# On the spectrum of the double-layer operator on locally-dilation-invariant
Lipschitz domains

Simon N. Chandler-Wilde, Raffael Hagger, Karl-Mikael Perfekt, Jani A.
Virtanen11footnotemark: 1 Department of Mathematics and Statistics, University
of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK, s.n.chandler-
<EMAIL_ADDRESS><EMAIL_ADDRESS>Seminar,
Christian-Albrechts-Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel,
Germany<EMAIL_ADDRESS>of Mathematical Sciences, Norwegian
University of Science and Technology (NTNU), NO-7491 Trondheim, Norway, karl-
<EMAIL_ADDRESS>
###### Abstract

We say that $\Gamma$, the boundary of a bounded Lipschitz domain, is locally
dilation invariant if, at each $x\in\Gamma$, $\Gamma$ is either locally
$C^{1}$ or locally coincides (in some coordinate system centred at $x$) with a
Lipschitz graph $\Gamma_{x}$ such that $\Gamma_{x}=\alpha_{x}\Gamma_{x}$, for
some $\alpha_{x}\in(0,1)$. In this paper we study, for such $\Gamma$, the
essential spectrum of $D_{\Gamma}$, the double-layer (or Neumann-Poincaré)
operator of potential theory, on $L^{2}(\Gamma)$. We show, via localisation
and Floquet-Bloch-type arguments, that this essential spectrum is the union of
the spectra of related continuous families of operators $K_{t}$, for
$t\in[-\pi,\pi]$; moreover, each $K_{t}$ is compact if $\Gamma$ is $C^{1}$
except at finitely many points. For the 2D case where, additionally, $\Gamma$
is piecewise analytic, we construct convergent sequences of approximations to
the essential spectrum of $D_{\Gamma}$; each approximation is the union of the
eigenvalues of finitely many finite matrices arising from Nyström-method
approximations to the operators $K_{t}$. Through error estimates with explicit
constants, we also construct functionals that determine whether any particular
locally-dilation-invariant piecewise-analytic $\Gamma$ satisfies the well-
known spectral radius conjecture, that the essential spectral radius of
$D_{\Gamma}$ on $L^{2}(\Gamma)$ is $<1/2$ for all Lipschitz $\Gamma$. We
illustrate this theory with examples; for each we show that the essential
spectral radius is $<1/2$, providing additional support for the conjecture. We
also, via new results on the invariance of the essential spectral radius under
locally-conformal $C^{1,\beta}$ diffeomorphisms, show that the spectral radius
conjecture holds for all Lipschitz curvilinear polyhedra.

##  1 Introduction

Given a bounded Lipschitz domain111For us, as, e.g., in [46], “domain” will
just mean “open set”; a domain need not be connected.
$\Omega_{-}\subset\mathbb{R}^{d},d\geq 2$, with boundary $\Gamma$ and outward-
pointing unit normal vector $n$, the interior and exterior Dirichlet and
Neumann problems for Laplace’s equation (posed in $\Omega_{-}$ and in
$\Omega_{+}:=\mathbb{R}^{d}\setminus\overline{\Omega_{-}}$, respectively), can
be reformulated as boundary integral equations involving the operators

$D_{\Gamma}\pm\textstyle{\frac{1}{2}}I\quad\text{ and }\quad
D^{\prime}_{\Gamma}\pm\textstyle{\frac{1}{2}}I$ (1.1)

(see, e.g., [63], [45, §5.9, §5.15.1]), where the _double-layer (or Neumann or
Neumann-Poincaré) operator_ $D_{\Gamma}$ and the _adjoint double-layer
operator_ $D_{\Gamma}^{\prime}$ are defined by

$D_{\Gamma}\phi(x)=\int_{\Gamma}\frac{\partial\Phi(x,y)}{\partial
n(y)}\phi(y)\,\mathrm{d}s(y)\quad\text{ and }\quad
D_{\Gamma}^{\prime}\phi(x)=\int_{\Gamma}\frac{\partial\Phi(x,y)}{\partial
n(x)}\phi(y)\,\mathrm{d}s(y),$ (1.2)

for $\phi\in{L^{2}(\Gamma)}$ and (almost all) $x\in\Gamma$, with the integrals
understood, in general, as Cauchy principal values. Here $\Phi(x,y)$ is the
fundamental solution for Laplace’s equation, defined by222Our sign convention
and normalisation are those of many authors (e.g. [61, 40]), but other authors
(e.g. [34]), use a fundamental solution that is the negative of ours.

$\Phi(x,y):=\frac{1}{2\pi}\log\left(\frac{1}{|x-y|}\right),\;\;d=2,\;\;\quad:=\frac{1}{(d-2)c_{d}}\frac{1}{|x-y|^{d-2}},\;\;d\geq
3,$ (1.3)

where $c_{d}$ is the surface measure of the unit sphere in $\mathbb{R}^{d}$.
Explicitly,

$D_{\Gamma}\phi(x)=\frac{1}{c_{d}}\int_{\Gamma}\frac{(x-y)\cdot
n(y)}{|x-y|^{d}}\phi(y)\,\mathrm{d}s(y)\quad\text{ and }\quad
D_{\Gamma}^{\prime}\phi(x)=\frac{1}{c_{d}}\int_{\Gamma}\frac{(y-x)\cdot
n(x)}{|x-y|^{d}}\phi(y)\,\mathrm{d}s(y),$ (1.4)

for $\phi\in{L^{2}(\Gamma)}$ and (almost all) $x\in\Gamma$.

Complementing (1.1),

$D_{\Gamma}^{\prime}-\lambda I,$ (1.5)

with $\lambda\in\mathbb{C}\setminus\\{-\frac{1}{2},\frac{1}{2}\\}$, arises as
the operator in the boundary integral equation reformulation of transmission
problems in electrostatics, where the Laplace equation $\Delta u=0$ holds in
$\Omega_{-}$ and $\Omega_{+}$ and the trace of $u$ or its normal derivative
jumps across $\Gamma$ (see, e.g., [45, §5.12]). In this context the case
$|\lambda|\geq 1/2$, especially with $\lambda$ real, is classically of
interest (e.g., [45, §5.12]); more recently the case where $\lambda$ is
complex with $|\lambda|<1/2$ has been studied intensively as a model of quasi-
static electromagnetic plasmonic problems (e.g., [1, 27, 58, 15]).

Motivated by these physical applications, and by questions in harmonic
analysis, there has been long-standing interest in the computation of the
spectrum and essential spectrum333Given a Banach space $Y$ and a bounded
linear operator $T\colon Y\to Y$ we denote the spectrum of $T$, the set of
$\lambda\in\mathbb{C}$ for which $T-\lambda I$ is not invertible, by
$\sigma(T;Y)$, and the essential spectrum, the set of $\lambda$ for which
$T-\lambda T$ is not Fredholm, by $\sigma_{\operatorname{ess}}(T;Y)$,
abbreviating these by $\sigma(T)$ and $\sigma_{\operatorname{ess}}(T)$ where
the Banach space $Y$ is clear from the context. of $D_{\Gamma}$ as an operator
on a variety of function spaces, especially for non-smooth domains (e.g., [38,
18, 34, 47, 12, 52, 1, 27, 51]). The largest part of this literature is
concerned specifically with the 2D/3D cases where $\Gamma$ is a (curvilinear)
polygon (e.g., [59, 10, 60, 48, 49]) or polyhedron (e.g., [55, 16, 47, 20, 26,
49, 27, 15]). In this paper we will study and compute the essential spectrum
of $D_{\Gamma}$ as an operator on $L^{2}(\Gamma)$ for a substantially larger
class of boundaries, namely for the case where the boundary $\Gamma$ is
locally dilation invariant in the sense of Definition 1.3 below. In 2D (3D)
this class includes polygons (polyhedra) but it also admits much wilder
boundary behaviour (e.g., Figure 1) as we discuss next in §1.1.

###  1.1 The spectral radius conjecture and the main question we address

Given a bounded linear operator $T\colon Y\to Y$ on a Banach space $Y$, we
define its spectral radius, $\rho(T;Y)$, and its essential spectral radius,
$\rho_{\operatorname{ess}}(T;Y)$, by

$\rho(T;Y):=\sup_{\lambda\in\sigma(T;Y)}|\lambda|\quad\mbox{and}\quad\rho_{\operatorname{ess}}(T;Y):=\sup_{\lambda\in\sigma_{\operatorname{ess}}(T;Y)}|\lambda|,$
(1.6)

abbreviating $\rho(T;Y)$ and $\rho_{\operatorname{ess}}(T;Y)$ by $\rho(T)$ and
$\rho_{\operatorname{ess}}(T)$, respectively, when the space $Y$ is clear from
the context. The analysis and computation we will carry out are motivated by
the so-called spectral radius conjecture.

This conjecture, in the explicit 1994 formulation of Kenig [34], is as
follows, where $L^{2}_{0}(\Gamma):=\\{\phi\in
L^{2}(\Gamma):\int_{\Gamma}\phi\,\mathrm{d}s=0\\}$.

######  Conjecture 1.1.

If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega_{-}$ and is
connected, the spectral radius of $D_{\Gamma}^{\prime}$ on $L^{2}_{0}(\Gamma)$
is $<\frac{1}{2}$, i.e.
$\rho(D^{\prime}_{\Gamma};L^{2}_{0}(\Gamma))<\frac{1}{2}$.

In Section 2 we will discuss the following alternative formulation of the
conjecture which makes sense regardless of the connectivity of $\Gamma$, and
show its equivalence with Conjecture 1.1.

######  Conjecture 1.2.

If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega_{-}$, the
essential spectral radius of $D_{\Gamma}$ on $L^{2}(\Gamma)$ is
$<\frac{1}{2}$, i.e.
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))<\frac{1}{2}$.

The spectral radius conjecture is very well studied, owing to its intrinsic
interest in harmonic analysis, its possible relevance for
computation444Notably, if the spectral radius conjecture holds then the
Neumann series for $(D^{\prime}_{\Gamma}\pm\frac{1}{2}I)\phi=g$, equivalently
the Neumann iteration
$\pm\frac{1}{2}\phi^{(n)}=g-D^{\prime}_{\Gamma}\phi^{(n-1)}$,
$n\in\mathbb{N}$, converges in $L^{2}_{0}(\Gamma)$., and its immediate role
within electrostatics, as well as in interpretations of electrodynamical
problems [26]. It originated in the setting of continuous functions in
$\overline{\Omega_{-}}$, dating all the way back to C. Neumann in the late
1800s, who treated convex domains, and to Radon [54], who famously analyzed
curves of bounded rotation. In a tour de force, Král [38] completely
characterized when the essential norm of $D_{\Gamma}$ is $<1/2$. This result
was extended into higher dimensions by Burago and Maz’ya [8] and Netuka [50].
For polyhedra in 3D, the essential norm can be $>1/2$. Nevertheless, Rathsfeld
[55] (and see [56]) and Grachev and Maz’ya [20] independently proved the
spectral radius conjecture in the continuous setting holds for general
polyhedra. These results were extended to locally conformal deformations of
polyhedra by Medková [44]. Even when specialised to the continuous setting,
the history is vast, and we refer to [64] for an in-depth survey.

As modern harmonic analysis developed, the natural setting for the double-
layer potential shifted toward $L^{2}(\Gamma)$. We make particular mention of
the demonstrations of $L^{2}$-boundedness of the Cauchy integral due to
Calderón [9] and Coifman, Mcintosh, and Meyer [13], and Verchota’s [63]
application of these results to study invertibility of
$D^{\prime}_{\Gamma}\pm\frac{1}{2}I$ on $L^{2}(\Gamma)$ and
$L^{2}_{0}(\Gamma)$ when $\Gamma$ is connected. Since then, a flurry of
activity and findings in this area have provided support for Kenig’s
conjecture, though a complete proof has proved elusive.

Indeed, to the best of our knowledge, Conjecture 1.1 has been established
(only) in the following cases: a) $\Omega_{-}$ is convex [18] (and see [12]
for extensions to locally convex domains); b) $\Omega_{-}$ has small Lipschitz
character555See, e.g., [11, Definition 3.1] for the definition of the
Lipschitz character of a Lipschitz domain. [47], a case which includes all
$C^{1}$ domains [19]; c) $\varepsilon$-regular Semmes–Kenig–Toro domains
$\Omega_{-}$ for sufficiently small $\varepsilon>0$ [30], including in
particular all domains whose gradient has vanishing mean oscillation [28]; d)
$\Omega_{-}$ is a polygon or curvilinear polygon in 2D [59, 60], or a
Lipschitz polyhedron in 3D [16].

(a)

(b)

Figure 1: Examples of bounded Lipschitz domains $\Omega_{-}$ with boundaries
$\Gamma\in\mathscr{D}_{\mathcal{A}}\subset\mathscr{D}$ that are piecewise
analytic locally dilation invariant.

Note that polygonal and polyhedral boundaries $\Gamma$ are locally invariant
under all dilations: at every point $x\in\Gamma$, $\Gamma$ locally coincides
with a graph $\Gamma_{x}$ such that (in some local coordinate system centred
at $x$) $\Gamma_{x}=\alpha\Gamma_{x}$ for all $\alpha>0$. In this paper we
will investigate domains where, locally, the dilation invariance
$\Gamma_{x}=\alpha_{x}\Gamma_{x}$ only holds for one $\alpha_{x}\in(0,1)$ (in
which case we say that $\Gamma$ is locally dilation invariant at $x$).
Precisely our focus will be on the following class of boundaries.

######  Definition 1.3 (Locally dilation invariant).

Given $\Gamma$, the boundary of a Lipschitz domain $\Omega_{-}$, we say that
$\Gamma\in\mathscr{D}$, the set of locally-dilation-invariant boundaries if,
at every $x\in\Gamma$, $\Gamma$ is either locally $C^{1}$ or locally coincides
(in some coordinate system centred at $x$) with a Lipschitz graph $\Gamma_{x}$
which is dilation invariant with respect to some $\alpha_{x}\in(0,1)$, i.e.
$\Gamma_{x}=\alpha_{x}\Gamma_{x}$.

Note that, already in 2D, $\mathscr{D}$ is a hugely larger class of domains
than that of the curvilinear polygons. Indeed, for a Jordan curve
$\Gamma\subset\mathbb{R}^{2}$ and $x\in\Gamma$, let

$v_{\Gamma}(x)=\int_{0}^{2\pi}|\\{x+re^{i\theta}\in\Gamma\,:\,r>0\\}|\,\frac{d\theta}{2\pi},$

where $\left\lvert\cdot\right\rvert$ in this equation denotes the counting
measure of a set. Then $D_{\Gamma}\phi$ is uniformly continuous on
$\Omega_{-}$ for every $\phi\in C(\Gamma)$ if and only if

$\sup_{x\in\Gamma}v_{\Gamma}(x)<\infty,$ (1.7)

see [7, 36, 37]. If $\Gamma$ is Lipschitz and locally dilation invariant at
$x$, then it is clear that $v_{\Gamma}(x)=\infty$, unless $\Gamma$ coincides
with two line segments around $x$. It follows that the only curves
$\Gamma\in\mathscr{D}$ satisfying (1.7) are curvilinear polygons. That is,
except for curvilinear polygons, our curves exhibit such wild boundary
behaviour that it is not possible to consider $D_{\Gamma}$ in the setting of
continuous boundary data.

In the context of studying the double-layer and related operators, the class
of domains $\mathscr{D}$ seems to have been first considered in [11], where
the essential numerical range666Recall that, for a bounded linear operator
$T\colon H\to H$ on a Hilbert space $H$, the numerical range of $T$ is
$W(T):=\left\\{\left\langle
T\phi,\phi\right\rangle:\left\lVert\phi\right\rVert=1\right\\}$ and its
essential numerical range is
$W_{\operatorname{ess}}(T):=\bigcap\limits_{K\text{
compact}}\overline{W(T+K)}$. of $D_{\Gamma}$ was studied and, in 2D and 3D,
examples of boundaries $\Gamma\in\mathscr{D}$ with large Lipschitz character
were constructed such that $D_{\Gamma}\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$
has arbitrarily large essential numerical radius,

$w_{\operatorname{ess}}(D_{\Gamma}):=\sup\limits_{z\in
W_{\operatorname{ess}}(D_{\Gamma})}|z|,$

and so also arbitrarily large essential norm,

$\|D_{\Gamma}\|_{\operatorname{ess}}:=\inf_{K\mbox{ \small
compact}}\|D_{\Gamma}-K\|,$

since, for any bounded operator $T$ on a Hilbert space [22, §1.3],

$\textstyle{\frac{1}{2}}\|T\|_{\operatorname{ess}}\leq
w_{\operatorname{ess}}(T)\leq\|T\|_{\operatorname{ess}}.$ (1.8)

The 2D examples in [11] (see, e.g., [11, Fig. 3] and cf. Figure 1) with
arbitrarily large $w_{\operatorname{ess}}(D_{\Gamma})$ are necessarily
examples for which (1.7) fails to hold, since for curvilinear polygons on
$L^{2}(\Gamma)$ it is well-known that

$\|D_{\Gamma}\|_{\operatorname{ess}}=w_{\operatorname{ess}}(D_{\Gamma})=\rho(D_{\Gamma};L^{2}(\Gamma))<\textstyle{\frac{1}{2}}$

(see [59, 10, 60, 48] for $\|D_{\Gamma}\|_{\operatorname{ess}}$ and
$\rho(D_{\Gamma};L^{2}(\Gamma))$; equality for
$w_{\operatorname{ess}}(D_{\Gamma})$ follows by (1.8) and since
$W_{\operatorname{ess}}(D_{\Gamma})\supset\sigma_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))$
[6]).

A natural question, prompted by the examples from [11] with
$w_{\operatorname{ess}}(D_{\Gamma})$ arbitrarily large and the spectral radius
conjecture, is the following:

> Given that there exist $\Gamma\in\mathscr{D}$ with
> $w_{\operatorname{ess}}(D_{\Gamma})\gg\frac{1}{2}$, in particular such
> examples in 2D, is there a $\Gamma\in\mathscr{D}$, in particular an example
> in 2D, with
> $\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))\geq\frac{1}{2}$?

Of course, a positive answer would provide a counterexample to Conjecture 1.2
and hence to the original spectral radius conjecture, Conjecture 1.1. The aim
of this paper is to address this question, through mathematical analysis and
computational methods supported by numerical analysis error estimates where
constants are made explicit. These will enable us to estimate
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))$ for a large class of
$\Gamma\in\mathscr{D}$ sufficiently accurately to determine whether or not
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))\geq\frac{1}{2}$.

###  1.2 Our main results and their significance

The first step in our analysis is Theorem 5.2 in §5.1, the localisation result
(cf. [16, 47, 15, 11]) that, for $\Gamma\in\mathscr{D}$ (and for any dimension
$d\geq 2$), there exists a finite set $F\subset\Gamma$ such that

$\sigma_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))=\bigcup_{x\in
F}\sigma_{\operatorname{ess}}(D_{\Gamma_{x}};L^{2}({\Gamma_{x}})).$ (1.9)

In the above formula $\Gamma_{x}$ is defined as above if $\Gamma$ is locally
dilation invariant at $x$, while if $\Gamma$ is locally $C^{1}$ at $x$ then
$\Gamma_{x}$ is the graph of a $C^{1}$ compactly supported function so that
[19] $D_{\Gamma_{x}}$ is compact and
$\sigma_{\operatorname{ess}}(D_{\Gamma_{x}};L^{2}({\Gamma_{x}}))=\\{0\\}$.

The localisation (1.9) reduces the computation of
$\sigma_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))$ to that of
$\sigma_{\operatorname{ess}}(D_{\Gamma_{x}};L^{2}({\Gamma_{x}}))$ for finitely
many $x$ for which $\alpha_{x}\Gamma_{x}=\Gamma_{x}$, for some
$\alpha_{x}\in(0,1)$. Computation of
$\sigma_{\operatorname{ess}}(D_{\Gamma_{x}};L^{2}({\Gamma_{x}}))$ for such $x$
is our focus in §4 where we study and compute spectral properties of
$D_{\Gamma}$ in the case that $\Gamma\subset\mathbb{R}^{d}$ is a dilation
invariant Lipschitz graph, meaning that there exists an $\alpha\in(0,1)$ such
that $\alpha\Gamma=\Gamma$ (an example is Figure 3). If
$\Gamma_{0}\subset\Gamma\setminus\\{0\\}$ is a particular relatively closed
and bounded Lipschitz subgraph of $\Gamma$ such that

$\Gamma=\bigcup_{j\in\mathbb{Z}}\alpha^{j}\Gamma_{0},$ (1.10)

and such that $(\alpha^{j}\Gamma_{0})\cap\Gamma_{0}$ has zero surface measure
for $j\neq 0$, we show that $D_{\Gamma}$ can be written as a discrete
$\ell_{1}$ convolution operator whose entries are bounded linear operators on
$L^{2}(\Gamma_{0})$ related to the discretization (1.10) of $\Gamma$. This
allows us to decompose $D_{\Gamma}$, by a Floquet-Bloch transform, into a
continuous family $(K_{t})_{t\in[-\pi,\pi]}$ of operators $K_{t}\colon
L^{2}(\Gamma_{0})\to L^{2}(\Gamma_{0})$, and, by standard results for such
convolutions (see [53, Thm. 2.3.25]), to characterise the essential spectrum
of $D_{\Gamma}$ as (Theorem 4.4)

$\sigma_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))=\sigma(D_{\Gamma};L^{2}(\Gamma))=\bigcup_{t\in[-\pi,\pi]}\sigma(K_{t};L^{2}(\Gamma_{0})).$
(1.11)

This characterisation is particularly useful when, apart from a singularity at
0, $\Gamma$ is $C^{1}$ (see, e.g., Figure 3), for then (Corollary 4.6) each
$K_{t}$ is compact, and so has a more easily computed discrete spectrum. The
characterisation (1.11) holds for every Lipschitz dilation invariant graph in
any dimension; in §4.3 we specialise to the 2D case where $\Gamma$, except at
$0$, is the graph of a real analytic function (see, e.g., Figure 3); we denote
by $\mathcal{A}$ this subset of the 2D dilation invariant graphs. We show, for
each $t\in[-\pi,\pi]$, that $K_{t}$ is unitarily equivalent to
$\tilde{K}_{t}$, a $2\times 2$ matrix of integral operators on $L^{2}(0,1)$
that have real analytic kernels. Further, the range of each of these integral
operators is a space of $1$-quasi-periodic real analytic functions. As a
consequence,

$\sigma(K_{t};L^{2}(\Gamma_{0}))=\sigma(\tilde{K}_{t};(C[0,1])^{2}),$

and this latter spectrum can be computed by approximating $\tilde{K}_{t}$ by a
$2N\times 2N$ matrix $\tilde{A}_{t,N}$ obtained by a simple midpoint-rule
based Nyström discretization. As is well-known (see [40, 62], Theorem 4.11),
the midpoint rule is exponentially convergent for periodic analytic functions,
so that it follows from Nyström-method spectral estimates for integral
operators with continuous kernels [3] that, for each $t$, the eigenvalues of
$A_{t,N}$ converge at an exponential rate to those of $\tilde{K}_{t}$ as
$N\to\infty$.

This leads (see Theorem 4.19) to a Nyström approximation,
$\sigma^{N}(D_{\Gamma})$, for
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$, which is
$\\{0\\}$ plus the union of the eigenvalues of finitely many $2N\times 2N$
matrices. Our first main result is to show, as Theorem 4.19, that
$\sigma^{N}(D_{\Gamma})\to\sigma(D_{\Gamma})$ in the Hausdorff metric as
$N\to\infty$; that this convergence is achievable is somewhat surprising given
that $D_{\Gamma}$ is neither compact nor self-adjoint. Our second, and more
substantial result (Theorem 4.21) is that we develop a fully discrete
algorithm to test whether, as an operator on $L^{2}(\Gamma)$,
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$. Precisely, we construct
(see Remark 4.22), for each $c>0$, a nonlinear functional
$S_{c}:\mathcal{A}\times\mathbb{N}\times\mathbb{N}\times\mathbb{N}\to\mathbb{R}$
with the properties that: a) the functional can be computed in finitely many
arithmetic operations and finitely many evaluations of elementary functions,
given finitely many sampled values of $f$ and its first and second derivatives
for real arguments, plus bounds on the analytic continuation of $f$ to a
neighbourhood of the real line that depends on $c$; b)
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ if, for some $c>0$ and
$m,M,N\in\mathbb{N}$, it holds that:

  1. i)

$\rho(A^{M}_{t_{k},N})<\frac{1}{2}$, for $k=1,\ldots,m$, where
$t_{k}:=(k-1/2)\pi/m$ and $A^{M}_{t_{k},N}$ is a specific approximation to
$A_{t_{k},N}$ depending on the parameter $M$;

  2. ii)

$S_{c}(\Gamma,m,M,N)<0$.

Conversely, if $\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$, then, for
all sufficiently small $c>0$, i) and ii) hold for all sufficiently large
$N,M,m\in\mathbb{N}$.

In §6 we bring these results together to address our question at the end of
§1.1. We restrict attention to the following 2D class of domains; this class
includes polygons and piecewise-analytic curvilinear polygons, but also wilder
boundary behaviour as illustrated in Figure 1.

######  Definition 1.4 (Piecewise analytic locally dilation invariant).

In the case $d=2$ we say that $\Gamma\in\mathscr{D}_{\mathcal{A}}$, the set of
piecewise-analytic locally-dilation-invariant boundaries, if
$\Gamma\in\mathscr{D}$ and $\Gamma$ is locally analytic (i.e., is locally the
graph of a real-analytic function) at all but finitely many $x\in\Gamma$.

Given $\Gamma\in\mathscr{D}_{\mathcal{A}}$, if $F\subset\Gamma$ is the finite
set of points at which $\Gamma$ is not locally analytic and is locally
dilation invariant, it is easy to see that $\Gamma_{x}\in\mathcal{A}$ for
$x\in F$ and that (1.9) holds for this set $F$. Thus (see Theorem 6.1)

$\Sigma^{N}(D_{\Gamma}):=\bigcup_{x\in
F}\sigma^{N}(D_{\Gamma_{x}})\to\sigma_{\operatorname{ess}}(D_{\Gamma})$ (1.12)

in the Hausdorff metric as $N\to\infty$, and note that
$\Sigma^{N}(D_{\Gamma})$ is the union of the eigenvalues of finitely many
$2N\times 2N$ matrices. Further, as we discuss in §6, it follows that
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ if these eigenvalues lie
within the disc of radius $\frac{1}{2}$, i.e. if

$R_{N}(D_{\Gamma}):=\max\\{|z|:z\in\Sigma^{N}(D_{\Gamma})\\}<\textstyle{\frac{1}{2}},$
(1.13)

and if also, for some $c>0$,

$\mathscr{S}_{c}(\Gamma,N):=\max_{x\in F}S_{c}(\Gamma_{x},N,N,N)<0,$ (1.14)

where $\mathscr{S}_{c}(\Gamma,N)$ can be computed in finitely many arithmetic
operations plus finitely many evaluations of elementary functions, given
inputs describing each $\Gamma_{x}$ as discussed above. Conversely, if
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ then, for all sufficiently
small $c>0$ and all sufficiently large $N$, $R_{N}(D_{\Gamma})<\frac{1}{2}$
and $\mathscr{S}_{c}(\Gamma,N)<0$. Thus, given inputs describing $\Gamma$, our
fully discrete algorithms enable us to test, for individual
$\Gamma\in\mathscr{D}_{\mathcal{A}}$, the validity of Conjecture 1.2, i.e.
whether or not $\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$, through
computation of the eigenvalues of finitely many finite matrices, plus finitely
many additional arithmetic operations.

In §5 we prove the localisation result (1.9). We also, in the spirit of
Medková’s study in the continuous setting [42, 44], consider the stability of
Conjecture 1.2 under locally conformal deformations. We prove that the
spectral radius conjecture is independent of such deformations, under the
assumption that the absolute value of the kernel of the double-layer potential
also defines a bounded operator on $L^{2}(\Gamma)$. While this additional
hypothesis may fail to include the wildest of boundaries, it applies to many
domains from $\mathscr{D}$. For example, as we discuss in §5.2, it is
satisfied by any dilation invariant Lipschitz graph $\Gamma$ whose generating
set $\Gamma_{0}$ is polygonal or polyhedral, in particular it holds if
$\Gamma$ is a polyhedron and, in 2D, if $\Gamma\in\mathscr{D}_{\mathcal{A}}$.
As one consequence (Corollary 5.7), Conjecture 1.2 holds for Lipschitz
curvilinear polyhedra, because [16] it holds for polyhedra; as another
(Corollary 5.11), if it holds for $\Gamma\in\mathscr{D}_{\mathcal{A}}$, then
it holds for any locally conformal deformation of $\Gamma$.

To illustrate the above results, and use them to test Conjecture 1.2 and
address our main question from §1.1, we include a range of numerical examples
in §6, for piecewise-analytic Lipschitz graphs $\Gamma\in\mathcal{A}$, and for
piecewise-analytic locally-dilation-invariant
$\Gamma\in\mathscr{D}_{\mathcal{A}}$ that are the boundaries of bounded
Lipschitz domains. For each example we plot an approximation
($\sigma^{N}(D_{\Gamma})$ or $\Sigma^{N}(D_{\Gamma})$, as appropriate) to
$\sigma_{\operatorname{ess}}(D_{\Gamma})$. Moreover, we employ the algorithms
described above to provide convincing numerical evidence that
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))<\frac{1}{2}$ in every
case we examine, including cases where $w_{\operatorname{ess}}(D_{\Gamma})$ is
significantly $>\frac{1}{2}$. These results are evidence that the spectral
radius conjecture holds for the class of 2D domains
$\mathscr{D}_{\mathcal{A}}$; we emphasise again that this conjecture has not
been studied previously for boundaries in this class, except for the special
case of curvilinear polygons.

Let us briefly summarise the remainder of the paper. In §2 we prove the
equivalence of Conjectures 1.1 and 1.2. In §3, as a key step to our main
results, we derive bounds for the spectral radii of general compact operators
in §3.1 (e.g., Corollary 3.5), specialising to the case of integral operators
approximated by the Nyström method in §3.2 (e.g., Theorem 3.8). In §4 we prove
the results noted above on the essential spectrum and essential spectral
radius of $D_{\Gamma}$ in the case when $\Gamma$ is a dilation-invariant
Lipschitz graph, with particular focus (§4.2–§4.4) on the 2D piecewise-
analytic case. In §5 we prove our localisation results. In §6 we bring the
earlier results together, in particular to study the spectral radius
conjecture for $\Gamma\in\mathscr{D}_{\mathcal{A}}$, and we illustrate our
theory by numerical examples.

##  2 Formulations of the spectral radius conjecture

Our results are related to the spectral radius conjecture of Kenig [34,
Problem 3.2.12], that
$\rho(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))<\frac{1}{2}$ if $\Gamma$ is the
boundary of a bounded Lipschitz domain. Conjecture 1.1, stated in the
introduction, is a version of this conjecture that avoids difficulties when
$\Gamma$ is not connected777By results of D. Mitrea [46, Theorem 4.1], if
$\Omega_{-}$ or $\Omega_{+}$ are not connected,
$\\{\frac{1}{2},-\frac{1}{2}\\}\cap\sigma(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))\neq\emptyset$.
Indeed, $D_{\Gamma}^{\prime}\pm\frac{1}{2}I$ are Fredholm of index zero on
$L^{2}(\Gamma)$ by part (2) of [46, Theorem 4.1], but by (4) of the same
theorem, and since $D_{\Gamma}(1)=-\frac{1}{2}$ so that
$D_{\Gamma}^{\prime}(L^{2}_{0}(\Gamma))\subset L^{2}_{0}(\Gamma)$ (see (2.1)
below), the codimension of $D_{\Gamma}^{\prime}-\frac{1}{2}I$ on
$L^{2}_{0}(\Gamma)$ is $\geq 1$ if $\Omega_{+}$ is not connected; that of
$D_{\Gamma}^{\prime}+\frac{1}{2}I$ is $\geq 1$ if $\Omega_{-}$ is not
connected.. In this section we show the equivalence between Conjecture 1.1 and
Conjecture 1.2, also stated in the introduction. Conjecture 1.2 concerns the
essential spectrum rather than the spectrum and makes sense whatever the
connectedness of $\Gamma$, $\Omega_{-}$, and $\Omega_{+}$.

Of course, because $D_{\Gamma}^{\prime}$ is the adjoint of $D_{\Gamma}$ (as an
operator on $L^{2}(\Gamma)$), and so shares the same essential spectrum, a
statement equivalent to Conjecture 1.2 is that
$\rho_{\operatorname{ess}}(D_{\Gamma}^{\prime};L^{2}(\Gamma))<\frac{1}{2}$.
Further, as bounded Lipschitz domains have only finitely many boundary
components, it is clear that Conjecture 1.2 is true if it is true whenever
$\Gamma$ is connected888Note that if $\Gamma_{1}$ and $\Gamma_{2}$ are
separate components of $\Gamma$, so that there is a positive distance between
$\Gamma_{1}$ and $\Gamma_{2}$, the double-layer operator from
$L^{2}(\Gamma_{1})$ to $L^{2}(\Gamma_{2})$ has a kernel that is bounded, so is
a Hilbert-Schmidt operator and hence compact.. Thus the equivalence of
Conjectures 1.1 and 1.2, i.e. that Conjecture 1.1 holds (as claimed, whenever
$\Gamma$ is connected) if and only if Conjecture 1.2 holds (as claimed,
whatever the topology of $\Gamma$), is implied by the following lemma.

######  Lemma 2.1.

Assume that $\Gamma$ is the boundary of a bounded Lipschitz domain
$\Omega_{-}$ and is connected. Then
$\rho(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))<\frac{1}{2}$ if and only if
$\rho_{\operatorname{ess}}(D_{\Gamma}^{\prime};L^{2}(\Gamma))<\frac{1}{2}$.

The main step in the proof of this lemma is the following theorem stated in
[18]999In [18, Theorem 1.1] slightly more is claimed, that the eigenvalues of
$D_{\Gamma}^{\prime}$ lie in $[-\frac{1}{2},\frac{1}{2})$. This is more than
is claimed in [33, Chapter IX, §11], and in fact $\frac{1}{2}$ is an
eigenvalue if $\Omega_{+}$ is not connected; see [46, Theorem 4.1].. As noted
in [18], the proof of this result is that in Kellogg’s classical book for the
case of smooth boundaries (see [33, Chapter IX, §11]), which carries over to
the Lipschitz case.

######  Theorem 2.2 (Theorem 1.1 in [18]).

If $\Gamma$ is the boundary of a bounded Lipschitz domain $\Omega_{-}$, the
eigenvalues of $D_{\Gamma}^{\prime}$, as an operator on $L^{2}(\Gamma)$, are
real and lie in $[-\frac{1}{2},\frac{1}{2}]$.

###### Proof of Lemma 2.1.

Let $\phi\in L^{2}(\Gamma)$ and let $\langle\cdot,\cdot\rangle$ denote the
inner product on $L^{2}(\Gamma)$. We first observe that, since
$D_{\Gamma}(1)=-\frac{1}{2}$, we have

$\langle
D_{\Gamma}^{\prime}\phi,1\rangle=\langle\phi,D_{\Gamma}(1)\rangle=-\tfrac{1}{2}\langle\phi,1\rangle,$
(2.1)

so $D_{\Gamma}^{\prime}(L^{2}_{0}(\Gamma))\subseteq L^{2}_{0}(\Gamma)$.

Now assume that $\lambda\in\mathbb{C}$ and $D_{\Gamma}^{\prime}-\lambda I$ is
invertible as an operator on $L^{2}_{0}(\Gamma)$. Let $P$ be orthogonal
projection from $L^{2}(\Gamma)$ onto the constants, so that $Q:=I-P$ is
projection onto $L^{2}_{0}(\Gamma)$. Then $(D_{\Gamma}^{\prime}-\lambda)Q+P$
is invertible as an operator on $L^{2}(\Gamma)$. It follows that

$D_{\Gamma}^{\prime}-\lambda
I=((D_{\Gamma}^{\prime}-\lambda)Q+P)+((D_{\Gamma}^{\prime}-\lambda)P-P)$ (2.2)

is Fredholm as an operator on $L^{2}(\Gamma)$. This implies that
$\sigma_{\operatorname{ess}}(D_{\Gamma}^{\prime};L^{2}(\Gamma))\subset\sigma(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))$,
which settles one direction.

Conversely, assume that
$\rho_{\operatorname{ess}}(D_{\Gamma}^{\prime};L^{2}(\Gamma))<\frac{1}{2}$.
Let $\lambda\in\mathbb{C}$ with $|\lambda|\geq\frac{1}{2}$. Then
$D_{\Gamma}^{\prime}-\lambda I$ is Fredholm of index 0 on $L^{2}(\Gamma)$, so
(see (2.2)) $(D_{\Gamma}^{\prime}-\lambda)Q+P$ is Fredholm of index zero on
$L^{2}(\Gamma)$, so that $D_{\Gamma}^{\prime}-\lambda I$ is Fredholm of index
zero on $L^{2}_{0}(\Gamma)$. Thus
$\lambda\in\sigma(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))$ if and only if
$\lambda$ is an eigenvalue of $D_{\Gamma}^{\prime}$. Hence, if
$\lambda\notin[-\frac{1}{2},\frac{1}{2}]$, $D_{\Gamma}^{\prime}-\lambda I$ is
invertible on both $L^{2}(\Gamma)$ and $L^{2}_{0}(\Gamma)$ by Theorem 2.2 (as
$L^{2}_{0}(\Gamma)\subseteq L^{2}(\Gamma)$, every eigenvalue on
$L^{2}_{0}(\Gamma)$ is also an eigenvalue on $L^{2}(\Gamma)$). But also, if
$\lambda=\pm\frac{1}{2}$, since $\Gamma$ is connected, Verchota’s results [63]
show that $D_{\Gamma}^{\prime}\pm\frac{1}{2}I$ is invertible on
$L^{2}_{0}(\Gamma)$. Thus $D_{\Gamma}^{\prime}-\lambda I$ is invertible on
$L^{2}_{0}(\Gamma)$ for $\left\lvert\lambda\right\rvert\geq\frac{1}{2}$, so
that, since the spectrum is closed,
$\rho(D_{\Gamma}^{\prime};L^{2}_{0}(\Gamma))<\frac{1}{2}$. ∎

##  3 Approximation of the spectral radius for compact operators

In this section we recall in §3.1 results from operator approximation theory
in Banach spaces related to the spectra of compact operators, and derive what
appear to be new general criteria for $\rho(T)<\rho_{0}$ when $T$ is compact
and $\rho_{0}>0$ (Lemma 3.2, Corollary 3.5). This leads, in §3.2, to results
relating to the approximation of integral operators with continuous kernels by
the Nyström method that will be key for the arguments in §4.2 and §4.3.
Notably, Theorem 3.8 provides criteria for $\rho(T)<\rho_{0}$ when $T$ is an
integral operator with a continuous kernel that requires the computation only
of the spectral radius of a finite matrix plus the norms of finitely many
finite matrix resolvents.

###  3.1 Operator approximation results

We recall first two standard results on the approximation of operators in
$\mathcal{L}(Y)$, the space of bounded linear operators on a Banach space $Y$.
The first is the basic perturbation estimate that, if $S,T\in\mathcal{L}(Y)$
and $S$ is invertible, then $T$ is invertible if $\left\lVert
S-T\right\rVert\left\lVert S^{-1}\right\rVert<1$, with

$\left\lVert T^{-1}\right\rVert^{-1}\geq\left\lVert
S^{-1}\right\rVert^{-1}-\left\lVert S-T\right\rVert.$ (3.1)

If $S\in\mathcal{L}(Y)$ is invertible, then
$\|S^{-1}\|^{-1}=\inf_{\|\phi\|=1}\|S\phi\|$, this sometimes called the lower
norm of $S$ (see, e.g., [41, Lemma 2.35]). The second estimate is as follows:

######  Lemma 3.1 (Theorem 4.7.7 of [23]).

Let $Y$ be a Banach space, $S\in\mathcal{L}(Y)$ and
$\lambda\notin\operatorname{\sigma}(S)\cup\\{0\\}$. If $T\in\mathcal{L}(Y)$ is
a compact operator that satisfies

$\left\lVert(T-S)T\right\rVert<\left\lvert\lambda\right\rvert\left\lVert(S-\lambda
I)^{-1}\right\rVert^{-1},$

then $T-\lambda I$ is invertible.

The following result is a consequence of the above estimates and the maximum
principle applied to the resolvent. Here $\mathbb{T}=\\{z:|z|=1\\}$ is the
unit circle in the complex plane, so that $\rho_{0}\mathbb{T}$ is the circle
of radius $\rho_{0}$.

######  Lemma 3.2.

Let $Y$ be a Banach space, $S,\widehat{S},T\in\mathcal{L}(Y)$, $\rho_{0}>0$,
$F\subset\rho_{0}\mathbb{T}$, and suppose that: $T$ is compact;
$\rho(\widehat{S})<\rho_{0}$; for every $\lambda\in\rho_{0}\mathbb{T}$ there
exists $\mu\in F$ such that

$\left\lVert(T-S)T\right\rVert<\rho_{0}\left(\left\lVert(\widehat{S}-\mu
I)^{-1}\right\rVert^{-1}-\left\lVert
S-\widehat{S}\right\rVert-|\lambda-\mu|\right).$ (3.2)

Then $\rho(T)<\rho_{0}$.

The idea is to choose $F$ to be a finite set101010We will also apply this
lemma later in the case that $F=\rho_{0}\mathbb{T}$, when (3.2) reduces to the
condition that, for every $\lambda\in\rho_{0}\mathbb{T}$,
$\|(T-S)T\|<\rho_{0}(\|(\widehat{S}-\lambda
I)^{-1}\|^{-1}-\|S-\widehat{S}\|)$., $S$ a finite rank approximation to $T$,
and $\widehat{S}$ a numerical approximation to $S$, in which case one can show
$\rho(T)<\rho_{0}$ by computing $\rho(\widehat{S})$ and $\|(\widehat{S}-\mu
I)^{-1}\|$ for finitely many $\mu$. Taking $F=\rho_{0}\mathbb{T}_{n}$, where
$\mathbb{T}_{n}$ is the $n$th roots of unity, we obtain:

######  Corollary 3.3.

Let $Y$ be a Banach space, $S,\widehat{S},T\in\mathcal{L}(Y)$, $\rho_{0}>0$,
$n\in\mathbb{N}$, and suppose that $T$ is compact,
$\rho(\widehat{S})<\rho_{0}$, and

$\left\lVert(T-S)T\right\rVert<\rho_{0}\left(\|(\widehat{S}-\mu
I)^{-1}\|^{-1}-\|S-\widehat{S}\|-2\rho_{0}\sin\left(\frac{\pi}{2n}\right)\right),\qquad\mu\in\rho_{0}\mathbb{T}_{n}.$
(3.3)

Then $\rho(T)<\rho_{0}$.

###### Proof of Lemma 3.2.

Suppose that the conditions of the lemma are satisfied. Then
$\rho(\widehat{S})<\rho_{0}$ and, for every $\lambda\in\rho_{0}\mathbb{T}$,
there exists $\mu\in F$ such that (3.2) holds. It follows from (3.1) that

$\left\lVert(T-S)T\right\rVert<\rho_{0}\left(\|(\widehat{S}-\lambda
I)^{-1}\|^{-1}-\|S-\widehat{S}\|\right),\qquad\lambda\in\rho_{0}\mathbb{T}.$
(3.4)

The resolvent map $\lambda\mapsto(\widehat{S}-\lambda I)^{-1}$ is analytic on
$|\lambda|>\rho(\widehat{S})$, which set contains all $\lambda$ with
$|\lambda|\geq\rho_{0}$. Thus, by the maximum principle,
$\|(\widehat{S}-\lambda I)^{-1}\|$ attains its maximum in
$|\lambda|\geq\rho_{0}$ on $\rho_{0}\mathbb{T}$. Thus (3.4) in fact holds for
all $\lambda$ with $|\lambda|\geq\rho_{0}$, so that $S-\lambda I$ is
invertible for all such $\lambda$ and, by (3.1),

$\left\lVert(T-S)T\right\rVert<\rho_{0}\left\lVert(S-\lambda
I)^{-1}\right\rVert^{-1},\qquad\mbox{if }|\lambda|\geq\rho_{0}.$ (3.5)

Since $T$ is compact, the result follows from Lemma 3.1. ∎

When Lemma 3.2 is used for computation with $F$ finite, it is desirable to
minimise the cardinality of $F$ since $\|(\widehat{S}-\mu I)^{-1}\|$ has to be
computed for every $\mu\in F$. One can choose $F=\rho_{0}\mathbb{T}_{n}$, with
points uniformly distributed on $\rho_{0}\mathbb{T}$, as in Corollary 3.3, but
$n$ needs to be at least large enough so that $\|(\widehat{S}-\mu
I)^{-1}\|^{-1}>2\rho_{0}\sin(\pi/(2n))$, for every $\mu\in F$. In many
applications, including in §4.2 and §4.3, $\|(\widehat{S}-\mu I)^{-1}\|$
varies significantly as $\mu$ moves around $\rho_{0}\mathbb{T}$ and it is more
efficient to vary the spacing of the points in $F$ approximately in proportion
to $\|(\widehat{S}-\mu I)^{-1}\|^{-1}$. The adaptive algorithm described in
the following lemma, which we will see implemented in Figure 4(b) below,
approximately achieves this.

######  Lemma 3.4.

Let $Y$ be a Banach space, $\widehat{S}\in\mathcal{L}(Y)$, and $\rho_{0}>0$.
Suppose that $\rho(\widehat{S})<\rho_{0}$, and recursively define
$\mu_{\ell}$, for $\ell\in\mathbb{N}$, by $\mu_{1}:=\rho_{0}$, and by

$\nu_{\ell}:=\left\lVert(\widehat{S}-\mu_{\ell}I)^{-1}\right\rVert^{-1}\quad\text{and}\quad\mu_{\ell+1}:=\mu_{\ell}e^{i\frac{\nu_{\ell}}{2\rho_{0}}},\quad\mbox{for
}\ell\in\mathbb{N}.$ (3.6)

Further, set $n_{*}\in\mathbb{N}$ to be the smallest integer such that
$\sum_{\ell=1}^{n_{*}}\nu_{\ell}\geq 4\pi\rho_{0}$, and set
$F:=\\{\mu_{1},\mu_{2},\ldots,\mu_{n_{*}+1}\\}$. Then, for every
$\lambda\in\rho_{0}\mathbb{T}$ there exists $\mu\in F$ such that

$\left\lVert(\widehat{S}-\lambda
I)^{-1}\right\rVert^{-1}\geq\left\lVert(\widehat{S}-\mu
I)^{-1}\right\rVert^{-1}-|\lambda-\mu|\geq
R_{*}:=\min_{\ell=1,\ldots,n_{*}}\left(\frac{\nu_{\ell}}{4}+\frac{\nu_{\ell+1}}{2}\right).$
(3.7)

###### Proof.

Arguing as in the proof of Lemma 3.2, $\|(\widehat{S}-\lambda I)^{-1}\|^{-1}$
is bounded below by some $c>0$ on $\rho_{0}\mathbb{T}$, so that $n_{*}$ is
well-defined with $n_{*}<1+4\pi\rho_{0}/c$. For $\ell=1,\ldots,n_{*}+1$,
$\mu_{\ell}=\rho_{0}e^{i\theta_{\ell}}$, with $\theta_{1}=0$,
$\theta_{n_{*}+1}\geq 2\pi$, and
$\theta_{\ell+1}-\theta_{\ell}=\nu_{\ell}/(2\rho_{0})$. Thus, if
$\lambda\in\rho_{0}\mathbb{T}$, then

$\lambda=\mu_{\ell}\exp(is\nu_{\ell}/(2\rho_{0}))=\mu_{\ell+1}\exp(i(s-1)\nu_{\ell}/(2\rho_{0})),$

for some $\ell\in\\{1,\ldots,n_{*}\\}$ and some $s\in[0,1]$. Since
$|e^{it}-1|=|\int_{0}^{t}e^{iu}\,\mathrm{d}u|\leq|t|$, for $t\in\mathbb{R}$,
it follows, using (3.1), that

$\displaystyle\|(\widehat{S}-\lambda I)^{-1}\|^{-1}$ $\displaystyle\geq$
$\displaystyle\nu_{\ell}-|\lambda-\mu_{\ell}|\geq\nu_{\ell}-\frac{s}{2}\nu_{\ell}=\left(1-\frac{s}{2}\right)\nu_{\ell}\quad\mbox{and}$
$\displaystyle\|(\widehat{S}-\lambda I)^{-1}\|^{-1}$ $\displaystyle\geq$
$\displaystyle\nu_{\ell+1}-|\lambda-\mu_{\ell+1}|\geq\nu_{\ell+1}-\frac{1-s}{2}\nu_{\ell}.$

In particular, $\nu_{\ell+1}\geq\nu_{\ell}/2$ and
$\nu_{\ell}\geq\nu_{\ell+1}-\nu_{\ell}/2$, so that
$s^{*}:=3/2-\nu_{\ell+1}/\nu_{\ell}\in[0,1]$. But, for $0\leq s\leq s*$,
$(1-s/2)\nu_{\ell}\geq(1-s^{*}/2)\nu_{\ell}=\nu_{\ell}/4+\nu_{\ell+1}/2$,
while, for $s^{*}\leq s\leq 1$,
$\nu_{\ell+1}-(1-s)\nu_{\ell}/2\geq\nu_{\ell+1}-(1-s^{*})\nu_{\ell}/2=\nu_{\ell}/4+\nu_{\ell+1}/2$,
and the bound (3.7) follows. ∎

The following corollary is immediate from the above lemma and Lemma 3.2.

######  Corollary 3.5.

Let $Y$ be a Banach space, $S,\widehat{S},T\in\mathcal{L}(Y)$, $\rho_{0}>0$,
and suppose that $T$ is compact, $\rho(\widehat{S})<\rho_{0}$, and

$\left\lVert(T-S)T\right\rVert<\rho_{0}\left(R_{*}-\|S-\widehat{S}\|\right),$
(3.8)

where $R_{*}$ is as defined in Lemma 3.4. Then $\rho(T)<\rho_{0}$.

We will apply the above results in the case when $S=T_{N}$, where
$(T_{N})_{N\in\mathbb{N}}$ is a collectively compact111111Recall, e.g., [2],
[40, §10.3], that a set $S\subset\mathcal{L}(Y)$ is said to be collectively
compact if $\\{T\phi:T\in S,\phi\in Y,\|\phi\|\leq 1\\}$ is relatively
compact. sequence of operators converging strongly to $T$ (we write $T_{N}\to
T$ for strong convergence). A standard, simple but important result (e.g., [2,
Cor. 1.9], [40, Thm. 10.10]) is that

$(T_{N})_{N\in\mathbb{N}}\mbox{ collectively compact},\quad T_{N}\to
T\quad\Rightarrow\quad\|(T-T_{N})T\|\to 0.$ (3.9)

A consequence of (3.9) is Theorem 3.6 below, which follows from [2, Thm 4.8]
and [2, Thm. 4.16] (or see [3]). This gives conditions on operators $T_{N}$
and $T$ that ensure convergence of $\sigma(T_{N})$ to $\sigma(T)$ in the
standard Hausdorff metric $d_{H}(\cdot,\cdot)$ (see, e.g., [24, §3.1.2]) on
the set $\mathbb{C}^{C}$ of compact subsets of $\mathbb{C}$. Given a sequence
$(A_{N})_{N\in\mathbb{N}}\subset\mathbb{C}^{C}$ and $A\in\mathbb{C}^{C}$ we
will write
$A_{N}\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}A$ if
$d_{H}(A,B)\to 0$, i.e. if $A_{N}$ converges to $A$ in the Hausdorff metric.
We recall (e.g., [24, Prop. 3.6]) that
$A_{N}\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}A$ if and only if
$(A_{N})_{N\in\mathbb{N}}$ is uniformly bounded and $\liminf A_{N}=\limsup
A_{N}=A$, where $\liminf A_{N}$ is the set of limits of sequences $(z_{N})$
such that $z_{N}\in A_{N}$ for each $N$, while $\limsup A_{N}$ is the set of
partial limits of such sequences.

######  Theorem 3.6 ([2]).

Let $Y$ be a Banach space and $T\in\mathcal{L}(Y)$ a compact operator.
Further, assume that $(T_{N})_{N\in\mathbb{N}}\subset\mathcal{L}(Y)$ is
collectively compact and converging strongly to $T$. Then
$\sigma(T_{N})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}\sigma(T)$.

###  3.2 The Nyström method

In this section, with a view to applications in §4.2 and §4.3, we apply the
results of §3.1 to the case where $Y=C(X)$, for some compact
$X\subset\mathbb{R}^{d-1}$, and where the operators are integral operators
that we approximate by the Nyström method. So suppose $d\geq 2$, let
$X\subset\mathbb{R}^{d-1}$ be a compact set of positive ($(d-1)$-dimensional)
Lebesgue measure, and $K\in\mathcal{L}(C(X))$ an integral operator with
continuous kernel $K(\cdot,\cdot)$, so that $K$ is compact. Thus, for $\phi\in
C(X)$ and $x\in X$,

$K\phi(x)=\int_{X}K(x,y)\phi(y)\,dy=J(K(x,\cdot)\phi),\quad\mbox{where}\quad
J(\psi):=\int_{X}\psi(y)\,dy,\quad\mbox{for }\psi\in C(X).$ (3.10)

In the Nyström method we approximate $K$ by replacing the integration
functional $J:C(X)\to\mathbb{C}$ by a sequence of numerical quadrature rules.
For each $N\in\mathbb{N}$ we choose points $x_{q,N}\in X$ and weights121212We
assume, for simplicity, that the weights $\omega_{q,N}$ are positive, but the
theory below applies, with minor changes, to the case of general real or
complex weights. $\omega_{q,N}\geq 0$, for $q=1,\ldots,N$, and define
$J_{N}:C(X)\to\mathbb{C}$ by

$J_{N}(\psi):=\sum_{q=1}^{N}\omega_{q,N}\psi(x_{q,N}),\qquad\psi\in C(X),$
(3.11)

and a Nyström approximation $K_{N}\in\mathcal{L}(C(X))$ to $K$ by

$K_{N}\phi(x):=J_{N}(K(x,\cdot)\phi))=\sum_{q=1}^{N}\omega_{q,N}K(x,x_{q,N})\phi(x_{q,N}),\qquad
x\in X,\;\;\phi\in C(X).$ (3.12)

We will assume that the sequence of quadrature rules is convergent, by which
we mean that

$J_{N}\to J,\quad\mbox{i.e.}\quad J_{N}\psi\to J\psi,\quad\mbox{for all
}\psi\in C(X).$ (3.13)

This implies (e.g., [2, Prop. 2.1, 2.2]) that $K_{N}\to K$ and that the
sequence $(K_{N})_{N\in\mathbb{N}}\subset\mathcal{L}(C(X))$ is collectively
compact, so that (3.9) holds and Theorem 3.6 is applicable. We will also
assume that, for each $N$, $J_{N}\phi=J\phi$ if $\phi\in C(X)$ is constant,
i.e. that

$\sum_{q=1}^{N}\omega_{q,N}=|X|,$ (3.14)

where $|X|$ denotes the Lebesgue measure of $X$.

Define $\widehat{K}_{N}:\mathbb{C}^{N}\to C(X)$ by

$\widehat{K}_{N}v(x):=\sum_{q=1}^{N}\omega_{q,N}K(x,x_{q,N})v_{q},$

for $x\in X$ and $v=(v_{1},...,v_{N})\in\mathbb{C}^{N}$. Moreover, let
$P_{N}\colon C(X)\to\mathbb{C}^{N}$ be defined by
$P_{N}\phi(q):=\phi(x_{q,N})$, for $q=1,...,N$, $\phi\in C(X)$, and define
$A_{N}\colon\mathbb{C}^{N}\to\mathbb{C}^{N}$ by $A_{N}:=P_{N}\hat{K}_{N}$, so
that the matrix entries of $A_{N}$ are given by

$A_{N}(p,q)=\omega_{q,N}K(x_{p,N},x_{q,N}),\qquad p,q=1,\ldots,N.$ (3.15)

In the following we will use $\|\cdot\|_{\infty}$ to denote all of: i) the
standard supremum norm on $C(X)$; ii) the standard infinity norm on
$\mathbb{C}^{N}$; iii) the induced operator norm of an operator on $C(X)$; iv)
the infinity norm of a square matrix.

######  Lemma 3.7.

The following inequalities and equalities hold for all $N\in\mathbb{N}$:

  * (i)

$\max_{1\leq p\leq
N}\sum_{q=1}^{N}\omega_{q,N}|K(x_{p,N},x_{q,N})|=\left\lVert
A_{N}\right\rVert_{\infty}\leq\left\lVert
K_{N}\right\rVert_{\infty}=\max_{x\in
X}\sum_{q=1}^{N}\omega_{q,N}|K(x,x_{q,N})|;$

  * (ii)

$\operatorname{\sigma}(K_{N})=\\{0\\}\cup\operatorname{\sigma}(A_{N})$;

  * (iii)

for $\lambda\in\mathbb{C}\setminus\operatorname{\sigma}(K_{N})$,

$\displaystyle\max\left(|\lambda|^{-1},\left\lVert(A_{N}-\lambda
I)^{-1}\right\rVert_{\infty}\right)\leq\left\lVert(K_{N}-\lambda
I)^{-1}\right\rVert_{\infty}$
$\displaystyle\leq|\lambda|^{-1}\left(1+\left\lVert
K_{N}\right\rVert_{\infty}\left\lVert(A_{N}-\lambda
I)^{-1}\right\rVert_{\infty}\right).$

###### Proof.

The first equality in (i) is the standard explicit formula for the infinity
norm of a matrix. The last equality is proved similarly, and then (i) is clear
(or see [23, Lemma 4.7.17]). That the spectra of $K_{N}$ and $A_{N}$ coincide
on $\mathbb{C}\setminus\\{0\\}$ is standard (e.g., [23, Lemma 4.7.18]), and
$0\in\operatorname{\sigma}(K_{N})$ since $K_{N}$ is compact and $C(X)$ is
infinite-dimensional, so (ii) holds. Part (iii) is a combination of [23, Lemma
4.7.18] and [2, Proposition 2.3], plus the facts that $0\in\sigma(K_{N})$ and
(e.g., [14, Th. 1.2.10]) $\|(K_{N}-\lambda
I)^{-1}\|\geq(\operatorname*{dist}(\lambda,\sigma(K_{N})))^{-1}$. ∎

Note that, by Lemma 3.7(i) and (3.14),

$\|A_{N}\|_{\infty}\leq\|K_{N}\|_{\infty}\leq K_{\rm
max}|X|,\quad\mbox{where}\quad K_{\rm max}:=\max_{x,y\in X}|K(x,y)|.$ (3.16)

Commonly, for computational efficiency or otherwise, and this is the case in
§4.2 and §4.3, we approximate $K(\cdot,\cdot)$ by another continuous kernel
$K^{{\dagger}}(\cdot,\cdot)$ in (3.12) and (3.15). Let
$K_{N}^{{\dagger}}\in\mathcal{L}(C(X))$ denote the operator defined by the
right hand side of (3.12) with $K$ replaced by $K^{{\dagger}}$, and similarly
$A_{N}^{{\dagger}}$ denote the matrix (3.15) with $K$ replaced by
$K^{{\dagger}}$. Lemma 3.7(i) and (3.14) imply that

$\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}\leq\|K_{N}-K_{N}^{{\dagger}}\|_{\infty}\leq
e^{{\dagger}}|X|,\quad\mbox{where}\quad e^{{\dagger}}:=\max_{x,y\in
X}|K(x,y)-K^{{\dagger}}(x,y)|.$ (3.17)

The following theorem (cf. Corollary 3.5) follows in large part from Lemma
3.2, Lemma 3.4, and Lemma 3.7.

######  Theorem 3.8.

Suppose that $\rho_{0}>0$, $N\in\mathbb{N}$, and
$\rho(A_{N}^{{\dagger}})<\rho_{0}$, and recursively define $\mu_{\ell}$, for
$\ell\in\mathbb{N}$, by $\mu_{1}:=\rho_{0}$, and by

$\nu_{\ell}:=\left\lVert(A_{N}^{{\dagger}}-\mu_{\ell}I)^{-1}\right\rVert_{\infty}^{-1}\quad\text{and}\quad\mu_{\ell+1}:=\mu_{\ell}e^{i\frac{\nu_{\ell}}{2\rho_{0}}},\quad\mbox{for
}\ell\in\mathbb{N}.$ (3.18)

Further, let $n_{N}\in\mathbb{N}$ be the smallest integer such that
$\sum_{\ell=1}^{n_{N}}\nu_{\ell}\geq 4\pi\rho_{0}$, and let

$R_{N}:=\min_{\ell=1,\ldots,n_{N}}\left(\frac{\nu_{\ell}}{4}+\frac{\nu_{\ell+1}}{2}\right).$
(3.19)

If

$\left\lVert(K-K_{N})K\right\rVert_{\infty}<\rho_{0}\left(\rho_{0}\left(1+\|K^{{\dagger}}_{N}\|_{\infty}R_{N}^{-1}\right)^{-1}-\|K_{N}-K_{N}^{{\dagger}}\|_{\infty}\right),$
(3.20)

or $\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}<R_{N}$ and

$\left\lVert(K-K_{N})K\right\rVert_{\infty}<\rho_{0}^{2}\left(1+\|K_{N}\|_{\infty}(R_{N}-\|A_{N}-A_{N}^{{\dagger}}\|_{\infty})^{-1}\right)^{-1},$
(3.21)

then $\rho(K)<\rho_{0}$. Conversely, if $\rho(K)<\rho_{0}$, provided
$e^{{\dagger}}$ defined by (3.17) is sufficiently small, (3.20), (3.21), and
$\rho(A_{N}^{{\dagger}})<\rho_{0}$ hold for all sufficiently large $N$.

###### Proof.

If $\rho_{0}>0$ and $\rho(A_{N}^{{\dagger}})<\rho_{0}$ then, by Lemma 3.4,
$\|(A_{N}^{{\dagger}}-\lambda I)^{-1}\|_{\infty}\leq R_{N}^{-1}$ for
$\lambda\in\rho_{0}\mathbb{T}$, so that $\|(K_{N}^{{\dagger}}-\lambda
I)^{-1}\|_{\infty}\leq
r^{-1}_{0}(1+\|K_{N}^{{\dagger}}\|_{\infty}R_{N}^{-1})$, by Lemma 3.7(iii).
Thus $\rho(K)<\rho_{0}$ if (3.20) holds, by Lemma 3.2 applied with $Y=C(X)$,
$T=K$, $S=K_{N}$, $\widehat{S}=K_{N}^{{\dagger}}$, and $F=\rho_{0}\mathbb{T}$.
Further, for $\lambda\in\rho_{0}\mathbb{T}$, the first of the above bounds and
(3.1) implies that
$R_{N}-\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}\leq\|(A_{N}-\lambda
I)^{-1}\|^{-1}_{\infty}$, so that, by Lemma 3.7(iii), $\|(K_{N}-\lambda
I)^{-1}\|_{\infty}\leq
r^{-1}_{0}(1+\|K_{N}\|_{\infty}(R_{N}-\|A_{N}-A_{N}^{{\dagger}}\|_{\infty})^{-1})$.
Thus $\rho(K)<\rho_{0}$ if (3.21) holds, by Lemma 3.2 applied with $Y=C(X)$,
$T=K$, $\widehat{S}=S=K_{N}$, and $F=\rho_{0}\mathbb{T}$.

To see the converse, note that, by Theorem 3.6,
$\sigma(K_{N})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}\sigma(K)$. Indeed
[2, Thm. 4.7], there exists $N_{0}\in\mathbb{N}$ such that $\|(K_{N}-\lambda
I)^{-1}\|_{\infty}$ is bounded uniformly in $\lambda$ and $N$ for $N\geq
N_{0}$ and $|\lambda|\geq\rho_{0}$, which implies, by (3.1) and (3.17), that
the same holds for $\|(K^{{\dagger}}_{N}-\lambda I)^{-1}\|_{\infty}$ if
$e^{{\dagger}}$ is sufficiently small. This implies, by Lemma 3.7(iii), that
the same holds for $\|(A^{{\dagger}}_{N}-\lambda I)^{-1}\|_{\infty}$. Noting
Lemma 3.7(ii), it follows that there exists $c>0$ such that, for all
sufficiently large $N$,
$\rho(A_{N}^{{\dagger}})=\rho(K_{N}^{{\dagger}})<\rho_{0}$ and $R_{N}\geq c$.
Thus, and applying (3.16) and (3.17), we see that, for some $c^{*}>0$, the
right hand sides of (3.20) and (3.21) are $\geq c^{*}$ for all sufficiently
large $N$ if $e^{{\dagger}}$ is sufficiently small. But also
$\|(K-K_{N})K\|\to 0$ by (3.9), so that (3.20) holds for all sufficiently
large $N$. ∎

######  Remark 3.9 (Computational cost as $N$ increases).

The argument in the above proof makes clear that, if $\rho(K)<\rho_{0}$ and
$e^{{\dagger}}$ is sufficiently small, then, for some $N_{0}\in\mathbb{N}$,
$\|(A_{N}^{{\dagger}}-\lambda I)^{-1}\|_{\infty}$ is bounded uniformly in
$\lambda$ and $N$ for $\lambda\in\rho_{0}\mathbb{T}$ and $N\geq N_{0}$. This
in turn implies that, for some $n_{\max}\in\mathbb{N}$, $n_{N}\leq n_{\max}$
for $N\geq N_{0}$. Thus the computational cost of evaluation of $R_{N}$ given
by (3.19) is $O(N^{3})$, the cost of inverting $n_{N}\leq n_{\max}$ order $N$
matrices by classical direct methods.

######  Remark 3.10 (Comparison of (3.20) and (3.21)).

Let $\mathrm{RHS}_{1}$ and $\mathrm{RHS}_{2}$ denote the right hand sides of
(3.20) and (3.21), respectively. If $K_{N}^{{\dagger}}=K_{N}$ (so
$A_{N}^{{\dagger}}=A_{N}$), then $\mathrm{RHS}_{2}=\mathrm{RHS}_{1}$. If
$K^{{\dagger}}_{N}\neq K_{N}$ with
$\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}<R_{N}$, then, where
$\mathcal{D}:=\rho_{0}^{-2}(R_{N}+\|K_{N}\|_{\infty}-\|A_{N}-A_{N}^{\dagger}\|_{\infty})(\mathrm{RHS}_{2}-\mathrm{RHS}_{1})$,

$\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle\rho_{0}^{-1}\|K_{N}-K_{N}^{{\dagger}}\|_{\infty}(\|K_{N}\|_{\infty}-\rho_{0}+R_{N}-\|A_{N}-A_{N}^{\dagger}\|_{\infty})\,+$
$\displaystyle\;\frac{\|K_{N}^{\dagger}\|_{\infty}(\|K_{N}-K_{N}^{\dagger}\|_{\infty}-\|A_{N}-A_{N}^{\dagger}\|_{\infty})+R_{N}(\|K_{N}-K_{N}^{\dagger}\|_{\infty}+\|K_{N}^{\dagger}\|_{\infty}-\|K_{N}\|_{\infty})}{R_{N}+\|K_{N}^{\dagger}\|_{\infty}},$

so that
$\mathrm{RHS}_{2}-\mathrm{RHS}_{1}>\rho_{0}\|K_{N}-K_{N}^{{\dagger}}\|_{\infty}(\|K_{N}\|_{\infty}-\rho_{0})/(R_{N}+\|K_{N}\|_{\infty}-\|A_{N}-A_{N}^{{\dagger}}\|_{\infty})$,
recalling (3.17). Thus (3.20) implies (3.21) if
$\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}<R_{N}$ and
$\rho_{0}\leq\|K_{N}\|_{\infty}$. Note that [2, Thm. 2.13]
$\|K_{N}\|_{\infty}\to\|K\|$ as $N\to\infty$ and it is
$\rho_{0}<\|K\|_{\infty}$ for which Theorem 3.8 is arguably of most interest,
as $\rho(K)\leq\|K\|_{\infty}$, and we may be able to estimate
$\|K\|_{\infty}$ sharply by other methods.

######  Remark 3.11 (The matrix case).

In §4.3 we will apply the above results, in particular Theorem 3.8, in a case
where $K$ is a $2\times 2$ matrix of integral operators on $C(X)$ with
continuous kernels, and $K_{N}$ is its Nyström method approximation defined by
approximating each integral operator in the $2\times 2$ matrix as in (3.12).
The matrix $A_{N}$ is then a $2N\times 2N$ matrix consisting of four $N\times
N$ blocks each defined as in (3.15). Parts (ii) and (iii) of Lemma 3.7 apply
in this case, as does (i) in a straightforwardly modified form, in particular
we still have that $\|A_{N}\|_{\infty}\leq\|K_{N}\|_{\infty}$. (Here
$\|A_{N}\|_{\infty}$ is the usual infinity norm of the matrix $A_{N}$ and
$\|K_{N}\|_{\infty}$ is the norm of $K_{N}$ as an operator on $(C(X))^{2}$,
which we equip with the norm defined by
$\|(\phi_{1},\phi_{2})\|_{\infty}:=\max\\{\|\phi_{1}\|_{\infty},\|\phi_{2}\|_{\infty}\\}$,
for $(\phi_{1},\phi_{2})\in C(X)^{2}$). Thus Theorem 3.8, which depends on
Lemma 3.7(ii) and (iii) and the general Banach space results of §3.1, still
applies. One way to see the validity of Lemma 3.7(ii) and (iii) in this matrix
case is to argue as follows. Choose $x^{*}\in\mathbb{R}^{d-1}$ such that
$X^{\prime}:=X+x^{*}$ does not intersect $X$. $K$ is equivalent, through an
obvious isometric isomorphism, to a matrix operator $K^{\prime}$ on
$C(X)\times C(X^{\prime})$, which is in turn equivalent, through another
obvious isometric isomorphism, to a single integral operator $\widetilde{K}$
on $C(X\cup X^{\prime})$. Lemma 3.7 applies to $\widetilde{K}$ and its Nyström
method approximation, so that (ii) and (iii) of this lemma (and (i) in
modified form) apply to $K$.

##  4 The Double-Layer Operator on Dilation invariant graphs

Let $A\subseteq\mathbb{R}^{d-1}$ be an open cone, $f\colon A\to\mathbb{R}$ a
Lipschitz continuous function, and consider the graph
$\Gamma=\left\\{(x,f(x)):x\in A\right\\}\subset\mathbb{R}^{d}$ in the case
that $\alpha\Gamma=\Gamma$, for some $\alpha\in(0,1)$; that is, in the case
that

$f(\alpha x)=\alpha f(x),\quad x\in A.$ (4.1)

We will term such graphs _dilation invariant_.

In this section, the largest of the paper, we study the spectrum and essential
spectrum of the double-layer (DL) operator $D_{\Gamma}\colon L^{2}(\Gamma)\to
L^{2}(\Gamma)$ given by (1.2) on such graphs. (The case $A=\mathbb{R}^{d-1}$
is of particular interest for later applications.) In §4.1 we show, for
general dimension $d\geq 2$, that (as operators on $L^{2}(\Gamma)$)
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$ (and, similarly,
that $W(D_{\Gamma})=W_{\operatorname{ess}}(D_{\Gamma})$) and, by Floquet-
Bloch-transform arguments, that $\sigma(D_{\Gamma})$ is the union of the
spectra of a family of operators $K_{t}:L^{2}(\Gamma_{0})\to
L^{2}(\Gamma_{0})$, for $t\in[-\pi,\pi]$, where $\Gamma_{0}$ is a particular
relatively closed and bounded subset of $\Gamma$. Moreover, helpful for the
later computation of $\sigma(K_{t})$, each $K_{t}$ is compact, and so has a
discrete spectrum, if $f\in C^{1}(A\setminus\\{0\\})$ (Corollary 4.6).

In the remaining subsections, §4.2–§4.4, we focus on the 2D case, considering
the Nyström approximation of spectral properties of $K_{t}$, combining the
general results of §3.2 with explicit estimates for the particular operators
$K_{t}$. The case $A=\mathbb{R}$, with $f$ real-analytic on
$\mathbb{R}\setminus\\{0\\}$, is treated in §4.3. It is this case that is
relevant, via, e.g., (1.12), to the computation of $\sigma(D_{\Gamma})$ and
the spectral radius conjecture when $\Gamma$ is the boundary of a bounded
Lipschitz domain. But this case is rather complex; the operator $K_{t}$ is
studied by reducing it to a $2\times 2$ operator matrix, corresponding to the
split of $\mathbb{R}\setminus\\{0\\}$ into the two half-axes $(-\infty,0)$ and
$(0,\infty)$. To get the main ideas across, and prove many of the results we
need in a simpler setting, we first study, in §4.2, the easier case
$A=(0,\infty)$ with $f$ real-analytic. Subsections §4.2 and §4.3 are concerned
with computation of the spectrum and spectral radius of $D_{\Gamma}$. In §4.4,
related to the question at the end of §1.1, we also compute lower bounds for
$W_{\operatorname{ess}}(D_{\Gamma})$, under the same assumptions on $\Gamma$
as in §4.2 and §4.3.

###  4.1 Floquet-Bloch transform results

Let $V_{\alpha}\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$ be dilation by $\alpha$,
that is,

$V_{\alpha}\phi(x)=\alpha^{(d-1)/2}\phi(\alpha x),\quad x\in\Gamma.$ (4.2)

$V_{\alpha}$ is unitary and commutes with $D_{\Gamma}$: noting that
$n(y)=n(\alpha y)$, for $y\in\Gamma$, we see that, for all $\phi\in
L^{2}(\Gamma)$ and almost every $x\in\Gamma$,

$\displaystyle D_{\Gamma}V_{\alpha}\phi(x)$
$\displaystyle=\frac{1}{c_{d}}\int_{\Gamma}\frac{(x-y)\cdot
n(y)}{|x-y|^{d}}\alpha^{(d-1)/2}\phi(\alpha y)\,\mathrm{d}s(y)$
$\displaystyle=\frac{\alpha^{(d-1)/2}}{c_{d}}\int_{\Gamma}\frac{(x-\alpha^{-1}y)\cdot
n(\alpha^{-1}y)}{|x-\alpha^{-1}y|^{d}}\phi(y)\,\alpha^{-(d-1)}\mathrm{d}s(y)$
$\displaystyle=\frac{\alpha^{(d-1)/2}}{c_{d}}\int_{\Gamma}\frac{(\alpha
x-y)\cdot n(y)}{|\alpha x-y|^{d}}\phi(y)\,\mathrm{d}s(y)$
$\displaystyle=V_{\alpha}D_{\Gamma}\phi(x).$

This already implies that the spectrum and the essential spectrum of
$D_{\Gamma}$ coincide, as implied by the following simple proposition (cf.
[11, Lemma 2.7]).

######  Proposition 4.1.

Let $H$ be a Hilbert space and $T\in\mathcal{L}(H)$. If $T$ commutes with a
sequence of unitary operators $(U_{j})_{j\in\mathbb{N}}$ that converges weakly
to $0$, then $\sigma(T)=\sigma_{\operatorname{ess}}(T)$.

###### Proof.

Assume there exists $\phi\in H\setminus\left\\{0\right\\}$ such that
$T\phi=0$. As $T$ and $U_{j}$ commute, also $TU_{j}\phi=0$ for all
$j\in\mathbb{N}$. In particular,
$\left\\{U_{j}\phi:j\in\mathbb{N}\right\\}\subseteq\ker(T)$. As the operators
$U_{j}$ are unitary and $U_{j}\to 0$ weakly, the sequence
$(U_{j}\phi)_{j\in\mathbb{N}}$ cannot have a convergent subsequence. Hence,
$\ker(T)$ is either trivial or infinite-dimensional. Similarly,
$\ker(T^{\prime})$ is either trivial or infinite-dimensional. Thus, if $T$ is
Fredholm, it is invertible. Considering $T-\lambda I$ instead of $T$ yields
the result. ∎

######  Corollary 4.2.

Let $\Gamma\subset\mathbb{R}^{d}$ be a dilation invariant graph. Then
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$,
$\overline{W(D_{\Gamma})}=W_{\operatorname{ess}}(D_{\Gamma})$, and
$\left\lVert D_{\Gamma}\right\rVert_{\operatorname{ess}}=\left\lVert
D_{\Gamma}\right\rVert$.

###### Proof.

Assume that $\phi,\psi\in L^{2}(\Gamma)$ have compact support and
$0\notin\operatorname{supp}\phi\cup\operatorname{supp}\psi$. Then

$\left\langle
V_{\alpha}^{j}\phi,\psi\right\rangle=\int_{\Gamma}\alpha^{j(d-1)/2}\phi(\alpha^{j}x)\overline{\psi(x)}\,\mathrm{d}s(x).$

If $\left\lvert j\right\rvert$ is sufficiently large, the integrand vanishes.
Because compactly supported functions are dense in $L^{2}(\Gamma)$, it follows
that $V_{\alpha}^{j}\to 0$ weakly as $\left\lvert j\right\rvert\to\infty$. The
equality of spectrum and essential spectrum follows from Proposition 4.1. The
results for the numerical range and norm follow from [11, Lemma 2.7]. ∎

To make use of standard Floquet-Bloch/Fourier transform results, it is
convenient to view $D_{\Gamma}$ as a discrete convolution operator. For
$j\in\mathbb{Z}$ let
$\Gamma_{j}:=\left\\{(\tilde{x},f(\tilde{x}))\in\Gamma:|\tilde{x}|\in[\alpha^{j+1},\alpha^{j}]\right\\}$.
We can identify $L^{2}(\Gamma_{j})$ with a closed subspace of $L^{2}(\Gamma)$
by extending by $0$. Let $P_{j}\colon L^{2}(\Gamma)\to L^{2}(\Gamma_{j})$
denote orthogonal projection. Clearly,

$P_{j}\phi(x)=\begin{cases}\phi(x)&\text{if }x\in\Gamma_{j},\\\
0&\text{otherwise,}\end{cases}$

and we note that $P_{j}V_{\alpha}=V_{\alpha}P_{j+1}$, so that also
$P_{j}V_{\alpha}^{k}=V_{\alpha}^{k}P_{j+k}$, $k\in\mathbb{Z}$. Let

$D_{j}:=V_{\alpha}^{j}P_{j}D_{\Gamma}|_{L^{2}(\Gamma_{0})}:L^{2}(\Gamma_{0})\to
L^{2}(\Gamma_{0}),$

so that

$D_{j}\phi(x)=\alpha^{j(d-1)/2}D_{\Gamma}\phi(\alpha^{j}x),\quad
x\in\Gamma_{0},\;\;\phi\in L^{2}(\Gamma_{0}).$ (4.3)

Let $\mathcal{G}:L^{2}(\Gamma)\to\ell^{2}(\mathbb{Z},L^{2}(\Gamma_{0}))$ be
the unitary operator131313This is a discretization operator in the sense,
e.g., of [41, §1.2.3].
$\phi\mapsto(V_{\alpha}^{j}P_{j}\phi)_{j\in\mathbb{Z}}$, and let
$\widetilde{D}_{\Gamma}:=\mathcal{G}D_{\Gamma}\mathcal{G}^{-1}$. It is a
straightforward calculation to see that the action of $\widetilde{D}_{\Gamma}$
is that of a discrete convolution: for
$\psi=(\psi_{n})_{n\in\mathbb{Z}}\in\ell^{2}(\mathbb{Z},L^{2}(\Gamma_{0}))$,
$\widetilde{D}_{\Gamma}\psi=((\widetilde{D}_{\Gamma}\psi)_{m})_{m\in\mathbb{Z}}$
where

$(\widetilde{D}_{\Gamma}\psi)_{m}=\sum_{n\in\mathbb{Z}}D_{m-n}\psi_{n},\quad
m\in\mathbb{Z}.$ (4.4)

The series in the above definition converges absolutely; indeed
$\widetilde{D}_{\Gamma}$ is an operator in the so-called Wiener algebra, i.e.
$\sum_{j\in\mathbb{Z}}\|D_{j}\|<\infty$ (e.g., [41, Defn. 1.43]), by the
following estimate.

######  Proposition 4.3.

The operators $D_{j}$ are Hilbert-Schmidt for $|j|\geq 2$ and satisfy the
Hilbert-Schmidt norm estimate

$\left\lVert
D_{j}\right\rVert_{\mathrm{HS}}\leq\frac{1}{c_{d}}\frac{|\Gamma_{0}|}{(\alpha-\alpha^{|j|})^{d-1}}\,\alpha^{|j|(d-1)/2},\quad|j|\geq
2,$ (4.5)

where $|\Gamma_{0}|$ denotes the surface measure of $\Gamma_{0}$.

###### Proof.

For $j\geq 2$ we have

$\int_{\Gamma_{j}}\int_{\Gamma_{0}}\left\lvert\frac{(x-y)\cdot
n(y)}{|x-y|^{d}}\right\rvert^{2}\,\mathrm{d}s(y)\,\mathrm{d}s(x)\leq\int_{\Gamma_{j}}\int_{\Gamma_{0}}\frac{1}{|x-y|^{2d-2}}\,\mathrm{d}s(y)\,\mathrm{d}s(x)$

and $|x-y|\geq\alpha-\alpha^{j}$. Hence

$\int_{\Gamma_{j}}\int_{\Gamma_{0}}\left\lvert\frac{(x-y)\cdot
n(y)}{|x-y|^{d}}\right\rvert^{2}\,\mathrm{d}s(y)\,\mathrm{d}s(x)\leq\frac{|\Gamma_{j}||\Gamma_{0}|}{(\alpha-\alpha^{j})^{2d-2}}=\frac{|\Gamma_{0}|^{2}\alpha^{j(d-1)}}{(\alpha-\alpha^{j})^{2d-2}}.$

Similarly, for $j\leq-2$,

$\int_{\Gamma_{j}}\int_{\Gamma_{0}}\left\lvert\frac{(x-y)\cdot
n(y)}{|x-y|^{d}}\right\rvert^{2}\,\mathrm{d}s(y)\,\mathrm{d}s(x)\leq\frac{|\Gamma_{0}|^{2}\alpha^{j(d-1)}}{(\alpha^{j+1}-1)^{2d-2}}=\frac{|\Gamma_{0}|^{2}\alpha^{-j(d-1)}}{(\alpha-\alpha^{-j})^{2d-2}}.$

Thus, recalling the standard characterisation of the Hilbert-Schmidt norm of
integral operators (e.g., [32, Ex. 11.11]),
$P_{j}D_{\Gamma}|_{L^{2}(\Gamma_{0})}$ is Hilbert-Schmidt, with Hilbert-
Schmidt norm bounded by the right hand side of (4.5). As unitary operators
preserve Hilbert-Schmidt norms, it follows that $D_{j}$ is Hilbert-Schmidt and
that (4.5) holds. ∎

Given a Hilbert space $H$ let $\mathcal{F}:\ell^{2}(\mathbb{Z},H)\to
L^{2}([-\pi,\pi],H)$ denote the operator, often termed in the general Hilbert
space case, e.g., [5], a Floquet-Bloch transform, that constructs a Fourier
series from coefficients, given by

$(\mathcal{F}\psi)(t)=(2\pi)^{-1/2}\sum_{j\in\mathbb{Z}}e^{ijt}\psi_{n},\quad
t\in[-\pi,\pi],$

for $\psi=(\psi_{j})_{j\in\mathbb{Z}}\in\ell^{2}(\mathbb{Z},H)$. It is
standard that this is a unitary operator (e.g., [14, Proof of Thm. 4.4.9])
that diagonalises discrete convolutions (see, e.g., [14, Thm. 4.49] for the
case when $H$ is finite-dimensional, [53, Thm. 2.3.25] for the general case).
In the case $H=L^{2}(\Gamma_{0})$, defining
$\widehat{D}:L^{2}([-\pi,\pi],H)\to L^{2}([-\pi,\pi],H)$ by
$\widehat{D}:=\mathcal{F}\widetilde{D}_{\Gamma}\mathcal{F}^{-1}$,
straightforward calculations yield that

$\langle\widehat{D}\phi,\psi\rangle_{L^{2}([-\pi,\pi],L^{2}(\Gamma_{0}))}=\int_{-\pi}^{\pi}\langle
K_{t}\phi(t),\psi(t)\rangle\,\mathrm{d}t,\quad\phi,\psi\in
L^{2}([-\pi,\pi],L^{2}(\Gamma_{0})),$ (4.6)

where

$K_{t}:=\sum\limits_{j=-\infty}^{\infty}e^{ijt}D_{j},\quad t\in\mathbb{R}.$
(4.7)

(The bounds of Proposition 4.10 imply that $K_{t}$ is well-defined by (4.7)
and depends continuously on $t$; indeed the mapping $t\mapsto K_{t}$ is
$C^{\infty}$.) The following characterisation follows immediately from (4.6),
the continuity of $t\mapsto K_{t}$, and Corollary 4.2; note that
$\operatorname*{conv}$ denotes the closed convex hull.

######  Theorem 4.4.

We have

$\displaystyle\|D_{\Gamma}\|_{\operatorname{ess}}$
$\displaystyle=\|D_{\Gamma}\|=\max\limits_{t\in[-\pi,\pi]}\left\lVert
K_{t}\right\rVert,$ $\displaystyle W_{\operatorname{ess}}(D_{\Gamma})$
$\displaystyle=\overline{W(D_{\Gamma})}=\operatorname*{conv}\left(\bigcup\limits_{t\in[-\pi,\pi]}W(K_{t})\right),$
$\displaystyle\sigma_{\operatorname{ess}}(D_{\Gamma})$
$\displaystyle=\operatorname{\sigma}(D_{\Gamma})=\bigcup\limits_{t\in[-\pi,\pi]}\operatorname{\sigma}(K_{t}).$

###### Proof.

That the essential spectrum, numerical range, and norm coincide with their
non-essential counterparts is Corollary 4.2. Since $D_{\Gamma}$ and
$\widehat{D}$ are unitarily equivalent, they have the same spectrum, numerical
range, and norm. The result thus follows from (4.6); see [53, Thm. 2.3.25] for
the case of the spectrum; the argument for the norm and numerical range are
similar. ∎

######  Remark 4.5 (Symmetry of $K_{t}$).

Where $K_{t}(\cdot,\cdot)$ denotes the kernel of $K_{t}$,
$K_{-t}(\cdot,\cdot)=\overline{K_{t}(\cdot,\cdot)}$, for $t\in[-\pi,\pi]$.
Thus $\|K_{-t}\|=\|K_{t}\|$, $W(K_{-t})=\\{\bar{z}:z\in W(K_{t})\\}$, and
$\sigma(K_{-t})=\\{\bar{z}:z\in\sigma(K_{t})\\}$, for $t\in[-\pi,\pi]$, so
that, where $w(K_{t}):=\sup_{z\in W(K_{t})}|z|$ is the numerical radius of
$K_{t}$,

$\|D_{\Gamma}\|_{\operatorname{ess}}=\max_{t\in[0,\pi]}\|K_{t}\|,\quad
w_{\operatorname{ess}}(D_{\Gamma})=\max_{t\in[0,\pi]}w(K_{t}),\quad\rho_{\operatorname{ess}}(D_{\Gamma})=\max_{t\in[0,\pi]}\rho(K_{t}).$

Our focus in the next subsections will be 2D cases where $f$ is analytic on
$A\setminus\\{0\\}$. In such cases, indeed whenever $\Gamma$ is locally
$C^{1}$ away from $0$, $K_{t}$ is compact for all $t$ as a consequence of
standard results on the double-layer operator on $C^{1}$ domains [19] and
Proposition 4.10.

######  Corollary 4.6.

Suppose that $f\in C^{1}(A\setminus\\{0\\})$ so that $\Gamma$ is locally
$C^{1}$ at each $x\in\Gamma\setminus\\{0\\}$. Then $K_{t}$ is compact for
every $t\in\mathbb{R}$.

###### Proof.

Let $Q=P_{-1}+P_{0}+P_{1}$. Then
$QD_{\Gamma}|_{L^{2}(\Gamma_{-1}\cup\Gamma_{0}\cup\Gamma_{1})}$ is the DL
operator on $\Gamma_{-1}\cup\Gamma_{0}\cup\Gamma_{1}$. It thus follows from
[19, Theorem 1.2] that
$QD_{\Gamma}|_{L^{2}(\Gamma_{-1}\cup\Gamma_{0}\cup\Gamma_{1})}$ is compact. As

$D_{j}=V_{\alpha}^{j}P_{j}D_{\Gamma}|_{L^{2}(\Gamma_{0})}=V_{\alpha}^{j}P_{j}QD_{\Gamma}|_{L^{2}(\Gamma_{0})},$

$D_{j}$ is compact for $j=-1,0,1$. The compactness of $K_{t}$ thus follows
from Proposition 4.3. ∎

###  4.2 The 2D case: One-sided infinite graphs

Figure 2: Graph of $f\colon\mathbb{R}_{+}\to\mathbb{R}$,
$f(x):=x\sin^{2}(\pi\log_{\alpha}(x))$, for $\alpha=\frac{3}{4}$

We continue to assume that $\Gamma$ is a dilation invariant graph, as defined
at the start of §4, but specialise now to the case where $d=2$ and the cone is
the half-axis $A=\mathbb{R}_{+}:=(0,\infty)$. Thus
$\Gamma=\left\\{(x,f(x)):x\in\mathbb{R}_{+}\right\\}$ and, for some
$\alpha\in(0,1)$, $f(\alpha x)=\alpha f(x)$, $x\in\mathbb{R}_{+}$. Our
starting point is Corollary 4.4 which expresses $\sigma(D_{\Gamma})$ as the
union of the spectra of the operators $K_{t}$, $t\in[-\pi,\pi]$. Recall that
these operators are compact if $f\in C^{1}(\mathbb{R}_{+})$. Our goal is to
apply the Nyström method and the results of §3.2 to compute spectral
properties of $K_{t}$ and hence of $D_{\Gamma}$ in the case that
$f\in\mathcal{A}(\mathbb{R}_{+})$, the space of functions
$\mathbb{R}_{+}\to\mathbb{R}$ that are real analytic (a prototypical example
is Figure 2). Our standing assumption through this subsection is that

$\Gamma=\left\\{(x,f(x)):x\in\mathbb{R}_{+}\right\\}\;\mbox{ where
}\;f\in\mathcal{A}(\mathbb{R}_{+})\;\mbox{ and, for some $\alpha\in(0,1)$,
}\;f(\alpha x)=\alpha f(x),\quad x\in\mathbb{R}_{+}.$ (4.8)

Notably, via approximations of the spectrum of $K_{t}$ for each $t$, we will
obtain (see Theorem 4.13) a Nyström approximation $\sigma^{N}(D_{\Gamma})$,
for $\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$, which is
$\\{0\\}$ plus the union over finitely many $t\in[-\pi,\pi]$ of the spectra of
$N\times N$ matrices $A^{M}_{t,N}$, where each $A^{M}_{t,N}$ is obtained via
Nyström discretisation of a unitary transformation, $\tilde{K}_{t}$, of
$K_{t}$. Our first main result, Theorem 4.13, is to show that
$\sigma^{N}(D_{\Gamma})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\sigma(D_{\Gamma})$ as $N\to\infty$. Our other, more substantial
result (Theorem 4.15) is to develop a fully discrete algorithm to test whether
the spectral radius conjecture holds for $\Gamma$, i.e. to test whether
$\sigma_{\operatorname{ess}}(D_{\Gamma})=\sigma(D_{\Gamma})<\frac{1}{2}$. This
algorithm, which derives from Theorem 3.8, requires the computation only of
the spectral radii of finitely many finite matrices plus the norms of finitely
many finite matrix resolvents.

Before we begin our analysis we note the following equivalences to (4.8) that
will play a key role in our calculations. Here, and throughout, the notations

$\Sigma_{c}:=\left\\{z\in\mathbb{C}:\operatorname{Im}z\in(-c,c)\right\\},\quad\Sigma_{0}:=\mathbb{R},$
(4.9)

for $c>0$ will be convenient.

######  Lemma 4.7.

Given $f:\mathbb{R}_{+}\to\mathbb{R}$ and $\alpha\in(0,1)$, define
$g:\mathbb{R}\to\mathbb{R}$ by

$g(x):=\alpha^{-x}f(\alpha^{x}),\quad x\in\mathbb{R},\quad\mbox{so that}\quad
f(x)=xg(\log_{\alpha}x),\quad x\in\mathbb{R}_{+}.$ (4.10)

Then the following are equivalent:

  1. i)

$f$ is real analytic on $\mathbb{R}_{+}$ and $f(\alpha x)=\alpha f(x)$, $x>0$;

  2. ii)

$g:\mathbb{R}\to\mathbb{R}$ is real analytic and $g(x+1)=g(x)$,
$x\in\mathbb{R}$;

  3. iii)

for some $c>0$, $g$ has an analytic extension to $\Sigma_{c}$ that satisfies
$g(z+1)=g(z)$, for $z\in\Sigma_{c}$, and $g$ and its derivatives $g^{\prime}$
and $g^{\prime\prime}$ are bounded in $\Sigma_{c}$.

Note that if $f$ satisfies our standing assumption (4.8), then, defining $g$
by (4.10),

$f^{\prime}(x)=g(\log_{\alpha}x)+\frac{g^{\prime}(\log_{\alpha}x)}{\log\alpha},\quad
f^{\prime\prime}(x)=\frac{g^{\prime}(\log_{\alpha}x)}{x\log\alpha}+\frac{g^{\prime\prime}(\log_{\alpha}x)}{x\log^{2}\alpha},\quad
x>0.$ (4.11)

It follows from (4.10), the first of (4.11), and the equivalence of i) and
iii), that $f$ is Lipschitz continuous on $[0,\infty)$ if we set $f(0):=0$.

To make use of the results from §3.2 it is convenient to make a change of
variables so that we work with integral operators on $[0,1]$ rather than
$\Gamma_{0}$. Introducing the unitary transformation $U\colon
L^{2}(\Gamma_{0})\to L^{2}(0,1)$ given by

$U\phi(s):=\phi(\alpha^{s},f(\alpha^{s}))(1+f^{\prime}(\alpha^{s})^{2})^{1/4}\alpha^{s/2}\left\lvert\log\alpha\right\rvert^{1/2},\quad
s\in[0,1],\;\;\phi\in L^{2}(\Gamma_{0}),$ (4.12)

define

$\tilde{K}_{t}:=UK_{t}U^{-1}=\sum_{j=-\infty}^{\infty}e^{ijt}UD_{j}U^{-1},\quad
t\in\mathbb{R}.$ (4.13)

Straightforward computations, starting from (4.13), (4.3), and (1.4), give
that

$\displaystyle\tilde{K}_{t}\phi(x)$
$\displaystyle=\int_{0}^{1}\tilde{K}_{t}(x,y)\phi(y)\,\mathrm{d}y,\quad
x\in[0,1],\;\;t\in\mathbb{R},$ (4.14)

where, for $t\in\mathbb{R}$, and $x,y\in\mathbb{R}$ with
$x-y\not\in\mathbb{Z}$,

$\tilde{K}_{t}(x,y)=\frac{1}{2\pi}\sum\limits_{j=-\infty}^{\infty}e^{ijt}\,\frac{p_{j}(x,y)}{1+q_{j}(x,y)^{2}}\left(\frac{1+f^{\prime}(\alpha^{x})^{2}}{1+f^{\prime}(\alpha^{y})^{2}}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert,$
(4.15)

with

$p_{j}(x,y):=\frac{(\alpha^{y}-\alpha^{x+j})f^{\prime}(\alpha^{y})+f(\alpha^{x+j})-f(\alpha^{y})}{(\alpha^{x+j}-\alpha^{y})^{2}},\quad
q_{j}(x,y):=\frac{f(\alpha^{x+j})-f(\alpha^{y})}{\alpha^{x+j}-\alpha^{y}}.$
(4.16)

By Taylor’s theorem applied to $F(t):=f((1-t)\alpha^{y}+t\alpha^{x+j})$, we
see that, for the same range of $x$ and $y$,

$\displaystyle q_{j}(x,y)$
$\displaystyle=\int_{0}^{1}f^{\prime}((1-t)\alpha^{y}+t\alpha^{x+j})\,\mathrm{d}t,$
(4.17) $\displaystyle p_{j}(x,y)$
$\displaystyle=\int_{0}^{1}f^{\prime\prime}((1-t)\alpha^{y}+t\alpha^{x+j})(1-t)\,\mathrm{d}t.$
(4.18)

Using (4.17) and (4.18) to extend the definitions of $q_{j}(x,y)$ and
$p_{j}(x,y)$ to
$\\{(x,y)\in\mathbb{R}^{2}:x-y\in\mathbb{Z}\\}$,141414Precisely, for
$j\in\mathbb{Z}$, $p_{j}(x,y)$ and $q_{j}(x,y)$ are given explicitly by (4.16)
for $y-x\neq j$, while (4.17) and (4.18) imply that
$p_{j}(x,y)=\frac{1}{2}f^{\prime\prime}(\alpha^{y})$,
$q_{j}(x,y)=f^{\prime}(\alpha^{y})$, for $y-x=j$. we see that, for each
$t\in\mathbb{R}$, each term in the sum (4.15) is continuous on
$\mathbb{R}^{2}$. Further, using (4.11) and the equivalence of i) and iii) in
Lemma 4.7, it is easy to see that $p_{j}(x,y)=O(1)$ as $j\to\infty$,
$=O(\alpha^{-j})$ as $j\to-\infty$, uniformly for $x$ and $y$ in compact
subsets of $\mathbb{R}$, so that the series (4.15) converges absolutely and
uniformly on compact subsets, so that $\tilde{K}_{t}(\cdot,\cdot)\in
C(\mathbb{R}^{2})$. Further, $f(\alpha x)=\alpha f(x)$ implies that
$\tilde{K}_{t}(x+1,y)=e^{-it}\tilde{K}_{t}(x,y)$ and
$\tilde{K}_{t}(x,y+1)=e^{it}\tilde{K}_{t}(x,y)$, $x,y\in\mathbb{R}$. Note that
$\tilde{K}_{t}(\cdot,\cdot)\in C(\mathbb{R}^{2})$ implies that
$\tilde{K}_{t}:L^{2}(0,1)\to C[0,1]$, that $\tilde{K}_{t}$ is compact as an
operator on both $C[0,1]$ and $L^{2}(0,1)$, and that the spectrum of
$\tilde{K}_{t}$ is the same on $C[0,1]$ as on $L^{2}(0,1)$, so that

$\sigma(K_{t};L^{2}(\Gamma_{0}))=\sigma(\tilde{K}_{t};L^{2}(0,1))=\sigma(\tilde{K}_{t};C[0,1]),\quad
t\in\mathbb{R}.$ (4.19)

We will use the above equivalence, and the Nyström method results from §3.2,
to compute the spectrum and spectral radius of $K_{t}$, in particular by
applying Theorem 3.8. As a step towards estimating the left hand side of
(3.20) in the case when $K=\tilde{K}_{t}$ and $K_{N}$ is its Nyström method
approximation, we first show that, under our standing assumption (4.8),
$\tilde{K}_{t}(x,\cdot)$ and $\tilde{K}_{t}(\cdot,y)$ can be extended to
bounded analytic functions on $\Sigma_{c}$ for some $c>0$. We will use the
notation $\left\lVert\cdot\right\rVert_{c}$ for the sup norm on $\Sigma_{c}$
and, for functions $h\colon\Sigma_{c}\times\Sigma_{d}\to\mathbb{C}$, the
notation

$\left\lVert
h\right\rVert_{c,d}:=\sup\limits_{\begin{subarray}{c}x\in\Sigma_{c}\\\
y\in\Sigma_{d}\end{subarray}}\left\lvert h(x,y)\right\rvert.$

In an extension of these notations, for
$h\colon\Sigma_{c}\times\Sigma_{c}\to\mathbb{C}$ we define

$\left\lVert
h\right\rVert_{c,0}:=\sup\limits_{\begin{subarray}{c}x\in\Sigma_{c}\\\
y\in\mathbb{R}\end{subarray}}\left\lvert
h(x,y)\right\rvert\quad\text{and}\quad\left\lVert
h\right\rVert_{0,c}:=\sup\limits_{\begin{subarray}{c}x\in\mathbb{R}\\\
y\in\Sigma_{c}\end{subarray}}\left\lvert h(x,y)\right\rvert,$

and, for functions $h:\Sigma_{c}\to\mathbb{C}$, we set
$\|h\|_{0}:=\|h\|_{\infty}:=\sup_{x\in\mathbb{R}}|h(x)|$, the ordinary sup
norm on $\mathbb{R}$.

Let us now estimate the norms $\|\tilde{K}_{t}\|_{c,0}$ and
$\|\tilde{K}_{t}\|_{0,c}$ under our standing assumption (4.8). Since $g$ is
real-valued on $\mathbb{R}$, it follows, using Lemma 2.1, that (4.21) holds
for all sufficiently small $c>0$.

######  Proposition 4.8.

Given (4.8), define $g$ by (4.10) so that, by Lemma 4.7, $g$ has an analytic
continuation to $\Sigma_{c}$, for some $c>0$, such that $g$, $g^{\prime}$, and
$g^{\prime\prime}$ are bounded on $\Sigma_{c}$ and $g(z+1)=g(z)$,
$z\in\Sigma_{c}$. Let
$\mathfrak{I}_{c}:=\|\operatorname{Im}g\|_{c}+\|\operatorname{Im}g^{\prime}\|_{c}/|\log\alpha|$,
and set

$\displaystyle\mathfrak{F}_{d}:=\|g\|_{d}+\|g^{\prime}\|_{d}/|\log\alpha|\quad\mbox{and}\quad\mathfrak{G}_{d}:=\|g^{\prime}\|_{d}+\|g^{\prime\prime}\|_{d}/|\log\alpha|,\quad\mbox{for
}d=0,c.$ (4.20)

If

$c\leq\arccos(\alpha)/\left\lvert\log\alpha\right\rvert\quad\mbox{and}\quad\mathfrak{I}_{c}<1,$
(4.21)

then $\tilde{K}_{t}(x,\cdot)$ and $\tilde{K}_{t}(\cdot,y)$ extend to bounded
analytic functions on $\Sigma_{c}$ for all $x,y,t\in\mathbb{R}$, and

$\displaystyle\|\tilde{K}_{t}\|_{c,0}$
$\displaystyle\leq\frac{\left(1+\mathfrak{F}_{c}^{2}\right)^{1/4}}{\pi\left(1-\mathfrak{I}_{c}^{2}\right)}\Bigg{[}\mathfrak{G}_{c}\,\frac{1+\alpha^{1/2}+\alpha^{-1/2}}{4\alpha^{2}}+|\log\alpha|(\mathfrak{F}_{c}+\mathfrak{F}_{0})\,\sum_{j=2}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}\Bigg{]},$
(4.22) $\displaystyle\|\tilde{K}_{t}\|_{0,c}$
$\displaystyle\leq\frac{\left(1+\mathfrak{F}_{0}^{2}\right)^{1/4}}{\pi\left(1-\mathfrak{I}_{c}^{2}\right)^{5/4}}\Bigg{[}\mathfrak{G}_{c}\,\frac{1+\alpha^{1/2}+\alpha^{-1/2}}{4\alpha^{2}}+2|\log\alpha|\,\mathfrak{F}_{c}\,\sum_{j=2}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}\Bigg{]}.$
(4.23)

Moreover, $\mathbb{R}\to C([0,1]\times[0,1])$,
$t\mapsto\tilde{K}_{t}(\cdot,\cdot)$ is continuous.

######  Remark 4.9 (Bound on the sum in (4.22) and (4.23)).

For fixed $\alpha\in(0,1)$, let $F(x):=\alpha^{x/2}/(\alpha-\alpha^{x})$, for
$x>0$, so that the $j$th term in the sum in (4.22) and (4.23) is $F(j)$. Since
$F$ is decreasing on $(1,\infty)$ we have that, for $n=2,3,\ldots$ and
$0<\alpha<1$,

$\displaystyle\sum_{j=n+1}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}$
$\displaystyle\leq$ $\displaystyle\mathcal{B}_{n}(\alpha)\quad\mbox{so that}$
(4.24)
$\displaystyle\sum_{j=2}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}$
$\displaystyle\leq$
$\displaystyle\mathcal{B}^{*}_{n}(\alpha):=\sum_{j=2}^{n}F(j)+\mathcal{B}_{n}(\alpha),$
(4.25)

where

$\mathcal{B}_{n}(\alpha):=\int_{n}^{\infty}F(x)\,\mathrm{d}x=\frac{1}{2\alpha^{1/2}}\int_{n}^{\infty}\frac{\mathrm{d}x}{\sinh\left(|\log\alpha|(x-1)/2\right)}=\frac{\log(\tanh((n-1)|\log\alpha|/4))}{\alpha^{1/2}\log\alpha}.$

We note, for later reference, that

$\mathcal{B}_{n}(\alpha)\sim
2\alpha^{n/2-1}/|\log\alpha|,\qquad\mbox{as}\qquad n\to\infty.$ (4.26)

###### Proof of Proposition 4.8.

Using (4.11) it follows that

$f(\alpha^{x})=\alpha^{x}g(x),\quad
f^{\prime}(\alpha^{x})=g(x)+\frac{g^{\prime}(x)}{\log\alpha},\quad
f^{\prime\prime}(\alpha^{x})=\frac{g^{\prime}(x)}{\alpha^{x}\log\alpha}+\frac{g^{\prime\prime}(x)}{\alpha^{x}\log^{2}\alpha},\quad
x\in\mathbb{R}.$ (4.27)

Note that the first of the bounds (4.21) implies that $c|\log\alpha|<\pi/2$,
so that the assumptions we have made on $g$ mean that (4.10) provides an
analytic extension of $f$ from $\mathbb{R}_{+}$ to the sector of the complex
plane

$G_{c}:=\\{re^{i\theta}:r>0,|\theta|<c|\log\alpha|\\}=\\{\alpha^{z}:z\in\Sigma_{c}\\},$

and the equations (4.11) hold for all $x$ in this sector. Further, with this
extension, (4.27) holds for all $x\in\Sigma_{c}$. Since (4.11) holds for all
$x\in G_{c}$, and noting that $G_{c}$ is convex, we see that the integrals
(4.17) and (4.18) are well-defined for all $x,y\in\Sigma_{c}$,
$j\in\mathbb{Z}$, and provide analytic continuations of $p_{j}$ and $q_{j}$ to
$\Sigma_{c}\times\Sigma_{c}$. Further, by the uniqueness of analytic
continuation, the equations (4.16) hold for all $x,y\in\Sigma_{c}$, with
$x+j\neq y$.

To complete the proof we will demonstrate that (4.15) provides, for each
$x,y\in\mathbb{R}$, analytic continuations of $\tilde{K}_{t}(\cdot,y)$ and
$\tilde{K}_{t}(x,\cdot)$ from $\mathbb{R}$ to $\Sigma_{c}$ which satisfy the
bounds (4.22) and (4.23), by showing that each term in (4.15) is well-defined
(i.e., that $f^{\prime}(\alpha^{y})^{2}\neq-1$, for $y\in\Sigma_{c}$, and
$1+(q_{j}(x,y))^{2}\neq 0$ for $x,y\in\Sigma_{c}$), and that the series (4.15)
converges absolutely and uniformly for $(x,y)\in\mathbb{R}\times\Sigma_{c}$
and $(x,y)\in\Sigma_{c}\times\mathbb{R}$.

Using (4.27) we see that

$\sup_{z\in
G_{c}}|\operatorname{Im}f^{\prime}(z)|=\sup_{y\in\Sigma_{c}}|\operatorname{Im}f^{\prime}(\alpha^{y})|\leq\mathfrak{I}_{c},$

so that

$\displaystyle\inf\limits_{y\in\Sigma_{c}}\left\lvert
1+f^{\prime}(\alpha^{y})^{2}\right\rvert$ $\displaystyle\geq
1-\sup\limits_{y\in\Sigma_{c}}\left\lvert\operatorname{Im}f^{\prime}(\alpha^{y})\right\rvert^{2}\geq
1-\mathfrak{I}_{c}^{2},$ (4.28)

and, using (4.17),

$\inf_{x,y\in\Sigma_{c}}\left|1+q_{j}(x,y)^{2}\right|\geq 1-\sup_{z\in
G_{c}}|\operatorname{Im}f^{\prime}(z)|^{2}\geq 1-\mathfrak{I}_{c}^{2}.$ (4.29)

Thus, where $T_{j}(x,y)$ denotes the $j$-th term in the sum (4.15), we see
that $T_{j}(x,y)$ is well-defined and analytic for $x,y\in\Sigma_{c}$,
$j\in\mathbb{Z}$. Moreover, we have $T_{j}(x+1,y)=e^{-it}T_{j+1}(x,y)$ and it
follows from $f(\alpha x)=\alpha f(x)$ that $T_{j}(x,y+1)=e^{it}T_{j-1}(x,y)$,
for $x,y\in\Sigma_{c}$, $j\in\mathbb{Z}$. Thus, to prove absolute and uniform
convergence of (4.15) and the bounds (4.22) and (4.23), it suffices to
restrict consideration to $x,y\in\Xi_{c}:=\\{z\in\mathbb{C}:$
$\operatorname{Re}z\in[0,1],$ $\operatorname{Im}z\in(-c,c)\\}$.

We see that $\sup_{x\in\Xi_{c}}|\alpha^{x/2}|\leq 1$ and, using (4.27), that

$\displaystyle\sup\limits_{x\in[0,1]}\left\lvert
1+f^{\prime}(\alpha^{x})^{2}\right\rvert\leq
1+\mathfrak{F}_{0}^{2},\quad\sup\limits_{x\in\Xi_{c}}\left\lvert
1+f^{\prime}(\alpha^{x})^{2}\right\rvert\leq 1+\mathfrak{F}_{c}^{2}.$

To obtain a bound on $|T_{j}(x,y)|$ for $(x,y)\in[0,1]\times\Xi_{c}$, and for
$(x,y)\in\Xi_{c}\times[0,1]$, it remains to bound $p_{j}(x,y)$. Let

$G_{c}^{*}:=\\{re^{i\theta}:\alpha^{2}\leq
r\leq\alpha^{-1},|\theta|<c|\log\alpha|\\}=\\{\alpha^{z}:\operatorname{Re}z\in[-1,2],\,\operatorname{Im}z\in(-c,c)\\}.$

It is clear that $(1-t)\alpha^{y}+t\alpha^{x+j}\in G_{c}^{*}$ for every
$t\in[0,1]$, $|j|\leq 1$, $x\in\Xi_{c}$, and $y\in[0,1]$ if and only if
$G_{c}^{*}$ is star-shaped with respect to every point in $[\alpha,1]$, which
holds if and only if $c$ satisfies the first of the bounds (4.21). Likewise,
$(1-t)\alpha^{y}+t\alpha^{x+j}\in G_{c}^{*}$ for every $t\in[0,1]$, $|j|\leq
1$, $x\in[0,1]$, and $y\in\Xi_{c}$ if and only if the first of (4.21) holds.
Thus, if (4.21) holds, $|j|\leq 1$, and either $(x,y)\in[0,1]\times\Xi_{c}$ or
$(x,y)\in\Xi_{c}\times[0,1]$, it follows from (4.11) and (4.18) that

$|p_{j}(x,y)|\leq\frac{1}{2}\sup_{z\in
G_{c}^{*}}|f^{\prime\prime}(z)|\leq\frac{\mathfrak{G}_{c}}{2\alpha^{2}|\log\alpha|}.$

On the other hand, if $|j|\geq 2$ and $x,y\in\Xi_{c}$, then it follows from
(4.16) (which we have observed above holds for all $x,y\in\Sigma_{c}$ with
$x+j\neq y$) that

$|p_{j}(x,y)|\leq\frac{|f^{\prime}(\alpha^{y})|+|q_{j}(x,y)|}{|\alpha^{x+j}-\alpha^{y}|}.$

Further, for $j\in\mathbb{Z}$,

$|q_{j}(x,y)|\leq\sup_{z\in G_{c}}|f^{\prime}(z)|\leq\mathfrak{F}_{c},\quad
x,y\in\Xi_{c},$

while $|f^{\prime}(\alpha^{y})|\leq\mathfrak{F}_{0}$, for $y\in[0,1]$,
$\leq\mathfrak{F}_{c}$, for $y\in\Xi_{c}$. Moreover, for $|j|\geq 2$,
$x,y\in\Xi_{c}$,

$|\alpha^{x-j}-\alpha^{y}|\geq\left|\alpha^{\operatorname{Re}x-j}-\alpha^{\operatorname{Re}y}\right|\geq\left\\{\begin{array}[]{cc}\alpha-\alpha^{j},&j\geq
2,\\\ \alpha^{j+1}-1,&j\leq-2.\end{array}\right.$

Putting these bounds together we see that, for $|j|\leq 1$,

$\displaystyle\sup\limits_{x\in\Xi_{c},y\in[0,1]}|T_{j}(x,y)|\leq\frac{\mathfrak{G}_{c}\left(1+\mathfrak{F}_{c}^{2}\right)^{1/4}\alpha^{j/2}}{2\alpha^{2}\left(1-\mathfrak{I}_{c}^{2}\right)},\qquad\sup\limits_{x\in[0,1],y\in\Xi_{c}}|T_{j}(x,y)|\leq\frac{\mathfrak{G}_{c}\left(1+\mathfrak{F}_{0}^{2}\right)^{1/4}\alpha^{j/2}}{2\alpha^{2}\left(1-\mathfrak{I}_{c}^{2}\right)^{5/4}},$

while, for $|j|\geq 2$,

$\displaystyle\sup\limits_{x\in\Xi_{c},y\in[0,1]}|T_{j}(x,y)|\leq\frac{|\log\alpha|(\mathfrak{F}_{0}+\mathfrak{F}_{c})\left(1+\mathfrak{F}_{c}^{2}\right)^{1/4}\alpha^{|j|/2}}{\left(1-\mathfrak{I}_{c}^{2}\right)(\alpha-\alpha^{|j|})},$
$\displaystyle\sup\limits_{x\in[0,1],y\in\Xi_{c}}|T_{j}(x,y)|\leq\frac{2|\log\alpha|\,\mathfrak{F}_{c}\left(1+\mathfrak{F}_{c}^{2}\right)^{1/4}\alpha^{|j|/2}}{\left(1-\mathfrak{I}_{c}^{2}\right)^{5/4}(\alpha-\alpha^{|j|})}.$

From these bounds on $|T_{j}(x,y)|$ it is clear that the series (4.15), with
$p_{j}$ and $q_{j}$ given by (4.17) and (4.18), converges absolutely and
uniformly for $(x,y)\in[0,1]\times\Xi_{c}$ and $(x,y)\in\Xi_{c}\times[0,1]$,
so that $\tilde{K}_{t}(x,\cdot)$ and $\tilde{K}_{t}(\cdot,y)$ are analytic in
$\Sigma_{c}$, for $x,y\in\mathbb{R}$, as required. The convergence of the
series is also uniform in $t$, which implies the continuity of
$t\mapsto\tilde{K}_{t}(\cdot,\cdot)$. Further, the above bounds on
$|T_{j}(x,y)|$ imply the bounds (4.22) and (4.23). ∎

Recalling (4.19) and that the kernel of $\tilde{K}_{t}$ is continuous, we will
approximate the spectrum and spectral radius of $K_{t}$, for $t\in\mathbb{R}$,
by approximating the spectrum and spectral radius of $\tilde{K}_{t}$,
considered as an operator on $C[0,1]$, using the Nyström method results of
§3.2. Define $\tilde{K}_{t,N}\colon C[0,1]\to C[0,1]$, for $N\in\mathbb{N}$
and $t\in\mathbb{R}$, by (cf. (3.12))

$\tilde{K}_{t,N}\phi(x):=J_{N}(\tilde{K}_{t}(x,\cdot)\phi)=\frac{1}{N}\sum_{q=1}^{N}\tilde{K}_{t}(x,x_{q,N})\phi(x_{q,N}),\quad
x\in[0,1],\;\;\phi\in C[0,1],$ (4.30)

where

$J_{N}\psi(x):=\sum_{q=1}^{N}\omega_{q,N}\phi(x_{q,N}),\quad\psi\in
C[0,1],\;\;N\in\mathbb{N},$

with $x_{q,N}:=\frac{1}{N}(q-\frac{1}{2})$, $\omega_{q,N}:=N^{-1}$, for
$q=1,\ldots,N$, $N\in\mathbb{N}$. Note that $J_{N}\psi$ is just the $N$-point
midpoint rule approximation to $J\psi:=\int_{0}^{1}\psi(x)\,\mathrm{d}x$, and
that this numerical quadrature rule sequence satisfies (3.13) and (3.14), so
that, for each $t\in\mathbb{R}$, $\tilde{K}_{t,N}\to\tilde{K}_{t}$ as
$N\to\infty$ and $(\tilde{K}_{t,N})_{N\in\mathbb{N}}$ is collectively compact.
This implies, from the general result (3.9), that
$\|(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\|_{\infty}\to 0$ as
$N\to\infty$. In our case this convergence is exponential (cf. [40, Ex.
12.11], [62, §19]).

######  Proposition 4.10.

Let $f$ and $c$ be as in Proposition 4.8. Then, for every $t\in\mathbb{R}$ and
$N\in\mathbb{N}$,

$\left\lVert(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\right\rVert_{\infty}\leq\frac{2e^{2\pi
c}\|\tilde{K}_{t}\|_{0,c}\|\tilde{K}_{t}\|_{c,0}}{e^{2\pi Nc}-1}.$

To prove this proposition we will use the following classical result.

######  Theorem 4.11 (Theorem 9.28 in [39]).

Let $\psi\colon\mathbb{R}\to\mathbb{C}$ be a $1$-periodic function that can be
extended to a bounded analytic function on $\Sigma_{c}$ for some $c>0$. Then

$\left\lvert J\psi-
J_{N}\psi\right\rvert=\left\lvert\int_{0}^{1}\psi(x)\,\mathrm{d}x-\frac{1}{N}\sum\limits_{q=1}^{N}\psi(x_{q,N})\right\rvert\leq\frac{2\left\lVert\psi\right\rVert_{c}}{e^{2\pi
Nc}-1},\qquad N\in\mathbb{N}.$

###### Proof of Proposition 4.10.

It is clear from the definitions of $\tilde{K}_{t}$ and $\tilde{K}_{t,N}$ that
it is enough to consider the case $t\in[-\pi,\pi]$. Assuming $t\in[-\pi,\pi]$,
let $M_{g_{t}}\colon C[0,1]\to C[0,1]$ be multiplication by
$g_{t}(x):=e^{ixt}$. Then $M_{g_{t}}$ is a surjective isometry with
$M_{g_{t}}^{-1}=M_{g_{-t}}$. Let
$\tilde{L}_{t}:=M_{g_{t}}\tilde{K}_{t}M_{g_{t}}^{-1}$ and
$\tilde{L}_{t,N}:=M_{g_{t}}\tilde{K}_{t,N}M_{g_{t}}^{-1}$. Then
$\tilde{L}_{t}$ is an integral operator with kernel
$\tilde{L}_{t}(x,y)=e^{i(x-y)t}\tilde{K}_{t}(x,y)$. As
$\tilde{K}_{t}(x+1,y)=e^{-it}\tilde{K}_{t}(x,y)$ and
$\tilde{K}_{t}(x,y+1)=e^{it}\tilde{K}_{t}(x,y)$, we get
$\tilde{L}_{t}(x+1,y)=\tilde{L}_{t}(x,y)=\tilde{L}_{t}(x,y+1)$ for all
$x,y\in\mathbb{R}$. Moreover, $\|\tilde{L}_{t}\|_{0,c}\leq e^{\pi
c}\|\tilde{K}_{t}\|_{0,c}$ and $\|\tilde{L}_{t}\|_{c,0}\leq e^{\pi
c}\|\tilde{K}_{t}\|_{c,0}$. For $\phi\in C[0,1]$,

$\psi(x):=\int_{0}^{1}\tilde{L}_{t}(x,y)\phi(y)\,\mathrm{d}y,\qquad
x\in\Sigma_{c},$

is an analytic and $1$-periodic extension of $\tilde{L}_{t}\phi$ from $[0,1]$
to $\Sigma_{c}$. Thus also $y\mapsto\tilde{L}_{t}(x,y)\psi(y)$ is analytic and
$1$-periodic for every $x\in\mathbb{R}$. By Theorem 4.11 we obtain that, for
$x\in\mathbb{R}$,

$|(\tilde{L}_{t}-\tilde{L}_{t,N})\tilde{L}_{t}\phi(x)|=|\tilde{L}_{t}\psi(x)-\tilde{L}_{t,N}\psi(x)|=|J(\tilde{L}_{t}(x,\cdot)\psi))-J_{N}(\tilde{L}_{t}(x,\cdot)\psi))|\leq\frac{2C_{t}(x)}{e^{2\pi
Nc}-1},$

where $C_{t}(x):=\sup_{y\in\Sigma_{c}}|\tilde{L}_{t}(x,y)\psi(y)|$. As
$\left\lvert\psi(y)\right\rvert\leq\sup_{z\in[0,1]}|\tilde{L}_{t}(y,z)|\left\lVert\phi\right\rVert_{\infty}$,
for $y\in\Sigma_{c}$, we get

$C_{t}(x)\leq\|\tilde{L}_{t}\|_{0,c}\|\tilde{L}_{t}\|_{c,0}\left\lVert\phi\right\rVert_{\infty}\leq
e^{2\pi
c}\|\tilde{K}_{t}\|_{0,c}\|\tilde{K}_{t}\|_{c,0}\left\lVert\phi\right\rVert_{\infty},$

for all $x\in[0,1]$ and $\phi\in C[0,1]$. Therefore,

$\left\lVert(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\right\rVert_{\infty}=\left\lVert(\tilde{L}_{t}-\tilde{L}_{t,N})\tilde{L}_{t}\right\rVert_{\infty}\leq\frac{2e^{2\pi
c}\|\tilde{K}_{t}\|_{0,c}\|\tilde{K}_{t}\|_{c,0}}{e^{2\pi Nc}-1}.\qed$

Let $A_{t,N}$ be the $N\times N$-matrix defined by (cf. (3.15))

$A_{t,N}(p,q):=\omega_{q,N}\tilde{K}_{t}(x_{p,N},x_{q,N})=\frac{1}{N}\tilde{K}_{t}(x_{p,N},x_{q,N}),\quad
p,q=1,\ldots,N.$ (4.31)

$\tilde{K}_{t}(\cdot,\cdot)$ is defined by the series (4.15) which we truncate
to evaluate numerically. For $M\in\mathbb{N}$ let

$\tilde{K}_{t}^{M}(x,y):=\frac{1}{2\pi}\sum\limits_{j=-M}^{M}e^{ijt}\,\frac{p_{j}(x,y)}{1+q_{j}(x,y)^{2}}\left(\frac{1+f^{\prime}(\alpha^{x})^{2}}{1+f^{\prime}(\alpha^{y})^{2}}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert,\qquad
x,y\in[0,1],$ (4.32)

where $p_{j}$ and $q_{j}$ are defined by (4.16), and consider the matrices
$A_{t,N}^{M}$ given by
$A_{t,N}^{M}(p,q):=\frac{1}{N}\tilde{K}_{t}^{M}(x_{p,N},x_{q,N})$,
$p,q=1,\ldots,N$, so that $A_{t,N}^{M}$ is an approximation to $A_{t,N}$
obtained by using finitely many terms in the series defining
$\tilde{K}_{t}(\cdot,\cdot)$. Similarly, define $\tilde{K}^{M}_{t,N}$ by
(4.30) with $\tilde{K}_{t,N}$ and $\tilde{K}_{t}(\cdot,\cdot)$ replaced by
$\tilde{K}^{M}_{t,N}$ and $\tilde{K}^{M}_{t}(\cdot,\cdot)$, respectively. For
$M,N\in\mathbb{N}$, with $M\geq 2$, and $t\in\mathbb{R}$, by (3.17),

$\left\lVert
A_{t,N}-A_{t,N}^{M}\right\rVert_{\infty}\leq\|\tilde{K}_{t,N}-\tilde{K}_{t,N}^{M}\|_{\infty}\leq\sup_{x,y\in[0,1]}|\tilde{K}_{t}(x,y)-\tilde{K}^{M}_{t}(x,y)|\leq
C_{1}(M),$ (4.33)

where, using the notations of (4.20) and Remark 4.9,

$C_{1}(M):=\frac{2|\log\alpha|\mathfrak{F}_{0}}{\pi}\left(1+\mathfrak{F}_{0}^{2}\right)^{1/4}\,\mathcal{B}_{M}(\alpha);$
(4.34)

this bound (4.33)-(4.34) (cf. (4.22)) is obtained as in Proposition 4.8 (set
$c=0$, only take the terms with $|j|\geq M+1$, and note that
$\mathfrak{I}_{0}=0$), and by using (4.24). Notice that, by (4.26),
$C_{1}(M)=O(\alpha^{M/2})$ as $M\to\infty$.

Our aim now is to estimate $\rho(D_{\Gamma};L^{2}(\Gamma))$ and
$\sigma(D_{\Gamma};L^{2}(\Gamma))$ by computing $\rho(A^{M}_{t,N})$ and
$\sigma(A^{M}_{t,N})$ for only finitely many $t$. For this purpose bounds on
the Lipschitz constants of $A^{M}_{t,N}$ and $\tilde{K}_{t,N}$ as functions of
$t$ will be helpful. It follows from (4.32) that, for $M,N\in\mathbb{N}$,
$t\in\mathbb{R}$, and $p,q=1,\ldots,N$,

$\displaystyle\left\lvert\frac{\partial}{\partial
t}A_{t,N}^{M}(p,q)\right\rvert$ $\displaystyle\leq$ $\displaystyle
B_{N}^{M}(p,q)$ $\displaystyle:=$ $\displaystyle\frac{1}{2\pi
N}\sum\limits_{j=-M}^{M}\frac{|j|\,\left|p_{j}(x_{p,N},x_{q,N})\right|}{1+q_{j}(x_{p,N},x_{q,N})^{2}}\left(\frac{1+f^{\prime}(\alpha^{x_{p,N}})^{2}}{1+f^{\prime}(\alpha^{x_{q,N}})^{2}}\right)^{1/4}\alpha^{\frac{x_{p,N}+x_{q,N}+j}{2}}\left\lvert\log\alpha\right\rvert,$

so that, for $s,t\in\mathbb{R}$, $M,N\in\mathbb{N}$,

$\left\lVert A^{M}_{t,N}-A^{M}_{s,N}\right\rVert_{\infty}\leq|s-t|\left\lVert
B^{M}_{N}\right\rVert_{\infty}.$ (4.36)

Note that, for all $M$ and $N$,

$\|B_{N}^{M}\|_{\infty}\leq
C_{2}:=\frac{1}{2\pi}\sup_{x,y\in[0,1]}\sum\limits_{j=-\infty}^{\infty}|j|\,\left|p_{j}(x,y)\right|\left(1+\|f^{\prime}\|_{\infty}^{2}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert,$
(4.37)

which is finite by the bounds on $p_{j}$ in the proof of Proposition 4.8.
Similarly, from (4.15), for $t\in\mathbb{R}$, $N\in\mathbb{N}$, $x,y\in[0,1]$,

$\displaystyle\left\lvert\frac{\partial}{\partial
t}\tilde{K}_{t,N}(x,y)\right\rvert\leq C_{2}\quad\mbox{so
that}\quad\|\tilde{K}_{t,N}-\tilde{K}_{s,N}\|_{\infty}\leq|s-t|C_{2},$ (4.38)

for $s,t\in\mathbb{R}$ and $N\in\mathbb{N}$, by (3.17). Further, by (3.16),
(4.23) with $c=0$, and (4.25) with151515Computations indicate that
$\mathcal{B}_{10}^{*}(\alpha)$ exceeds the left hand side of (4.25) by not
more than $1.4\%$ for $\alpha\in(0,1)$. $n=10$,

$\|\tilde{K}_{t,N}\|_{\infty}\leq
C_{3}:=\frac{\left(1+\mathfrak{F}_{0}^{2}\right)^{1/4}}{\pi}\Bigg{[}\mathfrak{G}_{0}\,\frac{1+\alpha^{1/2}+\alpha^{-1/2}}{4\alpha^{2}}+2|\log\alpha|\,\mathfrak{F}_{0}\,\mathcal{B}_{10}^{*}(\alpha)\Bigg{]},$
(4.39)

for $t\in\mathbb{R}$ and $N\in\mathbb{N}$. Provided $c$ is such that the
conditions of Lemma 4.7 iii) and (4.21) hold, we have also, by Propositions
4.8 and 4.10 and (4.25), that,

$\left\lVert(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\right\rVert_{\infty}\leq\frac{C_{4}}{e^{2\pi
Nc}-1},\qquad t\in\mathbb{R},\;\;N\in\mathbb{N},$ (4.40)

where $C_{4}:=2e^{2\pi c}C_{5}C_{6}$, and $C_{5}$ and $C_{6}$ denote the right
hand sides of (4.22) and (4.23), respectively, with the sum replaced by its
upper bound $\mathcal{B}^{*}_{10}(\alpha)$.

As noted above Proposition 4.10, $\\{\tilde{K}_{t,N}:N\in\mathbb{N}\\}$ is
collectively compact for each $t\in\mathbb{R}$. We will need, in the proof of
Theorem 4.13, the following stronger statement.

######  Lemma 4.12.

The sets $\\{\tilde{K}_{t,N}:t\in\mathbb{R},\,N\in\mathbb{N}\\}$ and
$\\{\tilde{K}^{M}_{t,N}:t\in\mathbb{R},\,M,N\in\mathbb{N}\\}$ are collectively
compact.

###### Proof.

Where $c$ is such that the conditions of Lemma 4.7 iii) and (4.21) hold,
Proposition 4.8 implies that, for $t\in\mathbb{R}$, $N\in\mathbb{N}$, and
$\phi\in C[0,1]$ with $\|\phi\|_{\infty}\leq 1$, $\tilde{K}_{t,N}\phi$ has an
analytic continuation from $[0,1]$ to $\Sigma_{c}$. Where $\mathcal{F}$
denotes this set of analytic continuations, it follows from (4.22) that
$\mathcal{F}$ is uniformly bounded on $\Sigma_{c}$, so that (e.g., [57, Thm.
14.6]) $\mathcal{F}$ is normal, so that
$\\{\tilde{K}_{t,N}\phi:t\in\mathbb{R},N\in\mathbb{N},\|\phi\|_{\infty}\leq
1\\}$ is relatively compact, i.e.,
$\\{\tilde{K}_{t,N}:t\in\mathbb{R},\,N\in\mathbb{N}\\}$ is collectively
compact. The same argument applies to the family $\tilde{K}^{M}_{t,N}$, on
noting, by inspection of the proof of Proposition 4.8, that, under the same
conditions on $c$, $\tilde{K}_{t}^{M}(\cdot,y)$ extends to an analytic
function on $\Sigma_{c}$ bounded by the right hand side of (4.22), for all
$t,y\in\mathbb{R}$, $M\in\mathbb{N}$. ∎

In the following result, which holds for every $\Gamma$ that satisfies our
standing assumption (4.8), we propose an approximation for the spectrum of
$D_{\Gamma}$ as the union of the spectra of finitely many finite matrices, and
show that this approximation converges in the Hausdorff metric. In this
theorem $\sigma(\tilde{K}^{M}_{t,N})$ denotes the spectrum of
$\tilde{K}^{M}_{t,N}$ either on $L^{2}(0,1)$ or on $C[0,1]$ (cf. (4.19)),
which coincides with $\\{0\\}\cup\sigma(A^{M}_{t,N})$ by Lemma 3.7(ii), and
$\sigma(D_{\Gamma})$ denotes the spectrum of $D_{\Gamma}$ on $L^{2}(\Gamma)$,
which coincides with the essential spectrum by Corollary 4.2.

######  Theorem 4.13.

Choose sequences
$(m_{N})_{N\in\mathbb{N}},(M_{N})_{N\in\mathbb{N}}\subset\mathbb{N}$ such that
$m_{N},M_{N}\to\infty$ as $N\to\infty$, and for each $N$, let
$T_{N}:=\\{\pm(k-1/2)\pi/m_{N}:k=1,\ldots,m_{N}\\}$. Then

$\sigma^{N}(D_{\Gamma}):=\bigcup\limits_{t\in
T_{N}}\sigma(\tilde{K}^{M_{N}}_{t,N})=\\{0\\}\cup\bigcup\limits_{t\in
T_{N}}\sigma(A^{M_{N}}_{t,N})\;\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\;\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$
(4.41)

as $N\to\infty$.

###### Proof.

Set $\Lambda:=\operatorname{\sigma}(D_{\Gamma})$, so that
$\Lambda=\bigcup_{t\in[-\pi,\pi]}\operatorname{\sigma}(\tilde{K}_{t})$ by
Theorem 4.4 and (4.19), and set $\Lambda_{N}:=\sigma^{N}(D_{\Gamma})$. We
first observe that $(\Lambda_{N})_{N\in\mathbb{N}}$ is uniformly bounded as
$\\{\|\tilde{K}^{M_{N}}_{t,N}\|_{\infty}:N\in\mathbb{N},\,t\in T_{N}\\}$ is
bounded, by (4.33) and (4.39). Thus, as noted above Theorem 3.6, to prove that
$\Lambda_{N}\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}\Lambda$ it is
enough to show that $\lim\inf\Lambda_{N}=\lim\sup\Lambda_{N}=\Lambda$.

Next, we note that if the sequences
$(\mathfrak{M}_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$,
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$, and
$(t_{k})_{k\in\mathbb{N}}\subset[-\pi,\pi]$ satisfy $N_{k}\to\infty$,
$\mathfrak{M}_{k}\to\infty$, $t_{k}\to t\in[-\pi,\pi]$, as $k\to\infty$, then
the sequence $(\tilde{K}^{\mathfrak{M}_{k}}_{t_{k},N_{k}})_{k\in\mathbb{N}}$
is collectively compact by Lemma 4.12, and converges strongly to
$\tilde{K}_{t}$ by (4.38) and (4.33), and since
$\tilde{K}_{t,N_{k}}\to\tilde{K}_{t}$, as noted above Proposition 4.10. Thus
also
$\sigma(\tilde{K}^{\mathfrak{M}_{k}}_{t_{k},N_{k}})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\sigma(\tilde{K}_{t})$, by Theorem 3.6.

For every $t\in[-\pi,\pi]$ we can choose a sequence $(t_{N})_{N\in\mathbb{N}}$
such that $t_{N}\in T_{N}$ for each $N$ and $t_{N}\to t$. Then, by the
observation just made,
$\Lambda_{N}\supset\sigma(\tilde{K}^{M_{N}}_{t_{N},N})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\sigma(\tilde{K}_{t})$ as $N\to\infty$, so that
$\lim\inf\Lambda_{N}\supset\lim\inf\sigma(\tilde{K}^{M_{N}}_{t_{N},N})=\sigma(\tilde{K}_{t})$.
Thus $\lim\inf\Lambda_{N}\supset\Lambda$.

If $\lambda\in\lim\sup\Lambda_{N}$ then there exists a sequence
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ and a sequence
$(\lambda_{k})_{k\in\mathbb{N}}$ such that $\lambda_{k}\to\lambda$ and, for
each $k$, $\lambda_{k}\in\Lambda_{N_{k}}$, so that
$\lambda_{k}\in\sigma(\tilde{K}^{\mathfrak{M}_{k}}_{t_{k},N_{k}})$, for some
$t_{k}\in T_{N_{k}}$, where $\mathfrak{M}_{k}:=M_{N_{k}}$. By passing to a
subsequence if necessary, we may assume that $t_{k}\to t\in[-\pi,\pi]$, so
that
$\sigma(\tilde{K}^{\mathfrak{M}_{k}}_{t_{k},N_{k}})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\sigma(\tilde{K}_{t})$, in particular
$\lim\inf\sigma(\tilde{K}^{\mathfrak{M}_{k}}_{t_{k},N_{k}})=\sigma(\tilde{K}_{t})$,
so that $\lambda\in\sigma(\tilde{K}_{t})\subset\Lambda$. Thus
$\lim\sup\Lambda_{N}\subset\Lambda\subset\lim\inf\Lambda_{N}$. But also
$\lim\inf\Lambda_{N}\subset\lim\sup\Lambda_{N}$, so the proof is complete. ∎

######  Remark 4.14 (Reduced computation expression for
$\sigma^{N}(D_{\Gamma})$).

Note that, by Remark 4.5 and Lemma 3.7(ii), $\sigma^{N}(D_{\Gamma})$, given by
(4.41), can be written as

$\sigma^{N}(D_{\Gamma})=\\{0\\}\cup\\{\lambda,\overline{\lambda}:\lambda\in\sigma(A^{M_{N}}_{t,N}),\,t\in
T_{N},\,t>0\\}.$

Our second main result, obtained by applying Theorem 3.8161616To obtain
(4.43), motivated by Remark 3.10 our starting point is (3.21) rather than
(3.20), since our interest will be to apply Theorem 4.15 in cases where
$\rho_{0}<\|D_{\Gamma}\|$., provides a criterion, given $\rho_{0}>0$, for
$\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$. Note that this criterion requires
computation of spectral quantities only for the $N\times N$ matrices
$A_{t,N}^{M}$ for finitely many $t\in[0,\pi]$.

######  Theorem 4.15.

Under our standing assumption (4.8), define $g$ by (4.10) so that, by Lemma
4.7, $g$ has an analytic continuation to $\Sigma_{c}$, for some $c>0$, such
that $g$, $g^{\prime}$, and $g^{\prime\prime}$ are bounded on $\Sigma_{c}$,
and, without loss of generality, assume that (4.21) holds. Suppose that
$\rho_{0}>0$, $m,M,N\in\mathbb{N}$, with $M\geq 2$. For $k=1,\ldots,m$, set
$t_{k}:=(k-1/2)\pi/m$, suppose that $\rho(A^{M}_{t_{k},N})<\rho_{0}$ for
$k=1,\ldots,m$, and recursively define $\mu_{k,\ell}$, for $\ell=1,2,\ldots$,
by $\mu_{k,1}:=\rho_{0}$, and by

$\nu_{k,\ell}:=\left\lVert(A_{t_{k},N}^{M}-\mu_{k,\ell}I)^{-1}\right\rVert_{\infty}^{-1}\quad\text{and}\quad\mu_{k,\ell+1}:=\mu_{k,\ell}e^{i\frac{\nu_{k,\ell}}{2\rho_{0}}},\quad\mbox{for
}\ell\in\mathbb{N}.$ (4.42)

Further, for $k=1,\ldots,m$, let $n_{k}$ denote the smallest integer such that
$\sum_{\ell=1}^{n_{k}}\nu_{k,\ell}\geq 4\pi\rho_{0}$, and let

$R_{m,M,N}:=\min_{\stackrel{{\scriptstyle
k=1,\ldots,m}}{{\ell=1,\ldots,n_{k}}}}\left(\frac{\nu_{k,\ell}}{4}+\frac{\nu_{k,\ell+1}}{2}\right).$

If

$\mathcal{L}_{c}(f,N):=\frac{C_{4}}{e^{2\pi
Nc}-1}<\mathcal{R}_{c}(f,m,M,N):=r^{2}_{0}\left(1+C_{3}\left(R_{m,M,N}-C_{1}(M)-\frac{\pi}{2m}\|B_{N}^{M}\|_{\infty}\right)^{-1}\right)^{-1},$
(4.43)

then $\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$. Conversely, if
$\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$, then, provided $m$ and $M$ are
sufficiently large, there exists $N_{0}\in\mathbb{N}$ such that (4.43) holds
and $\rho(A_{t,N}^{M})<\rho_{0}$ for $t\in[0,\pi]$ and all $N\geq N_{0}$.

###### Proof.

By Theorem 4.4, Remark 4.5, and (4.19), to show that
$\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$, it is enough to check that
$\rho_{C[0,1]}(\tilde{K}_{t})<\rho_{0}$ for every $t\in[0,\pi]$. So pick
$t\in[0,\pi]$. Then $|t-t_{k}|\leq\pi/(2m)$, for some $k\in\\{1,\ldots,m\\}$.
To conclude that $\rho_{C[0,1]}(\tilde{K}_{t})<\rho_{0}$ we apply Theorem 3.8
with $K=\tilde{K}_{t}$, $K_{N}=\tilde{K}_{t,N}$, $A_{N}=A_{t,N}$, and
$A_{N}^{{\dagger}}=A_{t_{k},N}^{M}$, noting that, with these definitions,
$\|K_{N}\|_{\infty}=\|\tilde{K}_{t,N}\|_{\infty}\leq C_{3}$,

$\|A_{N}-A_{N}^{{\dagger}}\|_{\infty}\leq\|A_{t,N}-A_{t,N}^{M}\|_{\infty}+\|A^{M}_{t,N}-A_{t_{k},N}^{M}\|_{\infty}\leq
C_{1}(M)+\frac{\pi}{2m}\|B_{N}^{M}\|_{\infty},$

by (4.33) and (4.36), and $\|(K-K_{N})K\|_{\infty}\leq C_{4}/(e^{2\pi Nc}-1)$,
by (4.40).

Conversely, if $\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$ then, by Theorem 4.4
and (4.19), $\rho_{C[0,1]}(\tilde{K}_{t})<\rho_{0}$ for $t\in[0,\pi]$. Arguing
as in the proof of Theorem 3.8, this implies that for every $t\in[0,\pi]$
there exists $N_{0}(t)$ such that $\|(\tilde{K}_{t,N}-\lambda I)^{-1}\|$ is
bounded uniformly in $\lambda$ and $N$ for $N\geq N_{0}(t)$ and
$|\lambda|\geq\rho_{0}$. This, combined with the estimate (4.38) and a
standard compactness argument, implies that, for some $N^{*}\in\mathbb{N}$,
and $c^{*}>0$, $\|(\tilde{K}_{t,N}-\lambda I)^{-1}\|^{-1}\geq c^{*}$ for all
$N\geq N^{*}$, $|\lambda|\geq\rho_{0}$, and $t\in[0,\pi]$. It follows, by
(4.33) and (3.1), and recalling Lemma 3.7(ii), that, for all $t\in[0,\pi]$ and
all sufficiently large $M$ and $N$,
$\rho(A_{t,N}^{M})=\rho_{C[0,1]}(\tilde{K}^{M}_{t,N})<\rho_{0}$ and
$\|(\tilde{K}^{M}_{t,N}-\lambda I)^{-1}\|^{-1}\geq c^{*}/2$, for
$|\lambda|\geq\rho_{0}$. Thus, and by Lemma 3.7(iii), $R_{m,N,M}\geq
3c^{*}/8$, for all $m\in\mathbb{N}$ and all sufficiently large $M$ and $N$. It
follows that (4.43) holds, for all sufficiently large $m$, $M$, and $N$, since
(4.34) and (4.26), and that $\|B_{N}^{M}\|_{\infty}$ is bounded independently
of $M$ and $N$ by (4.37), imply that the right hand side of (4.43) is positive
and bounded away from zero. ∎

###  4.3 The 2D case: two-sided infinite graphs

Figure 3: Graph of $f\colon\mathbb{R}\to\mathbb{R}$, $f(x):=\left\lvert
x\right\rvert\sin^{2}(\pi\log_{\alpha}\left\lvert x\right\rvert)$, in the case
$\alpha=\frac{2}{3}$

We now extend the results of the previous subsection, for the case when $d=2$
and $A=\mathbb{R}_{+}$, to the more involved case171717As discussed in the
introduction and at the beginning of §4, it is this case which is relevant to
the spectral radius conjecture. where $d=2$ and $A=\mathbb{R}$. Our goals and
methods are those of §4.2, but, where
$\dot{\mathbb{R}}:=\mathbb{R}\setminus\\{0\\}$, we assume now that
$f\in\mathcal{A}(\dot{\mathbb{R}})$, the space of functions
$\mathbb{R}\to\mathbb{R}$ that are real analytic on $\dot{\mathbb{R}}$ with
$f(0)=0$ (a prototypical example is Figure 3). Thus our standing assumption
through this subsection is that

$\Gamma=\left\\{(x,f(x)):x\in\mathbb{R}\right\\}\;\mbox{ where
}\;f\in\mathcal{A}(\dot{\mathbb{R}})\;\mbox{ and, for some $\alpha\in(0,1)$,
}\;f(\alpha x)=\alpha f(x),\quad x\in\mathbb{R}.$ (4.44)

Define $f_{\pm}:\mathbb{R}_{+}\to\mathbb{R}$ by $f_{\pm}(x):=f(\pm x)$,
$x\in\mathbb{R}_{+}$, and $g_{\pm}:\mathbb{R}\to\mathbb{R}$ (cf. (4.10)) by

$g_{\pm}(x):=\alpha^{-x}f_{\pm}(\alpha^{x}),\quad x\in\mathbb{R},\quad\mbox{so
that}\quad f_{\pm}(x)=xg_{\pm}(\log_{\alpha}x),\quad x\in\mathbb{R}_{+}.$
(4.45)

Note that the assumption $f\in\mathcal{A}(\dot{\mathbb{R}})$ is equivalent to
the assumption that $f_{\pm}\in\mathcal{A}(\mathbb{R}_{+})$ and $f(0)=0$. Note
also that Lemma 4.7 applies with $f$ and $g$ replaced by $f_{\pm}$ and
$g_{\pm}$. One simple consequence of these observations (see the discussion
below (4.11)) is that (4.44) implies $f\in C^{0,1}(\mathbb{R})$.

As in the one-sided case, our starting point is the formula for
$\sigma(D_{\Gamma})$ in Corollary 4.4 in terms of $\sigma(K_{t})$, for
$t\in[-\pi,\pi]$. Again, $K_{t}$ is compact for every $t\in\mathbb{R}$ by
Corollary 4.6, and, following the pattern of §4.2, our goal is to compute
$\sigma(K_{t})$ and $\rho(K_{t})$ by the results of §3.2. As in §4.2, to make
use of these results it is convenient to work with integral operators on
$[0,1]$ rather than $\Gamma_{0}$. Reflecting that $\Gamma_{0}$ has two
components, $\Gamma_{0}^{+}:=\\{(x,f(x))\in\Gamma_{0}:x>0\\}$ and
$\Gamma_{0}^{-}:=\\{(x,f(x))\in\Gamma_{0}:x<0\\}$, it is natural to work in
this case with a unitary operator $U:L^{2}(\Gamma_{0})\to(L^{2}(0,1))^{2}$,
defined by $U\psi:=(U_{+}P_{+}\psi,U_{-}P_{-}\psi)^{T}$, $\psi\in
L^{2}(\Gamma_{0})$, where $P_{\pm}$ is restriction to $\Gamma_{0}^{\pm}$, i.e.
$P_{\pm}\psi:=\psi|_{\Gamma_{0}^{\pm}}$, $\psi\in L^{2}(\Gamma_{0})$, and
$U_{\pm}:L^{2}(\Gamma_{0}^{\pm})\to L^{2}(0,1)$ is the unitary operator
defined by (cf. (4.12))

$U_{\pm}\phi(x):=\phi(\pm\alpha^{x},f_{\pm}(\alpha^{x}))(1+f_{\pm}^{\prime}(\alpha^{x})^{2})^{1/4}\alpha^{x/2}|\log\alpha|^{1/2},\qquad
x\in[0,1],\;\;\phi\in L^{2}(\Gamma_{0}^{\pm}).$

With $U$ as given above we define an operator $\tilde{K}_{t}$ on
$(L^{2}(0,1))^{2}$, which is unitarily equivalent to $K_{t}$, by (4.13). It is
easy to see (cf. (4.14)) that

$\tilde{K}_{t}=\begin{pmatrix}\tilde{K}^{-}_{t}&\tilde{L}^{-}_{t}\\\
\tilde{L}^{+}_{t}&\tilde{K}^{+}_{t}\end{pmatrix},\qquad t\in\mathbb{R},$
(4.46)

where the entries of $\tilde{K}_{t}$ are integral operators on $L^{2}(0,1)$
with continuous kernels. The kernels of $\tilde{K}_{t}^{\pm}$ are
$\tilde{K}^{\pm}_{t}(\cdot,\cdot)$, where (cf. (4.15))

$\tilde{K}^{\pm}_{t}(x,y):=\frac{1}{2\pi}\sum\limits_{j=-\infty}^{\infty}e^{ijt}\,\frac{p^{\pm}_{j}(x,y)}{1+q^{\pm}_{j}(x,y)^{2}}\left(\frac{1+f_{\pm}^{\prime}(\alpha^{x})^{2}}{1+f_{\pm}^{\prime}(\alpha^{y})^{2}}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert,\quad
x,y\in\mathbb{R},$ (4.47)

and $p_{j}^{\pm}$ and $q_{j}^{\pm}$ are defined by (4.16), (4.17), and (4.18),
but with $f$ replaced by $f_{\pm}$. The kernels of $\tilde{L}_{t}^{\pm}$ are
$\tilde{L}_{t}^{\pm}(\cdot,\cdot)$, where, for $x,y\in\mathbb{R}$,

$\tilde{L}^{\pm}_{t}(x,y):=\frac{1}{2\pi}\sum\limits_{j=-\infty}^{\infty}e^{ijt}\,\frac{\tilde{p}^{\pm}_{j}(x,y)}{1+\tilde{q}^{\pm}_{j}(x,y)^{2}}\left(\frac{1+f_{\pm}^{\prime}(\alpha^{x})^{2}}{1+f_{\mp}^{\prime}(\alpha^{y})^{2}}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert$
(4.48)

and

$\tilde{p}^{\pm}_{j}(x,y):=\frac{(\alpha^{y}+\alpha^{x+j})f_{\mp}^{\prime}(\alpha^{y})+f_{\pm}(\alpha^{x+j})-f_{\mp}(\alpha^{y})}{(\alpha^{x+j}+\alpha^{y})^{2}},\quad\tilde{q}^{\pm}_{j}(x,y):=\frac{f_{\pm}(\alpha^{x+j})-f_{\mp}(\alpha^{y})}{\alpha^{x+j}+\alpha^{y}}.$
(4.49)

Analogously to (4.19), we have that

$\sigma(K_{t};L^{2}(\Gamma_{0}))=\sigma(\tilde{K}_{t};(L^{2}(0,1))^{2})=\sigma(\tilde{K}_{t};(C[0,1])^{2}),\quad
t\in\mathbb{R}.$ (4.50)

As in the one-sided case (see the discussion around (4.30)), to estimate
spectral properties of $\tilde{K}_{t}$ as an operator on $(C[0,1])^{2}$ we
approximate $\tilde{K}^{\pm}_{t}$ by finite rank operators
$\tilde{K}^{\pm}_{t,N}$ given by

$\tilde{K}^{\pm}_{t,N}\phi(x):=\frac{1}{N}\sum_{n=1}^{N}\tilde{K}^{\pm}_{t}(x,x_{n,N})\phi(x_{n,N}),\quad\phi\in
C[0,1],\;x\in[0,1],\;N\in\mathbb{N}.$

Similarly, we approximate $\tilde{L}^{\pm}_{t}$ by $\tilde{L}^{\pm}_{t,N}$,
leading to finite rank approximations $\tilde{K}_{t,N}$, $N\in\mathbb{N}$, to
$\tilde{K}_{t}$, given by (4.46) with $\tilde{K}^{\pm}_{t}$ and
$\tilde{L}^{\pm}_{t}$ replaced by $\tilde{K}^{\pm}_{t,N}$ and
$\tilde{L}^{\pm}_{t,N}$. Arguing as below (4.30), we have that
$\tilde{K}_{t,N}\to\tilde{K}_{t}$ and $(\tilde{K}_{t,N})_{N\in\mathbb{N}}$ is
collectively compact, so that
$\|(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\|_{\infty}\to 0$ as
$N\to\infty$, where $\|\cdot\|_{\infty}$ here denotes the operator norm of an
operator on $(C[0,1])^{2}$ equipped with the norm $\|\cdot\|_{\infty}$ defined
by $\|\phi\|_{\infty}:=\max(\|\phi_{+}\|_{\infty},\|\phi_{-}\|_{\infty})$, for
$\phi=(\phi_{+},\phi_{-})^{T}\in(C[0,1])^{2}$ Indeed, we have the following
analogue of Proposition 4.10, in which our other norm notations are as defined
above Proposition 4.8.

######  Proposition 4.16.

Let $t\in\mathbb{R}$ and $c>0$ and assume that, for every $x,y\in\mathbb{R}$,
$\tilde{K}^{\pm}_{t}(x,\cdot)$, $\tilde{L}^{\pm}_{t}(x,\cdot)$,
$\tilde{K}^{\pm}_{t}(\cdot,y)$ and $\tilde{L}^{\pm}_{t}(\cdot,y)$ have
analytic continuations from $\mathbb{R}$ to $\Sigma_{c}$ that are bounded in
$\Sigma_{c}$. Then
$\|(\tilde{K}_{t}-\tilde{K}_{t,N})\tilde{K}_{t}\|_{\infty}\leq 2e^{2\pi
c}C^{*}/(e^{2\pi Nc}-1)$, where

$\displaystyle C^{*}$
$\displaystyle:=\max\bigg{\\{}\|\tilde{K}^{-}_{t}\|_{0,c}\|\tilde{K}^{-}_{t}\|_{c,0}+\|\tilde{L}^{-}_{t}\|_{0,c}\|\tilde{L}^{+}_{t}\|_{c,0}+\|\tilde{K}^{-}_{t}\|_{0,c}\|\tilde{L}^{-}_{t}\|_{c,0}+\|\tilde{L}^{-}_{t}\|_{0,c}\|\tilde{K}^{+}_{t}\|_{c,0},$
$\displaystyle\qquad\qquad\quad\|\tilde{L}^{+}_{t}\|_{0,c}\|\tilde{K}^{-}_{t}\|_{c,0}+\|\tilde{K}^{+}_{t}\|_{0,c}\|\tilde{L}^{+}_{t}\|_{c,0}+\|\tilde{L}^{+}_{t}\|_{0,c}\|\tilde{L}^{-}_{t}\|_{c,0}+\|\tilde{K}^{+}_{t}\|_{0,c}\|\tilde{K}^{+}_{t}\|_{c,0}\bigg{\\}}.$
(4.51)

###### Proof.

Notice first that

$(\tilde{K}_{t}-K^{N}_{t})\tilde{K}_{t}=\begin{pmatrix}(\tilde{K}^{-}_{t}-K^{-}_{t,N})\tilde{K}^{-}_{t}+(\tilde{L}^{-}_{t}-L^{-}_{t,N})\tilde{L}^{+}_{t}&(\tilde{K}^{-}_{t}-K^{-}_{t,N})\tilde{L}^{-}_{t}+(\tilde{L}^{-}_{t}-L^{-}_{t,N})\tilde{K}^{+}_{t}\\\
(\tilde{L}^{+}_{t}-L^{+}_{t,N})\tilde{K}^{-}_{t}+(\tilde{K}^{+}_{t}-K^{+}_{t,N})\tilde{L}^{+}_{t}&(\tilde{L}^{+}_{t}-L^{+}_{t,N})\tilde{L}^{-}_{t}+(\tilde{K}^{+}_{t}-K^{+}_{t,N})\tilde{K}^{+}_{t}\end{pmatrix},$

and denote the entries of this matrix by $A_{j,k}$, $j,k=1,2$. For each of
these terms we obtain an estimate similar to that in Proposition 4.10, by
arguing as in the proof of that proposition. Since also

$\|(\tilde{K}_{t}-K^{N}_{t})\tilde{K}_{t}\|_{\infty}=\max_{j=1,2}\sum_{k=1}^{2}\|A_{j,k}\|_{\infty},$

the result follows. ∎

Let us now estimate, under our standing assumption (4.44), the norms of the
kernels that appear in Proposition 4.16 (cf. Proposition 4.8).

######  Proposition 4.17.

Given (4.44), define $g_{\pm}$ by (4.45) so that, by Lemma 4.7, $g_{\pm}$ have
analytic continuations to $\Sigma_{c}$, for some $c>0$, such that $g_{\pm}$,
$g_{\pm}^{\prime}$, and $g_{\pm}^{{}^{\prime\prime}}$ are bounded on
$\Sigma_{c}$ and $g_{\pm}(z+1)=g_{\pm}(z)$, $z\in\Sigma_{c}$. Let

$\displaystyle\mathfrak{I}_{c,\pm}:=\|\operatorname{Im}g_{\pm}\|_{c}+\|\operatorname{Im}g_{\pm}^{\prime}\|_{c}/|\log\alpha|\quad\mbox{and}\quad\mathfrak{K}_{c,\pm}:=\alpha^{-1}|\log\alpha|\max(\|g_{\pm}\|_{c},\|g_{\mp}\|_{0}),$
(4.52)

and set

$\displaystyle\mathfrak{F}_{d,\pm}:=\|g_{\pm}\|_{d}+\|g_{\pm}^{\prime}\|_{d}/|\log\alpha|\quad\mbox{and}\quad\mathfrak{G}_{d,\pm}:=\|g_{\pm}^{\prime}\|_{d}+\|g_{\pm}^{\prime\prime}\|_{d}/|\log\alpha|,\quad\mbox{for
}d=0,c.$ (4.53)

If

$c\leq\arccos(\alpha)/\left\lvert\log\alpha\right\rvert,\quad\mathfrak{I}_{c,\pm}<1,\quad\mbox{and}\quad\|\operatorname{Im}g_{\pm}\|_{c}+\alpha
c\mathfrak{K}_{c,\pm}<\alpha^{2},$ (4.54)

then $\tilde{K}^{\pm}_{t}(x,\cdot)$, $\tilde{L}^{\pm}_{t}(x,\cdot)$,
$\tilde{K}^{\pm}_{t}(\cdot,y)$, and $\tilde{L}^{\pm}_{t}(\cdot,y)$ extend to
bounded analytic functions on $\Sigma_{c}$ for all $x,y,t\in\mathbb{R}$, and

$\displaystyle\|\tilde{K}^{\pm}_{t}\|_{c,0}$
$\displaystyle\leq\frac{\left(1+\mathfrak{F}_{c,\pm}^{2}\right)^{1/4}}{\pi\left(1-\mathfrak{I}_{c,\pm}^{2}\right)}\Bigg{[}\mathfrak{G}_{c,\pm}\,\frac{1+\alpha^{1/2}+\alpha^{-1/2}}{4\alpha^{2}}+|\log\alpha|(\mathfrak{F}_{c,\pm}+\mathfrak{F}_{0,\pm})\,\sum_{j=2}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}\Bigg{]},$
$\displaystyle\|\tilde{K}^{\pm}_{t}\|_{0,c}$
$\displaystyle\leq\frac{\left(1+\mathfrak{F}_{0,\pm}^{2}\right)^{1/4}}{\pi\left(1-\mathfrak{I}_{c,\pm}^{2}\right)^{5/4}}\Bigg{[}\mathfrak{G}_{c,\pm}\,\frac{1+\alpha^{1/2}+\alpha^{-1/2}}{4\alpha^{2}}+2|\log\alpha|\,\mathfrak{F}_{c,\pm}\,\sum_{j=2}^{\infty}\frac{\alpha^{j/2}}{\alpha-\alpha^{j}}\Bigg{]},$
$\displaystyle\|\tilde{L}^{\pm}_{t}\|_{c,0}$
$\displaystyle\leq\frac{1}{2\pi}\frac{\left\lvert\log\alpha\right\rvert\mathfrak{F}_{0,\mp}+\mathfrak{K}_{c,\pm}}{1-\alpha^{-4}\left(\left\lVert\operatorname{Im}g_{\pm}\right\rVert_{c}+\alpha
c\mathfrak{K}_{c,\pm}\right)^{2}}\,\left(1+\mathfrak{F}_{c,\pm}^{2}\right)^{1/4}\sum\limits_{j=-\infty}^{\infty}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}},$
(4.55) $\displaystyle\|\tilde{L}^{\pm}_{t}\|_{0,c}$
$\displaystyle\leq\frac{1}{2\pi}\frac{\left\lvert\log\alpha\right\rvert\mathfrak{F}_{c,\mp}+\mathfrak{K}_{c,\mp}}{1-\alpha^{-4}\left(\left\lVert\operatorname{Im}g_{\mp}\right\rVert_{c}+\alpha
c\mathfrak{K}_{c,\mp}\right)^{2}}\,\left(\frac{1+\mathfrak{F}_{0,\pm}^{2}}{1-\mathfrak{I}_{c,\mp}^{2}}\right)^{1/4}\sum\limits_{j=-\infty}^{\infty}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}.$
(4.56)

Moreover, the mappings $\mathbb{R}\to C([0,1]\times[0,1])$,
$t\mapsto\tilde{K}^{\pm}_{t}(\cdot,\cdot)$ and
$t\mapsto\tilde{L}^{\pm}_{t}(\cdot,\cdot)$, are continuous.

######  Remark 4.18 (Bound on the sum in (4.55) and (4.56)).

For fixed $\alpha\in(0,1)$, let $F(x):=(\alpha^{x/2}+\alpha^{-x/2})^{-1}$, for
$x\in\mathbb{R}$, so that the $j$th term in the sum in (4.55) and (4.56) is
$\alpha^{-2}F(j)$, and note that $F$ is even. Arguing as in Remark 4.9, since
$F$ is decreasing on $[0,\infty)$,

$\sum_{j=n+1}^{\infty}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}=\alpha^{-2}\sum_{j=n+1}^{\infty}F(j)\leq\mathcal{C}_{n}(\alpha):=\alpha^{-2}\int_{n}^{\infty}F(x)\,\mathrm{d}x=\frac{2\arctan(\alpha^{n/2})}{\alpha^{2}\left\lvert\log\alpha\right\rvert},$
(4.57)

for $0<\alpha<1$ and $n\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. Thus, for
$0<\alpha<1$ and $n\in\mathbb{N}_{0}$,

$\displaystyle\sum\limits_{j=-\infty}^{\infty}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}$
$\displaystyle\leq\mathcal{C}_{n}^{*}(\alpha):=\frac{1}{2\alpha^{2}}+2\alpha^{-2}\sum_{j=1}^{n}F(j)+2\mathcal{C}_{n}(\alpha)\quad\mbox{and}$
(4.58)
$\displaystyle\left\lvert\sum\limits_{j=-\infty}^{\infty}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}-\sum\limits_{j=-n}^{n}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}\right\rvert$
$\displaystyle\leq 2\mathcal{C}_{n}(\alpha).$ (4.59)

###### Proof of Proposition 4.17.

The results for $\tilde{K}_{t}^{\pm}(\cdot,\cdot)$ follow immediately from
Proposition 4.8, applied with $f$ replaced by $f_{\pm}$. It remains to
consider the off-diagonal entries $\tilde{L}^{\pm}_{t}$. We give the detail
for $\tilde{L}_{t}^{+}$; the results for $\tilde{L}_{t}^{-}$ follow by the
same argument with the roles of $f_{+}$ and $f_{-}$ reversed.

We extend the definition of $\tilde{L}^{+}_{t}$ via (4.48) to all
$x,y\in\Sigma_{c}$. The upcoming computations will show that this is well-
defined and the estimates (4.55) and (4.56) hold. Denote by $T_{j}(x,y)$ the
$j$th term in the sum (4.48), that is,

$T_{j}(x,y):=e^{ijt}\frac{\tilde{p}^{+}_{j}(x,y)}{1+\tilde{q}^{+}_{j}(x,y)^{2}}\left(\frac{1+f_{+}^{\prime}(\alpha^{x})^{2}}{1+f_{-}^{\prime}(\alpha^{y})^{2}}\right)^{1/4}\alpha^{\frac{x+y+j}{2}}\left\lvert\log\alpha\right\rvert,\quad
x,y\in\Sigma_{c}.$ (4.60)

We clearly have $T_{j}(x+1,y)=e^{-it}T_{j+1}(x,y)$ and the dilation
invariance, $f(\alpha x)=\alpha f(x)$, implies that
$T_{j}(x,y+1)=e^{it}T_{j-1}(x,y)$ for $x,y\in\Sigma_{c}$, $j\in\mathbb{Z}$.
Thus, to prove the uniform convergence of the series (4.48) and the bounds
(4.55) and (4.56), it suffices to restrict consideration to
$x,y\in\Xi_{c}=\\{z\in\mathbb{C}:$ $\operatorname{Re}z\in[0,1]$,
$\operatorname{Im}z\in(-c,c)\\}$. First notice that, for $x,y\in\Xi_{c}$,

$\left\lvert\alpha^{x+j}+\alpha^{y}\right\rvert\geq\alpha^{\operatorname{Re}x+j}\cos(\log\alpha\operatorname{Im}x)+\alpha^{\operatorname{Re}y}\cos(\log\alpha\operatorname{Im}y)\geq\alpha^{\operatorname{Re}x+j+1}+\alpha^{\operatorname{Re}y+1}\geq\alpha^{j+2}+\alpha^{2},$
(4.61)

by the first of (4.54), which implies that $\cos(t\log\alpha)\geq\alpha$, for
$-c\leq t\leq c$. Hence, for $x,y\in\Xi_{c}$ and $j\in\mathbb{Z}$, recalling
(4.45),

$\displaystyle\left\lvert\tilde{q}^{+}_{j}(x,y)\right\rvert=\left\lvert\frac{\alpha^{x+j}g_{+}(x)-\alpha^{y}g_{-}(y)}{\alpha^{x+j}+\alpha^{y}}\right\rvert\leq\frac{\alpha^{\operatorname{Re}x+j}|g_{+}(x)|+\alpha^{\operatorname{Re}y}|g_{-}(y)|}{\alpha^{\operatorname{Re}x+j+1}+\alpha^{\operatorname{Re}y+1}}.$

Using the estimate

$(a+b)/(c+d)\leq\max\left\\{a/c,b/d\right\\},$ (4.62)

which holds for all $a,b\geq 0$, $c,d>0$, it follows that

$\sup\limits_{x\in\Xi_{c},y\in[0,1]}\left\lvert\tilde{q}^{+}_{j}(x,y)\right\rvert\leq\mathfrak{K}_{c,+}/|\log\alpha|\quad\mbox{and}\quad\sup\limits_{x\in[0,1],y\in\Xi_{c}}\left\lvert\tilde{q}^{+}_{j}(x,y)\right\rvert\leq\mathfrak{K}_{c,-}/|\log\alpha|,\quad
j\in\mathbb{Z}.$

Noting that (4.27) holds with $f$ and $g$ replaced with $f_{\pm}$ and
$g_{\pm}$, we have also that
$|f_{\pm}^{\prime}(\alpha^{y})|\leq\mathfrak{F}_{0,\pm}$, for $y\in[0,1]$,
while $|f_{\pm}^{\prime}(\alpha^{y})|\leq\mathfrak{F}_{c,\pm}$, for
$y\in\Xi_{c}$. Thus

$\displaystyle\sup\limits_{x\in\Xi_{c},y\in[0,1]}\left\lvert\tilde{p}^{+}_{j}(x,y)\right\rvert$
$\displaystyle\leq\sup\limits_{x\in\Xi_{c},y\in[0,1]}\frac{|f_{-}^{\prime}(\alpha^{y})|+\left\lvert\tilde{q}^{+}_{j}(x,y)\right\rvert}{\left\lvert\alpha^{x+j}+\alpha^{y}\right\rvert}\leq\frac{\mathfrak{F}_{0,-}+\mathfrak{K}_{c,+}/|\log\alpha|}{\alpha^{j+2}+\alpha^{2}}\quad\mbox{and}$
$\displaystyle\sup\limits_{x\in[0,1],y\in\Xi_{c}}\left\lvert\tilde{p}^{+}_{j}(x,y)\right\rvert$
$\displaystyle\leq\frac{\mathfrak{F}_{c,-}+\mathfrak{K}_{c,-}/|\log\alpha|}{\alpha^{j+2}+\alpha^{2}}.$

Moreover, for $x,y\in\Xi_{c}$,

$\displaystyle\left\lvert\operatorname{Im}\tilde{q}^{+}_{j}(x,y)\right\rvert$
$\displaystyle=\left\lvert\operatorname{Im}\left(\frac{\alpha^{x+j}g_{+}(x)-\alpha^{y}g_{-}(y)}{\alpha^{x+j}+\alpha^{y}}\right)\right\rvert$
$\displaystyle=\frac{1}{\left\lvert\alpha^{x+j}+\alpha^{y}\right\rvert^{2}}\left\lvert\operatorname{Im}\left(\alpha^{2(\operatorname{Re}x+j)}g_{+}(x)-\alpha^{\bar{x}+j+y}g_{-}(y)+\alpha^{x+j+\bar{y}}g_{+}(x)-\alpha^{2\operatorname{Re}y}g_{-}(y)\right)\right\rvert.$

Thus, using (4.61), noting that
$|\operatorname{Im}(\alpha^{it}z)|\leq|\operatorname{Im}(z)|+|\sin(t\log\alpha)||z|\leq|\operatorname{Im}(z)|+|t\log\alpha||z|$,
for $t\in\mathbb{R}$ and $z\in\mathbb{C}$, and using (4.62) again to obtain
the last two inequalities, we see that for $x,y\in\Xi_{c}$, where
$r=\alpha^{\operatorname{Re}x+j}$ and $s=\alpha^{\operatorname{Re}y}$,

$\displaystyle\left\lvert\operatorname{Im}\tilde{q}^{+}_{j}(x,y)\right\rvert$
$\displaystyle\leq\frac{1}{\alpha^{2}(r+s)^{2}}\Big{(}r^{2}|\operatorname{Im}g_{+}(x)|+s^{2}|\operatorname{Im}g_{-}(y)|\,+$
$\displaystyle\hskip
17.22217ptrs\big{(}|\operatorname{Im}g_{-}(y)|+|\operatorname{Im}g_{+}(x)|+c|\log\alpha|(|g_{-}(y)|+|g_{+}(x)|)\big{)}\Big{)}$
$\displaystyle\leq\frac{r|\operatorname{Im}g_{+}(x)|+s|\operatorname{Im}g_{-}(y)|+c|\log\alpha|\max\\{s|g_{-}(y)|,r|g_{+}(x)|\\}}{\alpha^{2}(r+s)}$
$\displaystyle\leq\alpha^{-2}\left(\max\\{|\operatorname{Im}g_{+}(x)|,|\operatorname{Im}g_{-}(y)|\\}+c|\log\alpha|\max\\{|g_{-}(y)|,|g_{+}(x)|\\}\right).$

Thus

$\displaystyle\sup\limits_{x\in\Xi_{c},y\in[0,1]}\left\lvert\operatorname{Im}\tilde{q}^{+}_{j}(x,y)\right\rvert$
$\displaystyle\leq\alpha^{-2}\left(\|\operatorname{Im}g_{+}\|_{c}+\alpha
c\mathfrak{K}_{c,+}\right)\quad\mbox{and}$
$\displaystyle\sup\limits_{x\in[0,1],y\in\Xi_{c}}\left\lvert\operatorname{Im}\tilde{q}^{+}_{j}(x,y)\right\rvert$
$\displaystyle\leq\alpha^{-2}\left(\|\operatorname{Im}g_{-}\|_{c}+\alpha
c\mathfrak{K}_{c,-}\right)$

so that

$\displaystyle\inf\limits_{x\in\Xi_{c},y\in[0,1]}\left|1+\tilde{q}^{+}_{j}(x,y)^{2}\right|$
$\displaystyle\geq
1-\sup\limits_{x\in\Xi_{c},y\in[0,1]}\left\lvert\operatorname{Im}\tilde{q}^{+}_{j}(x,y)\right\rvert^{2}$
$\displaystyle\geq 1-\alpha^{-4}\left(\|\operatorname{Im}g_{+}\|_{c}+\alpha
c\mathfrak{K}_{c,+}\right)^{2}\quad\mbox{and}$
$\displaystyle\inf\limits_{x\in[0,1],y\in\Xi_{c}}\left|1+\tilde{q}^{+}_{j}(x,y)^{2}\right|$
$\displaystyle\geq 1-\alpha^{-4}\left(\|\operatorname{Im}g_{-}\|_{c}+\alpha
c\mathfrak{K}_{c,-}\right)^{2}.$

Combining these estimates, and using (4.28) with $f$ replaced by $f_{-}$, we
get

$\displaystyle\sup\limits_{x\in\Xi_{c},y\in[0,1]}\left\lvert
T_{j}(x,y)\right\rvert$
$\displaystyle\leq\frac{|\log\alpha|\mathfrak{F}_{0,-}+\mathfrak{K}_{c,+}}{1-\alpha^{-4}\left(\left\lVert\operatorname{Im}g_{+}\right\rVert_{c}+\alpha
c\mathfrak{K}_{c,+}\right)^{2}}\left(1+\mathfrak{F}_{c,+}^{2}\right)^{1/4}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}\quad\mbox{and}$
$\displaystyle\sup\limits_{x\in[0,1],y\in\Xi_{c}}\left\lvert
T_{j}(x,y)\right\rvert$
$\displaystyle\leq\frac{|\log\alpha|\mathfrak{F}_{c,-}+\mathfrak{K}_{c,-}}{1-\alpha^{-4}\left(\left\lVert\operatorname{Im}g_{-}\right\rVert_{c}+\alpha
c\mathfrak{K}_{c,-}\right)^{2}}\left(\frac{1+\mathfrak{F}_{0,+}^{2}}{1-\mathfrak{I}_{c,-}^{2}}\right)^{1/4}\frac{\alpha^{j/2}}{\alpha^{j+2}+\alpha^{2}}.$

The estimates (4.55) and (4.56) and the other results for $\tilde{L}_{t}^{+}$
follow. ∎

Let $\tilde{B}_{t,N}^{\pm}$ and $\tilde{C}^{\pm}_{t,N}$ be the $N\times N$
matrices defined by the right hand side of (4.31) with
$\tilde{K}_{t}(\cdot,\cdot)$ replaced by $\tilde{K}_{t}^{\pm}(\cdot,\cdot)$
and $\tilde{L}_{t}^{\pm}(\cdot,\cdot)$, respectively, and define the $2N\times
2N$ matrix $A_{t,N}$ by

$A_{t,N}:=\begin{pmatrix}B_{t,N}^{-}&C_{t,N}^{-}\\\
C_{t,N}^{+}&B_{t,N}^{+}\end{pmatrix},\qquad
t\in\mathbb{R},\;\;N\in\mathbb{N}.$

Recalling Remark 3.11, the matrix $A_{t,N}$ and the operator $K_{t,N}$ are
related by Lemma 3.7(ii) and (iii). As in §4.2, our goal is to estimate the
spectrum and spectral radius of $K_{t,N}$ via computing the spectra of
approximations to $A_{t,N}$ for some fixed $N$ and finitely many
$t\in[-\pi,\pi]$.

To define these approximations, proceeding analogously to §4.2, for
$M\in\mathbb{N}$ let $\tilde{K}_{t}^{\pm,M}(\cdot,\cdot)$ and
$\tilde{L}_{t}^{\pm,M}(\cdot,\cdot)$ be approximations to
$\tilde{K}_{t}^{\pm}(\cdot,\cdot)$ and $\tilde{L}_{t}^{\pm}(\cdot,\cdot)$,
respectively, given by (4.47) and (4.48) but with the infinite series replaced
by finite sums from $j=-M$ to $M$ (cf. (4.32)). Let $B_{t,N}^{\pm,M}$ and
$C_{t,N}^{\pm,M}$ be the corresponding approximations to the matrices
$B_{t,N}^{\pm}$ and $C_{t,N}^{\pm}$, so that

$B_{t,N}^{\pm,M}(p,q)=\frac{1}{N}\tilde{K}_{t}^{\pm,M}(x_{p,N},x_{q,N}),\quad
C_{t,N}^{\pm,M}(p,q)=\frac{1}{N}\tilde{L}_{t}^{\pm,M}(x_{p,N},x_{q,N}),\quad
p,q=1,\ldots,N,$

and let

$A_{t,N}^{M}:=\begin{pmatrix}B_{t,N}^{-,M}&C_{t,N}^{-,M}\\\
C_{t,N}^{+,M}&B_{t,N}^{+,M}\end{pmatrix},\qquad
t\in\mathbb{R},\;\;M,N\in\mathbb{N}.$ (4.63)

Similarly, define the operators $\tilde{K}^{\pm,M}_{t,N}$ and
$\tilde{L}^{\pm,M}_{t,N}$ as we defined $\tilde{K}^{\pm}_{t,N}$ and
$\tilde{L}^{\pm}_{t,N}$ above Proposition 4.16, but replacing the kernels
$\tilde{K}_{t}^{\pm}(\cdot,\cdot)$ and $\tilde{L}_{t}^{\pm}(\cdot,\cdot)$ in
their definitions by the respective approximations
$\tilde{K}_{t}^{\pm,M}(\cdot,\cdot)$ and $\tilde{L}_{t}^{\pm,M}(\cdot,\cdot)$,
and let

$\tilde{K}_{t,N}^{M}=\begin{pmatrix}\tilde{K}^{-,M}_{t,N}&\tilde{L}^{-,M}_{t,N}\\\
\tilde{L}^{+,M}_{t,N}&\tilde{K}^{+,M}_{t,N}\end{pmatrix},\qquad
t\in\mathbb{R},\;\;M,N\in\mathbb{N}.$ (4.64)

Then, similarly to (4.33), we have that, for $t\in\mathbb{R}$ and
$M,N\in\mathbb{N}$ with $M\geq 2$,

$\|A_{t,N}-A_{t,N}^{M}\|_{\infty}\leq\|\tilde{K}_{t,N}-\tilde{K}_{t,N}^{M}\|_{\infty}\leq
C_{1}(M),$ (4.65)

where, using the notations of Remarks 4.9 and 4.18 and Proposition 4.17, and
setting $\mathfrak{K}_{0}:=\mathfrak{K}_{0,+}=\mathfrak{K}_{0,-}$,

$\displaystyle C_{1}(M)$
$\displaystyle:=\frac{1}{\pi}\max\Big{\\{}\left(1+\mathfrak{F}_{0,+}^{2}\right)^{1/4}\big{(}2|\log\alpha|\mathfrak{F}_{0,+}\mathcal{B}_{M}(\alpha)+(|\log\alpha|\mathfrak{F}_{0,-}+\mathfrak{K}_{0})\mathcal{C}_{M}(\alpha)\big{)},$
$\displaystyle\hskip
53.81927pt\left(1+\mathfrak{F}_{0,-}^{2}\right)^{1/4}\big{(}2|\log\alpha|\mathfrak{F}_{0,-}\mathcal{B}_{M}(\alpha)+(|\log\alpha|\mathfrak{F}_{0,+}+\mathfrak{K}_{0})\mathcal{C}_{M}(\alpha)\big{)}\Big{\\}}.$
(4.66)

Note that by (4.26) and (4.57), $C_{1}(M)=O(\alpha^{M/2})$ as $M\to\infty$.

Arguing as in (4.2), we have also that, for $M,N\in\mathbb{N}$,
$t\in\mathbb{R}$, $p,q=1,\ldots,N$,