (see Section 2.2).

In this article we will study spectral properties of $H_{0}\pm V$, where
$V\geq 0$ is a potential that is defined both on vertices and edges, and
satisfies $V(\mathrm{e})=V(\overline{\mathrm{e}})$. Use the notation

(4) $H_{\pm}:=H_{0}\pm V\ .$

Since the Dirac delta functions on vertices and edges lie in $l^{2}(X)$, we
have that the range of $V$ coincides with its set of eigenvalues. From this it
is clear that the trace class norm of $V$ is

$\left\lVert
V\right\rVert_{1}=\sum_{x\in\mathcal{V}}|V(x)|+\frac{1}{2}\sum_{\mathrm{e}\in\mathcal{A}}|V(\mathrm{e})|\
.$

When $V$ is of trace class, there exists a unique
$\bm{\xi}(\lambda;H_{\pm},H_{0})$ in $L^{1}(\mathbb{R})$ that satisfies the
trace formula

$\text{Tr}(f(H_{\pm})-f(H_{0}))=\int_{\mathbb{R}}\mathrm{d}\lambda\,\bm{\xi}(\lambda;H_{\pm},H_{0})f^{\prime}(\lambda),$

for all $f\in C^{\infty}_{0}(\mathbb{R})$ (see the original work [Kre53], or
the monograph [Yaf92, Ch. 8]). The function $\bm{\xi}(\cdot;H_{\pm},H_{0})$ is
called the Spectral Shift Function (SSF) for the pair $(H_{\pm},H_{0})$. This
function can be defined in an abstract setting and is an important object in
the analysis of linear operators. For instance, it is related to the
scattering matrix $S(\lambda;H_{\pm},H_{0})$ by the Birman-Krein formula
[BK62, Yaf92]

${\rm det}S(\lambda;H_{\pm},H_{0})=e^{-2\pi
i\bm{\xi}(\lambda;H_{\pm},H_{0})},\quad a.e.\,\lambda\in\sigma_{{\rm
ac}}(H_{0}).$

It also gives the number of discrete eigenvalues of $H_{\pm}$ outside the
essential spectrum (see 1.6 for more details).

Our main goal in this article is to describe the SSF near the thresholds in
the spectrum of $H_{0}$. These types of results have been obtained in the
literature for different models. In particular for the Laplacian in
$\mathbb{R}^{d}$ perturbed by a decaying electric potential. These results are
related to Levinson’s theorem (see [Yaf10, Rob99] and references therein).
More recently, results for 2D and 3D magnetic Schrödinger and Dirac operators
were obtained [FR04, Tie11, BM18, BR20]. In the discrete case, some trace
formulas have been obtained for periodic graphs in settings very close to ours
[KS22, IK12].

One of the novelties of this work is related to the rich structure of the
spectrum of $H_{0}$, which is due to the remarkable form of the band
functions. In particular, we have the interaction of a flat band with a non-
constant band function at the maximal point of the last one, as well as the
existence of saddle points which implies the appearance of hyperbolic
thresholds. To the best of our knowledge, this is the first article where
these situations are studied.

###  1.1. Main results

Let $\mu\in\mathbb{Z}^{2}$, and define the edges $e_{1}=((0,0),(1,0))$,
$e_{2}=((0,0),(0,1))$. Take the following real-valued functions on
$\mathbb{Z}^{2}$:

(5) $v_{1}(\mu):=V(\mu)\quad;\quad
v_{2}(\mu):=V(\mu\mathrm{e}_{1})\quad\text{and}\quad
v_{3}(\mu):=V(\mu\mathrm{e}_{2})\ ,$

where for $\mathrm{e}=(x,y)$ we have the natural action

$\mu\mathrm{e}=(\mu+x,\mu+y)\ .$

Whenever is convenient, we will keep the notation $x,y$ for elements of
$\mathbb{Z}^{2}$ seen as a vertices and $\mu,\nu$ for elements of
$\mathbb{Z}^{2}$ seen as elements of the group acting on $X$.

To study the most basic properties of the SSF it will suffice to assume that

(6)
$\sum_{\mu\in\mathbb{Z}^{2},\langle\mu\rangle>N}|v_{j}(\mu)|=O(N^{-2\beta_{j}}),\quad
N\to\infty\ ,$

with $0<\beta_{j}\leq 1$ for $j=1,2,3$, where
$\langle\mu\rangle=(1+|\mu|^{2})^{1/2}$. For instance, it is easy to see that
this condition ensures the existence of the SSF, as it implies that $V$ is
summable. Moreover we have the following theorem:

######  Theorem 1.1.

Let us suppose that the perturbation $V$ is positive and each $v_{j}$
satisfies Eq. 6 with $\beta_{j}>0$. Then, on any compact set
$\mathcal{K}\subset\mathbb{R}\backslash\\{-m\\}$,

$\sup_{\lambda\in\mathcal{K}}\bm{\xi}(\lambda;H_{\pm},H_{0})<\infty\ ,$

i.e. the SSF is bounded away from $-m$.

######  Remark 1.2.

This theorem implies in particular that the SSF
$\bm{\xi}(\lambda;H_{\pm},H_{0})$ is bounded at the hyperbolic thresholds, and
the study of the SSF at hyperbolic thresholds seems to have not been carried
out before in any model. However, there is a growing interest in the
understanding of the spectral properties of discrete Hamiltonians inside their
continuous spectrum, where these kind of thresholds appear [IJ19, IJ21].

One consequence of 1.1 is that the only possible point of unbounded growth of
the SSF is $-m$. In our second main theorem we describe the explicit
asymptotic behavior of the SSF at that point, for potentials $V$ that shows a
power-like decay. More precisely, for $\gamma>0$ and for any multi-index
$\alpha$ we assume

(7) $|{\rm D}^{\alpha}v_{j}(\mu)|\leq
C_{\alpha}\langle\mu\rangle^{-\gamma-\rho|\alpha|};\qquad j=2,3$

where ${\rm D_{\mu_{j}}}v(\mu):=v(\mu+\delta_{j})-v(\mu)$, and ${\rm
D}^{\alpha}:={\rm D}_{\mu_{1}}^{\alpha_{1}}...{\rm D}_{\mu_{d}}^{\alpha_{d}}.$
Moreover,

(8) $\lim_{|\mu|\to\infty}|\mu|^{\gamma}v_{l}(\mu)=\Gamma_{l};\qquad l=2,3\ ,$

exist and at least one $\Gamma_{j}\in\mathbb{R}$ is not equal to $0$.

We define the matrix

$\Gamma:=\begin{pmatrix}\Gamma_{2}&0\\\\[5.0pt] 0&\Gamma_{3}\end{pmatrix}\ ,$

and the constant

(9)
$\mathcal{C}:=\pi\int_{\mathbb{T}^{2}}\rm{Tr}\left(\Big{(}\mathcal{A}(\xi)^{*}\Gamma\mathcal{A}(\xi)\Big{)}^{2/\gamma}\right)\mathrm{d}\xi\
,$

where

$\mathcal{A}:=\begin{pmatrix}b(\xi)r(\xi)^{-1/2}&0\\\\[5.0pt]
a(\xi)r(\xi)^{-1/2}&0\end{pmatrix}\ ,$

and $a$, $b$, $r=a^{2}+b^{2}$ are given by the analytic fibration of $H_{0}$
(see 2.1). Notice that in condition Eq. 8 we do not ask to $v_{2}$ and $v_{3}$
to have the same decaying rate. For instance, if $v_{2}$ decays faster that
$v_{3}$, and hence $\Gamma_{2}=0$, the constant $\mathcal{C}$ will not depend
on $v_{2}$.

######  Theorem 1.3.

Let $V\geq 0$ and suppose that $v_{1}$ satisfies Eq. 6 with $\beta_{1}>0$, and
that $v_{2}$ and $v_{3}$ satisfy Eqs. 7 and 8, with $\gamma>2$. Then,

(10)
$\bm{\xi}(\lambda;H_{-},H_{0})=\begin{cases}-\mathcal{C}|\lambda+m|^{-2/\gamma}(1+o(1))&\text{
if }\,\lambda\uparrow-m\ ,\\\ O(1)&\text{ if }\,\lambda\downarrow-m\
,\end{cases}$

and

(11) $\bm{\xi}(\lambda;H_{+},H_{0})=\begin{cases}O(|\ln(|\lambda+m|)|)&\text{
if }\,\lambda\uparrow-m\ ,\\\
\mathcal{C}|\lambda+m|^{-2/\gamma}(1+o(1))&\text{ if }\,\lambda\downarrow-m\
.\end{cases}$

######  Remark 1.4.

The asymptotic order ${-2}/{\gamma}$ of the SSF is clearly determined by the
perturbation $V$. This may be interpreted as the contribution of the constant
band at $-m$. The contribution of the non-constant band is instead encoded in
the constant $\mathcal{C}$. This constant contains an explicit interaction
between the perturbation and the whole non-constant band functions (see 2.1).
As far as we know, this is the first time that such a behavior has been
observed.

######  Remark 1.5.

1.3 is proved by a reduction of the SSF to an eigenvalue counting function,
and by using a general theorem on the spectral asymptotics for integral
operators with toroidal symbol. The statement and proof of this last theorem
appears in section Section 6. The result is, to the best of our knowledge, new
and may be of independent interest. The proof uses an appropriate Cwikel
estimate (Lemma 6.3).

######  Remark 1.6.

Since $V$ is compact, $\sigma_{\rm ess}(H)=\sigma_{\rm ess}(H_{0})$. Thus,
when $m>0$ we have that $(-m,m)$ is a gap in the essential spectrum of
$H_{\pm}$. Then, for $\lambda\in(-m,m)$ we can consider the function

$\mathcal{N}^{\pm}(\lambda)={\rm Rank}\mathds{1}_{(\lambda,m)}(H_{\pm})\ ,$

with $\mathds{1}_{\Omega}$ being the characteristic function over the Borel
set $\Omega$. Clearly, this function counts the number of discrete eigenvalues
of $H_{\pm}$ on the interval $(-m+\lambda,m)$. By 1.1 these functions are well
defined, as there are no accumulation of eigenvalues at $m$ from below.
Moreover, for $\lambda\in(-m,m)$

$\bm{\xi}(\lambda;H_{\pm},H_{0})=\pm\mathcal{N}^{\pm}(\lambda)+O(1)\ ,$

(see [Pus98]). Similarly, we can write analogous equations for the eigenvalues
of $H_{\pm}$ on the intervals $(-\infty,-\sqrt{m^{2}+8})$ and
$(\sqrt{m^{2}+8},\infty)$. In consequence, 1.1 and 1.3 gives the asymptotic
distribution of the discrete eigenvalues of $H_{\pm}$ (under conditions Eqs. 6
and 8). In order to reduce the length of the article and to maintain the
symmetry of the results, we plan to consider $\mathcal{N}^{\pm}(\lambda)$ for
$\gamma\in(0,2]$ elsewhere.

###  1.2. Notations and structure of the article

Throughout the article we will denote by $C$ different constants that appear
which are non-essential to our purposes. Whenever $\pm$ appears it refers to
two independent statements that are valid independently. Also for non-negative
real functions $f,g$ we note

$f\asymp g$

if there exists two non-negative constants $C_{1},C_{2}$ such that

$C_{1}f\leq g\leq C_{2}f\ .$

The rest of the paper is organized as follows. In Section 2 we recall the
integral representation and describe the spectrum of $H_{0}$. Then, in Section
3, we obtain an explicit expression for the LAP that will be the base of our
further investigations. In Section 4 we study the operators that appear in the
SSF representation given in Section 3. In Section 5 we prove our results for
hyperbolic thresholds. In Section 6 we obtain some abstract results concerning
the eigenvalue asymptotics for a class of integral operators. Finally, in
Section 7 we put together these results to obtain the proofs of our main
theorems in the parabolic and Dirac point case.

##  2\. Analysis of the unperturbed operator $H_{0}$

###  2.1. Integral decomposition

In this section we construct an unitary operator $\mathscr{U}:l^{2}(X)\to
L^{2}(\mathbb{T}^{2},\mathbb{C}^{3})$ in order to write $H_{0}$ as an analytic
fibered operator. We take the convention that
$\mathbb{T}^{2}=\mathbb{R}^{2}/[0,1]^{2}$ and recall that $\mathbb{T}^{2}$ is
the dual of $\mathbb{Z}^{2}$ by $\xi(\mu)=e^{2\pi i\,\xi\cdot\mu}$, for
$\xi\in\mathbb{T}^{2}$ and $\mu\in\mathbb{Z}^{2}$. The construction of
$\mathscr{U}$ is a particularization of the general construction obtained in
[Par17]. We define $\mathscr{U}:C_{c}(X)\to
L^{2}(\mathbb{T}^{2},\mathbb{C}^{3})$ by setting, for $f\in C_{c}(X)$ and
$\xi\in\mathbb{T}^{2}$,

$(\mathscr{U}f)(\xi)=\left(\sum_{\mu\in\mathbb{Z}^{2}}e^{-2\pi
i\xi\cdot\mu}f(\mu),\sum_{\mu\in\mathbb{Z}^{2}}e^{-2\pi
i\xi\cdot\mu}f(\mu\mathrm{e}_{1}),\sum_{\mu\in\mathbb{Z}^{2}}e^{-2\pi
i\xi\cdot\mu}f(\mu e_{2})\right).$

We denote by $\operatorname{Trig\,Pol}(\mathbb{T}^{2};\mathbb{C}^{3})$ the
subspace of $L^{2}(\mathbb{T}^{2};\mathbb{C}^{3})$ composed by functions
$\varphi$ that admit

$\varphi(\xi)=\sum_{\mu\in\mathbb{Z}^{2}}e^{2\pi
i\xi\cdot\mu}\varphi_{\mu}\text{ ; with
}\mathbf{0}\neq\varphi_{\mu}\in\mathbb{C}^{3}\text{ for only finitely many
}\mu\ .$

This is a conveniently dense space in $L^{2}(\mathbb{T}^{2};\mathbb{C}^{3})$
and coincides with $\mathscr{U}(C_{c}(X))$. To write the adjoint of
$\mathscr{U}$ define the index $\eta((x,y))=y-x$ and fix an orientation on the
graph by setting
$\mathcal{A}^{+}=\\{\mathrm{e}\in\mathcal{A}:\eta(e)=\delta_{i},i=1,2\\}$. Set
also the integer part of an edge by $[(x,y)]:=x$ . Therefore, for
$\varphi\in\operatorname{Trig\,Pol}(\mathbb{T}^{2};\mathbb{C}^{3})$

$\displaystyle(\mathscr{U}^{*}\varphi)(x)=$
$\displaystyle\int_{\mathbb{T}^{2}}\mathrm{d}\xi e^{2\pi i\xi\cdot
x}\varphi_{1}(\xi)\ ,$ $\displaystyle(\mathscr{U}^{*}\varphi)(\mathrm{e})=$
$\displaystyle\int_{\mathbb{T}^{2}}\mathrm{d}\xi e^{2\pi
i\xi\cdot\left[\mathrm{e}\right]}\varphi_{\imath(\mathrm{e})}(\xi)\quad{\text{
if }}\mathrm{e}\in\mathcal{A}^{+}\ ,$
$\displaystyle(\mathscr{U}^{*}\varphi)(\mathrm{e})=$
$\displaystyle-\int_{\mathbb{T}^{2}}\mathrm{d}\xi e^{2\pi
i\xi\cdot(\left[\mathrm{e}\right]+\eta(\mathrm{e}))}\varphi_{\imath(\mathrm{e})}(\xi)\quad{\text{
if }}\mathrm{e}\notin\mathcal{A}^{+}\ ,$

where $\imath(e)=j+1$ if there exist $\mu\in\mathbb{Z}^{2}$ such that either
$\mu\mathrm{e}_{j}=e$ or $\mu\overline{\mathrm{e}}_{j}=e$. Then $\mathscr{U}$
extends to a unitary operator, still denoted by $\mathscr{U}$, from $l^{2}(X)$
to $L^{2}(\mathbb{T}^{2},\mathbb{C}^{3})$.

Note that this definition of $\mathscr{U}$ correspond to the following choice
of Fourier transform in $\mathbb{Z}^{2}$:

$\mathscr{F}^{*}:l^{2}(\mathbb{Z}^{2})\to
L^{2}(\mathbb{T}^{2})\quad;\quad(\mathscr{F}^{*}f)(\xi)\equiv\hat{f}(\xi):=\sum_{\mu\in\mathbb{Z}^{2}}e^{-2\pi
i\xi\cdot\mu}f(\mu)\ .$

We stress the fact that this choice can be considered as not standard, but it
is rather natural in this context. We now reproduce, adapted to our setting,
the result from [Par17] that will be the starting point of our investigation.
To simplify the computations, we introduce for $\xi=(\xi_{1},\xi_{2})$

$\displaystyle a(\xi):=(-1+e^{-2\pi i\xi_{1}}),\quad b(\xi):=(-1+e^{-2\pi
i\xi_{2}})\ .$

######  Proposition 2.1 ([Par17, Prop. 3.5]).

The operator $H_{0}$ is, by conjugation by $\mathscr{U}$, unitarily equivalent
to a matrix-valued multiplication operator in
$L^{2}(\mathbb{T}^{2},\mathbb{C}^{3})$ given by the real-analytic function

$h_{0}(\xi)=\begin{pmatrix}m&a(\xi)&b(\xi)\\\ \overline{a(\xi)}&-m&0\\\
\overline{b(\xi)}&0&-m\end{pmatrix}\ .$

###  2.2. Spectral theory for $H_{0}$

To compute the spectrum of $\mathscr{U}H_{0}\mathscr{U}^{*}$ we will obtain
its band functions. Because for every $\xi\in\mathbb{T}^{2}$, $h_{0}(\xi)$ has
three eigenvalues, we will have three band functions. Note that, even if
$h_{0}(\xi)$ is real-analytic, in contrast with the case when the base is
$1$-dimensional, one can not a priori choose the band functions
$\\{\lambda_{j}\\}$ to be analytic. We will have however an explicit
expression for them and be able to compute
$\sigma(H_{0})=\bigcup_{j}\lambda_{j}(\mathbb{T}^{2})$.

One can see that for $\xi\in\mathbb{T}^{2}$, the characteristic polynomial
associated to $h_{0}(\xi)$ is given by

(12) $p_{\xi}(z)=(m-z)(m+z)^{2}+(m+z)\left(|b(\xi)|^{2}+|a(\xi)|^{2}\right)\
.$

We notice the identities

$|a(\xi)|^{2}=2(1-\cos(2\pi\xi_{1}))=4\sin^{2}(\pi\xi_{1})\text{ and
}|b(\xi)|^{2}=2(1-\cos(2\pi\xi_{2}))=4\sin^{2}(\pi\xi_{2})\ .$

There are three band functions:

$z_{0}(\xi)=-m\ ,\quad z_{\pm}(\xi)=\pm\sqrt{m^{2}+|a(\xi)|^{2}+|b(\xi)|^{2}}\
.$

This give us Eq. 2. For convenience we set
$r_{m}(\xi):=m^{2}+|a(\xi)|^{2}+|b(\xi)|^{2}\geq 0$ and assume the convention
$r_{0}\equiv r$.

To compute the thresholds in $\mathcal{T}$ notice that

(13) $\nabla
z_{\pm}(\xi)=\pm\frac{2\pi}{r_{m}(\xi)^{\frac{1}{2}}}(\sin(2\pi\xi_{1}),\sin(2\pi\xi_{2})).$

Then, both $z_{+}$ and $z_{-}$ have the same critical points and they give Eq.
3.

First assume $m>0$. From Eq. 13 we see that the thresholds
$\\{\pm\sqrt{m^{2}+8},m\\}$ are of _elliptic_ type. These thresholds are
situated at the edges of the bands. The thresholds $\\{\pm\sqrt{m^{2}+4}\\}$
are of _hyperbolic_ type and correspond to embedded thresholds (see [IJ19] for
this distinction for the Laplacian). Finally, there is $-m$ that corresponds
to the degenerated eigenvalue, but it coincides also with the maximum of
$z_{-}$.

When $m=0$ there is no gap in the spectrum and the thresholds are $\\{0,\pm
2,\pm\sqrt{8}\\}$. However, since $r((0,0))=0$, $z_{\pm}$ are not everywhere
differentiable, which indicates that $0$ is a different type of threshold. We
call this a _Dirac point_.

One of our focus of attention in the sequel will be the spectral properties
near the energy corresponding to a degenerated eigenvalue of $H_{0}$. From the
Floquet-Bloch theory described above, this eigenvalue appears due to the
existence of a flat band. However, one can also understand directly in
$l^{2}(X)$ its appearance: let suppose for simplicity that $m=0$. Let
$x^{\prime}\in\mathcal{V}$ be fixed and consider the closed path
$\Gamma_{x^{\prime}}\subset\mathcal{A}$ given by
$\\{(x^{\prime},x^{\prime}+\delta_{1}),(x^{\prime}+\delta_{1},x^{\prime}+\delta_{1}+\delta_{2}),(x^{\prime}+\delta_{1}+\delta_{2},x^{\prime}+\delta_{2}),(x^{\prime}+\delta_{2},x^{\prime})\\}$.
We define $f_{x^{\prime}}\in l^{2}(X)$ by $f_{x^{\prime}}(\mathrm{e})=1$ if
$\mathrm{e}\in\Gamma_{x^{\prime}}$ and such that it vanishes elsewhere.
Remember that by definition of $C^{1}(X)$, we have that
$f_{x^{\prime}}(\overline{\mathrm{e}})=-1$ for
$\mathrm{e}\in\Gamma_{x^{\prime}}$. Since $f_{x^{\prime}}$ vanishes in
vertices we have $(H_{0}f_{x^{\prime}})(\mathrm{e})=0$ for every $\mathrm{e}$.
Furthermore,

$\displaystyle(H_{0}f_{x^{\prime}})(x^{\prime})=$ $\displaystyle-
f_{x^{\prime}}(x^{\prime},x^{\prime}+\delta_{1})-f_{x^{\prime}}(x^{\prime},x^{\prime}+\delta_{2})-f_{x^{\prime}}(x^{\prime},x^{\prime}-\delta_{1})-f_{x^{\prime}}(x^{\prime},x^{\prime}-\delta_{2})$
$\displaystyle=$ $\displaystyle-1+1+0+0=0.$

Similar computations hold for $x^{\prime}+\delta_{1}$,
$x^{\prime}+\delta_{1}+\delta_{2}$ and $x^{\prime}+\delta_{2}$ while for the
other vertices $f_{x^{\prime}}$ vanishes in every edge around them. If follows
that $H_{0}f_{x^{\prime}}\equiv 0$. Repeating the procedure for different
$x^{\prime}\in\mathcal{V}$ we get an infinitely dimensional kernel.

##  3\. SSF representation

Let us start this section by giving some notation that we will use henceforth.

Denote by $\mathfrak{S}_{\infty}$ the class of compact operators. For
$K\in\mathfrak{S}_{\infty}$ self-adjoint and $r>0$, we set

$n_{\pm}(r;K):=\operatorname{Rank}\mathds{1}_{(r,\infty)}(\pm K)\ .$

Thus, the functions $n_{\pm}(\cdot;K)$ are respectively the counting functions
of the positive and negative eigenvalues of the operator $K$. If $K$ is
compact but not necessarily self-adjoint we will use the notation

$n_{*}(r;K):=n_{+}(r^{2};K^{*}K),\quad r>0;$

thus $n_{*}(\cdot;K)$ is the counting function of the singular values of $K$,
which we denote by $\\{s_{j}(K)\\}$. Since
$n_{+}(r^{2};K^{*}K)=n_{+}(r^{2};KK^{*})$

(14) $n_{*}(r;K)=n_{*}(r;K^{*}),\quad r>0.$

Besides, for $r_{1},r_{2}>0$, we have the Weyl inequalities

(15) $n_{\pm}(r_{1}+r_{2};K_{1}+K_{2})\leq
n_{\pm}(r_{1};K_{1})+n_{\pm}(r_{2};K_{2}),$

where $K_{j}$, $j=1,2$, are self-adjoint compact operators (see e.g. [BS87,
Theorem 9.2.9]), as well as the Ky Fan inequality

(16) $n_{*}(r_{1}+r_{2};K_{1}+K_{2})\leq
n_{*}(r_{1};K_{1})+n_{*}(r_{2};K_{2}),$

for compact but not necessarily self-adjoint $K_{j}$, $j=1,2$, (see e.g.
[BS87, Subsection 11.1.3]).

Denote by $\mathfrak{S}_{p}$, $p\in[1,\infty)$, the Schatten-von Neumann class
of compact operators, equipped with the norm

$\left\lVert K\right\rVert_{p}^{p}=\sum s_{j}(K)^{p}\ .$

Further, since for $K\in\mathfrak{S}_{p},$
$\|K\|_{p}=\left(-\int_{0}^{\infty}\,\mathrm{d}n_{*}(r;K)\,r^{p}\right)^{1/p}$,
the Chebyshev-type estimate

(17) $n_{*}(r;K)\leq r^{-p}\|K\|_{p}^{p}$

holds true for any $r>0$ and $p\in[1,\infty)$.

Now, define for $z\in\mathbb{C}^{+}$ the operator
$K(z)=V^{1/2}(H_{0}-z)^{-1}V^{1/2}.$ By 3.3 below, the norm limits

(18) $\lim_{\delta\downarrow 0}K(\lambda+i\delta)=:K(\lambda+i0)$

exist for any $\lambda\in\mathbb{R}\setminus\mathcal{T}$. Moreover, recalling
that $H_{\pm}=H_{0}\pm V$, by [Pus98, Theorem 1.1] we have that the SSF admits
the representative

(19)
$\bm{\xi}(\lambda;H_{\pm},H_{0})=\pm\frac{1}{\pi}\int_{\mathbb{R}}\frac{\mathrm{d}t}{1+t^{2}}n_{\mp}(1;{\rm
Re}\,K(\lambda+i0)+t\,{\rm Im}\,K(\lambda+i0)).$

_All the results of this article about the SSF will concern its representative
given by Eq. 19._

From this representation it is possible to obtain the following lemma, which
is a small modification of [Pus98, Lemma 2.1] that we state for the ease of
the reader.

######  Lemma 3.1.

Let ${\rm T}(\lambda)$ be a family of finite rank operators. Then, for any
$\epsilon\in(0,1)$

$\bm{\xi}(\lambda;H_{\pm},H_{0})\leq\pm n_{\mp}(1\mp\epsilon;{\rm
Re}\,K(\lambda+i0))+\frac{1}{\pi\varepsilon}||{\rm Im}\,K(\lambda+i0)-{\rm
T}(\lambda)||_{1}+\operatorname{Rank}{\rm T}(\lambda)\ ;$
$\bm{\xi}(\lambda;H_{\pm},H_{0})\geq\pm n_{\mp}(1\pm\epsilon;{\rm
Re}\,K(\lambda+i0))-\frac{1}{\pi\varepsilon}||{\rm Im}\,K(\lambda+i0)-{\rm
T}(\lambda)||_{1}-\operatorname{Rank}{\rm T}(\lambda)\ .$

Motivated by Lemma 3.1, we now look for a more explicit expression of the
operator $\mathscr{U}K(\lambda+i0)\mathscr{U}^{*}$. Throughout this section we
will consider a fixed $\lambda\in\sigma(H_{0})\backslash\mathcal{T}$, write
$z=\lambda+i\delta$ and study the limit as $\delta\downarrow 0$. Furthermore,
for simplicity we will assume $\lambda>0$ but, by symmetry, the results also
holds for the lower band.

We have that for $z\in\mathbb{C}\backslash\sigma(H_{0})$

$(h_{0}-z)^{-1}(\xi)=\frac{1}{p_{\xi}(z)}\begin{pmatrix}(m+z)^{2}&(m+z)a(\xi)&(m+z)b(\xi)\\\
(m+z)\overline{a(\xi)}&z^{2}-m^{2}-|b(\xi)|^{2}&\overline{a(\xi)}b(\xi)\\\
(m+z)\overline{b(\xi)}&a(\xi)\overline{b(\xi)}&z^{2}-m^{2}-|a(\xi)|^{2}\end{pmatrix},\
$

the characteristic polynomial ${p_{\xi}(z)}$ given in Eq. 12.

We are interested in studying, for $z$ with $\rm Im(z)>0$,
$\mathscr{U}V^{1/2}(H_{0}-z)^{-1}V^{1/2}\mathscr{U}^{*}.$ We can readily see
that

$\begin{split}\mathscr{U}V^{1/2}&(H_{0}-z)^{-1}V^{1/2}\mathscr{U}^{*}=\frac{\mathscr{U}V\mathscr{U}^{*}}{m+z}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-1&0\\\\[5.0pt] 0&0&-1\end{pmatrix}\\\
+\mathscr{U}V^{1/2}\mathscr{U}^{*}&\frac{1}{(m+z)(r_{m}(\xi)-z^{2})}\begin{pmatrix}(m+z)^{2}&(m+z)a(\cdot)&(m+z)b(\cdot)\\\\[5.0pt]
(m+z)\overline{a(\cdot)}&|a(\cdot)|^{2}&\overline{a(\cdot)}b(\cdot)\\\\[5.0pt]
(m+z)\overline{b(\cdot)}&a(\cdot)\overline{b(\cdot)}&|b(\cdot)|^{2}\end{pmatrix}\mathscr{U}V^{1/2}\mathscr{U}^{*}\
.\end{split}$

Let $g_{j}$ be defined by Eq. 5 but corresponding to the potential
$G:=V^{1/2}$. Now, for each $\xi^{\prime}\in\mathbb{T}^{2}$ and
$z\in\mathbb{C}$ define
$t_{z,m}(\xi^{\prime}):L^{2}(\mathbb{T}^{2};\mathbb{C}^{3})\to\mathbb{C}^{3}$
by

(20) $\begin{pmatrix}f_{1}\\\ f_{2}\\\
f_{3}\end{pmatrix}\mapsto\begin{pmatrix}(m+z)\displaystyle{\int_{\mathbb{T}^{2}}\mathrm{d}\eta\
\hat{g}_{1}(\xi^{\prime}-\eta)f_{1}(\eta)}\\\\[6.99997pt]
a(\xi^{\prime})\displaystyle{\int_{\mathbb{T}^{2}}\mathrm{d}\eta\
\hat{g}_{2}(\xi^{\prime}-\eta)f_{2}(\eta)}\\\\[6.99997pt]
b(\xi^{\prime})\displaystyle{\int_{\mathbb{T}^{2}}\mathrm{d}\eta\
\hat{g}_{3}(\xi^{\prime}-\eta)f_{3}(\eta)}\end{pmatrix}\ ,$

and $t^{\dagger}_{z,m}(\xi^{\prime}):\mathbb{C}^{3}\to
L^{2}(\mathbb{T}^{2};\mathbb{C}^{3})$ by

(21) $\begin{pmatrix}\omega_{1}\\\ \omega_{2}\\\
\omega_{3}\end{pmatrix}\mapsto\begin{pmatrix}\omega_{1}(m+z)\hat{g}_{1}(\cdot-\xi^{\prime})\\\
\omega_{2}\overline{a(\xi^{\prime})}\hat{g}_{2}(\cdot-\xi^{\prime})\\\
\omega_{3}\overline{b(\xi^{\prime})}\hat{g}_{3}(\cdot-\xi^{\prime})\end{pmatrix}\
.$

We also note $t_{z,0}(\xi^{\prime})=:t_{z}(\xi^{\prime})$ and notice that
$t^{\dagger}_{\overline{z},m}(\xi^{\prime})=t_{z,m}(\xi^{\prime})^{*}$. Then,
on can see that

(22)
$\mathscr{U}V^{1/2}(H_{0}-z)^{-1}V^{1/2}\mathscr{U}^{*}=\frac{\mathscr{U}V\mathscr{U}^{*}}{m+z}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-1&0\\\\[5.0pt]
0&0&-1\end{pmatrix}+\frac{1}{m+z}\int_{\mathbb{T}^{2}}\mathrm{d}\xi^{\prime}\
\frac{t_{z,m}^{\dagger}(\xi^{\prime})At_{z,m}(\xi^{\prime})}{(r_{m}(\xi^{\prime})-z^{2})}\
,$

where

$A=\begin{pmatrix}1&1&1\\\\[5.0pt] 1&1&1\\\\[5.0pt] 1&1&1\end{pmatrix}\ .$

In Eq. 22 define $T_{z,m}(\xi)=t_{z,m}^{\dagger}(\xi)At_{z,m}(\xi).$ As
before, we also denote $T_{z,0}(\xi)=:T_{z}(\xi)$.

From now on, we will assume that each $v_{j}$ satisfies Eq. 6 with
$0<\beta_{j}\leq 1$. Further, we set

(23)
$\widetilde{\beta}:=\min\\{\beta_{2},\beta_{3}\\};\qquad\beta:=\min\\{\beta_{1},\widetilde{\beta}\\}\
.$

######  Lemma 3.2.

For every $z\in\mathbb{C}$ and $\xi\in\mathbb{T}^{2}$ we have that
$T_{z,m}(\xi)\in\mathfrak{S}_{1}(L^{2}(\mathbb{T}^{2};\mathbb{C}^{3}))$.
Furthermore, $T_{z,m}$ is locally Hölder with exponent $\beta$ in $\xi$ in the
$\mathfrak{S}_{1}$-norm, and it depends analytically in $z$.

###### Proof.

First, it is easy to see that

(24)
$\begin{split}||t_{z,m}(\xi)||_{2}^{2}=&\int_{\mathbb{T}^{2}}\mathrm{d}\eta|m+z|^{2}|\hat{g}_{1}(\xi-\eta)|^{2}+|a(\xi)|^{2}|\hat{g}_{2}(\xi-\eta)|^{2}+|b(\xi)|^{2}|\hat{g}_{3}(\xi-\eta)|^{2}\\\
=&|m+z|^{2}\left\lVert\hat{g}_{1}\right\rVert^{2}+|a(\xi)|^{2}\left\lVert\hat{g}_{2}\right\rVert^{2}+|b(\xi)|^{2}\left\lVert\hat{g}_{3}\right\rVert^{2}\
,\end{split}$

which in particular gives us that

(25) $\ ||T_{z,m}(\xi)||_{1}=||t_{z,m}^{\dagger}(\xi)At_{z,m}(\xi)||_{1}\leq
C||t_{z,m}(\xi)||_{2}^{2}<\infty\ .$

Now, let us fix $z\in\mathbb{C}$ and $m\geq 0$. For $\xi$ and $\xi^{\prime}$
in $\mathbb{T}^{2}$ we have

(26)
$||T_{z,m}(\xi^{\prime})-T_{z,m}(\xi)||_{1}\leq||t_{z,m}^{\dagger}(\xi^{\prime})-t_{z,m}^{\dagger}(\xi)||_{2}||At_{z,m}(\xi^{\prime})||_{2}+||t_{z,m}^{\dagger}(\xi)A||_{2}||t_{z,m}(\xi^{\prime})-t_{z,m}(\xi)||_{2}.$

Using

(27)
$\begin{split}||t_{z,m}(\xi^{\prime})-t_{z,m}(\xi)||_{2}^{2}=&\int_{\mathbb{T}^{2}}\mathrm{d}\eta|m+z|^{2}|\hat{g}_{1}(\xi^{\prime}-\eta)-\hat{g}_{1}(\xi-\eta)|^{2}\\\
&+|a(\xi^{\prime})\hat{g}_{2}(\xi^{\prime}-\eta)-a(\xi)\hat{g}_{2}(\xi-\eta)|^{2}\\\
&+|b(\xi^{\prime})\hat{g}_{3}(\xi^{\prime}-\eta)-b(\xi)\hat{g}_{3}(\xi-\eta)|^{2}\
,\end{split}$

together with the fact that condition (6) ensures us (see [DDR19, Theorem
3.5])

(28)
$\int_{\mathbb{T}^{2}}\mathrm{d}\eta|\hat{g}_{j}(\eta+h)-\hat{g}_{j}(\eta)|^{2}=O(|h|^{2\beta_{j}}),\quad|h|\to
0,\quad 1\leq j\leq 3\ ,$

we can conclude that, for $|\xi^{\prime}-\xi|\to 0$,

(29) $\displaystyle||t_{z,m}(\xi^{\prime})-t_{z,m}(\xi)||_{2}^{2}\leq$
$\displaystyle
C\left(|m+z|^{2}\cdot|\xi^{\prime}-\xi|^{2\beta_{1}}+|a(\xi^{\prime})|^{2}|\xi^{\prime}-\xi|^{2\beta_{2}}\right.$
$\displaystyle\left.+|b(\xi^{\prime})|^{2}|\xi^{\prime}-\xi|^{2\beta_{3}}+|a(\xi^{\prime})-a(\xi)|^{2}\left\lVert\hat{g}_{2}\right\rVert^{2}+|b(\xi^{\prime})-b(\xi)|^{2}\left\lVert\hat{g}_{3}\right\rVert^{2}\right)$
(30) $\displaystyle\leq$ $\displaystyle C|\xi^{\prime}-\xi|^{2\beta}\ .$

Taking into account Eq. 26 we obtain the Hölder property. Finally, the
analyticity in $z$ can be directly observed in Eqs. 21 and 20. ∎

For $z\in\mathbb{C}\backslash\sigma(H_{0})$ we want to use the coarea formula

(31) $\int_{\mathbb{T}^{2}}\mathrm{d}\xi\
\frac{T_{z,m}(\xi)}{(r_{m}(\xi)-z^{2})}=\int_{m^{2}}^{M_{m}^{2}}\frac{\mathrm{d}\rho}{\rho-z^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}(\xi)|}\ ,$

where $\mathrm{d}\gamma=\mathrm{d}\gamma_{\rho}$ is the one dimensional
Hausdorff measure over the level curve $r_{m}^{-1}(\rho)$, and
$M_{m}:=\sqrt{m^{2}+8}$. To this end it is enough to show that

$\frac{\left\lVert T_{z,m}(\cdot)\right\rVert_{1}}{(r_{m}(\cdot)-z^{2})|\nabla
r_{m}(\cdot)|}\text{ is in }L^{1}(\mathbb{T}^{2})\ ,$

which easily follows from the fact that
$z\in\mathbb{C}\backslash\sigma(H_{0})$, the boundedness of $\left\lVert
T_{z,m}(\cdot)\right\rVert_{1}$ and

(32) $|\nabla
r_{m}(\xi_{1},\xi_{2})|^{2}=16\pi^{2}(\sin^{2}(2\pi\xi_{1})+\sin^{2}(2\pi\xi_{2}))\
.$

In Fig. 1 we show the level curves of $r_{m}$. One can readily see a
periodicity that will simplify some of our computations below.

Figure 1. This figure shows the level curves for $r_{m}$. The level curve
corresponding to the hyperbolic threshold is shown in red. Further, critical
points are marked with a cross.

The main result of this section is the following proposition which together
with Eq. 22 gives an explicit description of $K(\lambda+i0)$.

######  Proposition 3.3.

Let $\lambda\in\mathbb{R}\backslash\mathcal{T}$ and set $z=\lambda+i\delta$.
Then

(33)
$\begin{split}\int_{m^{2}}^{M^{2}_{m}}\frac{\mathrm{d}\rho}{\rho-z^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}(\xi)|}\longrightarrow\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}&\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}({\xi})|}\\\
-i\pi&\int_{r_{m}^{-1}(\lambda^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}({\xi})|}\ ,\end{split}$

as $\delta\downarrow 0$ in the $\mathfrak{S}_{1}$-norm.

We present some preparatory lemmata before given the proof. After making the
change $u^{2}=\rho$ in Eq. 31, let us set

(34)
$B_{m}(u,z):=\frac{2u}{u+z}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}({\xi})|}\ .$

Note that this operator-valued function admits a limit when $\delta\downarrow
0$ and so we can write $B(u,\lambda)$. Moreover, $B(\cdot,\lambda)$ satisfies
the following lemma:

######  Lemma 3.4.

Outside $\mathcal{T}$, the function $u\mapsto B_{m}(u,\lambda)$ is locally
Hölder with exponent $\beta$ (defined in Eq. 23).

###### Proof.

Let us consider an interval $I\Subset(m,\sqrt{m^{2}+4})$. For $u\in I$, and
taking the function $\arcsin:[-1,1]\to[-\frac{\pi}{2},\frac{\pi}{2}]$ we
define

$f_{u}(\xi_{1})=\frac{1}{\pi}\arcsin\left(\sqrt{\frac{u^{2}-m^{2}}{4}-\sin^{2}(\pi\xi_{1})}\right)\
,$

for
$|\xi_{1}|\leq\frac{1}{\pi}\arcsin\left(\sqrt{\frac{u^{2}-m^{2}}{4}}\right)=:\xi_{1}(u)$.
Since $(\xi,\pm f_{u}(\xi_{1}))$ parametrize $r_{m}^{-1}(u^{2})$ we obtain:

$\displaystyle\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}=\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\mathrm{d}\xi_{1}$
$\displaystyle\left(\frac{|\nabla
r(\xi_{1},f_{u}(\xi_{1}))|}{|\partial_{2}r(\xi_{1},f_{u}(\xi_{1}))|}\times\frac{T_{\lambda,m}(\xi_{1},f_{u}(\xi_{1})}{|\nabla
r(\xi_{1},f_{u}(\xi_{1}))|}\right.$ $\displaystyle\left.-\frac{|\nabla
r(\xi_{1},-f_{u}(\xi_{1}))|}{|\partial_{2}r(\xi_{1},-f_{u}(\xi_{1}))|}\times\frac{T_{\lambda,m}(\xi_{1},-f_{u}(\xi_{1})}{|\nabla
r(\xi_{1},-f_{u}(\xi_{1}))|}\right)$
$\displaystyle=\frac{1}{8\pi}\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\mathrm{d}\xi_{1}$
$\displaystyle\left(\frac{T_{\lambda,m}(\xi_{1},f_{u}(\xi_{1})}{\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}+\frac{T_{\lambda,m}(\xi_{1},-f_{u}(\xi_{1})}{\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}\right)\
,$

where

$F_{u}(\xi_{1})=\frac{u^{2}-m^{2}}{4}-\sin^{2}(\pi\xi_{1})\ .$

Now, let us consider $v<u$ in $I$ and bound $\|B(u,\lambda)-B(v,\lambda)\|$:

$\displaystyle\left\lVert\frac{2u}{u+\lambda}\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\mathrm{d}\xi_{1}\frac{T_{\lambda,m}(\xi_{1},f_{u}(\xi_{1}))}{\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}-\frac{2v}{v+\lambda}\int_{-\xi_{1}(v)}^{\xi_{1}(v)}\mathrm{d}\xi_{1}\frac{T_{\lambda,m}(\xi_{1},f_{v}(\xi_{1}))}{\sqrt{F_{v}(\xi_{1})(1-F_{v}(\xi_{1}))}}\right\rVert_{1}$
$\displaystyle\leq\frac{2u}{u+\lambda}\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\mathrm{d}\xi_{1}\frac{\left\lVert
T_{\lambda,m}(\xi_{1},f_{u}(\xi_{1}))-T_{\lambda,m}(\xi_{1},f_{v}(\xi_{1}))\right\rVert_{1}}{\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}$
$\displaystyle+\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\mathrm{d}\xi_{1}\left|\frac{2u}{(u+\lambda)\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}-\frac{2v}{(\lambda+v)\sqrt{F_{v}(\xi_{1})(1-F_{v}(\xi_{1}))}}\right|\lVert
T_{\lambda,m}(\xi_{1},f_{v}(\xi_{1}))\rVert_{1}$
$\displaystyle+\int_{\left[-\xi_{1}(u),\xi_{1}(u)\right]\backslash\left[-\xi_{1}(v),\xi_{1}(v)\right]}\mathrm{d}\xi_{1}\left\lVert\frac{2vT_{\lambda,m}(\xi_{1},f_{v}(\xi_{1}))}{(v+\lambda)\sqrt{F_{v}(\xi_{1})(1-F_{v}(\xi_{1}))}}\right\rVert_{1}\
.$

First we notice that

$u\to\int_{-\xi_{1}(u)}^{\xi_{1}(u)}\frac{\mathrm{d}\xi_{1}}{\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}$

is bounded on $I$ since

(35) $0<\min_{u\in
I}\\{\min_{|\xi_{1}|\leq\xi_{1}(u)}\\{F_{u}(\xi_{1})\\}\\}\text{ and
}\max_{u\in I}\\{\max_{|\xi_{1}|\leq\xi_{1}(u)}\\{F_{u}(\xi_{1})\\}\\}<1\ .$

Then, to treat the first term, if we take $I$ suitably small, one can use
Lemma 3.2 to obtain

$\left\lVert
T_{\lambda,m}(\xi_{1},f_{u}(\xi_{1}))-T_{\lambda,m}(\xi_{1},f_{v}(\xi_{1}))\right\rVert_{1}\leq
C(f_{u}(\xi_{1})-f_{v}(\xi_{1}))^{\beta}\ .$

One concludes by noticing that $f_{u}$ is Lipschitz in $u$, for $u\in I$, with
a uniform constant in $\xi_{1}$. For the second term, from Eq. 35, we have
that $\frac{2u}{(u+\lambda)\sqrt{F_{u}(\xi_{1})(1-F_{u}(\xi_{1}))}}\in
C^{1}(I)$. Using the same argument one can check that the norm in the third
term is uniformly bounded and then conclude by noticing that $u\to\xi_{1}(u)$
has a bounded derivative on $I$. ∎

As a direct consequence, by applying Sokhotski-Plemelj formula we obtain the
following.

######  Corollary 3.5.

Let $\lambda$ and $z=\lambda+i\delta$ be as before. Then

$\left\lVert\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,\lambda)}{u-z}-\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,\lambda)}{u-\lambda}-i\pi
B_{m}(\lambda,\lambda)\right\rVert_{1}=O(\delta)\text{ as }\delta\downarrow 0\
.$

The last lemma we need is the following.

######  Lemma 3.6.

Let $\lambda$ and $z=\lambda+i\delta$ be as before. Then

$\int_{m}^{M_{m}}\mathrm{d}u\left\lVert\frac{B_{m}(u,z)-B_{m}(u,\lambda)}{u-z}\right\rVert_{1}\rightarrow
0\text{ as }\delta\downarrow 0\ .$

###### Proof.

We start by noticing that

$\displaystyle\left\lVert
B_{m}(u,z)-B_{m}(u,\lambda)\right\rVert_{1}=\left\lVert\frac{2u}{u+z}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}(\xi)|}-\frac{2u}{u+\lambda}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right\rVert_{1}$ $\displaystyle\leq
2u\left(\left\lVert\left(\frac{1}{u+z}-\frac{1}{u+\lambda}\right)\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right\rVert_{1}+\left\lVert\frac{1}{u+\lambda}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)-T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right\rVert_{1}\right)$ (36)
$\displaystyle\leq\frac{2u\delta}{|u+z||u+\lambda|}\left\lVert\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right\rVert_{1}+\frac{2u}{|u+\lambda|}\left\lVert\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{z,m}(\xi)-T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right\rVert_{1}.$

We now treat each term of Eq. 36 separately. For the first term one only needs
Lemma 3.4 and check that

$\left(\sup_{u\in\left[m,M_{m}\right]}\frac{2u}{|u+z||u+\lambda|}\right)\left(\sup_{\xi\in\mathbb{T}^{2}}\left\lVert
T_{z,m}(\xi)\right\rVert_{1}\right)<\infty\ ,$

uniformly on $\delta$. For the second term, Lemma 3.2 gives us that

$\left\lVert B_{m}(u,z)-B_{m}(u,\lambda)\right\rVert_{1}\leq
C\delta\int_{r_{m}^{-1}(u^{2})}\frac{\mathrm{d}\gamma}{|\nabla
r_{m}(\xi)|}=:C\delta R(u)\ .$

Note that $R:\left[m;M_{m}\right]\setminus\mathcal{T}\to\mathbb{R}$ remains
bounded near $\\{m,M_{m}\\}$ (see Eq. 40 below). Furthermore, for
$\epsilon>0$, it satisfies
$R(\sqrt{m^{2}+4}\pm\epsilon)=O(\ln(\epsilon^{-1}))$. Hence $R\in
L^{1}(\left[m;M_{m}\right])$. The result follows by applying the dominated
convergence theorem once one notice that

$\frac{\delta R(u)}{\sqrt{(u-\lambda)^{2}+\delta^{2}}}\to 0\text{ as
}\delta\to 0,$

for all $u\notin\\{\lambda,\sqrt{m^{2}+4}\\}$. ∎

We have gathered all the result needed for proving 3.3.

###### Proof of 3.3.

Using a change of variable one can see that the result follows if

$\left\lVert\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,z)}{u-z}-\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,\lambda)}{u-\lambda}-i\pi
B_{m}(\lambda,\lambda)\right\rVert\to 0\text{ as }\delta\downarrow 0.$

This follows from the fact that it is bounded by

$\left\lVert\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,z)-B_{m}(u,\lambda)}{u-z}\right\rVert+\left\lVert\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,\lambda)}{u-z}-\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{B_{m}(u,\lambda)}{u-\lambda}-i\pi
B(\lambda,\lambda)\right\rVert$

once one takes into account Lemmas 3.6 and 3.5. ∎

As a consequence of 3.3, for $\lambda\in\mathbb{R}\backslash\mathcal{T}$ we
have

(37) $\mathscr{U}[{\rm
Im}\,K(\lambda+i0)]\mathscr{U}^{*}=\frac{-\pi}{m+\lambda}\int_{r_{m}^{-1}(\lambda^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}\ ,$

and

(38) $\begin{split}\mathscr{U}[{\rm
Re}\,K(\lambda+i0)]\mathscr{U}^{*}=&\frac{\mathscr{U}V\mathscr{U}^{*}}{m+\lambda}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-1&0\\\\[5.0pt] 0&0&-1\end{pmatrix}\\\
&+\frac{1}{m+\lambda}\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}({\xi})|}\ .\end{split}$

##  4\. Preliminary computations: parabolic thresholds and Dirac point case

Having in mind Lemma 3.1, in this section we study some asymptotic properties
of the operators given by Eqs. 37 and 38, for $\lambda$ close to $\pm m$. When
$m\neq 0$ and $|\lambda|<m$, the imaginary part obtained in Eq. 37 vanishes
and the p.v. part of Eq. 38 is just the integral over the interval $[m,M_{m}]$
. Hence, we start by considering in detail the case that $\lambda\to\pm m$
with $|\lambda|>m$. Several notations will depend on a $\pm$ subscript. We do
this to shorten notations and study both limits, $\pm m$, at the same time.
Notice that the case $m=0$ needs also to be considered. In that case, the
subscript $\pm$ will indicate if we are approaching $0$ by the positive or
negative side.

######  Lemma 4.1.

The following relation holds true

$\left|\left|\int_{r^{-1}_{m}(\lambda^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r_{m}(\xi)|}\right|\right|_{1}=O(\lambda(m+\lambda)),\quad\lambda\to\pm
m,|\lambda|>m.$

###### Proof.

First notice that from Eqs. 24 and 25, for all $\lambda$ and $\xi$

(39) $||T_{\lambda,m}(\xi)||_{1}\leq C\left((m+\lambda)^{2}+r(\xi)\right).$

Now, recalling that

$r(\xi)=2(1-\cos(2\pi\xi_{1}))+2(1-\cos(2\pi\xi_{2}))=4\sin^{2}(\pi\xi_{1})+4\sin^{2}(\pi\xi_{2})$

and comparing it with Eq. 32 one can easily deduce that for
$\xi\in[-1/4,1/4]\times[-1/4,1/4]$

(40) $\frac{1}{16\pi}|\nabla r(\xi)|^{2}\leq{r(\xi)}\leq|\nabla r(\xi)|^{2}\
.$

Furthermore, using

(41) $\nabla r_{m}(\xi)=\nabla r(\xi)\text{ and }\quad
r_{m}^{-1}(u^{2})=r^{-1}(u^{2}-m^{2})\ ,$

we see that for $\lambda$ near $\pm m$, the variable $\xi\in
r_{m}^{-1}(\lambda)=r^{-1}(\lambda-m^{2})$ is small in modulus. Thus from Eqs.
39 and 40

$\displaystyle\left|\left|\int_{r^{-1}_{m}(\lambda^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}\leq
C\int_{r^{-1}(\lambda^{2}-m^{2})}\mathrm{d}\gamma\,\frac{(m+\lambda)^{2}+r(\xi)}{r(\xi)^{1/2}}\leq$
$\displaystyle C\left((m+\lambda)^{2}+(\lambda^{2}-m^{2})\right)$
$\displaystyle=$ $\displaystyle C\lambda(m+\lambda)\ .\qed$

From Eq. 38, we define

(42)
${\mathcal{S}}_{0}(\lambda):=\text{p.v.}\int_{m}^{M_{m}}\mathrm{d}u\frac{2u}{u^{2}-\lambda^{2}}\int_{r_{m}^{-1}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}.$

Now, let us set

$B^{\pm}_{\lambda,m}(u):=\frac{2u}{(u\pm\lambda)}\int_{r^{-1}_{m}(u^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}\ ,$

together with

${\mathcal{S}}_{1}(\lambda):=\begin{cases}\displaystyle{\int_{\left[m,M_{m}\right]}\mathrm{d}u\frac{B^{\pm}_{\lambda,m}(u)}{u\mp\lambda}};&0\leq\pm\lambda<m\\\
\displaystyle{\int_{\left[m,\frac{m\pm\lambda}{2}\right]\cup\left[\frac{\pm
3\lambda-m}{2},M_{m}\right]}\mathrm{d}u\frac{B^{\pm}_{\lambda,m}(u)}{u\mp\lambda}};&\pm\lambda>m\end{cases}\
.$

######  Lemma 4.2.

$||{\mathcal{S}}_{0}(\lambda)-{\mathcal{S}}_{1}(\lambda)||_{1}=O(\lambda(m+\lambda)),\quad\lambda\to\pm
m.$

###### Proof.

Let us first notice that

(43)
${\mathcal{S}}_{0}(\lambda)-{\mathcal{S}}_{1}(\lambda)=\begin{cases}0;&|\lambda|<m\\\
\displaystyle{\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(B^{\pm}_{\lambda,m}(\rho\pm\lambda)-B^{\pm}_{\lambda,m}(-\rho\pm\lambda)\right)};&\pm\lambda>m\end{cases}\
.$

Thus, we only need to consider the cases where $|\lambda|>m$. To prove this
lemma we will write ${\mathcal{S}}_{0}-{\mathcal{S}}_{1}$ as the sum of five
terms, $R_{j}$, $j=0,1,2,3,4$, the first of which is

$R_{0}(\lambda):=\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{2\rho\pm
2\lambda}{\rho\pm 2\lambda}-\frac{-2\rho\pm 2\lambda}{-\rho\pm
2\lambda}\right)\int_{r^{-1}_{m}((\pm\lambda+\rho)^{2})}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{\nabla
r(\xi)}\ .$

Then, using Eqs. 39 and 40 we can see that

$\displaystyle||R_{0}(\lambda)||_{1}\leq$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\frac{\pm
4\lambda\rho}{(\rho\pm 2\lambda)(-\rho\pm
2\lambda)}\int_{r^{-1}((\pm\lambda+\rho)^{2}-m^{2})}\mathrm{d}\gamma\frac{||T_{\lambda,m}(\xi)||_{1}}{|\nabla
r(\xi)|}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\pm
3\lambda+m}\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\int_{r^{-1}((\pm\lambda+\rho)^{2}-m^{2})}\mathrm{d}\gamma\left(\frac{(m+\lambda)^{2}}{r(\xi)^{\frac{1}{2}}}+r(\xi)^{\frac{1}{2}}\right)$
$\displaystyle\leq$ $\displaystyle\frac{C}{\pm
3\lambda+m}\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\left((m+\lambda)^{2}+(\pm\lambda+\rho)^{2}-m^{2}\right)\
.$

From the last expression we can deduce that

(44) $||R_{0}(\lambda)||_{1}=\begin{cases}O((\lambda+m)^{2})&\text{ if }m\neq
0,\lambda\uparrow-m\\\ O((\lambda-m))&\text{ if }m\neq 0,\lambda\downarrow
m\\\ O(\lambda^{2})&\text{ if }m=0,\lambda\to 0\end{cases}\ .$

In order to write the decomposition of
${\mathcal{S}}_{0}(\lambda)-{\mathcal{S}}_{1}(\lambda)-R_{0}(\lambda)$ set

$\theta:=\frac{(\pm\lambda+\rho)^{2}-(\pm\lambda-\rho)^{2}}{4}=\pm\lambda\rho>0\
,$

and

$D_{\theta}(\xi_{1},\xi_{2})=\begin{cases}\frac{1}{2\pi}\Big{(}\arccos(-\theta+\cos(2\pi\xi_{1})),\arccos(-\theta+\cos(2\pi\xi_{2}))\Big{)}&\text{
if }\xi_{1}\geq 0\text{ and }\xi_{2}\geq 0\ ,\\\
\frac{1}{2\pi}\Big{(}-\arccos(-\theta+\cos(2\pi\xi_{1})),\arccos(-\theta+\cos(2\pi\xi_{2}))\Big{)}&\text{
if }\xi_{1}<0\text{ and }\xi_{2}\geq 0\ ,\\\
\frac{1}{2\pi}\Big{(}-\arccos(-\theta+\cos(2\pi\xi_{1})),-\arccos(-\theta+\cos(2\pi\xi_{2}))\Big{)}&\text{
if }\xi_{1}\leq 0\text{ and }\xi_{2}<0\ ,\\\
\frac{1}{2\pi}\Big{(}\arccos(-\theta+\cos(2\pi\xi_{1})),-\arccos(-\theta+\cos(2\pi\xi_{2}))\Big{)}&\text{
if }\xi_{1}\geq 0\text{ and }\xi_{2}<0\ .\end{cases}$

One can readily check that for $\rho\neq\lambda$, $D_{\theta}$ is injective on
$r^{-1}_{m}((\lambda-\rho)^{2})$ and satisfies
$D_{\theta}(r^{-1}_{m}((\lambda-\rho)^{2}))\subset
r^{-1}_{m}((\lambda+\rho)^{2}).$ Let us denote by $D_{\theta}^{*}d\gamma$ the
pullback measure induced by $D_{\theta}$ on $r_{m}^{-1}((\lambda-\rho)^{2})$
and set $\gamma_{R}:=r^{-1}_{m}((\lambda+\rho)^{2})\setminus
D_{\theta}(r^{-1}_{m}((\lambda-\rho)^{2}))$.

In consequence, we can write
$\mathcal{S}_{0}-\mathcal{S}_{1}-R_{0}=\sum_{i=1}^{4}R_{j}$ where

$\displaystyle R_{1}(\lambda):=$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{-2\rho\pm
2\lambda}{-\rho\pm
2\lambda}\right)\left(\int_{r^{-1}_{m}((\pm\lambda-\rho)^{2})}D_{\theta}^{*}\mathrm{d}\gamma\,\frac{(T_{\lambda,m}(D_{\theta}(\xi))-T_{\lambda,m}(\xi))}{|\nabla
r(D_{\theta}\xi)|}\right)$ $\displaystyle R_{2}(\lambda):=$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{-2\rho\pm
2\lambda}{-\rho\pm
2\lambda}\right)\left(\int_{\gamma_{R}}\mathrm{d}\gamma\,\frac{T_{\lambda,m}(\xi))}{|\nabla
r(\xi)|}\right)$ $\displaystyle R_{3}(\lambda):=$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{-2\rho\pm
2\lambda}{-\rho\pm
2\lambda}\right)\left(\int_{r^{-1}_{m}((\pm\lambda-\rho)^{2})}(D_{\theta}^{*}\mathrm{d}\gamma-\mathrm{d}\gamma)\frac{(T_{\lambda,m}(\xi))}{|\nabla
r(D_{\theta}\xi)|}\right)$ $\displaystyle R_{4}(\lambda):=$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{-2\rho\pm
2\lambda}{-\rho\pm
2\lambda}\right)\left(\int_{r^{-1}_{m}((\pm\lambda-\rho)^{2})}\mathrm{d}\gamma\,T_{\lambda,m}(\xi)\left(\frac{1}{|\nabla
r(D_{\theta}\xi)|}-\frac{1}{|\nabla r(\xi)|}\right)\right)\ .$

To estimate $R_{1}$ compute

(45)
$\begin{array}[]{ll}|D_{\theta}(\xi)-\xi|^{2}&=\displaystyle{\left|\int_{-\theta+\cos(2\pi\xi_{1})}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}}-\xi_{1}\right|^{2}+\left|\int_{-\theta+\cos(2\pi\xi_{2})}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{2}}}-\xi_{1}\right|^{2}}\\\\[15.00002pt]
&\leq\displaystyle{\left|\int_{\cos(2\pi\xi_{1})-\theta}^{\cos(2\pi\xi_{1})}\frac{\mathrm{d}t}{(1-t)^{1/2}(1+t)^{1/2}}\right|^{2}+\left|\int_{\cos(2\pi\xi_{2})-\theta}^{\cos(2\pi\xi_{2})}\frac{\mathrm{d}t}{(1-t)^{1/2}(1+t)^{1/2}}\right|^{2}}\\\\[15.00002pt]
&\leq C\theta,\end{array}$

since $\xi_{1}$, $\xi_{2}$ are small for $\lambda$ near $\pm m$.
Particularizing Eqs. 26, 24 and 29 we can also obtain

$\displaystyle||T_{\lambda,m}(D_{\theta}(\xi))-T_{\lambda,m}(\xi)||_{1}$
$\displaystyle\leq
C\left(2|m+\lambda|^{2}+r(D_{\theta}(\xi))+r(\xi)\right)^{\frac{1}{2}}$
$\displaystyle\left(|m+\lambda|\,|D_{\theta}(\xi)-\xi|^{\beta_{1}}+r(D_{\theta}(\xi))^{\frac{1}{2}}|D_{\theta}(\xi)-\xi|^{\widetilde{\beta}}+|D_{\theta}(\xi)-\xi|\right)$

and hence

$||R_{1}(\lambda)||_{1}\leq
C\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\frac{(4\lambda(\lambda+m)+2\rho^{2})^{\frac{1}{2}}}{\rho}\int_{r^{-1}((\pm\lambda+\rho)^{2}-m^{2})}\mathrm{d}\gamma\frac{|m+\lambda|\theta^{\frac{\beta_{1}}{2}}+r(\xi)^{\frac{1}{2}}\theta^{\frac{{\widetilde{\beta}}}{2}}+\theta^{\frac{1}{2}}}{|\nabla
r(\xi)|}\ .\\\ $

From the last expression we can estimate each term and deduce

(46) $R_{1}(\lambda)=\begin{cases}O(|\lambda+m|)&\text{ if }m\neq
0,\lambda\uparrow-m\\\ O(|\lambda-m|^{\frac{\beta_{1}}{2}})&\text{ if }m\neq
0,\lambda\downarrow m\\\ O(\lambda^{2})&\text{ if }m=0,\lambda\to
0\end{cases}\ .$

To estimate $R_{2}$ we need first an estimate on $\gamma_{R}$. Since
$D_{\theta}$ does not change the quadrant of $\xi\in
r^{-1}_{m}((\lambda-\rho)^{2})$, $\gamma_{R}$ is composed of 8 parts. More
precisely, if $(\xi_{1}^{-},0)\in r^{-1}_{m}((\lambda-\rho)^{2})$ and
$(\xi_{1}^{+},0)\in r^{-1}_{m}((\lambda+\rho)^{2})$, one of these parts is the
section of $r^{-1}_{m}((\lambda+\rho)^{2})$ going from
$D_{\theta}((\xi_{1}^{-},0))$ to $(\xi_{1}^{+},0)$. Hence, the convexity of
the level curves gives us

$\displaystyle\int_{\gamma_{R}}\mathrm{d}\gamma\leq$ $\displaystyle
C\left(|\arccos(1-\theta)|+\left|\arccos\left(1+\tfrac{m^{2}-(\pm\lambda+\rho)^{2}}{2}\right)-\arccos\left(1+\tfrac{m^{2}-\lambda^{2}-\rho^{2}}{2}\right)\right|\right)$
$\displaystyle\leq$ $\displaystyle C\theta^{\frac{1}{2}}\ .$

Using this estimate we compute

$\displaystyle\|R_{2}(\lambda)\|_{1}\leq$
$\displaystyle\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\left(\frac{-2\rho\pm
2\lambda}{-\rho\pm
2\lambda}\right)\left(\int_{\gamma_{R}}\mathrm{d}\gamma\,\frac{\|T_{\lambda,m}(\xi))\|_{1}}{|\nabla
r(\xi)|}\right)$ $\displaystyle\leq$ $\displaystyle
C|\lambda|^{\frac{1}{2}}\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho^{\frac{1}{2}}}\frac{2m\lambda+2\lambda^{2}+\rho^{2}\pm
2\lambda\rho}{(\lambda^{2}-m^{2}\pm 2\lambda\rho+\rho^{2})^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C|\lambda|^{\frac{1}{2}}\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\left(\frac{\rho^{2}\pm
2\lambda\rho+\lambda^{2}-m^{2}}{\rho}\right)^{\frac{1}{2}}\frac{2m\lambda+2\lambda^{2}+\rho^{2}\pm
2\lambda\rho}{(\lambda^{2}-m^{2}\pm 2\lambda\rho+\rho^{2})}$
$\displaystyle\leq$ $\displaystyle
C|\lambda|^{\frac{1}{2}}\left(1+\frac{\lambda+m}{\lambda-m}\right)\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho(\rho\pm
2\lambda)^{\frac{1}{2}}+\left(\frac{\lambda^{2}-m^{2}}{\rho}\right)^{\frac{1}{2}}\
.$

From this one gets

(47) $||R_{2}(\lambda)||_{1}=\begin{cases}O(|\lambda+m|)&\text{ if }m\neq
0,\lambda\uparrow-m\\\ O(1)&\text{ if }m\neq 0,\lambda\downarrow m\\\
O(\lambda^{2})&\text{ if }m=0,\lambda\to 0\end{cases}\ .$

Next, by Eqs. 25 and 24 we get

$\displaystyle||R_{3}(\lambda)||_{1}$ $\displaystyle\leq
C\int_{0}^{\frac{\pm\lambda-m}{2}}\frac{\mathrm{d}\rho}{\rho}\int_{r^{-1}((\pm\lambda-\rho)^{2}-m^{2})}|D_{\theta}^{*}\mathrm{d}\gamma-\mathrm{d}\gamma|\frac{(m+\lambda)^{2}+r(\xi)}{r(D_{\theta}(\xi))^{1/2}}$
(48) $\displaystyle\leq
C\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\frac{2\lambda^{2}+2m\lambda\mp
2\lambda\rho+\rho^{2}}{\rho((\pm\lambda+\rho)^{2}-m^{2})^{\frac{1}{2}}}\int_{r^{-1}((\pm\lambda-\rho)^{2}-m^{2})}|D_{\theta}^{*}\mathrm{d}\gamma-\mathrm{d}\gamma|\
.$

Let $(\xi_{1}(t),\xi_{2}(t))$ be a parametrization of the curve
$r^{-1}((\lambda-\rho)^{2}-m^{2})$. It is easy to see that

$\displaystyle|D_{\theta}^{*}\mathrm{d}\gamma$
$\displaystyle-\mathrm{d}\gamma|=\left|\left((\partial_{1}D_{\theta,1})(\xi(t))^{2}\xi_{1}^{\prime}(t)^{2}+(\partial_{2}D_{\theta,2})(\xi_{2}(t))^{2}\xi_{2}^{\prime}(t)^{2}\right)^{1/2}-(\xi_{1}^{\prime}(t)^{2}+\xi_{2}^{\prime}(t)^{2})^{1/2}\right|\mathrm{d}t\
,$

where $D_{\theta}(\xi)=(D_{\theta,1}(\xi),D_{\theta,2}(\xi))$. Notice that

$(\partial_{1}D_{\theta,1})(\xi)=\frac{\sin(2\pi\xi_{1})}{(1-(\cos(2\pi\xi_{1})-\theta)^{2})^{1/2}};\quad(\partial_{2}D_{\theta,2})(\xi)=\frac{\sin(2\pi\xi_{2})}{(1-(\cos(2\pi\xi_{2})-\theta)^{2})^{1/2}}.$

Then, setting

$Q_{\theta}(\xi):=1-\frac{\sin^{2}(2\pi\xi_{1})}{|1-(\cos(2\pi\xi_{1})-\theta)^{2}|}+1-\frac{\sin^{2}(2\pi\xi_{2})}{|1-(\cos(2\pi\xi_{2})-\theta)^{2}|}\
,$

it is easy to see that

(49) $|D_{\theta}^{*}\mathrm{d}\gamma-\mathrm{d}\gamma|\leq
Q_{\theta}(\xi)\mathrm{d}\gamma\ .$

Since $\theta$ is positive and $\xi$ is in a small neighborhood of $(0,0)$, we
can see that there exist positive constants $C_{1}$ and $C_{2}$ such that

(50) $Q_{\theta}(\xi)\leq
C\theta\left(\frac{1}{\sin^{2}(2\pi\xi_{1})+C_{1}\theta}+\frac{1}{\sin^{2}(2\pi\xi_{2})+C_{2}\theta}\right).$

Now, write $u^{2}=(\pm\lambda-\rho)^{2}$ and parametrize the part of
$r_{m}^{-1}(u^{2})$ in $[0,1/2]\times[0,1/2]$ by

$\xi_{1}(t)=\frac{1}{2\pi}\arccos\left(1-\frac{t}{2}\right);\quad\xi_{2}(t)=\frac{1}{2\pi}\arccos\left(\frac{t-u^{2}}{2}+1\right),$

for $t\in(0,u^{2})$. Notice that

$\xi^{\prime}_{1}(t)\asymp\frac{1}{\sqrt{t}},\quad\xi^{\prime}_{2}(t)\asymp\frac{1}{\sqrt{u^{2}-t}};\,\,\text{for}\,\,t\in(0,u^{2}),$

therefore

(51)
$\begin{split}&d\gamma\asymp\frac{\mathrm{d}t}{\sqrt{t}},\,\,\text{for}\,\,t\in(0,u^{2}/2),\\\
&\mathrm{d}\gamma\asymp\frac{\mathrm{d}t}{\sqrt{u^{2}-t}},\,\,\text{for}\,\,t\in(u^{2}/2,u^{2}).\end{split}$

Further,

(52)
$\begin{split}\sin^{2}(2\pi\xi_{1}(t))&=1-\left(1-\frac{t}{2}\right)^{2}\asymp
t,\,\,\text{for}\,\,t\,\,\text{near}\,\,0\\\
\sin^{2}(2\pi\xi_{2}(t))&=1-\left(\frac{t-u^{2}}{2}+1\right)^{2}\asymp
u^{2}-t,\,\,\text{for}\,\,t\,\,\text{near}\,\,u^{2}.\end{split}$

We can see from Eqs. 49, 50, 51 and 52 that

$\displaystyle\int_{r_{m}^{-1}((\pm\lambda-\rho)^{2})}|\mathrm{d}\gamma-D^{*}_{\theta}\mathrm{d}\gamma|\leq$
$\displaystyle\int_{r_{m}^{-1}((\pm\lambda-\rho)^{2})}Q_{\theta}(\xi)\mathrm{d}\gamma$
$\displaystyle\leq$ $\displaystyle
C\theta\left(\int_{0}^{u^{2}/2}\frac{\mathrm{d}t}{(t+A\theta)\sqrt{t}}+\int_{u^{2}/2}^{u^{2}}\frac{\mathrm{d}t}{(u^{2}-t+B\theta)(u^{2}-t)^{1/2}}\right)$
$\displaystyle\leq$ $\displaystyle C\theta^{\frac{1}{2}}\ .$

Here, by an abuse of notation, we are still calling
$r_{m}^{-1}((\pm\lambda-\rho)^{2})$ the part of this curve that is in
$[0,1/2]\times[0,1/2]$. The same estimates can be obtained for the parts of
this curve in the other quadrants.

Using this last estimate on Eq. 48 we obtain

$\displaystyle||R_{3}(\lambda)||_{1}\leq$ $\displaystyle
C(\pm\lambda)^{\frac{1}{2}}\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\frac{2\lambda^{2}+2m\lambda\mp
2\lambda\rho+\rho^{2}}{\rho^{\frac{1}{2}}((\pm\lambda+\rho)^{2}-m^{2})^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C(\pm\lambda)^{\frac{1}{2}}\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\frac{2\lambda(\lambda+m)+\tfrac{1}{4}(\pm\lambda-m)^{2}}{\rho^{\frac{1}{2}}(\lambda^{2}-m^{2})^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C(\pm\lambda)^{\frac{1}{2}}\left(\lambda(\lambda+m)+(\pm\lambda-m)^{2}\right)\
.$

Thus

(53) $||R_{3}(\lambda)||_{1}=\begin{cases}O(|\lambda+m|)&\text{ if }m\neq
0,\lambda\uparrow-m\\\ O(1)&\text{ if }m\neq 0,\lambda\downarrow m\\\
O(\lambda^{2})&\text{ if }m=0,\lambda\to 0\end{cases}\ .$

For $R_{4}$ we start by noticing

(54) $|\nabla r(D_{\theta}(\xi))|^{2}=|\nabla
r(\xi)|^{2}-16\pi^{2}\theta(-2\cos(2\pi\xi_{1})-2\cos(2\pi\xi_{2})+\theta^{2})\
,$

so we can deduce

$\left|\frac{1}{|\nabla r(D_{\theta}(\xi))|}-\frac{1}{|\nabla
r(\xi)|}\right|\leq C\frac{\theta}{|\nabla r(\xi)|^{2}|\nabla
r(D_{\theta}(\xi))|}.$

Also, from Lemmas 3.4, 39 and 40 we obtain,

$\displaystyle||R_{4}(\lambda)||_{1}\leq$ $\displaystyle
C\int_{0}^{\frac{\pm\lambda-m}{2}}\mathrm{d}\rho\frac{(\pm\lambda)\left((\lambda+m)^{2}+(\pm\lambda-\rho)^{2}-m^{2}\right)\left((\pm\lambda-\rho)^{2}-m^{2})\right)^{\frac{1}{2}}}{\left((\pm\lambda-\rho)^{2}-m^{2})\right)\left((\pm\lambda+\rho)^{2}-m^{2})\right)^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C\frac{(\pm\lambda)\left((\lambda+m)^{2}+\lambda^{2}-m^{2}\right)(\pm\lambda-m)}{(\pm\lambda-m)^{\frac{1}{2}}(\pm\lambda+3m)^{\frac{1}{2}}(\pm\lambda-m)^{\frac{1}{2}}(\pm\lambda+m)^{\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
C\frac{\pm\lambda^{2}(\lambda+m)}{(\pm\lambda+3m)^{\frac{1}{2}}(\pm\lambda+m)^{\frac{1}{2}}}\
.$

Finally we get

$||R_{4}(\lambda)||_{1}=\begin{cases}O(|\lambda+m|)&\text{ if }m\neq
0,\lambda\uparrow-m\\\ O(1)&\text{ if }m\neq 0,\lambda\downarrow m\\\
O(\lambda^{2})&\text{ if }m=0,\lambda\to 0\end{cases}\ ,$

which together with Eqs. 44, 46, 53 and 47 finishes the proof. ∎

For the next step notice that

$\mathcal{S}_{1}(\lambda)=\begin{cases}\displaystyle{\int_{\left[m^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}};&0\leq\pm\lambda<m\\\
\displaystyle{\int_{\left[m^{2},\left(\frac{m\pm\lambda}{2}\right)^{2}\right]\cup\left[\left(\frac{\pm
3\lambda-m}{2}\right)^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{T_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}};&\pm\lambda>m\end{cases}\ .$

Define
$\mathring{T}_{\lambda,m}(\xi)=T_{\lambda,m}(\xi)-T_{\lambda,m}(\xi_{0})$.
Taking into account Eqs. 24, 30 and 26 we obtain for $\xi$ small enough

(55) $||\mathring{T}_{\lambda,m}(\xi)||_{1}\leq
C(|m+\lambda||\xi|^{\beta}+r(\xi)^{\frac{1}{2}})(|\lambda+m|+r(\xi)^{1/2})\ .$

Set

$\mathcal{S}_{2}(\lambda):=\begin{cases}\displaystyle{\int_{\left[m^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}};&0\leq\pm\lambda<m\\\
\displaystyle{\int_{\left[m^{2},\left(\frac{m\pm\lambda}{2}\right)^{2}\right]\cup\left[\left(\frac{\pm
3\lambda-m}{2}\right)^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}};&\pm\lambda>m\end{cases}\ .$

Notice that, since $T_{\lambda,m}(\xi_{0})$ is of finite rank, for all $r>0$
we have

(56)
$n_{\pm}(r;{\mathcal{S}}_{1}(\lambda))=n_{\pm}(r;{{\mathcal{S}}_{2}(\lambda)})+O(1).$

Further, set

$\mathcal{S}_{3}(\lambda):=\begin{cases}\displaystyle{\int_{\left[m^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\pm
m,m}(\xi)}{|\nabla r(\xi)|}};&0\leq\pm\lambda<m\\\
\displaystyle{\int_{\left[m^{2},\left(\frac{m\pm\lambda}{2}\right)^{2}\right]\cup\left[\left(\frac{\pm
3\lambda-m}{2}\right)^{2},M^{2}_{m}\right]}\mathrm{d}\rho\frac{1}{\rho-\lambda^{2}}\int_{r_{m}^{-1}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\pm
m,m}(\xi)}{|\nabla r(\xi)|}};&\pm\lambda>m\end{cases}\ .$

######  Lemma 4.3.

For $\lambda$ near $\pm m$ and $\xi$ small enough we have

$||\mathring{T}_{\lambda,m}(\xi)-\mathring{T}_{\pm m,m}(\xi)||_{1}\leq
C(|\lambda\mp m|r(\xi)^{\frac{\beta_{1}}{2}})(|\pm
m+m|+|\lambda+m|+r(\xi)^{\frac{1}{2}})\ .$

###### Proof.

Let us start by defining

$s^{\pm}_{\lambda,m}(\xi):=t_{\lambda,m}(\xi)-t_{\lambda,m}(\xi_{0})-t_{\pm
m,m}(\xi)+t_{\pm m,m}(\xi_{0})\ .$

One can readily see that
$s^{\pm}_{\lambda,m}(\xi):L^{2}(\mathbb{T}^{2};\mathbb{C}^{3})\to\mathbb{C}^{3}$
is given by

$\begin{pmatrix}f_{1}\\\ f_{2}\\\
f_{3}\end{pmatrix}\mapsto\begin{pmatrix}(\lambda\mp
m)\displaystyle{\int_{\mathbb{T}^{2}}\mathrm{d}\eta\
(\hat{g}_{1}(\xi-\eta)-\hat{g}_{1}(-\eta))f_{1}(\eta)}\\\\[6.99997pt]
0\\\\[6.99997pt] 0\end{pmatrix}\ ,$

from which together with Eq. 28 we obtain

(57) $||s^{\pm}_{\lambda,m}(\xi)||_{2}\leq|\lambda\mp m|\,|\xi|^{\beta_{1}}\
.$

Noticing

$\displaystyle\mathring{T}_{\lambda,m}(\xi)-\mathring{T}_{\pm m,m}(\xi)$ (58)
$\displaystyle=$ $\displaystyle(s^{\pm}_{\lambda,m}(\xi)^{*}-t_{\pm
m,m}(\xi_{0})^{*})At_{\lambda,m}(\xi)+t_{\lambda,m}(\xi_{0})^{*}A(t_{\pm
m,m}(\xi)+s^{\pm}_{\lambda,m}(\xi))$ $\displaystyle+t_{\pm
m,m}(\xi)^{*}A(s^{\pm}_{\lambda,m}(\xi)+t_{\lambda,m}(\xi_{0}))+(s^{\pm}_{\lambda,m}(\xi)^{*}-t_{\lambda,m}(\xi)^{*})At_{\pm
m,m}(\xi_{0})$ $\displaystyle=$ $\displaystyle
s^{\pm}_{\lambda,m}(\xi)^{*}At_{\lambda,m}(\xi)+t_{\lambda,m}(\xi_{0})^{*}As^{\pm}_{\lambda,m}(\xi)+t_{\pm
m,m}(\xi)^{*}As^{\pm}_{\lambda,m}(\xi)+s^{\pm}_{\lambda,m}(\xi))At_{\pm
m,m}(\xi_{0})\ ,$

one concludes taking into account Eqs. 57 and 24. ∎

From this lemma one can easily see that

(59) $\\\
||{\mathcal{S}}_{2}(\lambda)-{\mathcal{S}}_{3}(\lambda)||_{1}=O(|\lambda+m|),\quad\lambda\rightarrow\pm
m\ .$

Using Eq. 55 we can deduce that there exists $\rho_{0}$ such that for all
$m^{2}<\rho\leq\rho_{0}$

(60)
$\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\left|\left|\mathring{T}_{\pm
m,m}(\xi)\right|\right|_{1}}{|\nabla r(\xi)|}\leq C\left(|\pm
m+m|^{2}(\rho-m^{2})^{\frac{\beta_{1}}{2}}+|\pm
m+m|(\rho-m^{2})^{\frac{1}{2}}+(\rho-m^{2})\right)\ ,$

thus

${\mathcal{S}}_{3}(\pm
m):=\int_{m^{2}}^{M_{m}^{2}}\frac{\mathrm{d}\rho}{\rho-m^{2}}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\pm
m,m}(\xi)}{|\nabla r(\xi)|}$

is well defined in the trace class.

######  Lemma 4.4.

For $m>0$

(61)
$||{\mathcal{S}}_{3}(\lambda)-{\mathcal{S}}_{3}(-m)||_{1}=O(|\lambda+m|\,|\ln(|\lambda+m|)|)\quad\lambda\to-m\
.$ (62)
$||{\mathcal{S}}_{3}(\lambda)-{\mathcal{S}}_{3}(0)||_{1}=O(|\lambda|),\quad\lambda\rightarrow
0\ .$ (63)
$||{\mathcal{S}}_{3}(\lambda)-{\mathcal{S}}_{3}(m)||_{1}=O(1),\quad\lambda\rightarrow
m\ .$

###### Proof.

We only give the proof of Eq. 61 as the others use similar ideas. Let us fix
$\rho_{0}$ such that Eq. 60 holds. Without loss of generality suppose
$\left(\frac{3\lambda+m}{2}\right)^{2}<\rho_{0}$. We start by comparing

$\left|\left|\int_{\rho_{0}}^{M_{m}^{2}}\mathrm{d}\rho\left(\frac{1}{\rho-\lambda^{2}}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}-\frac{1}{\rho-m^{2}}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{-m,m}(\xi)}{|\nabla
r(\xi)|}\right)\right|\right|_{1}\ .$

By Lemma 4.3 we see that it is bounded by

$\begin{split}&C|\lambda+m|\int_{\rho_{0}}^{M_{m}^{2}}\mathrm{d}\rho\left(\frac{\lambda-m}{(\rho-\lambda^{2})(\rho-m^{2})}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{||\mathring{T}_{\lambda,m}(\xi)||_{1}}{|\nabla
r(\xi)|}+\frac{1}{\rho-m^{2}}\int_{r^{-1}_{m}(\rho)}\frac{\mathrm{d}\gamma}{|\nabla
r(\xi)|}\right)\\\ \leq&C|\lambda+m|\ ,\end{split}$

where the last inequality follows by recalling the proof of Lemma 3.4. Next,
for $\lambda<-m<0$, the remainder term is given by

$\displaystyle\int_{\left[m^{2},\left(\frac{m-\lambda}{2}\right)^{2}\right]\cup\left[\left(\frac{3\lambda+m}{2}\right)^{2},\rho_{0}\right]}\mathrm{d}\rho\left(\frac{1}{\rho-\lambda^{2}}-\frac{1}{\rho-m^{2}}\right)\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}$ (64)
$\displaystyle-\int_{\left(\frac{m-\lambda}{2}\right)^{2}}^{\left(\frac{-3\lambda-m}{2}\right)^{2}}\frac{\mathrm{d}\rho}{\rho-m^{2}}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}$
$\displaystyle+\int_{m^{2}}^{\rho_{0}}\frac{\mathrm{d}\rho}{\rho-m^{2}}\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)-\mathring{T}_{-m,m}(\xi)}{|\nabla
r(\xi)|}\ .$

Notice that for $-m<\lambda<0$ the second term vanishes and the first integral
is over $\left[m^{2},\rho_{0}\right]$. First we see from Eq. 55 that,

$\displaystyle\left|\left|\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda,m}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}$ $\displaystyle\leq
C\int_{r^{-1}_{m}(\rho)}\mathrm{d}\gamma\frac{|\lambda+m|^{2}r(\xi)^{\frac{\beta_{1}}{2}}+|\lambda+m|r(\xi)^{\frac{1}{2}}+r(\xi)}{r(\xi)^{\frac{1}{2}}}$
$\displaystyle\leq
C\left(|\lambda+m|^{2}\,|\rho-m^{2}|^{\frac{\beta_{1}}{2}}+|\lambda+m|\,|\rho-m^{2}|^{\frac{1}{2}}+|\rho-m^{2}|\right)\
.$

To estimate the first term of Eq. 64 for $\lambda\uparrow-m$ , we use this
last inequality and that for
$\lambda\in\left[m^{2},\left(\frac{m-\lambda}{2}\right)^{2}\right]\cup\left[\left(\frac{3\lambda+m}{2}\right)^{2},\rho_{0}\right]$
we have that $|\lambda^{2}-\rho|\geq C|\lambda+m|$. So

$\displaystyle\int_{m^{2}}^{\left(\frac{m-\lambda}{2}\right)^{2}}\mathrm{d}\rho\frac{|\lambda^{2}-m^{2}|}{|\rho-\lambda^{2}|\,|\rho-m^{2}|}\left(|\lambda+m|^{2}\,|\rho-m^{2}|^{\frac{\beta_{1}}{2}}+|\lambda+m|\,|\rho-m^{2}|^{\frac{1}{2}}+|\rho-m^{2}|\right)$
$\displaystyle\leq
C\int_{m^{2}}^{\left(\frac{m-\lambda}{2}\right)^{2}}\mathrm{d}\rho\left(|\lambda+m|^{2}|\rho-m^{2}|^{\frac{\beta_{1}}{2}-1}+|\lambda+m|\,|\rho-m^{2}|^{\frac{1}{2}-1}+1\right)$
$\displaystyle\leq C|\lambda+m|\ ,$

and

$\displaystyle\int_{\left(\frac{m+3\lambda}{2}\right)^{2}}^{\rho_{0}}\frac{|\lambda^{2}-m^{2}|}{|\rho-\lambda^{2}|\,|\rho-m^{2}|}\left(|\lambda+m|^{2}\,|\rho-m^{2}|^{\frac{\beta_{1}}{2}}+|\lambda+m|\,|\rho-m^{2}|^{\frac{1}{2}}+|\rho-m^{2}|\right)$
$\displaystyle\leq
C\left(|\lambda+m|^{2}+|\lambda+m|+\int_{\left(\frac{m+3\lambda}{2}\right)^{2}}^{\rho_{0}}\frac{|\lambda^{2}-m^{2}|}{|\rho-\lambda^{2}|}\right)$
$\displaystyle\leq C(|\lambda+m|\,|\ln(|\lambda+m|)|)\ .$

Finally, the previous computation also gives us for $-m<\lambda<0$ (and hence
$0<\lambda^{2}<m^{2}<\rho$)

$\displaystyle\int_{m^{2}}^{\rho_{0}}\frac{|\lambda^{2}-m^{2}|}{|\rho-\lambda^{2}|\,|\rho-m^{2}|}\left(|\lambda+m|^{2}\,|\rho-m^{2}|^{\frac{\beta_{1}}{2}}+|\lambda+m|\,|\rho-m^{2}|^{\frac{1}{2}}+|\rho-m^{2}|\right)$
$\displaystyle\leq C(|\lambda+m|\,|\ln(|\lambda+m|)|)\ .\qed$

##  5\. Proof of the main results for hyperbolic thresholds

We turn now our attention to the _hyperbolic thresholds_. For ease of notation
we will set $m=0$ and consider the positive hyperbolic threshold $\tau=2$.

First, taking into account Eq. 38, we will need to study the operator

$\mathcal{R}(\lambda):=\text{p.v.}\int_{0}^{M_{0}}\mathrm{d}u\frac{2u}{u^{2}-\lambda^{2}}\int_{r^{-1}(u^{2})}\mathrm{d}\gamma\frac{{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}.$

Since $V$ is trace class, for all $r>0$

(65) $n_{\pm}(r;{\rm
Re}(K(\lambda+i0)))=n_{\pm}(r;\frac{1}{\lambda}\mathcal{R}(\lambda))+O(1).$

When integrating on $\mathbb{T}^{2}$, due to the periodicity, we will focus on
the rectangle $[0,1/2]\times[0,1/2]$. Let us take the critical points
$\xi_{a}=(1/2,0)$ and $\xi_{b}=(0,1/2)$ in $\mathbb{T}^{2}$. Notice that the
level curve defined by the threshold $\tau=2$, namely $r^{-1}(4)$, pass
through these points.

Define the triangles
$\triangle_{a}:=\\{\xi\in[0,1/2]\times[0,1/2]:\xi_{2}\leq\xi_{1}\\}$ and
$\triangle_{b}:=\\{\xi\in[0,1/2]\times[0,1/2]:\xi_{2}>\xi_{1}\\}$ and set

$\mathring{T}_{\lambda}(\xi):=\mathds{1}_{\triangle_{a}}(\xi)(T_{\lambda}(\xi)-T_{\lambda}(\xi_{a}))+\mathds{1}_{\triangle_{b}}(\xi)(T_{\lambda}(\xi)-T_{\lambda}(\xi_{b}))\
.$

For simplicity, all the following computations will be performed on the
triangle $\triangle_{a}$. We will also assume that $\lambda\geq 2$.

Denote by $\gamma_{\lambda}$ the part of the curve $r^{-1}(\lambda^{2})$ that
lies in $\triangle_{a}$. It admits the following parametrization:

(66)
$\xi_{1}(t)=\frac{1}{2\pi}\arccos\left(1-\frac{t}{2}\right);\quad\xi_{2}(t)=\frac{1}{2\pi}\arccos\left(1+\frac{t-\lambda^{2}}{2}\right)\
,$

for $t\in[\tfrac{\lambda^{2}}{2},4]$. We can easily compute

$\xi_{1}^{\prime}(t)=\frac{1}{2\pi
t^{1/2}(4-t)^{1/2}};\quad\xi_{2}^{\prime}(t)=-\frac{1}{2\pi(\lambda^{2}-t)^{1/2}(4+t-\lambda^{2})^{1/2}}\
,$

so

(67) $\mathrm{d}\gamma\leq C((4-t)^{-1/2})\mathrm{d}t\ .$

Most computations will make use of this explicit parametrization, but other
parts of $\mathbb{T}^{2}$ can be dealt with accordingly, as can be the case
$\lambda\leq 2$.

######  Lemma 5.1.

The following asymptotic relation holds true

$\left|\left|\int_{\gamma_{\lambda}}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}=O(1),\quad\lambda\to\tau.$

###### Proof.

For $\xi\in\triangle_{a}$, from Eqs. 26 and 30 we can see that

(68) $||\mathring{T}(\xi)||_{1}\leq C|\xi-\xi_{a}|^{\beta}\ .$

Further, we have that $|\nabla r(\xi)|^{2}\asymp
r(\xi-\xi_{a})\asymp|\xi-\xi_{a}|^{2}$ so

(69) $\left|\left|\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}\leq C|\xi-\xi_{a}|^{\beta-1}\ ,$

when $|\lambda-\tau|$ is small enough. Therefore

$\left|\left|\int_{\gamma_{\lambda}}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}\leq
C\int_{\gamma_{\lambda}}\mathrm{d}\gamma|\xi-\xi_{a}|^{\beta-1}\ .$

A direct inspection gives us that for $t\in[\tfrac{\lambda^{2}}{2},4]$

(70) $|\xi(t)-\xi_{a}|\leq C(4-t)^{\frac{1}{2}}\ .$

From this we obtain

$\int_{\gamma_{\lambda}}\mathrm{d}\gamma|\xi-\xi_{a}|^{\beta-1}\leq
C\int_{\tfrac{\lambda^{2}}{2}}^{4}\mathrm{d}t\frac{((4-t)^{\frac{1}{2}})^{\beta-1}}{(4-t)^{\frac{1}{2}}}\leq
C\int_{\tfrac{\lambda^{2}}{2}}^{4}\mathrm{d}t(4-t)^{\frac{\beta}{2}-1}=O(1)\text{
as }\lambda\to\tau\ .\qed$

Set

(71)
$\mathcal{R}_{1}(\lambda):=\text{p.v.}\int_{0}^{M_{0}}\mathrm{d}u\frac{2u}{u^{2}-\lambda^{2}}\int_{r^{-1}(u^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}.$

Denote by $\gamma_{\lambda,b}$ the part of the curve $r^{-1}(\lambda^{2})$
that lies in $\triangle_{b}$, then

$\int_{r^{-1}(\lambda^{2})}\mathrm{d}\gamma\frac{{{T}}_{\lambda}(\xi)-\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\gamma)|}={{T}}_{\lambda}(\xi_{a})\int_{\gamma_{\lambda}}\mathrm{d}\gamma\frac{1}{|\nabla
r(\gamma)|}+{{T}}_{\lambda}(\xi_{b})\int_{\gamma_{\lambda,b}}\mathrm{d}\gamma\frac{1}{|\nabla
r(\gamma)|}$

is an operator with rank at most twelve. Thus, we can use Lemma 3.1 together
with Eqs. 37, 65 and 5.1 to obtain that for any $\epsilon\in(0,1)$

(72) $\begin{split}\pm
n_{\mp}\left((1\pm\epsilon);\frac{1}{\lambda}\mathcal{R}_{1}(\lambda)\right)+O(1)\leq&\bm{\xi}(\lambda;H_{\pm},H_{0})\\\
\leq&\pm
n_{\mp}\left((1\mp\epsilon);\frac{1}{\lambda}\mathcal{R}_{1}(\lambda)\right)+O(1)\
.\end{split}$

as $\lambda\to\tau$.

######  Lemma 5.2.

There exists $u_{0}>0$ such that for $u,u^{\prime}$ in $(2-u_{0},2]$ or in
$[2,2+u_{0})$

$\left|\left|\int_{r^{-1}(u^{\prime
2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}-\int_{r^{-1}(u^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}=O(|u-u^{\prime}|^{\beta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}/2}\ln|u-u^{\prime}|^{-1}),\quad|u-u^{\prime}|\to
0.$

###### Proof.

Assume that $\sqrt{8}>u^{\prime}>u\geq\tau=2$. As before we consider the
triangle $\triangle_{a}$. Recall that we denote by $\gamma_{u}$ the part of
the curve $r^{-1}(u^{2})$ that lies in $\triangle_{a}$.

Set $\zeta:=(u^{\prime 2}-u^{2})/2$ and define the function

$D_{\zeta}(\xi_{1},\xi_{2})=\left(\frac{1}{2\pi}\arccos(\zeta+\cos(2\pi\xi_{1})),\xi_{2}\right)\
.$

Notice that $D_{\zeta}(\gamma_{u}^{\prime})\subset\gamma_{u}$ and

$\int_{\gamma_{u}}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}=\int_{\gamma_{u}^{\prime}}D_{\zeta}^{*}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(D_{\zeta}\xi)}{|\nabla
r(D_{\zeta}\xi)|}+\int_{\gamma_{u}\setminus
D_{\zeta}\gamma_{u}^{\prime}}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\ ,$

so we need to estimate

(73)
$\int_{\gamma_{u^{\prime}}}(D_{\zeta}^{*}\mathrm{d}\gamma-\mathrm{d}\gamma)\frac{\mathring{T}_{\lambda}(D_{\zeta}\xi)}{|\nabla
r(D_{\zeta}\xi)|}+\int_{\gamma_{u^{\prime}}}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(D_{\zeta}\xi)-\mathring{T}_{\lambda}(\xi)}{|\nabla
r(D_{\zeta}\xi)|}+\int_{\gamma_{u^{\prime}}}\mathrm{d}{\gamma}\,\mathring{T}_{\lambda}(\xi)\left(\frac{1}{|\nabla
r(D_{\zeta}\xi)|}-\frac{1}{|\nabla r(\xi)|}\right)$ $:=(I)+(II)+(III)\ ,$

and

$\int_{\gamma_{u}\setminus
D_{\zeta}\gamma_{u^{\prime}}}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}:=(IV)\ .$

We start by estimating (IV). It is easy to see that $\gamma_{u}\setminus
D_{\zeta}\gamma_{u}^{\prime}$ can be parametrized by Eq. 66 with
$t\in[4+u^{2}-u^{\prime 2},4]$. Then, by Eqs. 67, 69 and 70

(74) $\left|\left|\int_{\gamma_{u}\setminus
D_{\zeta}\gamma_{u^{\prime}}}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right|\right|_{1}\leq C\int_{4+u^{2}-u^{\prime
2}}^{4}\mathrm{d}t(4-t)^{\beta/2-1}\leq C|u^{\prime}-u|^{\beta/2}\\\ \ $

To estimate $(I)+(II)+(III)$ a partition of $\triangle_{a}$ is needed, namely
$\triangle_{a,1}:=\\{\xi\in\triangle_{a}:\xi_{1}\geq 3/8,\xi_{2}\leq 1/8\\}$
and $\triangle_{a,2}:=\triangle_{a}\setminus\triangle_{a,1}$. Accordingly we
write $\gamma_{u,j}$ for the part of the curve $r^{-1}(u^{2})$ that lies in
$\triangle_{a,j}$, $j=1,2$. The same for $(I_{j}),(II_{j}),(III_{j})$.

Let us start by studying $(I_{1}),(II_{1}),(III_{1})$. Consider first $(I)$ on
$\triangle_{a,1}$. We have that $|\mathrm{d}\gamma-
D_{\zeta}^{*}\mathrm{d}\gamma|\leq|\partial_{1}D_{\zeta,1}(\gamma)^{2}-1|\mathrm{d}\gamma$,
and

(75) $|\partial_{1}D_{\zeta,1}(\xi)^{2}-1|=\frac{|\zeta(2\cos
2\pi\xi_{1}+\zeta)|}{|\sin^{2}2\pi\xi_{1}-\zeta(2\cos 2\pi\xi_{1}+\zeta)|}\ .$

On $\triangle_{a,1}$ we have $-\zeta(2\cos 2\pi\xi_{1}+\zeta)>\tilde{C}\zeta$,
for a positive $\tilde{C}$ (when $\zeta$ is sufficiently small). Putting all
this together with 69

(76) $||(I_{1})||_{1}\leq C\int_{\gamma_{u^{\prime},1}}|\mathrm{d}\gamma-
D_{\zeta}^{*}\mathrm{d}\gamma||\xi-\xi_{a}|^{\beta-1}\leq
C\zeta\int_{\gamma_{u^{\prime},1}}\mathrm{d}\gamma\frac{|\xi-\xi_{a}|^{\beta-1}}{|\sin^{2}2\pi\xi_{1}+\tilde{C}\zeta|},$

and since

$\sin^{2}2\pi\xi_{1}(t)=\frac{t(4-t)}{4},$

by Eqs. 67 and 70

$||(I_{1})||_{1}\leq C\zeta\int_{u^{\prime
2}/2}^{4}\mathrm{d}t\frac{(4-t)^{\beta/2-1}}{4-t+\tilde{C}\zeta}\leq
C\zeta^{\beta/2}\int_{0}^{\infty}\mathrm{d}s\frac{s^{\beta/2-1}}{s+\tilde{C}}\
.$

Now, for $(II_{1})$, proceeding as in Eq. 45 we can show that

(77)
$||\mathring{T}_{\lambda}(D_{\zeta}\xi)-\mathring{T}_{\lambda}(\xi)||_{1}\leq
C|\xi-D_{\zeta}\xi|^{\beta}\leq C|\zeta|^{\frac{\beta}{2}}\ .$

Further

(78) $|\nabla r(D_{\zeta}\xi)|^{2}=|\nabla r(\xi)|^{2}-16\pi\zeta(\zeta+2\cos
2\pi\xi_{1}),$

and using the same parametrization as before

(79) $\begin{split}|\nabla
r(D_{\zeta}\xi(t))|^{2}&=4\pi^{2}t(4-t)+4\pi^{2}(4-u^{2}+t)(u^{2}-t)-16\pi^{2}\zeta(\zeta+(2-t))\\\
&\geq\tilde{B}(4-t)+\tilde{C}\zeta,\end{split}$

for some $\tilde{B},\tilde{C}>0$. Then,

$\displaystyle||(II_{1})||_{1}\leq$
$\displaystyle\left|\left|\int_{\gamma_{u^{\prime},1}}\mathrm{d}{\gamma}\frac{\mathring{T}_{\lambda}(D_{\zeta}\xi)-\mathring{T}_{\lambda}(\xi)}{|\nabla
r(D_{\zeta}\xi)|}\right|\right|_{1}\leq
C|\zeta|^{\beta/2}\int_{\gamma_{u^{\prime},1}}\frac{\mathrm{d}\gamma}{|\nabla
r(D_{\zeta}\xi)|}$ $\displaystyle\leq$ $\displaystyle
C|\zeta|^{\beta/2}\int_{u^{\prime
2}/2}^{4}\mathrm{d}t\frac{(4-t)^{-1/2}}{(4-t+{\tilde{C}}\zeta)^{1/2}}\leq
C\zeta^{\beta\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}/2}\ln(\zeta^{-1})\
.$

Finally consider $(III_{1})$. Since

$\left|\frac{1}{|\nabla r(\xi)|}-\frac{1}{|\nabla
r(D_{\zeta}\xi)|}\right|=\frac{\Big{|}|\nabla r(D_{\zeta}\xi)|^{2}-|\nabla
r(\xi)|^{2}\Big{|}}{|\nabla r(\xi)|\,|\nabla r(D_{\zeta}\xi)|(|\nabla
r(\xi)|+|\nabla r(D\xi)|)},$

from Eqs. 69, 78 and 79

$||(III_{1})||_{1}\leq C|\zeta|\int_{\gamma_{u^{\prime
2}}}\mathrm{d}\gamma\frac{|\xi-\xi_{a}|^{\beta-1}}{|\nabla r(D\gamma)|^{2}}\\\
\leq\zeta\int_{u^{\prime
2}/2}^{4}\mathrm{d}t\frac{(4-t)^{\beta/2-1}}{B(4-t)+\tilde{A}\zeta}\\\ \leq
C\zeta^{\beta/2}\ .$

Similar arguments can be used to deal with $(I_{2}),(II_{2}),(III_{2})$. In
fact, for these cases the computations are simpler because if $u_{0}$ is small
enough all the functions appearing on the integrals of Eq. 73 are Hölder
continuous (see Lemma 3.2). Moreover, the denominators are non-vanishing, so
the result follows immediately when we restrict to $\triangle_{a,2}$. ∎

For $\lambda>2$ we write principal value part of the operator
$\mathcal{R}_{1}(\lambda)$ (defined in Eq. 71) as follows

p.v.
$\displaystyle\int_{0}^{M_{0}}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}\int_{r^{-1}(u^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r({\xi})|}$ (80)
$\displaystyle=\int_{[0,(\lambda+2)/2]\cup[(3\lambda-2)/2,M_{0}]}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}\int_{r^{-1}(u^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}$
$\displaystyle+\int_{0}^{(\lambda-2)/2}\frac{\mathrm{d}\rho}{\rho}\left(\frac{2(\lambda+\rho)}{(2\lambda+\rho)}\int_{r^{-1}((\lambda+\rho)^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}-\frac{2(\lambda-\rho)}{(2\lambda-\rho)}\int_{r^{-1}((\lambda-\rho)^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}\right)$
$\displaystyle=:\tilde{\mathcal{R}}_{1}(\lambda)+{\mathcal{R}}_{1,pv}(\lambda)\
.$

Thus from Lemma 5.2 we can readily see that

(81) $||{\mathcal{R}}_{1,pv}(\lambda)||_{1}=o(|2-\lambda|),\quad\lambda\to 2\
.$

Set

$\mathcal{R}_{2}(\lambda):=\int_{[0,(\lambda+2)/2]\cup[(3\lambda-2)/2,M_{0}]}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}\int_{r^{-1}(\lambda^{2})}\mathrm{d}\gamma\frac{\mathring{T}_{\lambda}(\xi)}{|\nabla
r(\xi)|}.$

From Lemma 5.2 we can see that

(82) $||\tilde{\mathcal{R}}_{1}(\lambda)-\mathcal{R}_{2}(\lambda)||_{1}=O(1).$

Finally, the function

$\lambda\mapsto\int_{[0,(\lambda+2)/2]\cup[(3\lambda-2)/2,M_{0}]}\mathrm{d}u\frac{2u}{(u^{2}-\lambda^{2})}=\ln\left(\frac{(3\lambda+2)(M_{0}^{2}-\lambda^{2})}{\lambda^{2}(5\lambda-2)}\right)$

is bounded for $\lambda\to\tau=2$, so $\mathcal{R}_{2}(\tau)$ is well defined
and compact. Moreover, we notice that for $\xi\in\mathcal{U}_{a}$

(83)
$\begin{split}||\mathring{T}_{\lambda}(\xi)-\mathring{T}_{2}(\xi)||_{1}=&||[(t_{\lambda}(\xi)-t_{\lambda}(\xi_{a}))^{*}-(t_{2}(\xi)-t_{2}(\xi_{a}))^{*}]At_{\lambda}(\xi)\\\
&+[(t_{2}(\xi)-t_{2}(\xi_{a}))^{*}]A(t_{\lambda}(\xi)-t_{2}(\xi))\\\
&+[(t_{\lambda}(\xi_{a})-t_{2}(\xi_{a}))^{*}]A(t_{2}(\xi)-t_{\lambda}(\xi_{a}))\\\
&+t_{2}(\xi_{2})^{*}A[(t_{\lambda}(\xi)-t_{\lambda}(\xi_{2}))-(t_{2}(\xi)-t_{2}(\xi_{a}))]||_{1}\\\
\leq&C|\lambda-2||\xi-\xi_{2}|^{\beta}\ .\end{split}$

An analogous computations holds when $\xi\in\mathcal{U}_{b}$. From Eq. 83,
together with Lemma 5.2 we obtain that

(84) $\mathcal{R}_{1}(\lambda)\to\mathcal{R}_{1}(\tau),\quad\lambda\to\tau=2\
,$

in the trace class norm.

In consequence, putting together with Eqs. 72, 80, 81, 78 and 84 we conclude
that if $V$ satisfies Eq. 6 then

$\bm{\xi}(\lambda;H_{\pm},H_{0})=O(1),\quad\lambda\to 2.$

##  6\. Eigenvalue Asymptotics for integral operators with toroidal kernel

Denote by $S_{\rho}^{\gamma}(\mathbb{Z}^{d})$ the class of symbols given by
the functions $v:\mathbb{Z}^{d}\to\mathbb{C}$ that satisfies for any multi-
index $\alpha$

(85) $|{\rm D}^{\alpha}v(\mu)|\leq
C_{\alpha}\langle\mu\rangle^{-\gamma-\rho|\alpha|},$

where ${\rm D_{\mu_{j}}}v(\mu):=v(\mu+\delta_{j})-v(\mu)$, and ${\rm
D}^{\alpha}:={\rm D}_{\mu_{1}}^{\alpha_{1}}...{\rm D}_{\mu_{d}}^{\alpha_{d}}$.

_Condition 1: For $\gamma>d$, $\rho>0$ assume $\\{v_{k}\\}_{k=1}^{N}\in
S_{\rho}^{\gamma}(\mathbb{Z}^{d})$. We suppose also that_

(86) $v_{k}(\mu)=v_{0}(\mu)(\Gamma_{k}+o(1)),\quad|\mu|\to\infty,$

_for a function $v_{0}:\mathbb{Z}^{d}\to\mathbb{C}$ which, viewed as a
multiplication operator, satisfies_

(87)
$n_{\pm}(\lambda;v_{0})=\lambda^{-d/\gamma}(c_{\pm}+o(1)),\quad\lambda\downarrow
0,$

_with $c_{\pm}>0$. _

Let $\\{B_{k}\\}_{k=1}^{N}$ be a family of functions in
$L^{2}(\mathbb{T}^{d})$. Define

$\Psi:=\sum_{k=1}^{N}B_{k}\mathscr{F}{v_{k}}\mathscr{F}^{*}\overline{B}_{k}\
.$

This is a compact operator on $L^{2}(\mathbb{T}^{d})$ and has integral kernel
given by

(88) $\sum_{k=1}^{N}B_{k}(\xi)\widehat{v_{k}}(\xi-\eta)\overline{B}_{k}(\eta)\
.$

######  Theorem 6.1.

Assume Condition 1 and $\Psi$ as above. Then

$n_{\pm}(\lambda;\Psi)=\mathcal{C}_{B\pm}\lambda^{-d/\gamma}(1+o(1)),\quad\lambda\downarrow
0,$

where

$\mathcal{C}_{B\pm}=c_{\pm}\int_{\mathbb{T}^{d}}\mathrm{d}\xi\Big{(}\sum_{k=1}^{N}\Gamma_{k}|B_{k}(\xi)|^{2}\Big{)}^{d/\gamma}\
.$

######  Remark 6.2.

Using the ideas of [BS70], 6.1 can be easily extended to cover more general
kernels. In particular, in Eq. 88 one could consider matrix-valued $B$ or
introduce another hermitian matrix-valued function depending on $\xi$ and
$\eta$. Here we limit ourselves to setting we were aiming to apply it.
Although 6.1 bears resemblance with [BS70, Theorem 1.], we would like to
stress out that it does not follows directly from it or it’s related results
(see for instance [BS77a, BKS91]).

Set $\blacksquare:=[0,1)^{d}$. In order to prove the following lemma we need
to use the spaces of compact operators $\mathfrak{S}_{p,w}$ defined for
$0<p<\infty$ by

$\mathfrak{S}_{p,w}:\\{K\in\mathfrak{S}_{\infty}:s_{n}(K)=O(n^{-1/p})\\}$

with the quasi-norm

(89)
$||K||_{p,w}:=\sup_{n}\\{n^{1/p}s_{n}(K)\\}=\left(\sup_{s>0}\\{s^{p}n_{*}(s;K)\\}\right)^{1/p}\
.$

These spaces satisfy the "weakened triangle inequality"
$||K_{1}+K_{2}||_{p,w}\leq 2^{1/p}(||K||_{p,w}+||K||_{p,w})$ and the "weakened
Hölder inequality"

(90) $||K_{1}K_{2}||_{r,w}\leq c(p,q)||K_{1}||_{p,w}||K_{2}||_{q,w},$

for $r^{-1}=p^{-1}+q^{-1}$ and $c(p,q)=(p/r)^{1/p}(q/r)^{1/q}$ (see [Chapter
11][BS87]).

In order to prove 6.1 we need the following Cwikel–type estimate.

######  Lemma 6.3.

Assume that $|v(\mu)|\leq C\langle\mu\rangle^{-\gamma}$. Then, if $p,q\in
L^{2}(\blacksquare)$, there exists a positive constant $C(v,d)$ such that

$||p\mathscr{F}v\mathscr{F}^{*}q||_{d/\gamma,w}\leq
C(v,d)||p||_{L^{2}(\blacksquare)}||q||_{L^{2}(\blacksquare)}$

###### Proof.

By Eq. 90

(91)
$||p\mathscr{F}v\mathscr{F}^{*}q||_{d/\gamma,w}\leq||p\mathscr{F}|v|^{1/2}||_{2d/\gamma,w}|||v|^{1/2}\mathscr{F}^{*}q||_{2d/\gamma,w},$

Then it is enough to prove $||p\mathscr{F}|v|^{1/2}||_{2d/\gamma,w}\leq
C||p||_{L^{2}(\blacksquare)}$. First notice that by the min-max principle the
singular values satisfies

(92)
$s_{n}(p\mathscr{F}|v|^{1/2})^{2}=\lambda_{n}(p\mathscr{F}|v|\mathscr{F}^{*}\overline{p})\leq
C\lambda_{n}(p\mathscr{F}w\mathscr{F}^{*}\overline{p}),$

where $w(\mu)=\langle\mu\rangle^{-\gamma}$, and $\\{\lambda_{n}(K)\\}$ denotes
the non increasing sequence of eigenvalues of $0\leq
K\in\mathfrak{S}_{\infty}$.

Set $g(x):=\langle x\rangle^{-\gamma}$. Since $\gamma>d$, we have that $g\in
L^{1}(\mathbb{R}^{d})$ and $\hat{g}\in C_{0}(\mathbb{R}^{d})$. Here we use the
standard notation $\hat{g}$ for

(93) $\hat{g}(y):=\int_{\mathbb{R}^{d}}\mathrm{d}x\,\mathrm{e}^{-2\pi iy\cdot
x}g(x)\ .$

Further, $\hat{g}$ is smooth in $\mathbb{R}^{d}\setminus\\{0\\}$ and decay at
infinity faster than $|y|^{-n}$ for any $n\in\mathbb{Z}_{+}$ (same for its
derivatives). Then, we can use the Poisson summation formula (see for instance
[Gra08]) to obtain

$\displaystyle\sum_{\mu\in\mathbb{Z}^{d}}\omega(\mu)e^{-2\pi i\mu\cdot x}$
$\displaystyle=\sum_{\mu\in\mathbb{Z}^{d}\setminus{0}}\check{\hat{g}}(\mu)e^{-2\pi
i\mu\cdot x}+\omega(0)$ (94)
$\displaystyle=\sum_{\mu\in\mathbb{Z}^{d}}\check{{g}}(-x+\mu)-\check{{g}}(0)+\omega(0).$

Set $G(x)={g}(x)\sum_{\mu\in(-2,2)^{d}}e^{2\pi i\mu\cdot x}$. Then, it is
clear that
$\hat{G}(x)=\sum_{\mu\in\mathbb{Z}^{d}\setminus(-2,2)^{d}}\check{{g}}(-x+\mu)$
is smooth in $\mathbb{R}^{d}\setminus(-2,2)^{d}$. Further,
$\phi(x):=\sum_{\mu\in\mathbb{Z}^{d}\setminus(-2,2)^{d}}\hat{{g}}(x-\mu)-\check{{g}}(0)+\omega(0)$
is smooth in $(-2,2)^{d}$. To see this take $x\in(-2,2)^{d}$. Then $x-\mu\neq
0$ for all $\mu\in\mathbb{Z}^{d}\setminus(-2,2)^{d}$ and so, each function
$\hat{{g}}(\cdot-\mu)$ is smooth in $(-2,2)^{d}$ when
$\mu\in\mathbb{Z}^{d}\setminus(-2,2)^{d}$. Moreover, since each partial
derivative of $\hat{g}$ decays fast, using the Lebesgue dominated convergence
theorem permit us to derivate term by term the series of $\phi$.

Further, from Eq. 94 we make the decomposition

(95)
$p\mathscr{F}\omega\mathscr{F}^{*}\overline{p}=p\mathscr{F}G\mathscr{F}^{*}\overline{p}+p\Phi\overline{p},$

where $\Phi$ is the operator in $L^{2}(\blacksquare)$ with integral kernel
$\phi(\xi-\eta)$. Notice that the non-zero eigenvalues of
$p\mathscr{F}G\mathscr{F}^{*}\overline{p}$ are the same as those of the
operator
$\mathds{1}_{\blacksquare}p\mathscr{F}G\mathscr{F}^{*}\overline{p}\mathds{1}_{\blacksquare}$
in $L^{2}(\mathbb{R}^{d})$. Then, by Eqs. 92 and 95

$\displaystyle||p\mathscr{F}|v|^{1/2}||_{2d/\gamma,w}\leq$ $\displaystyle
C||p\mathscr{F}|w|^{1/2}||_{2d/\gamma,w}$ $\displaystyle=$ $\displaystyle
C||p\mathscr{F}|w|\mathscr{F}^{*}\overline{p}||_{d/\gamma,w}^{1/2}$
$\displaystyle\leq$ $\displaystyle
C\left(||\mathds{1}_{\blacksquare}p\mathscr{F}G\mathscr{F}^{*}\overline{p}\mathds{1}_{\blacksquare}||_{d/\gamma,w}+||p\Phi\overline{p}||_{d/\gamma,w}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
C\left(||\mathds{1}_{\blacksquare}p\mathscr{F}|G|^{1/2}||_{2d/\gamma,w}+||p\Phi\overline{p}||^{1/2}_{d/\gamma,w}\right)\
.$

The following definition is taken from [BKS91, Subsection 5.6] and we recall
it for convenience of the reader. Let $\phi\in L_{\rm
loc}^{2}(\mathbb{R}^{d})$ and for $\mu\in\mathbb{Z}^{d}$ define
$a_{\phi}(\mu)=(\int_{\blacksquare+\mu}|\phi(x)|^{2}dx)^{1/2}$. For $\alpha>0$
we introduce the spaces $l_{\alpha,w}(\mathbb{Z}^{d},L^{2}(\blacksquare))$ as
the set of functions $\phi\in L_{\rm loc}^{2}(\mathbb{R}^{d})$ that satisfies

$\\#\\{\mu\in\mathbb{Z}^{d}:|a_{\phi}(\mu)|>t\\}=O(t^{-\alpha})\ .$

We want to show that $G^{1/2}\in
l_{2d/\gamma,w}(\mathbb{Z}^{d},L^{2}(\blacksquare))$. For this it is enough to
notice that the sequence $(\int_{\blacksquare+\mu}\langle
x\rangle^{-\gamma}dx)^{1/2}\leq C\langle\mu\rangle^{-\gamma/2}$ and

(96) $\\#\\{\mu:\langle\mu\rangle^{-\gamma/2}>t\\}\asymp t^{-2d/\gamma}\ .$

Then, since $2d/\gamma<2$, by [BKS91, Subsection 5.7]

(97) $||\mathds{1}_{\blacksquare}p\mathscr{F}|G|^{1/2}||_{2d/\gamma,w}\leq
C\Big{|}\Big{|}|G|^{1/2}\Big{|}\Big{|}_{l_{2d/\gamma,w}}\Big{|}\Big{|}\mathds{1}_{\blacksquare}p\Big{|}\Big{|}_{L^{2d/\gamma}(\mathbb{R}^{d})}\
.$

On the other side, since $\phi$ is smooth in $B_{0}(\sqrt{d})$, it is in any
Besov Space $B_{p,\infty}^{\alpha}((-1,1)^{d})$ (notice that
$(-1,1)^{d}=\blacksquare-\blacksquare$). By [BS87, Theorem 6, page 274] or
[BS77, Theorem 6.1]

(98) $||p\Phi\overline{p}||_{d/\gamma,w}\leq
C||p||^{2}_{L^{2}(\blacksquare)}.$

Since $2d/\gamma<2$, we can conclude from Eqs. 91, 95, 97, 98 and 89 that

$\displaystyle|p\mathscr{F}v\mathscr{F}^{*}q||_{d/\gamma,w}$
$\displaystyle\leq
C\left(||p||_{L^{2}(\blacksquare)}+||p||_{L^{2d/\gamma}(\blacksquare)}\right)\left(||q||_{L^{2}(\blacksquare)}+||q||_{L^{2d/\gamma}(\blacksquare)}\right)$
$\displaystyle\leq C||p||_{L^{2}(\blacksquare)}||q||_{L^{2}(\blacksquare)}$

∎

Lemma 6.3 enable us to adapt the proof of [BS70, Theorem 1] to our case.

######  Lemma 6.4.

Let $X,Y\subset\blacksquare$ such that they do not have interior points in
common. Then, if $v\in S_{\rho}^{\gamma}(\mathbb{Z}^{d})$

$n_{*}\Big{(}s;\mathds{1}_{X}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y}\Big{)}=o(s^{-d/\gamma}),\quad
s\downarrow 0.$

###### Proof.

By [RT09, Theorem 4.3.6] the kernel of the operator
$\mathscr{F}v\mathscr{F}^{*}$ is smooth in $\mathbb{T}^{d}$ when $x\neq y$.
Suppose first that ${\rm dist}(X,Y)>0$ and consider smooth functions
$\nu_{X},\nu_{Y}$ such that $\nu_{X}=1$ in a neighborhood $O_{X}$ of $X$,
$\nu_{X}=0$ in a neighborhood $O_{Y}$ of $Y$ and ${\rm dist}(O_{X},O_{Y})>0$.
Similarly for $\nu_{Y}$. Then
$=\mathds{1}_{X}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y}=\mathds{1}_{X}\nu_{X}\mathscr{F}v\mathscr{F}^{*}\nu_{Y}\mathds{1}_{Y}$
and the kernel of $\nu_{X}\mathscr{F}v\mathscr{F}^{*}\nu_{Y}$ is smooth in
$\mathbb{T}^{2}$. Then, invoking again [BS87, Theorem 6, page 274], for any
$r>1$

$n_{*}(s;\nu_{X}\mathscr{F}v\mathscr{F}^{*}\nu_{Y})=o(s^{-r}),\quad
s\downarrow 0.$

Suppose now that ${\rm dist}(X,Y)=0$, fix $\epsilon>0$ and set
$X_{\epsilon}:=\\{x\in X:{\rm dist}(x,Y)>\epsilon\\}$. Write
$\mathds{1}_{X}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y}=\mathds{1}_{X_{\epsilon}}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y}+\mathds{1}_{X\setminus
X_{\epsilon}}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y}.$

By Lemma 6.3

(99) $s^{d/\gamma}n_{*}(s;\mathds{1}_{X\setminus
X_{\epsilon}}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y})\leq
C||\mathds{1}_{X\setminus
X_{\epsilon}}||_{L^{2}(\blacksquare)}^{\gamma/d}=o(\epsilon),$

uniformly for $s>0$. Further, by the first part of this proof, since
$\gamma>d$,

(100)
$n_{*}(s;\mathds{1}_{X_{\epsilon}}\mathscr{F}v\mathscr{F}^{*}\mathds{1}_{Y})=o(s^{-d/\gamma}),\quad
s\downarrow 0.$

Putting Eq. 16 together with Eqs. 100 and 99 we conclude the proof. ∎

######  Lemma 6.5.

Let $\\{\square_{j}\\}_{j=1}^{l}$ be a partition of $\blacksquare$ into finite
cubes of equal size. Let $\Psi_{\rm{diag}}:L^{2}(\blacksquare)\to
L^{2}(\blacksquare)$ be the integral operator with kernel

$\sum_{j=1}^{l}\mathds{1}_{\square_{j}}(x)\Big{(}\sum_{k=1}^{N}|B_{k,j}|^{2}\widehat{v}_{k}(x-y)\Big{)}\mathds{1}_{\square_{j}}(y),$

where $B_{k,j}\in\mathbb{C}$. Then, if $v_{k}$ satisfy the conditions of 6.1
we have

$n_{\pm}(\lambda;\Psi_{\rm{diag}})=C_{\rm{diag\pm}}\lambda^{-d/\gamma}(1+o(1)),\quad\lambda\downarrow
0,$

with
$C_{\rm{diag}\pm}=c_{\pm}\sum_{j=1}^{l}|\square_{j}|\Big{(}\sum_{k=1}^{N}|B_{k,j}|^{2}\Gamma_{k}\Big{)}^{d/\gamma}$
and $|\square_{j}|$ is the Lebesgue measure of $\square_{j}$.

###### Proof.

To simplify the notation assume $N=1$. Evidently
$\Psi_{\rm{diag}}=\bigoplus_{j}^{l}|B_{1,j}|^{2}\Psi_{j,\rm{diag}}$, where the
integral kernel of $\Psi_{j,\rm{diag}}$ is
$\mathds{1}_{\square_{j}}(x)\widehat{v}_{1}(x-y)\mathds{1}_{\square_{j}}(y)$.
Therefore

$n_{\pm}(\lambda;\Psi_{\rm{diag}})=\sum_{j}^{l}n_{\pm}(\lambda;|B_{1,j}|^{2}\Psi_{j,\rm{diag}}).$

Now, since the kernel
$\widehat{v}_{1}(x-y)=\sum_{j=1}^{l}\mathds{1}_{\square_{j}}(x)\widehat{v}_{1}(x-y)\mathds{1}_{\square_{j}}(y)+\sum_{j\neq
i}^{m}\mathds{1}_{\square_{j}}(x)\widehat{v}_{1}(x-y)\mathds{1}_{\square_{i}}(y)$,
we have that

$\mathcal{F}^{*}v_{1}\mathcal{F}=\bigoplus_{j}\Psi_{j,\rm{diag}}+R_{l},$

where $\sum_{j\neq
i}^{m}\mathds{1}_{\square_{j}}(x)\widehat{v}_{1}(x-y)\mathds{1}_{\square_{i}}(y)$
is the kernel of $R_{l}$.

By Eqs. 15 and 6.4, for any $\delta\in(0,1)$

$\displaystyle n_{\pm}(\lambda;\mathcal{F}^{*}v_{1}\mathcal{F})$
$\displaystyle\leq
n_{\pm}\Big{(}\lambda(1-\delta);\bigoplus_{j}\Psi_{j,\rm{diag}}\Big{)}+o(\lambda^{-d/\gamma})$

and

$\displaystyle n_{\pm}(\lambda;\mathcal{F}^{*}v_{1}\mathcal{F})$
$\displaystyle\geq
n_{\pm}\Big{(}\lambda(1+\delta);\bigoplus_{j}\Psi_{j,\rm{diag}}\Big{)}+o(\lambda^{-d/\gamma}).$

We can take the square $\square_{0}=[0,1/m)^{d}$, for some
$m\in\mathbb{Z}_{0}$, and assume that each $\square_{j}$ is just a translation
of $\square_{0}$. Notice that in this case $l=m^{d}$ and
$|\square_{j}|=1/m^{d}$. Moreover, each operator $\Psi_{j,\rm{diag}}$ is
unitary equivalent with each other, so the two previous inequalities imply
that for any $1\leq j\leq m^{d}$ and $\delta\in(0,1)$

(101) $n(\lambda(1+\delta);v_{1})+o(\lambda^{-d/\gamma})\leq
m^{d}n_{\pm}(\lambda;\Psi_{j,\rm{diag}})\leq
n(\lambda(1-\delta);v_{1})+o(\lambda^{-d/\gamma}).$

By Eq. 86 for all $\epsilon>0$ there exists $M$ such that $|\mu|>M$ implies

$|v_{1}(\mu)-\Gamma_{1}v_{0}(\mu)|<\epsilon|v_{0}(\mu)|.$

Then, for any $\delta\in(0,1)$

$\displaystyle
n_{+}\left(\lambda(1+\delta);v_{0}^{\pm}(\mu)\left(\Gamma_{1}|B_{1,j}|^{2}-\epsilon|B_{1,j}|^{2}\right)\right)+o(\lambda^{-d/\gamma})$
$\displaystyle\leq$
$\displaystyle|\square_{j}|^{-1}n_{\pm}\left(\lambda;|B_{1,j}|^{2}\Psi_{j,\rm{diag}}\right)$
$\displaystyle\leq$ $\displaystyle
n_{+}\left(\lambda(1-\delta);v_{0}^{\pm}(\mu)\left(\Gamma_{1}|B_{1,j}|^{2}+\epsilon|B_{1,j}|^{2}\right)\right)+o(\lambda^{-d/\gamma}).$

Here $v_{0}^{\pm}$ are the positive and negative parts of $v_{0}$.

Now, multiplying by $\lambda^{-d/\gamma}$and taking into account Eq. 87, the
limit when $\lambda$ goes to zero is

$\displaystyle
c_{\pm}\left((|\Gamma_{k}|-\epsilon)|B_{1,j}|^{2}\right)^{d/\gamma}$
$\displaystyle\leq|\square_{j}|^{-1}\lim_{\lambda\downarrow
0}\lambda^{d/\gamma}n_{\pm}\left(\lambda;|B_{1,j}|^{2}\Psi_{j,\rm{diag}}\right)$
$\displaystyle\leq
c_{\pm}\left((|\Gamma_{k}|+\epsilon)|B_{1,j}|^{2}\right)^{d/\gamma}$

Taking the limit when $\epsilon\downarrow 0$ we finish the proof. ∎

Proof of 6.1

###### Proof.

Let $\epsilon>0$ and for any $1\leq k\leq N$ take a step function in
$\blacksquare$ of the form

$B_{k,\epsilon}(x)=\sum_{j=1}^{l}B_{k,\epsilon,j}\mathds{1}_{\square_{\epsilon,j}}(x),$

$||B-B_{k,\epsilon}||_{L^{2}(\blacksquare)}^{2}<\epsilon$, and where
$\\{\square_{\epsilon,j}\\}$ are cubes chosen as in the previous lemma. Set
$\Psi_{\epsilon}$ as the operator with integral kernel
$\sum_{k=1}^{N}B_{k,\epsilon}(x)\widehat{v_{k}}(x-y)\overline{B_{k,\epsilon}(y)}$.
Using Lemma 6.3 we obtain

(102) $||\Psi-\Psi_{\epsilon}||_{\frac{d}{\gamma},w}\leq C\epsilon.$

Now, let $\Psi_{\epsilon,\rm{diag}}$ be the operator with integral kernel

$\sum_{j=1}^{l}\sum_{k=1}^{M}\mathds{1}_{\square_{\epsilon,j}}(x)\Big{(}|B_{k,\epsilon,j}|^{2}\widehat{v}_{k}(x-y)\Big{)}\mathds{1}_{\square_{\epsilon,j}}(y),$

then the difference $\Psi_{\epsilon}-\Psi_{\epsilon,\rm{diag}}$ is the
operator with kernel

$\sum_{i\neq
j}^{l}\sum_{k=1}^{N}\mathds{1}_{\square_{\epsilon,i}}(x)\Big{(}B_{k,\epsilon,i}\widehat{v}_{k}(x-y)\overline{B_{k,\epsilon,j}}\Big{)}\mathds{1}_{\square_{\epsilon,j}}(y).$

Thus, applying Lemma 6.4

$\lim_{s\downarrow
0}s^{d/\gamma}n_{*}\Big{(}s;\Psi_{\epsilon}-\Psi_{\epsilon,\rm{diag}}\Big{)}=0.$

Using the last inequality together with Eq. 16 gives that for any $\delta>0$

$n_{\pm}\Big{(}(1+\delta)\lambda;\Psi_{\epsilon,\rm{diag}}\Big{)}+o(\lambda^{d/\gamma})\leq
n_{\pm}(\lambda;\Psi)\leq
n_{\pm}\Big{(}(1-\delta)\lambda;\Psi_{\epsilon,\rm{diag}}\Big{)}+o(\lambda^{d/\gamma}),\quad\lambda\downarrow
0.$

Finally, use Lemma 6.5

$\displaystyle\lim_{\lambda\downarrow
0}\lambda^{d/\gamma}n_{\pm}(\lambda;\Psi)$
$\displaystyle=\lim_{\epsilon\downarrow
0}c_{\pm}\sum_{j=1}^{l}|\square_{\epsilon,j}|\Big{(}\sum_{k=1}^{N}|B_{k,\epsilon,j}|^{2}\Gamma_{k}\Big{)}^{d/\gamma}$
$\displaystyle=c_{\pm}\int_{\blacksquare}\mathrm{d}\xi\Big{(}\sum_{k=1}^{N}\Gamma_{k}|B_{k}(\xi)|^{2}\Big{)}^{d/\gamma}$

∎

##  7\. Proofs of the main results for parabolic thresholds and Dirac point
cases

###  7.1. Proof for the bounded part of the SSF

The boundedness of the SSF near the hyperbolic thresholds was proved in
Section 5. Further, for the elliptic thresholds we will only make explicit
computations for $m$, the case of $\pm M_{m}$ being analogous. Set

$B:=\mathscr{U}V\mathscr{U}^{*}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-1&0\\\\[5.0pt] 0&0&-1\end{pmatrix}\ ,$

and $\mathcal{Q}(\lambda):=B+{\mathcal{S}}_{3}(\lambda)$. Then, starting from
Lemma 3.1, using Eqs. 37, 38, 56 and 59 and Lemma 4.2, together with the Weyl
inequalities Eq. 15 and the Chebyshev-type estimate Eq. 17 we obtain

(103) $\displaystyle\pm
n_{\mp}\left((1\pm\epsilon);\frac{\mathcal{Q}(\lambda)}{\lambda+m}\right)+O(1)$
$\displaystyle\leq\bm{\xi}(\lambda;H_{\pm},H_{0})$ (104) $\displaystyle\leq\pm
n_{\mp}\left((1\mp\epsilon);\frac{\mathcal{Q}(\lambda)}{\lambda+m}\right)+O(1),$

for $|\lambda|\to m$.

The following fomula is valid for $\lambda\leq 0$ when $m>0$

$\displaystyle\mathcal{Q}(\lambda)=$
$\displaystyle\mathscr{U}G\mathscr{U}^{*}\frac{1}{r+m^{2}-\lambda^{2}}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-|b|^{2}+\lambda^{2}-m^{2}&\overline{a}b\\\\[5.0pt]
0&\overline{b}a&-|a|^{2}+\lambda^{2}-m^{2}\end{pmatrix}\mathscr{U}G\mathscr{U}^{*}$
$\displaystyle=$
$\displaystyle\mathscr{U}G\mathscr{U}^{*}\frac{1}{r+m^{2}-\lambda^{2}}\left(\begin{pmatrix}0&0&0\\\\[5.0pt]
0&\lambda^{2}-m^{2}&0\\\\[5.0pt]
0&0&\lambda^{2}-m^{2}\end{pmatrix}+\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-|b|^{2}&\overline{a}b\\\\[5.0pt]
0&\overline{b}a&-|a|^{2}\end{pmatrix}\right)\mathscr{U}G\mathscr{U}^{*}\ .$

Moreover, it is easy to see that

(105) $\begin{pmatrix}-|b|^{2}&\overline{a}b\\\\[5.0pt]
\overline{b}a&-|a|^{2}\end{pmatrix}=-\begin{pmatrix}b&\overline{a}\\\\[5.0pt]
-a&\overline{b}\end{pmatrix}\begin{pmatrix}1&0\\\\[5.0pt]
0&0\end{pmatrix}\begin{pmatrix}\overline{b}&-\overline{a}\\\\[5.0pt]
a&b\end{pmatrix}\leq 0.$

Therefore, for $0\geq\lambda>-m$, the operator $\mathcal{Q}(\lambda)$ is
nonpositive. Thus, Eq. 103 immediately implies the second assertion in Eq. 10
when $m>0$.

Next, let $k$ be $-m,0$ or $m$, and let us set

$l_{k}(\lambda)=\left\\{\begin{array}[]{cl}|\ln(|\lambda+m|)|&\text{if}\,\,k=-m\\\
1&\text{if}\,\,k=0,m\ .\end{array}\right.$

Thus, from Lemma 4.4 and Eqs. 103, 104, 15 and 17

(106) $\displaystyle\pm
n_{\mp}\left((1\pm\epsilon);\frac{\mathcal{Q}(k)}{\lambda+m}\right)+O(l_{k}(\lambda))$
$\displaystyle\leq\bm{\xi}(\lambda;H_{\pm},H_{0})$ (107) $\displaystyle\leq\pm
n_{\mp}\left((1\mp\epsilon);\frac{\mathcal{Q}(k)}{\lambda+m}\right)+O(l_{k}(\lambda)),$

for $\lambda\to\pm m$. Now, if $m>0$, by a change of variable we obtain that

(108)
${\mathcal{S}}_{3}(-m)={\mathcal{S}}_{3}(0)=\int_{0}^{M_{0}^{2}}\frac{\mathrm{d}\rho}{\rho}\int_{r^{-1}(\rho)}\mathrm{d}\gamma\frac{\mathring{T}_{0}(\gamma)}{|\nabla
r(\gamma)|}=\int_{\mathbb{T}^{2}}\mathrm{d}\xi\frac{T_{0}(\xi)}{r(\xi)}\ ,$

and so

(109)
$\mathcal{Q}(-m)=\mathcal{Q}(0)=\mathscr{U}G\mathscr{U}^{*}\frac{1}{r}\begin{pmatrix}0&0&0\\\\[5.0pt]
0&-|b|^{2}&\overline{a}b\\\\[5.0pt]
0&\overline{b}a&-|a|^{2}\end{pmatrix}\mathscr{U}G\mathscr{U}^{*}\ .$

We already know that this operator is non-positive, then the second statement
in Eq. 10 for $m=0$ and the first statement in Eq. 11 follow from Eq. 107.

###  7.2. Proof for the unbounded part of the SSF

Now we turn to the other two statements of 1.3. Take into account Eqs. 106,
107, 108 and 109. By an abuse of notation, we write
$\mathcal{Q}(0)=\mathscr{U}G\mathscr{U}^{*}\displaystyle{\frac{1}{r}}\begin{pmatrix}-|b|^{2}&\overline{a}b\\\\[5.0pt]
\overline{b}a&-|a|^{2}\end{pmatrix}\mathscr{U}G\mathscr{U}^{*}.$ Then, if we
define

${q}=\mathscr{U}G\mathscr{U}^{*}\frac{1}{\sqrt{r}}\begin{pmatrix}b&\overline{a}\\\\[5.0pt]
-a&\overline{b}\end{pmatrix}\begin{pmatrix}1&0\\\\[5.0pt] 0&0\end{pmatrix},$

we have that $\mathcal{Q}(0)=-qq^{*}$, and for all $s>0$

$n_{\mp}(s;\mathcal{Q}(0))=n_{\pm}(s;q^{*}q).$

In particular we have

(110) $\displaystyle\pm
n_{\pm}\left((1\pm\epsilon);\frac{q^{*}q}{\lambda+m}\right)+O(|\ln(|\lambda+m|)|)$
$\displaystyle\leq\bm{\xi}(\lambda;H_{\pm},H_{0})$ (111) $\displaystyle\leq\pm
n_{\pm}\left((1\mp\epsilon);\frac{q^{*}q}{\lambda+m}\right)+O(|\ln(|\lambda+m|)|),$

for $\pm\lambda\downarrow\mp m$.

Now, set $V_{eff}:\mathbb{Z}^{2}\to M_{n}(\mathbb{C}^{2})$ to be

$V_{eff}(\mu):=\begin{pmatrix}v_{2}(\mu)&0\\\\[10.00002pt]
0&v_{3}(\mu)\end{pmatrix}\ .$

The positive eigenvalues of the operator
$q^{*}q=\mathcal{A}^{*}\mathscr{F}V_{eff}\mathscr{F}^{*}\mathcal{A}$ coincide
with the ones of the integral operator in $L^{2}(\mathbb{T}^{2};\mathbb{C})$
with kernel

$\frac{\overline{b(\xi)}}{\sqrt{r(\xi)}}\widehat{v_{2}}(\xi-\eta)\frac{{b(\eta)}}{\sqrt{r(\eta)}}+\frac{\overline{a(\xi)}}{\sqrt{r(\xi)}}\widehat{v_{3}}(\xi-\eta)\frac{{a(\eta)}}{\sqrt{r(\eta)}}\
.$

This kernel is of the form Eq. 88, then it defines an operators $\Psi$ as in
Section 6. To finish the proof it is enough to use 6.1 in the case $d=2$. To
this end, it only remains to show that $v_{2}$ and $v_{3}$ satisfy _Condition
1_. First, Eq. 86 correspond to condition Eq. 8 by considering
$v_{0}(\mu)=\langle\mu\rangle^{-\gamma}$. Second,
$\langle\mu\rangle^{-\gamma}$ satisfies Eq. 87 as is shown by Eq. 96. Finally,
that $v_{2}$, $v_{3}$ are in $S_{\rho}^{\gamma}(\mathbb{Z}^{2})$ is just
condition Eq. 7.

Acknowledgments: P. Miranda was supported by the Chilean Fondecyt Grant
$1201857$. D. Parra was supported by the Chilean Fondecyt Grant $3210686$.
They would also like to deeply thank Lilia Simeonova who made available the
notes of the deceased G. Raikov, upon which the Section 6 was developed. The
main ideas were proposed by G. Raikov but any potential error is solely
attributed to the first two authors. We would also like to thank the anonymous
referee for the careful reading and useful suggestions.

## References

  * [AT15] Colette Anné and Nabila Torki-Hamza “The Gauss-Bonnet operator of an infinite graph” In _Anal. Math. Phys._ 5.2, 2015, pp. 137–159 DOI: 10.1007/s13324-014-0090-0
  * [BK62] M.. Birman and M.. Krein “On the theory of wave operators and scattering operators” In _Dokl. Akad. Nauk SSSR_ 144, 1962, pp. 475–478
  * [BKS91] M.. Birman, G.. Karadzhov and M.. Solomyak “Boundedness conditions and spectrum estimates for the operators $b(X)a(D)$ and their analogs” In _Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90)_ 7, Adv. Soviet Math. Amer. Math. Soc., Providence, RI, 1991, pp. 85–106
  * [BM18] Vincent Bruneau and Pablo Miranda “Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian” In _J. Funct. Anal._ 274.9, 2018, pp. 2499–2531 DOI: 10.1016/j.jfa.2017.10.007
  * [BR20] Vincent Bruneau and Georgi Raikov “Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians” In _Ann. Henri Poincaré_ 21.5, 2020, pp. 1451–1488 DOI: 10.1007/s00023-020-00904-6
  * [BS70] M.. Birman and M.. Solomjak “Asymptotics of the spectrum of weakly polar integral operators” In _Izv. Akad. Nauk SSSR Ser. Mat._ 34, 1970, pp. 1142–1158
  * [BS77] M.. Birman and M.. Solomjak “Estimates for the singular numbers of integral operators” In _Uspehi Mat. Nauk_ 32.1(193), 1977, pp. 17–84271
  * [BS77a] M.. Birman and M.. Solomjak “Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols” In _Vestnik Leningrad. Univ._ , 1977, pp. 13–21169
  * [BS87] M.. Birman and M.. Solomjak “Spectral theory of selfadjoint operators in Hilbert space” Translated from the 1980 Russian original by S. Khrushchëv and V. Peller, Mathematics and its Applications (Soviet Series) D. Reidel Publishing Co., Dordrecht, 1987, pp. xv+301
  * [DDR19] Radouan Daher, Julio Delgado and Michael Ruzhansky “Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds” In _Monatsh. Math._ 189.1, 2019, pp. 23–49 DOI: 10.1007/s00605-018-1253-0
  * [Eck45] Beno Eckmann “Harmonische Funktionen und Randwertaufgaben in einem Komplex” In _Comment. Math. Helv._ 17, 1945, pp. 240–255
  * [FR04] Claudio Fernández and Georgi Raikov “On the singularities of the magnetic spectral shift function at the Landau levels” In _Ann. Henri Poincaré_ 5.2, 2004, pp. 381–403 DOI: 10.1007/s00023-004-0173-9
  * [Gra08] Loukas Grafakos “Classical Fourier analysis” 249, Graduate Texts in Mathematics Springer, New York, 2008, pp. xvi+489
  * [IJ19] Kenichi Ito and Arne Jensen “Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacians” In _J. Funct. Anal._ 277.4, 2019, pp. 965–993 DOI: 10.1016/j.jfa.2019.05.018
  * [IJ21] Kenichi Ito and Arne Jensen “Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions” In _Integral Equations Operator Theory_ 93.3, 2021, pp. Paper No. 3228 DOI: 10.1007/s00020-021-02648-2
  * [IK12] Hiroshi Isozaki and Evgeny Korotyaev “Inverse problems, trace formulae for discrete Schrödinger operators” In _Ann. Henri Poincaré_ 13.4, 2012, pp. 751–788 DOI: 10.1007/s00023-011-0141-0
  * [Kre53] M.. Krein “On the trace formula in perturbation theory” In _Mat. Sbornik N.S._ 33(75), 1953, pp. 597–626
  * [KS22] Evgeny Korotyaev and Natalia Saburova “Trace formulas for Schrödinger operators on periodic graphs” In _J. Math. Anal. Appl._ 508.2, 2022, pp. Paper No. 12588833 DOI: 10.1016/j.jmaa.2021.125888
  * [Par17] D. Parra “Spectral and scattering theory for Gauss–Bonnet operators on perturbed topological crystals” In _Journal of Mathematical Analysis and Applications_ 452.2, 2017, pp. 792–813 DOI: http://dx.doi.org/10.1016/j.jmaa.2017.03.002
  * [Pus98] A. Pushnitski “Representation for the spectral shift function for perturbations of a definite sign” In _St.Petersburg Math. J._ 6, 1998, pp. 1181–1194
  * [Rob99] Didier Robert “Semiclassical asymptotics for the spectral shift function” In _Differential operators and spectral theory_ 189, Amer. Math. Soc. Transl. Ser. 2 Amer. Math. Soc., Providence, RI, 1999, pp. 187–203 DOI: 10.1090/trans2/189/15
  * [RT09] M. Ruzhansky and V. Turunen “Pseudo-Differential Operators and Symmetries: Background Analysis and Advanced Topics”, Pseudo-Differential Operators Springer Basel AG, 2009 URL: http://books.google.fr/books?id=DDpz%5C%5C%5C%5C_MfxZrUC
  * [Tie11] Rafael Tiedra de Aldecoa “Asymptotics near $\pm m$ of the spectral shift function for Dirac operators with non-constant magnetic fields” In _Comm. Partial Differential Equations_ 36.1, 2011, pp. 10–41 DOI: 10.1080/03605301003758369
  * [Yaf10] D.. Yafaev “Mathematical scattering theory. Analytic theory” Analytic theory 158, Mathematical Surveys and Monographs American Mathematical Society, Providence, RI, 2010, pp. xiv+444 DOI: 10.1090/surv/158
  * [Yaf92] D.. Yafaev “Mathematical scattering theory” General theory, Translated from the Russian by J. R. Schulenberger 105, Translations of Mathematical Monographs American Mathematical Society, Providence, RI, 1992, pp. x+341 DOI: 10.1090/mmono/105