$\left|\frac{\partial}{\partial t}B_{t,N}^{\pm,M}(p,q)\right|\leq
B_{N}^{\pm,M}(p,q)\quad\mbox{and}\quad\left|\frac{\partial}{\partial
t}C_{t,N}^{\pm,M}(p,q)\right|\leq C_{N}^{\pm,M}(p,q),$ (4.67)

where $B_{N}^{\pm,M}(p,q)$ is defined by (4.2) but with $p_{j}$, $q_{j}$, and
$f$ replaced by $p_{j}^{\pm}$, $q_{j}^{\pm}$ and $f_{\pm}$, respectively
(compare (4.15) and (4.47)). Similarly, $C_{N}^{\pm,M}(p,q)$ is defined by the
right hand side of (4.2) with $p_{j}$ and $q_{j}$ replaced by
$\tilde{p}_{j}^{\pm}$ and $\tilde{q}_{j}^{\pm}$, respectively, and with $f$
replaced by $f_{\pm}$ in the numerator, by $f_{\mp}$ in the denominator
(compare (4.15) and (4.48)). Thus, where

$B_{N}^{M}:=\begin{pmatrix}B_{N}^{-,M}&C_{N}^{-,M}\\\
C_{N}^{+,M}&B_{N}^{+,M}\end{pmatrix},\qquad M,N\in\mathbb{N},$ (4.68)

(4.36) holds (with the above definitions of $A_{t,N}$, $A_{t,N}^{M}$ and
$B_{N}^{M}$) for all $s,t\in\mathbb{R}$, $M,N\in\mathbb{N}$. Note also,
arguing as in (4.37) and (4.38), that $\|B_{N}^{M}\|_{\infty}$ is bounded
uniformly for $M,N\in\mathbb{N}$, and that
$\|\tilde{K}_{t,N}-\tilde{K}_{s,N}\|_{\infty}=O(|s-t|)$ as $|s-t|\to 0$,
uniformly for $s,t\in\mathbb{R}$, $N\in\mathbb{N}$. Further, similarly to
(4.39) and (4.66), $\|\tilde{K}_{t,N}\|_{\infty}\leq C_{3}$ for
$t\in\mathbb{R}$ and $N\in\mathbb{N}$, where181818Computations indicate that
$\mathcal{C}_{10}^{*}(\alpha)$ exceeds the left hand side of (4.58) by not
more than $2.1\%$ for $\alpha\in(0,1)$.

$\displaystyle C_{3}$
$\displaystyle:=\frac{1}{2\pi}\max\Big{\\{}\left(1+\mathfrak{F}_{0,+}^{2}\right)^{1/4}\big{(}\mathfrak{G}_{0,+}\mathfrak{R}(\alpha)+4|\log\alpha|\mathfrak{F}_{0,+}\mathcal{B}^{*}_{10}(\alpha)+(|\log\alpha|\mathfrak{F}_{0,-}+\mathfrak{K}_{0})\mathcal{C}^{*}_{10}(\alpha)\big{)},$
$\displaystyle\hskip
49.51373pt\left(1+\mathfrak{F}_{0,-}^{2}\right)^{1/4}\big{(}\mathfrak{G}_{0,-}\mathfrak{R}(\alpha)+4|\log\alpha|\mathfrak{F}_{0,-}\mathcal{B}^{*}_{10}(\alpha)+(|\log\alpha|\mathfrak{F}_{0,+}+\mathfrak{K}_{0})\mathcal{C}^{*}_{10}(\alpha)\big{)}\Big{\\}},$
(4.69)

$\mathfrak{R}(\alpha):=(1+\alpha^{1/2}+\alpha^{-1/2})/(2\alpha^{2})$,
$\mathcal{B}^{*}_{10}$ and $\mathcal{C}^{*}_{10}$ are defined by (4.25) and
(4.58), and the other notations are defined in Proposition 4.17. Provided $c$
is such that the conditions of Proposition 4.17 hold, we have also (cf.
(4.40)), by Propositions 4.16 and 4.17, that (4.40) holds with

$C_{4}:=2e^{2\pi c}\widetilde{C}^{*},$ (4.70)

where $\widetilde{C}^{*}$ is defined by the right hand side of (4.51), but
with the norms on that right hand side replaced by the upper bounds in
Proposition 4.17. Additionally, we replace the infinite sums in these upper
bounds by the bounds $\mathcal{B}_{10}^{*}(\alpha)$ and
$\mathcal{C}_{10}^{*}(\alpha)$.

The following result, which holds for every $\Gamma$ that satisfies our
standing assumption (4.44), is identical to Theorem 4.13, except that
$\tilde{K}_{t,N}^{M_{N}}$ and $A_{t,N}^{M_{N}}$ are defined here by (4.64) and
(4.63), respectively. In this theorem $\sigma(\tilde{K}^{M}_{t,N})$ denotes
the spectrum of $\tilde{K}^{M}_{t,N}$ either on $(L^{2}(0,1))^{2}$ or on
$(C[0,1])^{2}$ (cf. (4.19)), which coincides with
$\\{0\\}\cup\sigma(A^{M}_{t,N})$ by Lemma 3.7(ii) and Remark 3.11, and
$\sigma(D_{\Gamma})$ denotes the spectrum of $D_{\Gamma}$ on $L^{2}(\Gamma)$,
which coincides with the essential spectrum by Corollary 4.2. The proof of
this result follows that of Theorem 4.13, noting that the argument of Lemma
4.12 applies to each of the operator families $\tilde{K}_{t,N}^{\pm,M}$ and
$\tilde{L}_{t,N}^{\pm,M}$ so that (where $\tilde{K}_{t,N}^{M}$ is defined by
(4.64)) $\\{\tilde{K}_{t,N}^{M}:t\in[-\pi,\pi],M,N\in\mathbb{N}\\}$ is
collectively compact.

######  Theorem 4.19.

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, as $N\to\infty$,

$\sigma^{N}(D_{\Gamma}):=\bigcup\limits_{t\in
T_{N}}\sigma(\tilde{K}^{M_{N}}_{t,N})=\\{0\\}\cup\bigcup\limits_{t\in
T_{N}}\sigma(A^{M_{N}}_{t,N})\;\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\;\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma}).$
(4.71)

######  Remark 4.20.

Using Remark 3.11, we see that Remark 4.14 applies also to
$\sigma^{N}(D_{\Gamma})$ given by (4.71).

Our second main result of this subsection, obtained by applying Theorem 3.8,
noting Remark 3.11, is proved in the same way as the analogous result, Theorem
4.15, in the one-sided case. In this theorem $A_{t,N}^{M}$, $B_{N}^{M}$,
$C_{1}(M)$, $C_{3}$, and $C_{4}$, are defined by (4.63), (4.68), (4.66),
(4.69), and (4.70), respectively.

######  Theorem 4.21.

Under our standing assumption (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, without loss of generality, assume that
(4.54) 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
(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

$\mathcal{L}_{c}(f,N):=\frac{C_{4}}{e^{2\pi
Nc}-1}<\mathcal{R}_{c}(f,m,M,N):=r^{2}_{0}\left(1+C_{3}\left(R_{m,M,N}-C_{1}(M)-\frac{\pi}{2m}\|B_{N}^{M}\|_{\infty}\right)^{-1}\right)^{-1},$
(4.72)

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 (4.72) holds
and $\rho(A_{t,N}^{M})<\rho_{0}$ for $t\in[0,\pi]$ and all $N\geq N_{0}$.

######  Remark 4.22.

The condition (4.72) can be written as

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

where $S_{c}(\Gamma,m,M,N):=\mathcal{L}_{c}(f,N)-\mathcal{R}_{c}(f,m,M,N)$.
For any $c>0$ such that the conditions of the first sentence of the above
theorem are satisfied (and these conditions are necessarily satisfied for all
sufficiently small $c>0$), we note that:

i) The proof of the above theorem, which follows that of Theorem 4.15, shows
that, provided $\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$, $R_{m,M,N}$ is
positive and bounded away from zero for all sufficiently large
$m,M,N\in\mathbb{N}$, so that the same holds for $\mathcal{R}_{c}(f,m,M,N)$.
Since also $\mathcal{L}_{c}(f,N)\to 0$ as $N\to\infty$,
$S_{c}(\Gamma,m,M,N)<0$ for all sufficiently large $m$, $M$, and $N$, if
$\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$.

ii) For fixed $m,M,N\in\mathbb{N}$, evaluation of the functionals
$\mathcal{L}(f,N)$ and $\mathcal{R}_{c}(f,m,M,N)$ requires inputs relating to
the functions $f_{\pm}$ defined by (4.45), namely: the constant
$\alpha\in(0,1)$ and the bounds $\|g_{\pm}\|_{0}$, $\|g_{\pm}^{\prime}\|_{0}$,
and $\|g_{\pm}^{\prime\prime}\|_{0}$ on $g$ defined by (4.45) (to evaluate
$C_{1}(M)$ and $C_{3}$); the bounds $\|g_{\pm}\|_{c}$,
$\|g_{\pm}^{\prime}\|_{c}$, and $\|g_{\pm}^{\prime\prime}\|_{c}$ (to evaluate
$C_{4}$); the values $f_{\pm}(\alpha^{x_{p,N}+j})$,
$f_{\pm}^{\prime}(\alpha^{x_{p,N}})$, and
$f_{\pm}^{\prime\prime}(\alpha^{x_{p,N}})$, for $p=1,\ldots,N$ and
$j=-M,\ldots,M$ (to compute the entries of the matrices $A_{t_{k},N}^{M}$, for
$k=1,\ldots,m$).

iii) If exact values of $\|g_{\pm}\|_{0}$, $\|g_{\pm}^{\prime}\|_{0}$,
$\|g_{\pm}^{\prime\prime}\|_{0}$, $\|g_{\pm}\|_{c}$,
$\|g_{\pm}^{\prime}\|_{c}$, and $\|g_{\pm}^{\prime\prime}\|_{c}$ are not
available, it is enough to replace them with upper bounds which leads to upper
bounds $\widehat{C}_{1}(M)$, $\widehat{C}_{3}$ and $\widehat{C}_{4}$ for
$C_{1}(M)$, $C_{3}$ and $C_{4}$, respectively (with $\widehat{C}_{1}(M)\to 0$
as $M\to\infty$ at the same rate as $C_{1}(M)$). Let
$\widehat{\mathcal{L}}_{c}(f,N)$ and $\widehat{\mathcal{R}}_{c}(f,m,M,N)$ be
defined by (4.43) with $C_{1}(M)$, $C_{3}$, $C_{4}$, replaced by their upper
bounds, so that $\mathcal{L}_{c}(f,N)\leq\widehat{\mathcal{L}}_{c}(f,N)$ and
$\widehat{\mathcal{R}}_{c}(f,m,M,N)\leq\mathcal{R}_{c}(f,m,M,N)$. Then the
conclusions of Theorem 4.21 hold with $\mathcal{L}_{c}(f,N)$ and
$\mathcal{R}_{c}(f,m,M,N)$ replaced by $\widehat{\mathcal{L}}_{c}(f,N)$ and
$\widehat{\mathcal{R}}_{c}(f,m,M,N)$, respectively; in particular
$S_{c}(\Gamma,m,M,N)<0$ if
$\widehat{\mathcal{L}}_{c}(f,N)<\widehat{\mathcal{R}}_{c}(f,m,M,N)$ and,
provided $\rho(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$,
$\widehat{\mathcal{L}}_{c}(f,N)<\widehat{\mathcal{R}}_{c}(f,m,M,N)$ for all
sufficiently large $m$, $M$, and $N$.

###  4.4 The 2D case: lower bounds for the numerical range

This paper is motivated by the question at the end of §1.1 which notes that
there exist (e.g., [11, Fig. 3]) 2D examples of $\Gamma\in\mathcal{D}$ with
$w_{\operatorname{ess}}(D_{\Gamma})\gg\frac{1}{2}$ and asks: is
$\rho_{L^{2}(\Gamma),\operatorname{ess}}\geq\frac{1}{2}$ for any of these
examples? We will explore this in §6 where we will see that
$\rho_{L^{2}(\Gamma),\operatorname{ess})}<\frac{1}{2}$ for each example we
treat, supporting Conjecture 1.2. We will also see that
$w_{\operatorname{ess}}(D_{\Gamma})>\frac{1}{2}$ for at least some of these
examples, indeed that $W_{\operatorname{ess}}(D_{\Gamma})\supset
B_{R}:=\\{z\in\mathbb{C}:|z|<R\\}$ for $R$ substantially larger than
$\frac{1}{2}$.

As the route to obtain these estimates for
$W_{\operatorname{ess}}(D_{\Gamma})$, in this subsection we obtain lower
bounds191919By a lower bound for $W(D_{\Gamma})$ we mean, simply, a set
$S\subset\mathbb{C}$ such that $S\subset W(D_{\Gamma})$. for
$W(D_{\Gamma})=W_{\operatorname{ess}}(D_{\Gamma})$ in the case when $\Gamma$
is a dilation invariant graph, precisely in the 2D case when either (4.8) or
(4.44) applies. It is enough to restrict attention to the one-sided case (4.8)
as one easily sees (e.g., [11, §2.3]) that if $\Gamma$ satisfies (4.44) and
$\widetilde{\Gamma}:=\\{(x,f(x)):x>0\\}$, then $\widetilde{\Gamma}$ satisfies
(4.8) and

$W(D_{\widetilde{\Gamma}})\subset W(D_{\Gamma}).$ (4.73)

We obtain lower bounds for $W(D_{\Gamma})$ in the case that (4.8) applies via
the Nyström method that we used in §4.2 to approximate $\sigma(D_{\Gamma})$.
Firstly, by Corollary 4.4, and since, for $t\in[-\pi,\pi]$, $\tilde{L}_{t}$,
defined in the proof of Proposition 4.10, is unitarily equivalent to
$\tilde{K}_{t}$ defined by (4.13), which is unitarily equivalent to $K_{t}$,
we have that $W(\tilde{L}_{t})\subset W(D_{\Gamma})$. For
$p\in\mathbb{N}_{0}$, let $\mathcal{T}_{p}\subset L^{2}(0,1)$ denote the set
of trigonometric polynomials of degree at most $p$. An orthonormal basis of
$\mathcal{T}_{p}$ is given by $\left\\{e_{-p},\ldots,e_{p}\right\\}$, where
$e_{j}(x):=e^{2\pi ijx}$. Consider the restricted numerical range

$W_{p}(\tilde{L}_{t}):=\left\\{\langle\tilde{L}_{t}\phi,\phi\rangle:\phi\in\mathcal{T}_{p},\left\lVert\phi\right\rVert_{L^{2}(0,1)}=1\right\\}.$

It is clear that $W_{p}(\tilde{L}_{t})\subset W(\tilde{L}_{t})$ for
$t\in[-\pi,\pi]$, and that (e.g., [11, §2.3])
$W_{p}(\tilde{L}_{t})=W(T^{p,t})$, where $T^{p,t}=(T^{p,t}_{jk})_{j,k=-p}^{p}$
is the $(2p+1)\times(2p+1)$ matrix with
$T^{p,t}_{jk}:=\langle\tilde{L}_{t}e_{k},e_{j}\rangle$, $j,k=-p,\dots,p$.
Recalling from the proof of Proposition 4.10 that $\tilde{L}_{t}$ has kernel
$\tilde{L}_{t}(x,y)=e^{i(x-y)t}\tilde{K}_{t}(x,y)$, we will approximate
$T^{p,t}$ by $T^{p,t,N}$, where

$T^{p,t,N}_{jk}:=\frac{1}{N^{2}}\sum\limits_{m,n=1}^{N}e^{i(x_{m,N}-x_{n,N})t}\tilde{K}_{t}(x_{m,N},x_{n,N})e_{k}(x_{n,N})\overline{e_{j}(x_{m,N})},\qquad
j,k=-p,\ldots,p,$ (4.74)

recalling that $x_{m,N}:=\tfrac{1}{N}\left(m-\tfrac{1}{2}\right)$,
$m=1,\ldots,N$, and then approximate $T^{p,t,N}$ further by $T^{p,t,N,M}$,
defined by (4.74) but with $\tilde{K}_{t}$, given by the infinite sum (4.15),
replaced by $\tilde{K}_{t}^{M}$, given by the finite sum (4.32). The following
result, which uses the notation (4.34), will enable us to estimate the
difference between $W_{p}(\tilde{L}_{t})=W(T^{p,t})$ and $W(T^{p,t,N,M})$.

######  Proposition 4.23.

Let $f$ and $c>0$ be as in Proposition 4.8, in particular satisfying (4.21),
so that $\tilde{L}_{t}(x,\cdot)$ and $\tilde{L}_{t}(\cdot,y)$ are analytic and
bounded in $\Sigma_{c}$ for all $x,y,t\in\mathbb{R}$, let
$\tilde{L}^{M}_{t}(x,y):=e^{i(x-y)}\tilde{K}^{M}(x,y)$, for $x,y\in\mathbb{R}$
and $M\in\mathbb{N}$, and suppose that $g$ is a bounded, analytic,
$1$-periodic function on $\Sigma_{c}$. Then, for $M,N\in\mathbb{N}$ and
$t\in[-\pi,\pi]$,

$\left\lvert\langle\tilde{L}_{t}g,g\rangle-\frac{1}{N^{2}}\sum\limits_{m,n=1}^{N}\tilde{L}^{M}_{t}(x_{m,N},x_{n,N})g(x_{n,N})\overline{g(x_{m,N})}\right\rvert\leq
2\left\lVert g\right\rVert_{0}\left\lVert g\right\rVert_{c}e^{\pi
c}\frac{\|\tilde{K}_{t}\|_{0,c}+\|\tilde{K}_{t}\|_{c,0}}{e^{2\pi
Nc}-1}+C_{1}(M).$

###### Proof.

$\displaystyle\left\lvert\langle\tilde{L}_{t}g,g\rangle-\frac{1}{N^{2}}\sum\limits_{m,n=1}^{N}\tilde{L}_{t}(x_{m,N},x_{n,N})g(x_{n,N})\overline{g(x_{m,N})}\right\rvert$
$\displaystyle\leq$ $\displaystyle\int_{0}^{1}\left\lvert
g(x)\right\rvert\left\lvert\int_{0}^{1}\tilde{L}_{t}(x,y)g(y)\,\mathrm{d}y-\frac{1}{N}\sum\limits_{n=1}^{N}\tilde{L}_{t}(x,x_{n,N})g(x_{n,N})\right\rvert\,\mathrm{d}x$
$\displaystyle+\frac{1}{N}\sum\limits_{n=1}^{N}\left\lvert\int_{0}^{1}\overline{g(x)}\tilde{L}_{t}(x,x_{n,N})\,\mathrm{d}x-\frac{1}{N}\sum\limits_{m=1}^{N}\overline{g(x_{m,N})}\tilde{L}_{t}(x_{m,N},x_{n,N})\right\rvert\left\lvert
g(x_{n,N})\right\rvert$ $\displaystyle\leq$ $\displaystyle\left\lVert
g\right\rVert_{0}\frac{2\|\tilde{L}_{t}\|_{0,c}\left\lVert
g\right\rVert_{c}}{e^{2\pi Nc}-1}+\frac{2\left\lVert
g\right\rVert_{c}\|\tilde{L}_{t}\|_{c,0}}{e^{2\pi Nc}-1}\left\lVert
g\right\rVert_{0},$

by Theorem 4.11, noting that $g$ and, from the proof of Proposition 4.10,
$\tilde{L}_{t}(x,\cdot)$, for $x\in\mathbb{R}$, and $\tilde{L}_{t}(\cdot,y)$,
for $y\in\mathbb{R}$, are all $1$-periodic. The claimed result now follows
immediately from (4.33) and by noting that, from the proof of Proposition
4.10, $\|\tilde{L}_{t}\|_{0,c}\leq e^{\pi c}\|\tilde{K}_{t}\|_{0,c}$ and
$\|\tilde{L}_{t}\|_{c,0}\leq e^{\pi c}\|\tilde{K}_{t}\|_{c,0}$. ∎

Clearly, $g\in\mathcal{T}^{p}$ with $\left\lVert g\right\rVert_{L^{2}(0,1)}=1$
if and only if

$g=\sum\limits_{j=-p}^{p}c_{j}e_{j}\quad\text{with}\quad\sum\limits_{j=-p}^{p}\left\lvert
c_{j}\right\rvert^{2}=1,$ (4.75)

in which case

$\left\lVert g\right\rVert_{c}\leq\sum\limits_{j=-p}^{p}\left\lvert
c_{j}\right\rvert\left\lVert
e_{j}\right\rVert_{c}\leq\sum\limits_{j=-p}^{p}\left\lvert
c_{j}\right\rvert\left\lVert
e_{j}\right\rVert_{c}\leq\sum\limits_{j=-p}^{p}\left\lvert c_{j}\right\rvert
e^{2\pi pc}\leq\sqrt{2p+1}\;e^{2\pi pc}.$

The following corollary follows immediately from this observation and
Proposition 4.23, noting that, if $g$ and $c_{-p},\ldots,c_{p}$ are related by
(4.75), then the summation in Proposition 4.23 coincides with
$\sum_{j,k=-p}^{p}T_{j,k}^{p,t,N,M}c_{k}\overline{c_{j}}\in W(T^{p,t,N,M})$.

######  Corollary 4.24.

For $p\in\mathbb{N}_{0}$, $M,N\in\mathbb{N}$, $t\in[-\pi,\pi]$, where $c>0$ is
as in Proposition 4.23,

$d_{H}(W_{p}(\tilde{L}_{t}),W(T^{p,t,N,M}))\leq C_{7}(p,N,M):=2(2p+1)e^{\pi
c(2p+1)}\frac{\|\tilde{K}_{t}\|_{0,c}+\|\tilde{K}_{t}\|_{c,0}}{e^{2\pi
Nc}-1}+C_{1}(M).$

We can approximate $W(T^{p,t,N,M})$ by a standard method that dates back to
[31]. Choose $n\in\mathbb{N}$ and, for $\ell=0,\ldots,n$, let
$\theta_{\ell}:=2\pi\ell/n$, let $\lambda_{\ell}$ denote the largest
eigenvalue of $\operatorname{Re}(e^{-i\theta_{\ell}}T^{p,t,N,M})$ and
$x_{\ell}$ an associated unit eigenvector, and let $z_{\ell}:=\langle
T^{p,t,N,M}x_{\ell},x_{\ell}\rangle$, for $\ell=0,1,\ldots,n-1$, and
$z_{n}:=z_{0}$. Then
$W_{n}:=\operatorname*{conv}(\\{z_{0},\ldots,z_{n}\\})\subset W(T^{p,t,N,M})$
and [31]
$W_{n}\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}} \end{picture}W(T^{p,t,N,M})$ as
$n\to\infty$. The following simple corollary of the above results, in which we
use the notations just introduced, will enable us to show that $B_{R}\subset
W(\tilde{L}_{t})\subset W(D_{\Gamma})$ for concrete values of $R>0$ in the
examples we treat in §6.

######  Corollary 4.25.

Suppose that $p\in\mathbb{N}_{0}$, $M,N,n\in\mathbb{N}$, $t\in[-\pi,\pi]$,
that $c>0$ is as in Proposition 4.23, and that $0$ is an interior point of
$W_{n}$, in which case $z_{\ell}=\exp(i\gamma_{\ell})|z_{\ell}|$,
$\ell=0,\ldots,n$, with
$\gamma_{0}\leq\gamma_{1}\leq\ldots\leq\gamma_{n}=\gamma_{0}+2\pi$. Then,
where
$\theta_{\max}:=\max_{\ell=1,\ldots,n}(\gamma_{\ell}-\gamma_{\ell-1})\in(0,2\pi]$
and $R_{\min}:=\min_{\ell=0,\ldots,n}|z_{\ell}|>0$, if
$R^{*}:=R_{\min}\cos(\theta_{\max}/2)-C_{7}(p,t,N,M)>0$, then
$\overline{B_{R^{*}}}\subset W(D_{\Gamma})$.

###### Proof.

Under the above assumptions, $\overline{B_{R}}\subset W_{n}$, where
$R:=R_{\min}\cos(\theta_{\max}/2)$. Since $W_{n}\subset W(T^{p,t,N,M})$ and
$W_{p}(\tilde{L}_{t})\subset W(\tilde{L}_{t})\subset W(D_{\Gamma})$, it
follows that $\overline{B_{R^{*}}}\subset W(D_{\Gamma})$ by Corollary 4.24. ∎

##  5 Localization and deformation

Let $\Omega_{-}\subset\mathbb{R}^{d}$, $d\geq 2$, be a Lipschitz domain with
boundary $\Gamma=\partial\Omega_{-}$ and outward-pointing unit normal vector
$n_{y}=n^{\Gamma}_{y}$ at almost every $y\in\Gamma$. If
$\psi\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a $C^{1}$-diffeomorphism, then
$\psi(\Omega_{-})$ is a Lipschitz domain with boundary $\psi(\Gamma)$ by [29,
Theorem 4.1]. Note that this is no longer true if $\psi$ is only assumed to be
a bi-Lipschitz map (see [21, Lemma 1.2.1.4], and the discussion in [29]). As
before, $D_{\Gamma}$ will be the double-layer operator on $\Gamma$. To
simplify notation we will use $\sim$ to denote equality up to compact
operators, and abbreviate $D_{\Gamma}-\lambda I$ as $D_{\Gamma}-\lambda$. For
$x\in\mathbb{R}^{d}$ and $r>0$, $B(x,r):=\\{y\in\mathbb{R}^{d}:|y-x|<r\\}$.

###  5.1 Localization without deformation

To start, we will consider domains with locally-dilation-invariant boundaries
$\Gamma\in\mathscr{D}$; recall Definition 1.3. Note that being locally
dilation invariant at $x$ is equivalent to the existence of an isometry
$\psi_{x}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\psi_{x}(x)=0$ and
$\psi_{x}\big{(}B(x,\delta(x))\cap\Gamma\big{)}=B(0,\delta(x))\cap\Gamma_{x}$,
for some $\delta(x)>0$, where $\Gamma_{x}$ is the graph of a Lipschitz
continuous function with $\alpha\Gamma_{x}=\Gamma_{x}$. We also assume that
$\psi_{x}$ preserves the orientation of the outer normal vector field, that
is, the outward-pointing normal vector field on $B(x,\delta(x))\cap\Gamma$ is
mapped to the upwards-pointing normal vector field on
$B(0,\delta(x))\cap\Gamma_{x}$. Similarly, $\Gamma$ is locally $C^{1}$ at $x$
if there is a $\delta(x)>0$ and an Euclidean map
$\psi_{x}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\psi_{x}(x)=0$ and
$\psi_{x}(B(x,\delta(x))\cap\Gamma)=B(0,\delta(x))\cap\Gamma_{x}$, where
$\Gamma_{x}$ is the graph of a $C^{1}$-function with compact support. Again,
we assume that $\psi_{x}$ preserves the orientation of the outer normal vector
field. We will use extensively below the notation $\delta(x)$, for
$x\in\Gamma$, which will have one of the above meanings, depending on whether
$\Gamma$ is locally dilation invariant at $x$ or is $C^{1}$ at $x$.

Note that $D_{\Gamma}$ is compact for every $C^{1}$ domain [19], as we
recalled in §1.2; this implies that also $D_{\widetilde{\Gamma}}$ is compact
if $\widetilde{\Gamma}$ is the graph of a $C^{1}$-function with compact
support. Consequently, if $\Gamma$ consists of different parts that are
separated by $C^{1}$ areas, we are able to localize using sharp cut-off
functions. In the following we equip $\Gamma$ with the topology induced from
$\mathbb{R}^{d}$ and the standard surface measure. In particular, $\chi_{E}$
denotes the characteristic function of a subset $E\subset\Gamma$ that is
measurable with respect to the surface measure on $\Gamma$.

######  Lemma 5.1.

Let $E\subset\Gamma$ be a measurable subset and assume that $\Gamma$ is
locally $C^{1}$ at every $x\in\partial E$. Then $D_{\Gamma}$ essentially
commutes with $\chi_{E}$, i.e., the commutator
$[D_{\Gamma},\chi_{E}]:=D_{\Gamma}\chi_{E}-\chi_{E}D_{\Gamma}$ is compact.

###### Proof.

Since $\partial E$ is a compact subset of $\Gamma$, there is an
$\varepsilon>0$ such that

$(\partial E+B(0,\varepsilon))\cap\Gamma\subset\bigcup\limits_{x\in\partial
E}(B(x,\delta(x))\cap\Gamma).$

Set $E_{o}:=(\partial E+B(0,\tfrac{\varepsilon}{2}))\cap\Gamma$. We know that
$\chi_{A}D_{\Gamma}\chi_{B}$ is compact, even Hilbert–Schmidt, whenever $A$
and $B$ have positive distance from each other. We have

$\displaystyle E=(E\cap E_{o})\cup(E\setminus E_{o})\quad\mbox{and}\quad
E^{c}=(E^{c}\cap E_{o})\cup(E^{c}\setminus E_{o}),$

where $E^{c}:=\Gamma\setminus E$. By inspection, we see that the four sets on
the right of these equalities, if non-empty, have pairwise positive distance
from each other except for $E\cap E_{o}$ and $E^{c}\cap E_{o}$. We thus have

$\displaystyle\chi_{E}D_{\Gamma}\chi_{E^{c}}$ $\displaystyle\sim\chi_{E\cap
E_{o}}D_{\Gamma}\chi_{E^{c}\cap
E_{o}}=\chi_{E}\chi_{E_{o}}D_{\Gamma}\chi_{E_{o}}\chi_{E^{c}},$
$\displaystyle\chi_{E^{c}}D_{\Gamma}\chi_{E}$
$\displaystyle\sim\chi_{E^{c}\cap E_{o}}D_{\Gamma}\chi_{E\cap
E_{o}}=\chi_{E^{c}}\chi_{E_{o}}D_{\Gamma}\chi_{E_{o}}\chi_{E}.$

The operator $\chi_{E_{o}}D_{\Gamma}\chi_{E_{o}}$ is compact because
$\overline{E}_{o}$ is $C^{1}$ everywhere by construction. We conclude that
$[D_{\Gamma},\chi_{E}]=\chi_{E^{c}}D_{\Gamma}\chi_{E}-\chi_{E}D_{\Gamma}\chi_{E^{c}}$
is compact as well. ∎

In particular, this shows that if $\Gamma$ is locally $C^{1}$ at some point
$x\in\Gamma$, then $D_{\Gamma}$ is not Fredholm, since if
$E=B(x,\tfrac{\delta(x)}{2})$, then $\chi_{E}D_{\Gamma}\chi_{E}$ is compact,
and Lemma 5.1 implies that
$D_{\Gamma}\sim\chi_{E}D_{\Gamma}\chi_{E}+\chi_{E^{c}}D_{\Gamma}\chi_{E^{c}}$.
Hence

$0\in\sigma_{\operatorname{ess}}(\chi_{E}D_{\Gamma}\chi_{E})\subset\sigma_{\operatorname{ess}}(D_{\Gamma}).$
(5.1)

More generally, if $E_{1},\ldots,E_{N}$ are measurable subsets of $\Gamma$
such that $\Gamma=\cup_{j=1}^{N}E_{j}$, $E_{j}\cap E_{k}=\emptyset$ for $j\neq
k$ and $\Gamma$ is locally $C^{1}$ at every $x\in\cup_{j=1}^{N}\partial
E_{j}$, then $D_{\Gamma}\sim\sum_{j=1}^{N}\chi_{E_{j}}D_{\Gamma}\chi_{E_{j}}$
and thus

$\sigma_{\operatorname{ess}}(D_{\Gamma})=\bigcup\limits_{j=1}^{N}\sigma_{\operatorname{ess}}(\chi_{E_{j}}D_{\Gamma}\chi_{E_{j}}).$
(5.2)

Equation (5.2) is a localisation result. We have also the following, more
substantial localisation result (cf. [16, 47, 15, 11]) for the case where
$\Gamma\in\mathscr{D}$.

######  Theorem 5.2.

Let $\Omega_{-}\subset\mathbb{R}^{d}$ be a bounded Lipschitz domain such that
$\Gamma=\partial\Omega_{-}\in\mathscr{D}$, and pick
$x_{1},\ldots,x_{N}\in\Gamma$ for which
$\Gamma\subset\bigcup\limits_{j=1}^{N}B(x_{j},\delta(x_{j}))$. Then

$\sigma_{\operatorname{ess}}(D_{\Gamma})=\bigcup\limits_{x\in\Gamma}\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})=\bigcup\limits_{j=1}^{N}\sigma_{\operatorname{ess}}(D_{\Gamma_{x_{j}}}).$

###### Proof.

Let $x\in\Gamma$. If $\Gamma$ is locally $C^{1}$ at $x$, then, by (5.1) and
since $D_{\Gamma_{x}}$ is compact,
$\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})=\left\\{0\right\\}\subset\sigma_{\operatorname{ess}}(D_{\Gamma})$.
So assume that $\Gamma$ is locally dilation invariant at $x$. Then there is an
$\alpha\in(0,1)$ such that $\alpha\Gamma_{x}=\Gamma_{x}$. Let
$\eta\colon\Gamma\to[0,1]$ be a Lipschitz continuous function with support in
$B(x,\delta(x))\cap\Gamma$ that is equal to $1$ in a neighbourhood of $x$. Let
$\psi_{x}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ be an isometry with
$\psi_{x}(x)=0$ and
$\psi_{x}(B(x,\delta(x))\cap\Gamma)=B(0,\delta(x))\cap\Gamma_{x}$. Define
$\Psi_{x}\colon L^{2}(\Gamma_{x})\to L^{2}(\Gamma)$ by

$\Psi_{x}\phi(y)=\begin{cases}\phi(\psi_{x}(y))&\text{if }y\in
B(x,\delta(x))\cap\Gamma,\\\ 0&\text{otherwise,}\end{cases}$

which has adjoint $\Psi_{x}^{\prime}:L^{2}(\Gamma)\to L^{2}(\Gamma_{x})$ given
by

$\Psi_{x}^{\prime}\phi(y)=\begin{cases}\phi(\psi_{x}^{-1}(y))&\text{if }y\in
B(0,\delta(x))\cap\Gamma_{x},\\\ 0&\text{otherwise}.\end{cases}$

$\Psi_{x}$ and $\Psi_{x}^{\prime}$ are partial isometries with initial spaces
$L^{2}\big{(}B(0,\delta(x))\cap\Gamma_{x}\big{)}$ and
$L^{2}\big{(}B(x,\delta(x))\cap\Gamma\big{)}$, respectively. For
$m\in\mathbb{N}$ the composition
$\Psi_{x}^{\prime}\eta^{m}\Psi_{x}:L^{2}(\Gamma_{x})\to L^{2}(\Gamma_{x})$ is
the operation of multiplication by $(\eta^{\prime})^{m}$, where
$\eta^{\prime}(y):=\eta(\psi^{-1}_{x}(y))$, for $y\in
B(x,\delta(x))\cap\Gamma_{x}$, $\eta^{\prime}(y):=0$, $y\in\Gamma_{x}\setminus
B(x,\delta(x))$.

Let $V_{\alpha}$ be the unitary dilation as defined in (4.2) but with $\Gamma$
replaced by $\Gamma_{x}$. Then

$V_{\alpha}^{n}\eta^{\prime}V_{\alpha}^{-n}\phi(y)=\begin{cases}\eta(\psi_{x}^{-1}(\alpha^{n}y))\phi(y)&\text{if
}\alpha^{n}y\in B(0,\delta(x))\cap\Gamma_{x},\\\
0&\text{otherwise},\end{cases}$

for $n\in\mathbb{N}$, $y\in\Gamma_{x}$ and $\phi\in L^{2}(\Gamma_{x})$. By
dominated convergence, it follows that
$V_{\alpha}^{n}\eta^{\prime}V_{\alpha}^{-n}\phi\to\phi$ as $n\to\infty$ for
all $\phi\in L^{2}(\Gamma_{x})$. (We can see, similarly, that, for every
$m\in\mathbb{N}$,
$V_{\alpha}^{n}(\eta^{\prime})^{m}V_{\alpha}^{-n}\phi\to\phi$ as $n\to\infty$
for all $\phi\in L^{2}(\Gamma_{x})$.) Using that $\psi_{x}$ is an isometry, we
further observe that $\Psi_{x}^{\prime}\eta
D_{\Gamma}\eta\Psi_{x}=\eta^{\prime}D_{\Gamma_{x}}\eta^{\prime}$. This
implies, since also $D_{\Gamma_{x}}V_{\alpha}=V_{\alpha}D_{\Gamma_{x}}$, that

$V_{\alpha}^{n}\Psi_{x}^{\prime}\eta
D_{\Gamma}\eta\Psi_{x}V_{\alpha}^{-n}=V_{\alpha}^{n}\eta^{\prime}D_{\Gamma_{x}}\eta^{\prime}V_{\alpha}^{-n}=(V_{\alpha}^{n}\eta^{\prime}V_{\alpha}^{-n})D_{\Gamma_{x}}(V_{\alpha}^{n}\eta^{\prime}V_{\alpha}^{-n})\to
D_{\Gamma_{x}}$ (5.3)

in the strong operator topology.

Now assume that $D_{\Gamma}-\lambda$ is Fredholm and let $A\colon
L^{2}(\Gamma)\to L^{2}(\Gamma)$ be a Fredholm regularizer of
$D_{\Gamma}-\lambda$, so that the products $A(D_{\Gamma}-\lambda)$ and
$(D_{\Gamma}-\lambda)A$ are compact perturbations of the identity. Then, for
any $\phi\in L^{2}(\Gamma_{x})$,

$\left\lVert\phi\right\rVert\leq\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta
A\eta(D_{\Gamma}-\lambda)\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert+\left\lVert
V_{\alpha}^{n}(I-\Psi_{x}^{\prime}\eta
A\eta(D_{\Gamma}-\lambda)\eta\Psi_{x})V_{\alpha}^{-n}\phi\right\rVert.$ (5.4)

The first term can be estimated as

$\displaystyle\left\lVert V_{\alpha}^{n}\Psi_{x}^{\prime}\eta
A\eta(D_{\Gamma}-\lambda)\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert$
$\displaystyle\leq\left\lVert
A\right\rVert\left\lVert\eta(D_{\Gamma}-\lambda)\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert$
$\displaystyle=\left\lVert A\right\rVert\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta(D_{\Gamma}-\lambda)\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert,$

where we used that $V_{\alpha}$ is an isometry, $\Psi_{x}^{\prime}$ is an
isometry on the range of the multiplication operator $\eta$, and
$\left\lVert\eta\right\rVert\leq 1$. By (5.3),

$\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta(D_{\Gamma}-\lambda)\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert\to\left\lVert(D_{\Gamma_{x}}-\lambda)\phi\right\rVert,$

as $n\to\infty$. The second term in (5.4) can be estimated as

$\displaystyle\left\lVert V_{\alpha}^{n}(I-\Psi_{x}^{\prime}\eta
A\eta(D_{\Gamma}-\lambda)\eta\Psi_{x})V_{\alpha}^{-n}\phi\right\rVert$
$\displaystyle\quad\leq\left\lVert V_{\alpha}^{n}(I-\Psi_{x}^{\prime}\eta
A(D_{\Gamma}-\lambda)\eta^{2}\Psi_{x})V_{\alpha}^{-n}\phi\right\rVert+\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta
A[\eta,D_{\Gamma}]\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert$
$\displaystyle\quad\leq\left\lVert V_{\alpha}^{n}(I-\eta^{\prime
3})V_{\alpha}^{-n}\phi\right\rVert+\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta(A(D_{\Gamma}-\lambda)-I)\eta^{2}\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert+\left\lVert
V_{\alpha}^{n}\Psi_{x}^{\prime}\eta
A[\eta,D_{\Gamma}]\eta\Psi_{x}V_{\alpha}^{-n}\phi\right\rVert.$

Note that both $A(D_{\Gamma}-\lambda)-I$ and $[\eta,D_{\Gamma}]$ are compact.
We also observe that for any compact operator $K\colon L^{2}(\Gamma_{x})\to
L^{2}(\Gamma_{x})$ we have $V_{\alpha}^{n}KV_{\alpha}^{-n}\to 0$ in the strong
operator topology, since $V_{\alpha}^{-n}\to 0$ in the weak operator topology.
Therefore, and using that $V_{\alpha}^{n}\eta^{\prime 3}V_{\alpha}^{-n}\to I$,
we obtain that the second term in (5.4) tends to zero as $n\to\infty$. We
conclude that $\left\lVert\phi\right\rVert\leq\left\lVert
A\right\rVert\left\lVert(D_{\Gamma_{x}}-\lambda)\phi\right\rVert$ for every
$\phi\in L^{2}(\Gamma_{x})$. The same argument also shows that
$\left\lVert\phi\right\rVert\leq\left\lVert
A\right\rVert\left\lVert(D_{\Gamma_{x}}-\lambda)^{\prime}\phi\right\rVert$ for
every $\phi\in L^{2}(\Gamma_{x})$. It follows that $D_{\Gamma_{x}}-\lambda$ is
invertible and thus Fredholm.

Conversely, choose $x_{1},\ldots,x_{N}\in\Gamma$ such that
$\Gamma\subset\cup_{i=1}^{N}B(x_{i},\delta(x_{i}))$, and suppose that
$D_{\Gamma_{x_{j}}}-\lambda$ is Fredholm for $j=1,\ldots,N$, and let $A_{j}$
be a Fredholm regularizer of $D_{\Gamma_{x_{j}}}-\lambda$. For every $j$ let
$\psi_{x_{j}}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ be an isometry with
$\psi_{x_{j}}(x_{j})=0$ and
$\psi_{x_{j}}(B(x_{j},\delta(x_{j}))\cap\Gamma)=B(0,\delta(x_{j}))\cap\Gamma_{x_{j}}$.
Choose Lipschitz continuous functions $\eta_{1},\ldots,\eta_{N}$ such that
$\left\\{\eta_{j}^{2}:j=1,\ldots,N\right\\}$ is a partition of unity of
$\Gamma$ subordinate to the sets $B(x_{j},\delta(x_{j}))$. Now note that,
since $D_{\Gamma_{x_{j}}}$ essentially commutes with $\eta_{j}^{\prime}$,
$A_{j}$ essentially commutes with $\eta_{j}^{\prime}$ as well. This implies
that

$\displaystyle\sum\limits_{j=1}^{N}\Psi_{x_{j}}\eta_{j}^{\prime}A_{j}\Psi_{x_{j}}^{\prime}\eta_{j}(D_{\Gamma}-\lambda)$
$\displaystyle\sim\sum\limits_{j=1}^{N}\Psi_{x_{j}}A_{j}\eta_{j}^{\prime}\Psi_{x_{j}}^{\prime}(D_{\Gamma}-\lambda)\eta_{j}=\sum\limits_{j=1}^{N}\Psi_{x_{j}}A_{j}\Psi_{x_{j}}^{\prime}\eta_{j}(D_{\Gamma}-\lambda)\eta_{j}$
$\displaystyle=\sum\limits_{j=1}^{N}\Psi_{x_{j}}A_{j}\eta_{j}^{\prime}(D_{\Gamma_{x_{j}}}-\lambda)\eta_{j}^{\prime}\Psi_{x_{j}}^{\prime}\sim\sum\limits_{j=1}^{N}\Psi_{x_{j}}\eta_{j}^{\prime}A_{j}(D_{\Gamma_{x_{j}}}-\lambda)\eta_{j}^{\prime}\Psi_{x_{j}}^{\prime}$
$\displaystyle\sim\sum\limits_{j=1}^{N}\Psi_{x_{j}}\eta_{j}^{\prime
2}\Psi_{x_{j}}^{\prime}=\sum\limits_{j=1}^{N}\eta_{j}^{2}=I.$

Similarly,
$(D_{\Gamma}-\lambda)\sum\limits_{j=1}^{N}\eta_{j}\Psi_{x_{j}}A_{j}\eta_{j}^{\prime}\Psi_{x_{j}}^{\prime}\sim
I$. Thus $D_{\Gamma}-\lambda$ is Fredholm. ∎

###  5.2 Deformations

Let $\beta\in(0,1)$. We will now consider the situation where a Lipschitz
domain $\Omega_{-}$ with boundary $\Gamma$ is deformed by a
$C^{1,\beta}$-diffeomorphism $\psi\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ that
is conformal or anti-conformal at a point $x\in\Gamma$, meaning that the
Jacobian $J\psi(x)$ of $\psi$ at $x$ lies in $\mathbb{R}O(d)$, where
$\mathbb{R}O(d):=\left\\{\lambda A\in\mathbb{R}^{d\times
d}:\lambda\in\mathbb{R},A\text{ orthogonal}\right\\}$. Due to the invariance
of the double-layer potential under transformations from $\mathbb{R}O(d)$, we
will for our purposes be able to assume that $J\psi(x)=I$. Let
$D_{\Gamma}(\cdot,\cdot)$ denote the kernel of $D_{\Gamma}$. Likewise, $V_{j}$
will be the integral operator with kernel $V_{j}(\cdot,\cdot)$, $|D_{\Gamma}|$
will be the integral operator with kernel $|D_{\Gamma}(\cdot,\cdot)|$, and so
on. We write

$\displaystyle D_{\psi(\Gamma)}(\psi(z),\psi(y))$
$\displaystyle=D_{\Gamma}(z,y)$
$\displaystyle+\underbrace{D_{\psi(\Gamma)}(\psi(z),\psi(y))-\frac{(z-y)\cdot
n^{\Gamma}(y)}{c_{d}|\psi(z)-\psi(y)|^{d}}n^{\psi(\Gamma)}(\psi(y))\cdot
J\psi(y)n^{\Gamma}(y)}_{V_{1}(z,y)}$
$\displaystyle+\underbrace{D_{\Gamma}(z,y)(n^{\psi(\Gamma)}(\psi(y))\cdot
J\psi(y)n^{\Gamma}(y)-1)}_{V_{2}(z,y)}$
$\displaystyle+\underbrace{D_{\Gamma}(z,y)n^{\psi(\Gamma)}(\psi(y))\cdot
J\psi(y)n^{\Gamma}(y)\left[\frac{|z-y|^{d}}{|\psi(z)-\psi(y)|^{d}}-1\right]}_{V_{3}(z,y)}.$
(5.5)

The term $V_{1}$ is weakly singular, satisfying an estimate of the form
$|V_{1}(z,y)|\leq C_{\beta}|z-y|^{1+\beta-d}$ for $\beta\in(0,1)$ and almost
every $z,y\in\Gamma$, see [43, Lemma 6]. (In order to apply Medková’s results
from [43], note that the reduced boundary
$\partial_{\ast}\Omega_{-}\subset\Gamma$ has full surface measure for a
Lipschitz domain $\Omega_{-}$, cf. [17, p. 209].) Therefore the operator
$V_{1}\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$ is compact.

We view $V_{2}$ as the composition of $D_{\Gamma}$ with a multiplication
operator, noting that by [43, Lemma 3] we have

$\lim_{\rho\to 0^{+}}\operatorname*{ess\,sup}\limits_{y\in\Gamma\cap
B(x,\rho)}|n^{\psi(\Gamma)}(\psi(y))\cdot J\psi(y)n^{\Gamma}(y)-1|=0.$

In particular, for every $\varepsilon>0$, there is $\rho>0$ such that the
operator norm of

$V_{2}\colon L^{2}(\Gamma\cap B(x,\rho))\to L^{2}(\Gamma)$

is smaller than $\varepsilon$.

For a bounded Lipschitz domain with boundary $\Gamma$, let
$\mathscr{A}_{2}=\mathscr{A}_{2}(\Gamma)$ denote the algebra of integral
operators $K$ such that $|K|\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$ is bounded.
We will work within the class of domains such that
$D_{\Gamma}\in\mathscr{A}_{2}$ in this subsection. The preceding
considerations show that $D_{\psi(\Gamma)}\in\mathscr{A}_{2}(\psi(\Gamma))$ if
$D_{\Gamma}\in\mathscr{A}_{2}(\Gamma)$. Since the relationship between
$\Gamma$ and $\psi(\Gamma)$ is symmetric, we see that

$D_{\psi(\Gamma)}\in\mathscr{A}_{2}(\psi(\Gamma))\textrm{ if and only if
}D_{\Gamma}\in\mathscr{A}_{2}(\Gamma).$ (5.6)

Note that $D_{\Gamma}\in\mathscr{A}_{2}$ for any $C^{1,\beta}$-boundary
$\Gamma$, since $D_{\Gamma}$ has a weakly singular kernel in this case. It is
also known that $|D_{\widetilde{\Gamma}}|\colon L^{2}(\widetilde{\Gamma})\to
L^{2}(\widetilde{\Gamma})$ is bounded for any of the following graphs
$\widetilde{\Gamma}$:

  * (i)

a 2D wedge (modelling a polygonal corner) [48];

  * (ii)

a 3D wedge (modelling a polyhedral edge) [51];

  * (iii)

a 3D polyhedral cone [16];

  * (iv)

a 3D smooth cone (such as a circular cone) [52].

Furthermore, this statement may be extended to dilation-invariant domains
which are built from such graphs.

######  Lemma 5.3.

Suppose that $\widetilde{\Gamma}$ is a dilation-invariant graph, i.e.,
$\widetilde{\Gamma}=\alpha\widetilde{\Gamma}$, for some $\alpha\in(0,1)$, and
suppose that, at every $x\in\widetilde{\Gamma}\setminus\left\\{0\right\\}$,
$\widetilde{\Gamma}$ is either locally $C^{1,\beta}$ for some $\beta\in(0,1)$,
or coincides locally with a graph of type (i), (ii), (iii), or (iv). Then
$|D_{\widetilde{\Gamma}}|$ is bounded on $L^{2}(\widetilde{\Gamma})$.

###### Proof.

By hypothesis, we know that $\left\lvert D_{j}\right\rvert\colon
L^{2}(\widetilde{\Gamma}_{0})\to L^{2}(\widetilde{\Gamma}_{0})$ is bounded for
$j=-1,0,1$, where $D_{j}$ and $\widetilde{\Gamma}_{0}$ are as in §4.1. The
lemma therefore follows from Proposition 4.3 and (4.4). ∎

A simple localization argument now demonstrates that the condition
$D_{\Gamma}\in\mathscr{A}_{2}$ is quite general.

######  Theorem 5.4.

Suppose that at every $x\in\Gamma$, $\Gamma$ is locally $C^{1,\beta}$, for
some $\beta\in(0,1)$, or coincides locally with a graph satisfying the
hypothesis of Lemma 5.3. Then $D_{\Gamma}\in\mathscr{A}_{2}(\Gamma)$.

Returning to the decomposition of $D_{\psi(\Gamma)}(\psi(z),\psi(y))$, we
note, for the third term $V_{3}$, the following consequence of the fact that
$J\psi(x)=I$.

######  Lemma 5.5.

Suppose that $D_{\Gamma}\in\mathscr{A}_{2}$. Then
$|V_{3}|(z,y)=|D_{\Gamma}|(z,y)Q(z,y)$, where $Q\in
L^{\infty}(\Gamma\times\Gamma)$ is non-negative and $\lim_{\rho\to
0^{+}}\operatorname*{ess\,sup}_{y,z\in\Gamma\cap B(x,\rho)}Q(z,y)=0$, where
the essential supremum is taken with respect to the product measure on
$\Gamma\times\Gamma$.

Let $\eta_{\rho}\colon\Gamma\to[0,1]$ be a Lipschitz continuous function with
$\operatorname{supp}\eta_{\rho}\subset B(x,\rho)$ such that $\eta_{\rho}$ is
equal to $1$ in a neighbourhood of $x$. Furthermore, denote the operator of
composition with $\psi$ by

$C_{\psi}\colon L^{2}(\psi(\Gamma))\to L^{2}(\Gamma),\quad
C_{\psi}\phi(z)=\phi(\psi(z)),$

and let $\eta_{\rho}^{\prime}=\eta_{\rho}\circ\psi^{-1}$. Note also that we
have the change of variables formula

$\int_{\psi(\Gamma)}\phi(y)\,\mathrm{d}s_{\psi(\Gamma)}(y)=\int_{\Gamma}(1+T(y))\phi(\psi(y))\,\mathrm{d}s_{\Gamma}(y),$
(5.7)

where $T\in L^{\infty}(\Gamma)$ and $\lim\limits_{y\to x}T(y)=0$. The operator

$\eta_{\rho}^{\prime}D_{\psi(\Gamma)}\eta_{\rho}^{\prime}\colon
L^{2}(\psi(\Gamma))\to L^{2}(\psi(\Gamma))$

is then similar to

$C_{\psi}\eta_{\rho}^{\prime}D_{\psi(\Gamma)}\eta_{\rho}^{\prime}C_{\psi^{-1}}\colon
L^{2}(\Gamma)\to L^{2}(\Gamma),$

which, by (5.7), is an integral operator with kernel
$(1+T(y))\eta_{\rho}(z)D_{\psi(\Gamma)}(\psi(z),\psi(y))\eta_{\rho}(y)$. By
the previous calculations, if $D_{\Gamma}\in\mathscr{A}_{2}$, we therefore
find that

$C_{\psi}\eta_{\rho}^{\prime}D_{\psi(\Gamma)}\eta_{\rho}^{\prime}C_{\psi^{-1}}\sim\eta_{\rho}D_{\Gamma}\eta_{\rho}+\eta_{\rho}\tilde{V}\eta_{\rho},$
(5.8)

where $\tilde{V}$ is an integral operator induced by a kernel of the form
$\tilde{V}(z,y)=D_{\Gamma}(z,y)\tilde{Q}(z,y)$ with $\tilde{Q}\in
L^{\infty}(\Gamma\times\Gamma)$ such that

$\lim_{\rho\to 0^{+}}\operatorname*{ess\,sup}\limits_{y,z\in\Gamma\cap
B(x,\rho)}|\tilde{Q}(z,y)|=0.$ (5.9)

For $\phi_{1},\phi_{2}\in L^{2}(\Gamma)$ with support in $B(x,\rho)$ we have

$\left\lvert\left\langle\tilde{V}\phi_{1},\phi_{2}\right\rangle\right\rvert\leq\int_{\Gamma}\int_{\Gamma}\left\lvert\phi_{2}(z)\tilde{V}(z,y)\phi_{1}(y)\right\rvert\,\mathrm{d}s(y)\,\mathrm{d}s(z)\leq\operatorname*{ess\,sup}\limits_{y,z\in\Gamma\cap
B(x,\rho)}|\tilde{Q}(z,y)|\left\lVert\left\lvert
D_{\Gamma}\right\rvert\right\rVert\left\lVert\phi_{1}\right\rVert\left\lVert\phi_{2}\right\rVert.$

Since $D_{\Gamma}\in\mathscr{A}_{2}$, (5.9) implies that

$\lim_{\rho\to 0^{+}}\|\eta_{\rho}\tilde{V}\eta_{\rho}\|=0.$ (5.10)

Taking powers, we conclude that

$\lim_{\rho\to
0^{+}}\|(\eta_{\rho}D_{\Gamma}\eta_{\rho})^{n}-C_{\psi}(\eta_{\rho}^{\prime}D_{\psi(\Gamma)}\eta_{\rho}^{\prime})^{n}C_{\psi^{-1}}\|_{\operatorname{ess}}=0$
(5.11)

for every $n\geq 1$.

We now prove the main theorem of this section. When $R=1/2$, it says the
following: if, for every point $x$, $\Gamma$ is locally obtained by deforming
a domain that satisfies the spectral radius conjecture, then $\Gamma$
satisfies the spectral radius conjecture as well.

######  Theorem 5.6.

Let $R>0$ and $\beta>0$. Suppose that for every $x\in\Gamma$ there exists a
$C^{1,\beta}$-diffeomorphism $\psi_{x}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$,
conformal or anti-conformal at $x$, and a Lipschitz cut-off function
$\eta_{x}\colon\mathbb{R}^{d}\to[0,1]$ such that $\eta_{x}$ is equal to $1$ in
a neighbourhood of $x$, $D_{\psi_{x}(\Gamma)}\in\mathscr{A}_{2}$, and

$\sigma_{\operatorname{ess}}(\eta_{x}^{\prime}D_{\psi_{x}(\Gamma)}\eta_{x}^{\prime})\subset
B(0,R),$

where $\eta_{x}^{\prime}=\eta_{x}\circ\psi_{x}^{-1}$. Then
$D_{\Gamma}\in\mathscr{A}_{2}$ and
$\sigma_{\operatorname{ess}}(D_{\Gamma})\subset B(0,R)$.

###### Proof.

As noted previously, the condition on $\Gamma$, even for just one $x$, implies
that $D_{\Gamma}\in\mathscr{A}_{2}$.

Fix $x\in\Gamma$. Since the commutator
$[D_{\psi_{x}(\Gamma)},\eta_{x}^{\prime}]$ is compact, we see that
$(\eta_{x}^{\prime}D_{\psi_{x}(\Gamma)}\eta_{x}^{\prime})^{n}-\eta_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\eta_{x}^{\prime n}$ is compact for every $n\geq 1$.
By the hypothesis, choose $n=n(x)$ sufficiently large so that

$\|(\eta_{x}^{\prime}D_{\psi_{x}(\Gamma)}\eta_{x}^{\prime})^{n}\|_{\operatorname{ess}}^{\frac{1}{n}}=\|\eta_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\eta_{x}^{\prime
n}\|_{\operatorname{ess}}^{\frac{1}{n}}<R\|C_{\psi_{x}}\|^{-\frac{1}{n}}\|C_{\psi_{x}^{-1}}\|^{-\frac{1}{n}}.$

If we choose a new cut-off function
$\tilde{\eta}_{x}\colon\mathbb{R}^{d}\to[0,1]$ such that
$\operatorname{supp}\tilde{\eta}_{x}\subset\left\\{y\in\mathbb{R}^{d}:\eta_{x}(y)=1\right\\}$,
note that, since
$(\eta_{x}^{\prime}D_{\psi_{x}(\Gamma)}\eta_{x}^{\prime})^{n}-\eta_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\eta_{x}^{\prime n}$ is compact, we have

$\|(\tilde{\eta}_{x}^{\prime}D_{\psi_{x}(\Gamma)}\tilde{\eta}_{x}^{\prime})^{n}\|_{\operatorname{ess}}^{\frac{1}{n}}=\|\tilde{\eta}_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\tilde{\eta}_{x}^{\prime
n}\|_{\operatorname{ess}}^{\frac{1}{n}}\leq\|\eta_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\eta_{x}^{\prime
n}\|_{\operatorname{ess}}^{\frac{1}{n}}<R\|C_{\psi_{x}}\|^{-\frac{1}{n}}\|C_{\psi_{x}^{-1}}\|^{-\frac{1}{n}},$

so that

$\|C_{\psi_{x}}(\tilde{\eta}_{x}^{\prime}D_{\psi_{x}(\Gamma)}\tilde{\eta}_{x}^{\prime})^{n}C_{\psi_{x}^{-1}}\|_{\operatorname{ess}}^{\frac{1}{n}}\leq\|\eta_{x}^{\prime
n}D_{\psi_{x}(\Gamma)}^{n}\eta_{x}^{\prime
n}\|_{\operatorname{ess}}^{\frac{1}{n}}\|C_{\psi_{x}}\|^{\frac{1}{n}}\|C_{\psi_{x}^{-1}}\|^{\frac{1}{n}}<R.$

Hence, by making the support of $\eta_{x}$ sufficiently small and applying
(5.11), we obtain that

$\|(\eta_{x}D_{\Gamma}\eta_{x})^{n}\|_{\operatorname{ess}}^{\frac{1}{n}}=\|\eta_{x}^{n}D_{\Gamma}^{n}\eta_{x}^{n}\|_{\operatorname{ess}}^{\frac{1}{n}}<R.$

Now choose $\delta(x)$ sufficiently small so that $\eta_{x}\equiv 1$ in a
neighbourhood of $B(x,\delta(x))\cap\Gamma$. By compactness, we may pick
points $x_{1},\ldots,x_{N}$ so that
$\Gamma\subset\cup_{j=1}^{N}B(x_{j},\delta(x_{j}))$. Let

$r:=\max_{j=1,\ldots,N}\|(\eta_{x_{j}}D_{\Gamma}\eta_{x_{j}})^{n(x_{j})}\|_{\operatorname{ess}}^{\frac{1}{n(x_{j})}}<R$

and let $\left\\{\tau_{j}:j=1,\ldots,N\right\\}$ be a Lipschitz partition of
unity of $\Gamma$ subordinate to the sets $B(x_{j},\delta(x_{j}))$. Choose $m$
to be a large integer such that $n(x_{j})$ divides $m$ for all $j$. Then

$D^{m}_{\Gamma}=\sum\limits_{j=1}^{N}\tau_{j}D_{\Gamma}^{m}=\sum\tau_{j}\eta_{x_{j}}^{2m}D_{\Gamma}^{m}$

and therefore, recalling that $[D_{\Gamma},\eta_{x_{j}}]$ is compact,

$\displaystyle\|D_{\Gamma}^{m}\|_{\operatorname{ess}}$
$\displaystyle=\Bigg{\|}\sum\limits_{j=1}^{N}\tau_{j}(\eta_{x_{j}}D_{\Gamma}\eta_{x_{j}})^{m}\Bigg{\|}_{\operatorname{ess}}\leq\sum\limits_{j=1}^{N}\left\lVert(\eta_{x_{j}}D_{\Gamma}\eta_{x_{j}})^{m}\right\rVert_{\operatorname{ess}}\leq\sum\limits_{j=1}^{N}\left\lVert(\eta_{x_{j}}D_{\Gamma}\eta_{x_{j}})^{n(x_{j})}\right\rVert^{\frac{m}{n(x_{j})}}_{\operatorname{ess}}$
$\displaystyle\leq Nr^{m}<NR^{m}.$

Therefore $\sigma_{\operatorname{ess}}(D_{\Gamma})\subset B(0,R)$. ∎

Referring back to Lemma 5.3, all graphs $\widetilde{\Gamma}$ of type (i)-(iii)
satisfy (the graph version of) the spectral radius conjecture, results which
can also be found in the corresponding references. Graphs of type (iv) that
are convex also satisfy the spectral radius conjecture, by [18]. If
$\eta\colon\mathbb{R}^{3}\to[0,1]$ is a compactly supported Lipschitz
function, we thus have that

$\|(\eta
D_{\widetilde{\Gamma}}\eta)^{n}\|_{\operatorname{ess}}=\|\eta^{n}D_{\widetilde{\Gamma}}^{n}\eta^{n}\|_{\operatorname{ess}}\leq\|D_{\widetilde{\Gamma}}^{n}\|_{\operatorname{ess}}<2^{-n}$

for $n$ sufficiently large. Applying Theorem 5.6 we obtain the following.

######  Corollary 5.7.

Let $d=3$ and let $\Gamma$ be the boundary of a Lipschitz domain. Assume that
for every $x\in\Gamma$ there exists a $C^{1,\beta}$-diffeomorphism
$\psi_{x}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$, conformal or anti-conformal
at $x$, such that $\psi_{x}(\Gamma)$ is locally a subset of the boundary of a
polyhedral cone or a convex smooth cone. Then $D_{\Gamma}$ satisfies the
spectral radius conjecture.

The class of domains considered in Corollary 5.7 encompasses all domains in 3D
that may reasonably be referred to as a Lipschitz curvilinear polyhedra. From
Theorem 5.6 we of course also obtain the analogous corollary for 2D domains, a
well known result; the corresponding class of curves precisely describes the
$C^{1,\beta}$-curvilinear polygons in 2D.

###  5.3 Possible extensions and further questions

Theorem 5.6 in particular shows that the spectral radius conjecture only
depends on the local behaviour of $\Gamma$, under the extraneous assumption
that $D_{\Gamma}\in\mathscr{A}_{2}$. To achieve the same result for a general
Lipschitz domain, we need a different tool to treat the term $V_{3}$ which
does not rely on the introduction of absolute values. Since
$V_{3}(z,y)=\widetilde{Q}(z,y)D_{\Gamma}(z,y)$, for a specific kernel
$\widetilde{Q}(z,y)$, we would like an answer to the following question.

######  Question 5.8.

Describe suitable classes of kernels $B$ such that the kernel
$B(z,y)D_{\Gamma}(z,y)$ defines a bounded operator on $L^{2}(\Gamma)$, and
estimate the norm of the corresponding operator.

When $B$ is of the form $B(z,y)=a(z)-a(y)$, for some function $a$ on $\Gamma$,
the resulting operator is the commutator $[a,D_{\Gamma}]$. Commutators of
singular integral operators are very well studied, but we have been unable to
identify or correctly apply existing results to the kernel multiplier of
interest to us, namely,

$B(z,y)=\eta_{\rho}(z)\left(\frac{|z-y|^{d}}{|\psi(z)-\psi(y)|^{d}}-1\right)\eta_{\rho}(y).$

Other versions of Question 5.8 also seem interesting. For example, one could
ask for the stronger property that $B(z,y)$ be a kernel multiplier of each of
the Riesz transforms of $\Gamma$. We also note the similarity between the term
$V_{3}$ and the kernel of the Clifford–Cauchy integral operator, as presented
in [4, Consequence 3.6].

Of course, one would like to know that not only the spectral radius conjecture
is local, but that the entire essential spectrum is as well.

######  Question 5.9.

For a domain $\Omega_{-}$ with boundary $\Gamma$, is the essential spectrum
local and invariant under locally conformal deformations?

We can give a positive answer to this question if we, in addition to the
hypotheses of Theorem 5.6, assume that we do not have too many singular
points, by which we mean points where the boundary is not $C^{1}$. In the
statement, we let $\chi_{x,\rho}:=\chi_{B(x,\rho)}$ and
$\chi_{x,\rho}^{\prime}:=\chi_{x,\rho}\circ\psi_{x}^{-1}$ for $x\in\Gamma$ and
$\rho>0$.

######  Theorem 5.10.

Let $\beta>0$ and assume that $\Gamma$ has at most countably many singular
points. Further suppose that for every $x\in\Gamma$ there exists a
$C^{1,\beta}$-diffeomorphism $\psi_{x}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$,
conformal or anti-conformal at $x$, such that
$D_{\psi_{x}(\Gamma)}\in\mathscr{A}_{2}$. Then $D_{\Gamma}\in\mathscr{A}_{2}$
and

$\sigma_{\operatorname{ess}}(D_{\Gamma})\subset\bigcup\limits_{x\in\Gamma}\sigma_{\operatorname{ess}}(\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime})$

for arbitrary $\rho_{x}>0$. Moreover, there exist $\rho_{x}>0$ such that

$\sigma_{\operatorname{ess}}(D_{\Gamma})=\bigcup\limits_{x\in\Gamma}\sigma_{\operatorname{ess}}(\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime}).$

###### Proof.

By (5.6), $D_{\psi_{x}(\Gamma)}\in\mathscr{A}_{2}$ for just one $x\in\Gamma$
already implies $D_{\Gamma}\in\mathscr{A}_{2}$.

Let $x\in\Gamma$, $\rho_{x}>0$ and $\lambda\in\mathbb{C}$. If $\lambda=0$,
then
$0\in\sigma_{\operatorname{ess}}(\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime})$
either by (5.1) (if $\chi_{x,\rho_{x}}^{\prime}\equiv 1$) or by construction
(otherwise). So assume that $\lambda\neq 0$ and suppose that
$\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime}-\lambda$
is Fredholm for every $x\in\Gamma$.

Fix $x$ for a moment. As $\Gamma$ has at most countably many singular points
and $\psi_{x}$ is a diffeomorphism, we know that
$\chi_{x,\tilde{\rho}_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\tilde{\rho}_{x}}^{\prime}-\lambda$
is Fredholm for all but at most countably many
$\tilde{\rho}_{x}\in(0,\rho_{x}]$. In this case the essential norm of the
Fredholm regularizer $A_{x,\tilde{\rho}_{x}}$ of
$\chi_{x,\tilde{\rho}_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\tilde{\rho}_{x}}^{\prime}-\lambda$
is bounded by

$\left\lVert
A_{x,\tilde{\rho_{x}}}\right\rVert_{\operatorname{ess}}\leq\max\left\\{\left\lVert
A_{x,\rho_{x}}\right\rVert_{\operatorname{ess}},\left\lvert\lambda\right\rvert^{-1}\right\\}.$

Therefore, by (5.8) and (5.10), for every $x\in\Gamma$ we can choose
$\tilde{\rho}_{x}$ sufficiently small such that
$\chi_{x,\tilde{\rho}_{x}}D_{\Gamma}\chi_{x,\tilde{\rho}_{x}}-\lambda$ is also
Fredholm, and such that $\partial B(x,\tilde{\rho}_{x})$ contains none of the
singularities of $\Gamma$.

By compactness, we may choose finitely many points $x_{j}$ and corresponding
radii $\tilde{\rho}_{x_{j}}$ such that the balls
$B(x_{j},\tilde{\rho}_{x_{j}})$ cover $\Gamma$. Define recursively

$E_{1}:=B(x_{1},\tilde{\rho}_{x_{1}})\cap\Gamma,\quad
E_{j+1}=\big{(}B(x_{j+1},\tilde{\rho}_{x_{j+1}})\cap\Gamma\big{)}\setminus\bigcup\limits_{k=1}^{j}E_{k}.$

Now we can apply (5.2) to obtain that $D_{\Gamma}-\lambda$ is Fredholm.

Conversely, assume that $D_{\Gamma}-\lambda$ is Fredholm and fix $x\in\Gamma$.
By (5.1), we must have $\lambda\neq 0$. As we only have countably many
singularities, $\chi_{x,\rho}D_{\Gamma}\chi_{x,\rho}-\lambda$ is also Fredholm
for all but at most countably many $\rho>0$ by (5.2). Now note that the
situation between $D_{\Gamma}$ and $D_{\psi_{x}(\Gamma)}$ is symmetric. Hence,
the same argument as above shows that
$\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime}-\lambda$
is also Fredholm for a sufficiently small $\rho_{x}>0$. ∎

In particular, we may extend Theorem 5.2 to domains that are only
_approximately_ locally dilation invariant in the following sense.

######  Corollary 5.11.

Let $\beta>0$ and assume that $\Gamma$ has at most countably many singular
points. Denote the set of singular points by $\mathcal{S}$. Suppose that for
every $x\in\mathcal{S}$ there exists a $C^{1,\beta}$-diffeomorphism
$\psi_{x}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ such that

  * (i)

$\psi_{x}$ is conformal or anti-conformal at $x$,

  * (ii)

$\psi_{x}(\Gamma)$ is locally $C^{1,\beta}$ or locally dilation invariant at
$\psi_{x}(x)$,

  * (iii)

$D_{\psi_{x}(\Gamma)}\in\mathscr{A}_{2}$ (e.g., a domain with local behaviour
as in Theorem 5.4).

Then there exist $\rho_{x}>0$ such that
$\sigma_{\operatorname{ess}}(D_{\Gamma})=\left\\{0\right\\}\cup\bigcup\limits_{x\in\mathcal{S}}\sigma_{\operatorname{ess}}(\chi_{x,\rho_{x}}^{\prime}D_{\psi_{x}(\Gamma)}\chi_{x,\rho_{x}}^{\prime})$.

##  6 Synthesis and numerical examples

In this final section we bring earlier results together to study the case
where $\Gamma\in\mathscr{D}_{\mathcal{A}}$ (recall Definition 1.4), meaning
that $\Gamma$ is the boundary of a bounded Lipschitz domain
$\Omega_{-}\subset\mathbb{R}^{2}$ that is locally dilation invariant and is
piecewise analytic. We then illustrate the various results of the paper by
several examples. Regarding our first aim, the following result is immediate
from Theorems 5.2, 4.19, and 4.21.

######  Theorem 6.1.

Suppose that $\Gamma\in\mathscr{D}_{\mathcal{A}}$ and let $F\subset\Gamma$ be
the finite set of points at which $\Gamma$ is not locally analytic but is
locally dilation invariant, so that $\Gamma$ coincides locally near $x\in F$
with $\Gamma_{x}$, a dilation invariant graph (in some rotated coordinate
system centred at $x$). Then, where $\sigma^{N}$ is as defined in Theorem
4.19, $\Sigma^{N}(D_{\Gamma}):=\cup_{x\in
F}\sigma^{N}(D_{\Gamma_{x}})\begin{picture}(30.0,15.0)\put(13.0,16.0){\makebox(0.0,0.0)[]{\tiny\rm
H}} \put(15.0,5.0){\makebox(0.0,0.0)[]{$\to$}}
\end{picture}\sigma_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))$ as
$N\to\infty$. Further, suppose that $c>0$ is small enough such that the
conditions of the first sentence of Theorem 4.21 are satisfied for
$\Gamma=\Gamma_{x}$, $x\in F$. Then, for every $\rho_{0}>0$, where
$R_{N}(D_{\Gamma})$ is defined by (1.13),
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$ if: i)
$R_{N}(D_{\Gamma})<\rho_{0}$; and ii) for some $m,M,N\in\mathbb{N}$, with
$M\geq 2$,

$\mathscr{S}_{c}(\Gamma,m,M,N):=\max_{x\in F}S_{c}(\Gamma_{x},m,M,N)<0,$ (6.1)

where $S_{c}(\cdot,\cdot,\cdot,\cdot)$ is as defined in Remark 4.22.
Conversely, if $\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))<\rho_{0}$,
then, for all sufficiently large $N$, $R_{N}(D_{\Gamma})<\rho_{0}$, and, if
$m,M\in\mathbb{N}$ are also sufficiently large, then
$\mathscr{S}_{c}(\Gamma,m,M,N)<0$.

###### Proof.

Since $\Gamma\in\mathscr{D}_{\mathcal{A}}\subset\mathscr{D}$, $\Gamma$ is
locally $C^{1}$ at $x$ if $x\in\Gamma\setminus F$, while, if $x\in F$,
$\Gamma_{x}$ satisfies (4.44) (in some local coordinate system centred at
$x$). Thus, and by Theorem 5.2,
$\sigma_{\operatorname{ess}}(D_{\Gamma})=\cup_{x\in\Gamma}\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})=\cup_{x\in
F}\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})$, since
$\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})=\\{0\\}$ if $x\in\Gamma\setminus
F$, and $0\in\sigma_{\operatorname{ess}}(D_{\Gamma_{x}})$ if $x\in F$, by
Theorem 4.4 and Corollary 4.6. The result thus follows from Theorems 4.19 and
4.21. ∎

The following examples illustrate the above result and the results of §4 and
§5. In each example, whether $\Gamma$ is a dilation invariant graph or
$\Gamma\in\mathscr{D}_{\mathcal{A}}$, we demonstrate that
$\rho_{\operatorname{ess}}(D_{\Gamma};L^{2}(\Gamma))<\frac{1}{2}$, providing
new evidence in support of the spectral radius conjecture.

Example 1. We first consider an example where $\Gamma$ satisfies (4.8), with
$f:\mathbb{R}_{+}\to\mathbb{R}$ given by

$f(x):=x\sin^{2}(\pi\log_{\alpha}(x)),\quad x>0,$ (6.2)

for some $\alpha\in(0,1)$, so that (recall (4.10)) $g(x)=\sin^{2}(\pi
x)=(1-\cos(2\pi x))/2$, for $x\in\mathbb{R}$. As $\alpha$ increases in $(0,1)$
the graph of $f$ (see Figure 2) becomes increasingly oscillatory and its
Lipschitz character increases; elementary calculations give that the maximum
and minimum of $f^{\prime}$ are
$f^{\prime}_{\max}=\cos^{2}(\theta_{\alpha}/2)+\pi\sin(\theta_{\alpha})/|\log\alpha|>1$
and $f^{\prime}_{\min}=1-f^{\prime}_{\max}<0$, where
$\theta_{\alpha}:=\arctan(2\pi/|\log\alpha|)$; note that
$f^{\prime}_{\max}=\pi/|\log\alpha|+1/2+O(1-\alpha)$ as $\alpha\to 1^{-}$. To
apply Theorem 4.15 we need to compute $C_{1}(M)$, $C_{3}$, and $C_{4}$; see
(4.34), (4.39), and the definition below (4.40). This requires computation of
quantities that are defined in Proposition 4.8 in terms of202020In this
example, and the other examples below, we are able to compute these norms
exactly. We can, instead, just compute upper bounds; the theory and algorithm
apply essentially unchanged – see Remark 4.22 iii).

$\displaystyle\|\operatorname{Im}g^{(j)}\|_{c}=\frac{(2\pi)^{j}}{2}\sinh(2\pi
c),\;\;j=0,1,\;\;c\geq 0,$ $\displaystyle\|g\|_{c}=\cosh^{2}(\pi
c),\;\;\|g^{(j)}\|_{c}=\frac{(2\pi)^{j}}{2}\cosh(2\pi c),\;\;j=1,2,\;\;c\geq
0.$

Theorem 4.15 applies for all $c>0$ such that (4.21) applies, i.e. provided

$c\leq\frac{\arccos(\alpha)}{\left\lvert\log\alpha\right\rvert}\quad\text{and}\quad
c<\frac{1}{2\pi}\operatorname{arsinh}\left(\frac{2|\log\alpha|}{2\pi+|\log\alpha|}\right).$
(6.3)

(a) $\mathcal{L}_{c}(f,N)$, $\mathcal{R}_{c}(f,m,M,N)$, for
$\rho_{0}=\frac{1}{2}$, and the maximum of the spectral radii of
$A_{t_{k},N}^{M}$, $k=1,\ldots,m$, plotted against $N$ for $m=16,000$ and
$M=100$. The right-hand scale (in red) is to be used for $r_{\max}^{N}$
(plotted in red). The $\mathcal{R}_{c}(f,m,M,N)$ value is not shown for $N=16$
because it is negative.

(b) $\sigma^{N}(D_{\Gamma})=\\{0\\}\cup\bigcup_{t\in
t_{N}}\operatorname{\sigma}(A_{t,N}^{M_{N}})$, for $N=512$, $m_{N}=16,000$,
$M_{N}=100$, an approximation to
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$ by Theorem 4.13.
Also shown are the boundary of $W_{100}$ (for $p=10$, $N=512$, $M=100$ and
$t=\pi/18$) in black, $R^{*}\mathbb{T}\subset W_{100}\subset W(D_{\Gamma})$ in
blue (see Corollary 4.25 for definitions), $\frac{1}{2}\mathbb{T}$ in red. The
crosses on $\frac{1}{2}\mathbb{T}$ are $\mu_{k,\ell}$ for $k=16,000$ and
$\ell=1,\ldots,n_{k}=67$ (see Theorem 4.15)

Figure 4: Numerical results for Example 1; $f$ given by (6.2),
$\alpha=\frac{3}{4}$

In Figure 4 we plot results for the case $\alpha=\frac{3}{4}$ (see Figure 2),
when $f^{\prime}_{\max}\approx 11.43$ and the above conditions reduce to
$c\leq\arccos(3/4)/\log(4/3)\approx 2.51$ and
$c<\operatorname{arsinh}\left(2\log(4/3)/(2\pi+\log(4/3))\right)/(2\pi)$
$\approx 0.0139$. Choosing $c=0.013$ we plot in Figure 4(a), for the case
$\rho_{0}=\frac{1}{2}$, $\mathcal{L}_{c}(f,N)$ and $\mathcal{R}_{c}(f,m,M,N)$,
given by (4.43), against $N$, for $N=2^{j}$, $j=3,4,\ldots,9$, choosing
$m=16,000$ and $M=100$. We see that with these choices
$\mathcal{R}_{c}(f,m,M,N)$ is positive and bounded away from zero for
sufficiently large $N$, while, as is clear from its definition,
$\mathcal{L}_{c}(f,N)$ decreases exponentially with $N$; note that (4.43) is
satisfied for $N=512$. To apply Theorem 4.15 to conclude that
$\rho(D_{\Gamma})=\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ we also
need to check that $\rho(A^{M}_{t,N})<\frac{1}{2}$, for $M=100$, N = 512, and
$t=t_{k}=(k-1/2)/m$, $k=1,\ldots,m$, with $m=16,000$; equivalently, that
$r^{N}_{\max}:=\max_{z\in\sigma^{N}(D_{\Gamma})}|z|<\frac{1}{2}$, where
$\sigma^{N}(D_{\Gamma})$ is as defined by (4.41) with $N=512$, $m_{N}=16,000$,
$M_{N}=100$. For these parameter values $\sigma^{N}(D_{\Gamma})$, an
approximation to $\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$
by Theorem 4.13, is plotted in Figure 4(b), and $r^{N}_{\max}\approx
0.4837<\frac{1}{2}$. Our calculations are in standard double-precision
floating-point arithmetic rather than exact arithmetic212121In all these
examples the spectral radii of the matrices $A_{t,N}^{M}$ as well as the norms
of the $(A_{t,N}^{M}-\lambda_{l}I)^{-1}$ are computed using standard Matlab
routines; our codes are available at https://github.com/Raffael-Hagger/DLP,
but this is convincing evidence, by application of Theorem 4.15, that
$\rho(D_{\Gamma})<\frac{1}{2}$.

In Figure 4(b) we also plot, for the parameter values $p=10$, $N=512$,
$M=100$, $n=100$, and $t=\pi/18$, the bounded domain $W_{n}$, which222222In
our computations of $W_{n}$ we neglect the factor $e^{i(x_{m,N}-x_{n,N})t}$
when using (4.74) with $\tilde{K}_{t}$ replaced by $\tilde{K}_{t}^{M}$. The
resulting matrix is unitarily equivalent to $T^{p,t,N,M}$ as defined in §4.4,
so has the same numerical range. is contained in and is an approximation to
$W(T^{p,t,N,M})$ (these notations defined in §4.4). Where $R^{*}$ is as
defined in Corollary 4.25, we also plot the circle $R^{*}\mathbb{T}$ of radius
$R^{*}\approx 1.163$; by Corollary 4.25 it is guaranteed that
$\overline{B_{R^{*}}}\subset
W(D_{\Gamma})=W_{\operatorname{ess}}(D_{\Gamma})$, so that
$w(D_{\Gamma})=w_{\operatorname{ess}}(D_{\Gamma})\geq R^{*}$ is significantly
larger than $\frac{1}{2}$ for this example. We note that, for these parameter
values, $C_{7}(p,N,M)\leq 3.964\times 10^{-5}$ (see Corollaries 4.24 and
4.25).

In Figure 4(b) we additionally plot, to illustrate the application of Theorem
4.15 and the adaptive definition of the parameters defined by (4.42), the
points $\mu_{k,\ell}$ at which the resolvent of $A_{t_{k},N}^{M}$ is
calculated when $\rho_{0}=\frac{1}{2}$, $N=512$, $M=100$, $m=16,000$, for the
case $k=16,000$ (so that $t_{k}\approx\pi$) and $\ell=1,\ldots,n_{k}=67$. This
value of $k$ is fairly typical; $n_{k}$ varies in the range $[67,110]$ as a
function of $k$. It is clear that the adaptive algorithm of Theorem 4.15 (and
see Corollary 3.5) is significantly more efficient than the uniform grid on
$\rho_{0}\mathbb{T}$ of Corollary 3.3 (recall the discussion above Lemma 3.4).
For $k=16,000$ we have $\min_{\ell}\nu_{k,\ell}\approx 0.0101$ and
$\max_{\ell}\nu_{k,\ell}\approx 0.2216$ (see (4.42) for this notation). The
ratio of these maximum and minimum values, which is approximately the ratio of
the maximum to the minimum spacing of the points $\mu_{k,\ell}$, is about
$20.2$.

Example 2. We now turn to examples where we can apply the theory of §4.3.
First we consider the case of a cone, that is
$\Gamma=\\{(x,f(x)):x\in\mathbb{R}\\}$ where

$f(x):=\mu\left\lvert x\right\rvert,\quad x\in\mathbb{R},$ (6.4)

for some $\mu\in\mathbb{R}$. This clearly satisfies (4.44), for any
$\alpha\in(0,1)$, and, where $f_{\pm}$ and $g_{\pm}$ are defined by (4.45),
$f_{\pm}(x)=\mu x$, $g_{\pm}(x)=\mu$, for $x>0$. For this example the spectrum
and spectral radius are known, viz.

$\sigma_{\operatorname{ess}}(\Gamma)=\sigma(D_{\Gamma})=\\{0\\}\cup\left\\{\pm\frac{\sin(\arctan(|\mu|)(1-iy))}{2\sin(\pi(1-iy)/2)}:y\in\mathbb{R}\right\\}\;\;\mbox{so
that}\;\;\rho(D_{\Gamma})=\frac{|\mu|}{2\sqrt{1+\mu^{2}}}$ (6.5)

(see, e.g., [48]) and, since $D_{\Gamma}$ is diagonalised by the Mellin
transform when $\Gamma$ is a 2D cone, $D_{\Gamma}$ is normal so that
$W_{\operatorname{ess}}(D_{\Gamma})=\overline{W(D_{\Gamma})}=\operatorname*{conv}(\sigma(D_{\Gamma}))$
and $\|D_{\Gamma}\|=w(D_{\Gamma})=\rho(D_{\Gamma})$.

To make comparison of these known results with the methods of §4.3 we can
choose, in principle, any $\alpha\in(0,1)$, but the choice of $\alpha$ affects
the choice of $c$ via (4.54), and thus the rate of decrease with $N$ of
$\mathcal{L}_{c}(f,N)$, defined by (4.72). The operators
$\tilde{L}^{\pm}_{t}$, given by (4.48), and thus the matrices
$A_{t_{k},N}^{M}$, given by (4.63), also depend on $\alpha$, while the
operators $\tilde{K}^{\pm}_{t}$, given by (4.47), vanish in this case. It is
worth noting that the blocks of $A_{t_{k},N}^{M}$ corresponding to
$\tilde{L}_{t}^{\pm}$ are Toeplitz matrices in this example. The conditions
(4.54) reduce to $c\leq\arccos(\alpha)/|\log\alpha|$ and
$|\mu|c\leq\alpha^{2}/|\log\alpha|$. As the right hand sides of these
inequalities are increasing on $(0,1)$, it is beneficial to choose $\alpha$
closer to $1$ in order to be able to choose a larger $c$. We select
$\alpha=\frac{7}{8}$, so that $\arccos(\alpha)/|\log\alpha|\approx 3.78$ and
$\alpha^{2}/|\log\alpha|\approx 5.73$, and then take $c=0.57$, so that (4.54)
is satisfied for $|\mu|\leq 10$. For $\mu=10$ we see from Figure 5(a) that,
with $M=200$, $m=2,000$ and $N=16$, the inequality (4.72) is satisfied and
$\max_{k=1,\ldots,m}\rho(A_{t_{k},N}^{M})\approx 0.4975<\frac{1}{2}$, so that
$\rho(D_{\Gamma})<\frac{1}{2}$ by Theorem 4.21, in agreement with (6.5) which
gives $\rho(D_{\Gamma})=5/\sqrt{101}\approx 0.4975$. We see in Figure 5(b)
that the approximations $\sigma^{N}(D_{\Gamma})$ to $\sigma(D_{\Gamma})$,
given by Theorem 4.19, agree closely, for the given parameter values, with the
expected lemniscates given by (6.5) for different values of $\mu$.

(a) $\mathcal{L}_{c}(f,N)$, $\mathcal{R}_{c}(f,m,M,N)$, for
$\rho_{0}=\frac{1}{2}$, and the maximum of the spectral radii of
$A_{t_{k},N}^{M}$, $k=1,\ldots,m$, plotted against $N$ for $m=2,000$ and
$M=200$ in the case $\mu=10$.

(b) $\sigma^{N}(D_{\Gamma})=\\{0\\}\cup\bigcup_{t\in
T_{N}}\operatorname{\sigma}(A_{t,N}^{M})$ for $N=16$, $m_{N}=2,000$,
$M_{N}=200$, and $\mu=1$ (black), $\mu=2$ (blue), $\mu=5$ (green), $\mu=10$
(red). The yellow dashed lines indicate the lemniscates (6.5), i.e. the
(exact) spectrum of $D_{\Gamma}$ in each case.

Figure 5: Numerical results for Example 2; $f$ given by (6.4),
$\alpha=\frac{7}{8}$

Example 3. In this example we take

$f(x):=\left\lvert x\right\rvert\sin^{2}(\pi\log_{\alpha}\left\lvert
x\right\rvert),\quad x\in\mathbb{R},$ (6.6)

and $\Gamma$ satisfies (4.44), but now just for one $\alpha\in(0,1)$. Note
that $f_{\pm}$ and $g_{\pm}$, given by (4.45), are the same as $f$ and $g$ in
Example 1, so that $g_{\pm}$ satisfy the same bounds as $g$ in Example 1, and
$\Gamma$ has Lipschitz character $f^{\prime}_{\max}$ as defined in that
example. Thus the conditions (4.54) reduce in this example to (6.3) plus the
condition that $\sinh(2\pi c)/2+c|\log\alpha|\cosh^{2}(\pi c)<\alpha^{2}$. We
choose $\alpha=\frac{2}{3}$ (see Figure 3), for which $c=0.019$ satisfies this
condition and (4.54), and $f_{\max}^{\prime}\approx 8.26$. We see from Figure
6(a) that, with $M=60$, $m=10,000$ and $N=256$, the inequality (4.72) is
satisfied and $\max_{k=1,\ldots,m}\rho(A_{t_{k},N}^{M})<\frac{1}{2}$, so that
$\rho(D_{\Gamma})<\frac{1}{2}$ by Theorem 4.21. Figure 6(b) plots an
approximation $\sigma^{N}(D_{\Gamma})$ to $\sigma(D_{\Gamma})$ given by
Theorem 4.19, which is contained in the circle of radius $\frac{1}{2}$ (in
red). By contrast, by (4.73), and arguing as in Example 1, the numerical range
$W(D_{\Gamma})$ contains at least the closed disc of radius $R^{*}\approx
0.8179$ (shown in blue in Figure 6(b)), where $R^{*}$ is as given in Corollary
4.25.

(a) $\mathcal{L}_{c}(f,N)$, $\mathcal{R}_{c}(f,m,M,N)$, for
$\rho_{0}=\frac{1}{2}$, and the maximum of the spectral radii of
$A_{t_{k},N}^{M}$, $k=1,\ldots,m$, plotted against $N$ for $m=10,000$ and
$M=60$. $\mathcal{R}_{c}(f,m,M,N)$ values are not shown for $N=2,4,$ and $8$
because they are negative.

(b) $\sigma^{N}(D_{\Gamma})=\\{0\\}\cup\bigcup_{t\in
t_{N}}\operatorname{\sigma}(A_{t,N}^{M_{N}})$, for $N=256$, $m_{N}=10,000$,
$M_{N}=60$, an approximation to
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$ by Theorem 4.19.
By Theorem 6.1 this is also an approximation to
$\sigma_{\operatorname{ess}}(D_{\Gamma})$ for
$\Gamma\in\mathscr{D}_{\mathcal{A}}$ as in Figure 1(b). Also shown are the
boundary of $W_{100}$ (for $p=10$, $N=256$, $M=60$ and $t=\pi/13$) in black,
$R^{*}\mathbb{T}\subset W_{100}\subset W(D_{\Gamma})$ in blue (see Corollary
4.25), $\frac{1}{2}\mathbb{T}$ in red.

Figure 6: Numerical results for Example 3; $f$ given by (6.6),
$\alpha=\frac{2}{3}$.

Figure 6 is also, by Theorem 6.1, relevant to the bounded Lipschitz domain
$\Omega_{-}$ shown in Figure 1(b). This has boundary
$\Gamma\in\mathscr{D}_{\mathcal{A}}$ which is $C^{1}$ except at the single
point $x=0$ where it coincides locally with the graph of $f$ given by (6.6).
It follows by the above calculations and Theorem 6.1 that
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ for
$\Gamma=\partial\Omega_{-}$ and that Figure 6(b) is also a plot of the
approximation $\Sigma^{N}(D_{\Gamma})$, defined in Theorem 6.1, to the
essential spectrum of $D_{\Gamma}$ for $\Gamma=\partial\Omega_{-}$. While
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$,
$W_{\operatorname{ess}}(D_{\Gamma})$, for $\Gamma=\partial\Omega_{-}$,
contains the closed disc of radius $R^{*}\approx 0.8179$ shown in Figure 6(b),
by the above result for the graph $\Gamma$ given by (6.6), and a localisation
result [11, Theorem 3.2] (and see [11, Theorem 3.17]) for the essential
numerical range, analogous to Theorem 5.2. Thus also, by (1.8),
$\|D_{\Gamma}\|_{\operatorname{ess}}\geq
w_{\operatorname{ess}}(D_{\Gamma})\geq R^{*}$.

Example 4. In this example we define $f$ by (6.2) for $x>0$ and set $f(x):=0$
for $x\leq 0$, so that $\Gamma$ and $f$ satisfy (4.44). Recalling (4.45), we
see that $f_{+}$ and $g_{+}$ are the same as $f$ and $g$ in Example 1, while
$f_{-}=0$ and $g_{-}=0$. The graph $\Gamma$ has Lipschitz character
$L=(f_{\max}^{\prime}-f_{\min}^{\prime})/2=f_{\max}^{\prime}-1/2$, where
$f_{\max}^{\prime}$ and $f_{\min}^{\prime}$ are as defined in Example 1. The
conditions (4.54) reduce in this example to the same conditions as in Example
3, and again we choose $\alpha=\frac{2}{3}$ and $c=0.019$, so that $\Gamma$
has Lipschitz character $L=f_{\max}^{\prime}-1/2\approx 7.76$. Similarly to
the previous example, we see from Figure 7(a) that, with $M=50$, $m=5,000$,
and $N=256$, the inequality (4.72) is satisfied and
$\max_{k=1,\ldots,m}\rho(A_{t_{k},N}^{M})<\frac{1}{2}$, so that
$\rho(D_{\Gamma})<\frac{1}{2}$ by Theorem 4.21. The approximation
$\sigma^{N}(D_{\Gamma})$ to $\sigma(D_{\Gamma})$, given by Theorem 4.19 and
plotted in Figure 7(b), is contained in the circle of radius $\frac{1}{2}$,
while the numerical range $W(D_{\Gamma})$ contains at least the closed disc of
radius $R^{*}\approx 0.8179$ shown in Figure 7(b).

Figure 7 is also, by Theorem 6.1, relevant to the bounded Lipschitz domain
$\Omega_{-}$ shown in Figure 1(a). This has boundary
$\Gamma\in\mathscr{D}_{\mathcal{A}}$ which is $C^{1}$ except at the single
point $x=0$ where it coincides locally with the graph of the function $f$
described above. It follows by the above calculations and Theorem 6.1 that
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$ for
$\Gamma=\partial\Omega_{-}$ and that Figure 6(b) is also a plot of the
approximation $\Sigma^{N}(D_{\Gamma})$, defined in Theorem 6.1, to the
essential spectrum of $D_{\Gamma}$ for $\Gamma=\partial\Omega_{-}$. Arguing as
in the previous example, while
$\rho_{\operatorname{ess}}(D_{\Gamma})<\frac{1}{2}$,
$W_{\operatorname{ess}}(D_{\Gamma})$, for $\Gamma=\partial\Omega_{-}$,
contains the closed disc of radius $R^{*}\approx 0.8179$ plotted in Figure
7(b), and $\|D_{\Gamma}\|_{\operatorname{ess}}\geq
w_{\operatorname{ess}}(D_{\Gamma})\geq R^{*}$.

######  Remark 6.2 (Symmetry of the spectrum and essential spectrum).

For Example 2 it is immediate from (6.5) that $\sigma(D_{\Gamma})$ is
symmetric with respect to the origin; if $z\in\sigma(D_{\Gamma})$ then
$-z\in\sigma(D_{\Gamma})$. We conjecture, based on the numerical results for
Examples 3 and 4 and similar calculations, that this same symmetry holds in 2D
whenever $\Gamma$ is a dilation invariant graph satisfying (4.44). If this
conjecture is true, then, for all $\Gamma\in\mathscr{D}_{\mathcal{A}}$,
$\sigma_{\operatorname{ess}}(D_{\Gamma})$ is symmetric with respect to the
origin by Theorem 6.1. We note that symmetry results in the 2D case of this
sort are proved for $D_{\Gamma}:L^{2}(\Gamma)\to L^{2}(\Gamma)$ when $\Gamma$
is the boundary of a bounded $C^{2}$ domain $\Omega_{-}$ in [35, Proposition
6], and for $D^{\prime}_{\Gamma}$ as an operator on the natural energy space
for general Lipschitz $\Omega_{-}$ [25, Theorem 2.1] (in both these cases the
relevant spectrum lies in $[-1,1]$). In 3D, even when $\Gamma$ is a
polyhedron, the spectrum and essential spectrum of $D_{\Gamma}$ need not be
symmetric with respect to the origin (see, e.g., [15]).

(a) $\mathcal{L}_{c}(f,N)$, $\mathcal{R}_{c}(f,m,M,N)$, for
$\rho_{0}=\frac{1}{2}$, and the maximum of the spectral radii of
$A_{t_{k},N}^{M}$, $k=1,\ldots,m$, plotted against $N$ for $m=5000$ and
$M=50$. $\mathcal{R}_{c}(f,m,M,N)$ values are not shown for $N=2,4$ and $8$
because they are negative.

(b) $\sigma^{N}(D_{\Gamma})=\\{0\\}\cup\bigcup_{t\in
t_{N}}\operatorname{\sigma}(A_{t,N}^{M_{N}})$, for $N=256$, $m_{N}=5000$,
$M_{N}=50$, an approximation to
$\sigma(D_{\Gamma})=\sigma_{\operatorname{ess}}(D_{\Gamma})$ by Theorem 4.19.
By Theorem 6.1 this is also an approximation to
$\sigma_{\operatorname{ess}}(D_{\Gamma})$ for
$\Gamma\in\mathscr{D}_{\mathcal{A}}$ as in Figure 1(a). Also shown are the
boundary of $W_{100}$ (for $p=10$, $N=256$, $M=50$, and $t=\pi/13$) in black,
$R^{*}\mathbb{T}\subset W_{100}\subset W(D_{\Gamma})$ in blue (see Corollary
4.25), $\frac{1}{2}\mathbb{T}$ in red.

Figure 7: Numerical results for Example 4; $f_{+}=f$ given by (6.2),
$f_{-}=0$, $\alpha=\frac{2}{3}$

Acknowledgements. We are grateful to Marko Lindner (Hamburg University of
Technology), Marco Marletta (Cardiff), and Euan Spence (Bath) for discussions
regarding this project, and to the anonymous referee for their careful reading
of the manuscript and associated feedback. The 2nd and 4th authors were
supported by the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 844451 and by
Engineering and Physical Sciences Research Council (EPSRC) Grant EP/T008636/1.

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