On Malyshev’s method of automorphic functions

in diffraction by wedges

Dedicated to the memory of Vadim Malyshev

A.I. Komech

Faculty of Mathematics, Vienna University

<EMAIL_ADDRESS>

A.E. Merzon111The research supported by CONACYT-México and CIC-UMSNH, México.

Institute of Physics and Mathematics
University of Michoacan de San Nicolas de Hidalgo, Morelia, Mexico

<EMAIL_ADDRESS>
###### Abstract

We describe Malyshev’s method of automorphic functions in application to
boundary value problems in angles and to diffraction by wedges. We give a
consize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller,
G.E. Shilov and others.

MSC: 35J25; 30F10; 35J05; 35A30; 11F03; 35Q15; 78A45.

Keywords: elliptic equation; Helmholtz equation; boundary value problem; plane
angle; Fourier transform; analytic function; Riemann surface; characteristics;
covering map; automorphic function; Riemann–Hilbert problem; diffraction;
wedge; limiting absorption principle; limiting amplitude principle; limiting
amplitude: the Sommerfeld radiation condition.

###### Contents

  1. 1 Introduction
  2. 2 Diffraction by wedges and radar/sonar detection
  3. 3 Stationary diffraction and boundary value problems in angles
  4. 4 Malyshev’s method of automorphic functions
    1. 4.1 Reduction to undetermined algebraic equation on the Riemann surface
      1. 4.1.1 Fourier–Laplace transform
      2. 4.1.2 Undetermined algebraic equation on the Riemann surface
    2. 4.2 Method of automorphic functions
      1. 4.2.1 Covering maps
      2. 4.2.2  Shift equation
    3. 4.3 Reduction to the Riemann–Hilbert problem
  5. 5 Nonconvex angles of magnitude $\Phi>\pi$
  6. 6 Time-dependent diffraction by wedge
  7. 7 Limiting amplitude principle
  8. 8 The Sommerfeld diffraction theory and related results

##  1 Introduction

Vadim Malyshev was a very talented and versatile mathematician. He owns
significant results in the field of probability theory and Gibbs fields,
Markov processes and Euclidean quantum field theory. He also possessed
outstanding organizational skills, in particular, he founded the successful
and respected mathematical journal “Markov Processes and Related Fields”.

In 1970, V. Malyshev invented the method of automorphic functions [36], and
applied to random walks on the lattice in the quarter of plane. Later on, he
applied the method to queueing systems and analytic combinatorics [16]. In
1972–2022, the method was extended to boundary value problems for partial
differential equations in angles [22, 32] and to diffraction by wedges [30].
The main steps of Malyshev’s method are as follows:

I. Undetermined algebraic equation on the Riemann surface and analytic
continuation.

II. Elimination of one unknown function using covering automorphisms.

III. The reduction to the Riemann–Hilbert problem.

Malyshev’s method played the crucial role in the progress in the theory of
diffraction by wedges with general boundary conditions since 1972. The problem
was stated by M.I.Vishik in the Summer of 1967. In 1969–1971, one of the
authors (AK) tried to solve this problem while preparing his PhD Thesis. As
the result of these three- year efforts, the problem has been reduced to an
undetermined algebraic equation on the Riemann surface [23], though next steps
remained obscure. Fortunately, at the end of 1971, AK received the impetus
from his friend Alexander Shnirelman who noticed something similar in
Malyshev’s book [36], which he had recently reviewed by request of M.I.
Vishik. AK did not understand this book completely, but discovered two pages
which could have contained a creative idea. The book, opened on these pages,
lied on his desk for about two or three months, when AK pinned down two lines
with the key idea of automorphicity. The remaining work took about six months…

The extension of the research to diffraction problems was done in an intensive
collaboration of both authors, and took about 50 years. The main results of
the collaboration were the limiting absorption principle [44, 45], proof of
the completeness of Ursell’s trapping modes [31] , the extension to the
nonconvex angles [25, 30], and the Sommerfeld representation [24]. Moreover,
our general methods [30] allowed us to reproduce the formulas obtained by
Sommerfeld, Sobolev and Keller [28, 29, 46]. The identifications justify these
formulas as the limiting amplitudes in diffraction.

In the present, we give a consize survey of the development of Malyshev’s
method of automorphic functions since 1972 in the context of i) boundary value
problems in angles for elliplic partial differential equations, and ii) theory
of stationary and time-dependent diffraction by wedges. We focus on principal
ideas omitting nonessential technical details. All the details can be found in
[30].

##  2 Diffraction by wedges and radar/sonar detection

The radar or sonar emits the incident wave, which generates the reflected and
diffracted waves (the latter in green color) as shown in Fig. 1. Here $W$
denotes a conducting wedge (for example, the edge of an airplane wing), and
Q$=\mathbb{R}^{2}\setminus W$ is an angle of magnitude $\Phi$. The incident
wave reaches the wedge and generates the reflected and diffracted waves. The
diffracted wave is defined as the total wave minus the incident and reflected
waves.

The reflected wave is defined by geometric optics, and is absorbed by the
ground. On the other hand, the diffracted wave spreads in all directions, and
only this part of radiation returns to the radar which allows to detect the
airplane location.

Figure 1: Incident, reflected, and diffracted waves (the latter in green
color).

##  3 Stationary diffraction and boundary value problems in angles

The stationary diffraction by wedge is described by the boundary value problem
for the Helmholz equation in an angle $Q\subset\mathbb{R}^{2}$ of magnitude
$\Phi\in(0,2\pi]$:

$\left\\{\begin{array}[]{rcll}(\Delta+\omega^{2})u(x)&=&0,&x\in Q\\\ \\\
B_{l}u(x)&=&f_{l}(x),&x\in\Gamma_{l},\quad l=1,2\end{array}\right|,$ (3.1)

where $\Gamma_{1}$ and $\Gamma_{2}$ denote the sides of the angle, the
functions $f_{l}$ are defined by the incident wave, and $\omega\in\mathbb{R}$
is its frequency, see Fig. 1 and (7.60). The operators $B_{l}$ in the boundary
conditions correspond to the material properties of the wedge (conductor,
insulator, ferromagnetic, etc).

The relation of stationary problem (3.1) to time-dependent diffraction is
highly nontrivial. The key issue is that for $\omega\in\mathbb{R}$, the
problem admits an infinite number of linearly independent solutions. We
discuss this issue in detail in Section 7.

The stationary diffraction problem (3.1) with the Dirichlet and Neumann
boundary conditions ($B_{l}=1$ or $B_{l}=\frac{\partial}{\partial n}$) was
solved in 1896–1912 for $\Phi=2\pi$, by A. Sommerfeld [60]–[65] (the detailed
exposition and comments can be found in [48]). The extension to all
$\Phi\in(0,2\pi)$ was obtained in 1920 by H.S. Carslaw [8], in 1932–1937 by
V.I. Smirnov and S.L. Sobolev [55, 56, 57, 58, 59], and in 1951 by J.B. Keller
and A. Blank [21]. In 1958, G.D. Malujinetz solved the problem for all
$\Phi\in(0,2\pi)$ with the impedance (Leontovich) boundary condition
$\frac{\partial u(x)}{\partial n}+ib_{l}u(x)=f_{l}(x),\,\,x\in\Gamma_{l}$; see
[34, 35]. The detailed exposition of all these results can be found in [2] and
[30].

The  mixed boundary value problems of type

$\left\\{\begin{array}[]{rcll}Au(x)&=&0,&x\in Q\\\ \\\
B_{l}u(x)&=&f_{l}(x),&x\in\Gamma_{l},\quad l=1,2\end{array}\right|,\qquad
A=\sum_{|\alpha|\leq m}a_{\alpha}\partial^{\alpha},\quad
B_{l}=\sum_{|\alpha|\leq n_{l}}b_{l\alpha}\partial^{\alpha}$ (3.2)

were considered in 1958 by S.L. Sobolev [59] and in 1960–1961 by G.E. Shilov
[53, 54] in the quadrant $x_{1}>0$, $x_{2}>0$ for the case of hyperbolic
operator $A$ in the variable $x_{2}$ and with the Cauchy initial conditions at
$x_{2}=0$.

For strongly-elliptic second order operators $A$ and general differential
boundary operators $B_{l}$, the problem (3.2) was solved in 1972 in convex
angles $Q$ of magnitude $\Phi\in(0,\pi)$, see [22, 23]. Strong ellipticity
means that

$|\hat{A}(z)|\geq\varkappa(|z|^{2}+1),\qquad z\in\mathbb{R}^{2},$ (3.3)

where the symbol $\hat{A}(z):=\sum_{|\alpha|\leq 2}a_{\alpha}(-iz)^{\alpha}$
and $\varkappa>0$. In particular, the operator $A=-\Delta+1$ with the symbol
$\hat{A}(z)=z^{2}+1$ is strongly elliptic, and also the Helmholtz operator
$H=\Delta+\omega^{2}$ from (3.1) is strongly ellipltic for ${\rm Im{\hskip
1.42262pt}}\omega\neq 0$. The method [22, 23] relies on the Malyshev ideas of
automorphic functions [36] which is presented in the next section.

The extension of this result to nonconvex angles of magnitudes
$\Phi\in(\pi,2\pi)$, was done in 1992 by the authors [25].

Let us note that the Helmholtz operator $A=\Delta+\omega^{2}$ is not strongly
elliptic if $\omega\in\mathbb{R}$ since its symbol has the form
$\hat{A}(z)=-z^{2}+\omega^{2}$. Problem (3.2) for the Helmholtz operator in
convex angles was solved in 1972–1977 by A.E. Merzon [44, 45], who proved that
for real $\omega\in\mathbb{R}$, the problem admits only a finite number of
solutions satisfying the limiting absorption principle:

$u_{\omega}(x)=\lim_{\varepsilon\to 0+}u_{\omega+i\varepsilon}(x),\qquad x\in
Q,$ (3.4)

where $u_{\omega+i\varepsilon}$ denotes suitable solution to (3.1) with
$\omega+i\varepsilon$ instead of $\omega$.

The application of these results to time-dependent diffraction by wedges was
done in 2006–2019 by the authors [26, 27, 30], where, in particular, the
limiting amplitude principle (7.54) was established as well as (3.4).

Another approach to the construction of solutions to (3.2) has been suggested
by Maz’ya and Plamenevskii [37, 38]. However, this approach is applicable only
to equations with real coefficients that is not sufficient for application to
the diffraction problems.

Many works published since 1980’ concern a wide spectrum of properties of
solutions to the boundary problems of type (3.2) in different regions with
angles, see Grisvard [18], Costabel and Stephan [10], Dauge [12], Bernard [3,
4], Nazarov and Plamenevskii [49], Bonnet-Ben Dhia and Joly [6], Bonnet-Ben
Dhia, Dauge and Ramdani [5], Meister with collaborators [39]–[43], Penzel and
Teixeira [50], Castro and Kapanadze [9], and others. The detailed survey can
be found in [30].

Note that Malyshev’s method plays an important role in the theory of Queueing
Systems and Analytic Combinatorics [16]. Another important area of application
of Malyshev’s method is the linear theory of water waves. In particular, the
method was applied in 1996–2002 by the authors together with P.N. Zhevandrov
to trapped modes on a sloping beach. As the result, the long-standing problem
of the completeness of the Ursell’s modes has been solved [31, 32, 47]. This
progress is due to the fact that the method allows one to obtain all solutions
of the boundary value problems in angles.

We expect that the method can give a valuable progress in diffraction by
ferromagnetic wedges which is a challenging open problem of radar detection.
In this case, the operators $B_{l}$ in (3.2) are nonlocal pseudodifferential
operators.

##  4 Malyshev’s method of automorphic functions

In this section, we present basic steps of the method [22] which relies on
Malyshev’s ideas of automorphic functions [36].

Note that in the case of rational angles $\Phi=\pi/n$ and the Dirichlet and
Neumann boundary conditions, the boundary value problem (3.1) can be easily
solved by reflections in the sides of the angle. This method was well known at
least since the Gauss theory of electrostatics [17]. For $\Phi\neq\pi/n$ the
reflections do not give a solution, and for irrational $\Phi/\pi$, the method
suggested the reflections on a “Riemann surface” formed by the reflected
angles. This was the original step of the Sommerfeld approach which leaded him
to the famous “Sommerfeld integral representation” for solutions [60]. The
reflection on the Riemann surface and the theory of branching solutions to the
wave equation have been developed later by Sobolev [57] and [58, Chapter XII].

Very surprisingly, the method of automorphic functions [22, 36] also relies on
the reflections on a suitable Riemann surface $V$. However, in this approach,
$V$ is the surface in the Fourier space, contrary to the original ideas of
Sommerfeld. Namely, $V$ is the Riemann surface of complex characteristics of
the elliptic operator $A$:

$V=\\{z\in\mathbb{C}^{2}:\hat{A}(z)=0\\}.$ (4.5)

######  Remark 4.1.

The main idea of the Malyshev approach is the invariance of the Cauchy data of
solutions under covering maps of the Riemann surface $V$, see Remark 4.7.

In [22], the problem (3.2) with strongly–elliptic operators $A$ in convex
angles $Q$ is solved in the following steps:
1\. Reduction to an undetermined algebraic equation with two unknown functions
on the Riemann surface $V$. 2\. Elimination of one unknown function using its
invariance with respect to the covering map of the Riemann surface.
3\. Reduction of the obtained equation with one unknown function to the
Riemann–Hilbert problem on $V$.

Below in this section, we describe some details.

###  4.1 Reduction to undetermined algebraic equation on the Riemann surface

As an example, we consider the Dirichlet boundary value problem in the
quadrant $Q=\mathbb{R}^{+}\times\mathbb{R}^{+}$ :

$\left\\{\begin{array}[]{rcl}Au(x_{1},x_{2})&=&0\\\ \\\
u(x_{1},0)&=&f_{1}(x_{1}),\,\,u(0,x_{2})=f_{2}(x_{2})\end{array}\right|,\,\,\quad
x_{1}>0,\,\,\,x_{2}>0.$ (4.6)

####  4.1.1 Fourier–Laplace transform

We assume that the solution $u(x)\in C^{2}(\overline{Q})$ and is bounded by a
polynomial:

$|u(x)|+|\nabla u(x)|\leq C(1+|x|)^{p},\qquad x\in\mathbb{R}^{2}.$ (4.7)

Denote $\mathbb{C}^{+}=\\{\zeta\in\mathbb{C}:{\rm Im{\hskip
1.42262pt}}\zeta>0\\}$ and $Z^{+}=\mathbb{C}^{+}\times\mathbb{C}^{+}$, and
consider the complex Fourier–Laplace transform of solution

$\hat{u}(z)=\int_{0}^{\infty}\int_{0}^{\infty}e^{izx}u(x)dx_{1}dx_{2},\qquad
z=(z_{1},z_{2})\in Z^{+}.$ (4.8)

By (4.7), this integral is absolutely convergent and hence it is an analytic
function of two complex variables (this is a particular case of the
Paley–Wiener Theorem). Let us denote the Neumannn data of the solution as

$\varphi_{1}(x_{1})=\partial_{2}u(x_{1},0),\,\,\,x_{1}\geq
0;\qquad\varphi_{2}(x_{2})=\partial_{1}u(0,x_{2}),\,\,\,x_{2}\geq 0.$ (4.9)

It is well known that the solution $u(x)$ can be expressed via the Dirichlet
and Neumannn data $f_{1},f_{2},\varphi_{1},\varphi_{2}$ by the Green integral
formula [11]. In our case, it is useful to obtain this formula in the Fourier
transform. For this purpose, multiply the first equation in (3.2) by $e^{izx}$
and integrate over $Q$. Integrating by parts, we immediately obtain

$0=\int_{0}^{\infty}\int_{0}^{\infty}e^{izx}Au(x)dx_{1}dx_{2}=\hat{A}(z)\hat{u}(z)+F(z),\qquad
z\in Z^{+},$ (4.10)

where

$F(z)=P_{1}(z)\hat{f}_{1}(z_{1})+P_{2}(z)\hat{f}_{2}(z_{2})+S_{1}(z)\hat{\varphi}_{1}(z_{1})+S_{2}(z)\hat{\varphi}_{2}(z_{2}),\qquad
z\in Z^{+},$ (4.11)

and the functions $P_{l}$ and $S_{l}$ are polynomials.

####  4.1.2 Undetermined algebraic equation on the Riemann surface

Rewrite (4.10) as

$\hat{A}(z)\hat{u}(z)=-F(z),\qquad z\in Z^{+}.$ (4.12)

Now (4.5) implies the identity

$F(z)=0,\qquad z\in V^{+}:=V\cap Z^{+}$ (4.13)

since all the functions $\hat{A}(z),\hat{u}(z),F(z)$ are analytic in the
domain $Z^{+}$!

######  Remark 4.2.

Note that the set of complex characteristics $V$ is nonempty even for strongly
elliptic operators (3.3), though its intersection with the real plane
$\mathbb{R}^{2}$ is empty; see Example 4.5 below.

The identity (4.13) can be rewritten as undetermined linear algebraic equation

$S_{1}(z)\varphi_{1}(z_{1})+S_{2}(z)\varphi_{2}(z_{2})=G(z),\qquad z\in V^{+}$
(4.14)

with two unknown functions $\varphi_{1}(z_{1})$, $\varphi_{2}(z_{2})$, and
with known right-hand side:

$G(z):=-P_{1}(z)\hat{f}_{1}(z_{1})-P_{2}(z)\hat{f}_{2}(z_{2}),\qquad z\in
V^{+}.$ (4.15)

######  Remark 4.3.

The identity (4.12) implies the formula for the solution

$u(x)=-\Big{[}{\cal F}^{-1}\frac{F(z)}{\hat{A}(z)}\Big{]}(x),\qquad x\in Q,$
(4.16)

where ${\cal F}^{-1}$ denotes the inverse to the Fourier–Laplace transform
(4.8), and the right hand side is well defined due to (3.3). The formula
(4.16) can be transformed into the well known Green formula which expresses
the solution $u(x)$ via its Cauchy data.

###  4.2 Method of automorphic functions

####  4.2.1 Covering maps

Denote

$\psi_{1}(z)=\varphi_{1}(z_{1}),\quad\psi_{2}(z)=\varphi_{2}(z_{2}),\qquad\hat{g}_{1}(z)=\hat{f}_{1}(z_{1}),\quad\hat{g}_{2}(z)=\hat{f}_{2}(z_{2}).$
(4.17)

Now (4.14) becomes

$S_{1}(z)\psi_{1}(z)+S_{2}(z)\psi_{2}(z)=G(z),\qquad z\in V^{+},$ (4.18)

where

$G(z):=-P_{1}(z)\hat{g}_{1}(z)-P_{2}(z)\hat{g}_{2}(z).$ (4.19)

Of course, this equation is not equivalent to (4.14). To keep the equivalence,
we need an additional characterisation of the functions $\psi_{l}(z)$. This is
the key observation of Malyshev that the functions are automorphic with
respect to an appropriate groups of transformation of the Riemann surface $V$.

First, consider the coordinate projections $p_{l}:V\to\mathbb{C}$ defined by

$p_{1}(z_{1},z_{2})=z_{1},\qquad p_{2}(z_{1},z_{2})=z_{2}.$ (4.20)

These projections are two-sheeted since, for example,
$p_{1}(z_{1},z_{2})=z_{1}$ means that $z_{2}$ is the root of the quadratic
equation $\hat{A}(z_{1},z_{2})=0$. Accordingly, the inverse maps
$p_{l}^{-1}:\mathbb{C}\to V$ are double-valued: for
$z_{1},z_{2}\in\mathbb{C}$,

$p_{1}^{-1}(z_{1})=\\{\zeta_{1}^{-},\zeta_{1}^{+}\\},\qquad
p_{2}^{-1}(z_{2})=\\{\zeta_{2}^{-},\zeta_{2}^{+}\\},$ (4.21)

and at the branching points of $p_{l}^{-1}$, the two points
$\zeta_{l}^{\pm}\in V$ coincide.

######  Definition 4.4.

Covering maps $h_{1},h_{2}:V\to V$ are defined as follows: for any
$z_{1},z_{2}\in\mathbb{C}$,

$h_{1}\zeta_{1}^{\pm}=\zeta_{1}^{\mp},\qquad\qquad
h_{2}\zeta_{2}^{\pm}=\zeta_{2}^{\mp}.$ (4.22)

######  Example 4.5.

For the strongly-elliptic operator $A=-\Delta+1$, the corresponding Riemann
surface $V\\!\\!:z_{1}^{2}+z_{2}^{2}+1=0$ is shown in Fig. 2 in projection
onto the plane ${\rm Im{\hskip 1.42262pt}}z_{1},{\rm Im{\hskip
1.42262pt}}z_{2}$. It is easy to see that this projection does not cover the
circle $|{\rm Im{\hskip 1.42262pt}}z_{1}|^{2}+|{\rm Im{\hskip
1.42262pt}}z_{2}|^{2}<1$, and it covers twice each point with $|{\rm Im{\hskip
1.42262pt}}z_{1}|^{2}+|{\rm Im{\hskip 1.42262pt}}z_{2}|^{2}>1$. The surface
consists of two sheets shown in Fig. 2, and glued along the cuts.

Thus, $h_{1}$ permutes the points $\zeta_{1}^{\pm}\in V$ with the identical
projections $z_{1}=p_{1}\zeta_{1}^{\pm}$, and similarly, $h_{2}$ permutes the
points $\zeta_{2}^{\pm}\in V$ with the identical projections
$z_{2}=p_{2}\zeta_{2}^{\pm}$ (see Fig. 2):

$p_{1}h_{1}\zeta_{1}^{\pm}=p_{1}\zeta_{1}^{\mp}=z_{1},\qquad
p_{2}h_{2}\zeta_{2}^{\pm}=p_{2}\zeta_{2}^{\mp}=z_{2}.$ (4.23)

The maps $h_{l}:V\to V$ with $l=1,2$ define the corresponding automorphisms of
the ring of (meromorphic) functions $\psi(z)$ on the Riemann surface $V$:

$\psi^{h_{l}}(z):=\psi(h_{l}z),\qquad z\in V.$ (4.24)

Figure 2: Riemann surface $V\\!\\!:z_{1}^{2}+z_{2}^{2}+1=0$ in projection onto
the plane ${\rm Im{\hskip 1.42262pt}}z_{1},{\rm Im{\hskip 1.42262pt}}z_{2}$.

Figure 2 shows that

$\begin{array}[]{ll}p_{1}\zeta_{1}^{+}=z_{1}=p_{1}\zeta_{1}^{-},&{\rm
so}\,\,\,\psi_{1}(\zeta_{1}^{+})=\varphi_{1}(z_{1})=\psi_{1}(\zeta_{1}^{-}),\\\
\\\ p_{2}\zeta_{2}^{+}=z_{2}=p_{2}\zeta_{2}^{-},&{\rm
so}\,\,\,\psi_{2}(\zeta_{2}^{+})=\varphi_{2}(z_{2})=\psi_{2}(\zeta_{2}^{-}).\end{array}$

Now it is clear that the functions $\psi_{l}(z):=\varphi_{l}(z_{l})$ with
$l=1,2$ are invariant with respect to the automorphisms $h_{l}$:

$\,\,\,\,\,\psi_{l}^{h_{l}}(z)=\psi_{l}(z),\qquad\,\,\,\,\,\,\,z\in V^{+}.$
(4.25)

In other words, the functions $\psi_{l}$ are automorphic, and the
automorphisms defined by $h_{l}$ belong to the corresponding Galois groups of
extensions of the ring of functions of $z_{l}$.

####  4.2.2  Shift equation

Applying formally $h_{1}$ to (4.18), and using (4.25) with $l=1$, we get  a
new equation for the same unknown functions:

$S_{1}^{h_{1}}(z)\psi_{1}(z)+S_{2}^{h_{1}}(z)\psi_{2}^{h_{1}}(z)=G^{h_{1}}(z).$
(4.26)

The problem is that $\psi_{2}^{h_{1}}(z)=\psi_{2}(h_{1}z)$ and
$G^{h_{1}}(z)=G(h_{1}z)$ are not defined generally for $z\in V^{+}$ since
$V^{+}$ is not invariant with respect to the covering map $h_{1}$. In
particular, we have by (4.19),

$G^{h_{1}}(z):=-P_{1}^{h_{1}}(z)\hat{g}_{1}(z)-P_{2}^{h_{1}}(z)\hat{g}_{2}^{h_{1}}(z),$
(4.27)

where $\hat{g}_{2}^{h_{1}}(z)=\hat{g}_{2}(h_{1}z)$ is not defined generally
for $z\in V^{+}$. To save the situation, consider the case $f_{2}=0$. Then
$\hat{g}_{2}=0$, and now the right hand side of equation (4.26) is well
defined for $z\in V^{+}$. It is important that in this case
$\psi_{2}^{h_{1}}(z)$ is also well defined [30, Ch. 14]. The case $f_{1}=0$
can be considered similarly.

######  Remark 4.6.

The function $\psi_{2}(z)$ admits an analytic continuation outside the region
$V^{+}_{l}:=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{2}>0\\}$ on the Riemann
surface $V$, see [30, Ch. 14]. Let us stress that this is analytic
continuation along the surface $V$.

Now we can eliminate the function $\psi_{1}$ from (4.18) and (4.26). As a
result, we obtain an algebraic equation with a shift for one unknown function

$R_{1}(z)\psi_{2}^{h_{1}}(z)-R_{2}(z)\psi_{2}(z)=H(z),\quad z\in V^{+}.$
(4.28)

Finally, using (4.25) with l=2, we get

$R_{1}(z)\psi_{2}^{h}(z)-R_{2}(z)\psi_{2}(z)=H(z),\quad z\in V^{+};\qquad
h=h_{2}h_{1}.$ (4.29)

######  Remark 4.7.

The elimination of unknown functions using their invariance with respect to
suitable “reflections” is the main idea of Malyshev’s method.

###  4.3 Reduction to the Riemann–Hilbert problem

Let us illustrate the reduction of equation (4.28) to the Riemann–Hilbert
problem for a particular case of strongly-ellipltic operator $A=-\Delta+1$.
Its symbol is $\hat{A}(z)=z^{2}+1$, so
$V=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}^{2}+z_{2}^{2}=-1\\}$ and the
covering maps are

$h_{1}(z_{1},z_{2})=(z_{1},-z_{2}),\qquad h_{2}(z_{1},z_{2})=(-z_{1},z_{2}).$
(4.30)

Introduce the coordinate $w$ on the universal covering $\hat{V}=\mathbb{C}$ of
the surface $V$ by

$z_{1}=i\cos w,\qquad z_{2}=i\sin w.$ (4.31)

The maps (4.30) can be lifted to $\hat{V}$ as

$\hat{h}_{1}w=-w,\qquad\hat{h}_{2}w=-w+\pi.$ (4.32)

Now $h=w+\pi$, so (4.29) becomes

$\tilde{R}_{1}(w)\tilde{\psi}_{2}(w+\pi)-\tilde{R}_{2}(w)\tilde{\psi}_{2}(w)=\tilde{H}(w).$
(4.33)

where $\tilde{R}_{1}$, etc, denote the liftings of the corresponding functions
to the universal covering. The equation (4.33) holds for an appropriate region
of $w\in\mathbb{C}$. Restricted to the strip ${\rm Re{\hskip
1.42262pt}}w\in[0,\pi]$, this equation is the Riemann–Hilbert problem which
can be solved in quadratures [30, Chs 17 and 18]. Let us recall some details.

The function $z=e^{2iw}$ analytically transforms the strip to the plane with
the cut $[0,\infty)$. Denote the function
$\check{\psi}_{2}(t)=\tilde{\psi}_{2}(w)$, $\check{H}(t)=\tilde{H}(w)$ and
$\check{R}_{k}(t)=\tilde{R}_{k}(w)$ for $k=1,2$. Then relation (4.33) becomes

$\check{R}_{1}(t)\check{\psi}_{2}(t-i0)-\check{R}_{2}(t)\check{\psi}_{2}(t+i0)=\check{H}(t),\qquad
t>0.$ (4.34)

As the first step of the Riemann–Hilbert method, one must solve the
corresponding homogeneous problem:

$\check{R}_{1}(t)T(t-i0)-\check{R}_{2}(t)\check{T}(t+i0)=0,\qquad t>0.$ (4.35)

Equivalently,

$\frac{T(t+i0)}{T(t-i0)}=q(t):=\frac{\check{R}_{1}(t)}{\check{R}_{2}(t)},\qquad
t>0.$ (4.36)

The solution to this equation depends on zeros of the functions
$\check{R}_{1}(t)$ and $\check{R}_{2}(t)$ for $t>0$. Let us consider the
simplest case when such zeros do not exist, and moreover,

$q(0)=q(\infty)=1.$ (4.37)

Then the equation is equivalent to

$\log T(t+i0)-\log T(t-i0)=\log q(t),\qquad t>0.$ (4.38)

The solution is given by the Cauchy-type integral

$\log T(t)=\frac{1}{2\pi i}\int_{0}^{\infty}\frac{\log q(s)}{t-s}ds,\qquad
t\in\mathbb{C}\setminus[0,\infty).$ (4.39)

It is important that $T(t)$ is analytic and nonvanishing in the region
$\mathbb{C}\setminus[0,\infty)$. Now the nonhomogeneous problem (4.34) can be
solved as follows. First, (4.34) and (4.35) imply

$\frac{\check{\psi}_{2}(t-i0)}{T(t-i0)}-\frac{\check{\psi}_{2}(t+i0)}{T(t+i0)}=\frac{\check{H}(t)}{\check{R}_{1}(t)T(t-i0)},\qquad
t>0.$ (4.40)

Therefore, similarly to (4.38),

$\frac{\check{\psi}_{2}(t)}{T(t)}=-\frac{1}{2\pi
i}\int_{0}^{\infty}\frac{\check{H}(s)}{\check{R}_{1}(s)T(s-i0)(t-s)}ds,\qquad
t\in\mathbb{C}\setminus[0,\infty)$ (4.41)

since the function $\frac{\check{\psi}_{2}(t)}{T(t)}$ is analytic in
$\mathbb{C}\setminus[0,\infty)$.

Thus, we have calculated the function $\check{\psi}_{2}(t)$. Now $\psi_{2}(z)$
can be obtained from the relation (4.26). Hence, the functions
$\hat{\varphi}_{1}(z_{1})$ and $\hat{\varphi}_{2}(z_{2})$ are known. It
remains to substitute the obtained functions into the formula (4.11) for the
function $F$. Then the solution to (3.2) is expressed by (4.16), which can be
reduced to the integral of Sommerfeld type [24].

######  Remark 4.8.

Equation (4.28) is obtained using the invariance (4.25) with $l=1$, while
(4.29) uses also $l=2$. Note that the equation (4.28) reads now

$\tilde{T}_{1}(w)\tilde{\psi}_{2}(-w)-\tilde{T}_{2}(w)\tilde{\psi}_{2}(w)=\tilde{H}(w),$
(4.42)

which provisionally cannot be reduced to a nonsingular Riemann–Hilbert
problem, see [33]. Thus, both invariance conditions (4.25) are necessary for
the reduction.

######  Remark 4.9.

For the random walks studied in [36, 16], the corresponding Riemann surface
and the covering maps $h_{l}$ can be more complicated than for 2-nd order
elliptic operators which requires more sophisticated methods of the Galois
theory.

##  5 Nonconvex angles of magnitude $\Phi>\pi$

The extension of the theory outlined above to the case of nonconvex angle $Q$
differs drastically from the convex one. As an example, consider the Dirichlet
boundary value problem in the angle
$Q=\mathbb{R}^{2}\setminus\mathbb{R}^{+}\times\mathbb{R}^{+}$ :

$\left\\{\begin{array}[]{rcll}Au(x)&=&0,\qquad x\in Q&\\\ \\\
u(x_{1},0)&=&f_{1}(x_{1}),\,\,x_{1}>0;&u(0,x_{2})=f_{2}(x_{2}),\,\,x_{2}>0.\end{array}\right|.$
(5.43)

Note that the relations (4.10) and (4.12), (4.16) remain true in this case,
but now the function (4.11) is changed to its negative:

$F(z)=-P_{1}(z)\hat{f}_{1}(z_{1})-P_{2}(z)\hat{f}_{2}(z_{2})-S_{1}(z)\hat{\varphi}_{1}(z_{1})-S_{2}(z)\hat{\varphi}_{2}(z_{2}),\qquad
z\in\mathbb{R}^{2}.$ (5.44)

On the other hand, the key relation (4.13) is not well defined in contrast to
the case when the support of $u$ belongs to a convex angle. This is due to the
fact that the Fourier–Laplace transform (4.8) of the function $u$ with the
support in a nonconvex angle generally does not admit an analytical
continuation to a region of $\mathbb{C}^{2}$. Nevertheless, the function
(5.44) in this case is analytic in the same region
$Z^{+}=\mathbb{C}^{+}\times\mathbb{C}^{+}$ as the function (4.11).

The answer to this riddle was found in [25]. First, the function (5.44) admits
the splitting

$F(z)=\gamma_{1}(z)+\gamma_{2}(z),\quad\gamma_{1}(z)=-P_{1}(z)\hat{f}_{1}(z_{1})-S_{1}(z)\hat{\varphi}_{1}(z_{1}),\quad\gamma_{2}(z)=-P_{2}(z)\hat{f}_{2}(z_{2})-S_{2}(z)\hat{\varphi}_{2}(z_{2}),$
(5.45)

where the functions $\gamma_{l}(z)$ are analytic in the regions

$V_{l}^{+}=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{l}>0\\},\qquad l=1,2.$
(5.46)

Second, as shown in [25] (see also [30, Theorem 20.1]), each function
$\gamma_{l}$ admits an analytic continuation from $V_{l}^{+}$ to the region
$V^{-}:=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{1}<0,\,{\rm Im{\hskip
1.42262pt}}z_{2}<0\\}$, and the following identity holds :

$\gamma_{1}(z)+\gamma_{2}(z)=0,\qquad z\in V^{-}.$ (5.47)

This identity formally coincides with the undetermined equation (4.14), and it
allows us to calculate both unknown functions $\hat{\varphi}_{l}$ by methods
of Sections 4.2 and 4.3.

##  6 Time-dependent diffraction by wedge

The time-dependent diffraction by a wedge $W$ is described by the solution of
the wave equation in the plane angle $Q=\mathbb{R}^{2}\setminus W$ with
appropriate boundary conditions. For example, consider the Dirichlet boundary
conditions

$\left\\{\begin{array}[]{rcl}\ddot{u}(x,t)&=&\Delta u(x,t),\quad x\in Q\\\ \\\
u(x,t)&=&0,\quad x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\quad
t\in\mathbb{R}.$ (6.48)

The incident wave is defined by the initial condition

$u(x,t)=u^{in}(x,t),\qquad
u^{in}(x,t):=f(kx-\omega_{0}t)e^{i(kx-\omega_{0}t)},\qquad t<0,$ (6.49)

where the frequency $\omega_{0}\in\mathbb{R}$ and $k\in\mathbb{R}^{2}$ is the
wave vector. The incident wave $u^{in}(x,t)$ must be a solution to (6.48) for
$t<0$:

$\left\\{\begin{array}[]{rcl}\ddot{u}^{in}(x,t)&=&\Delta u^{in}(x,t),\quad
x\in Q\\\ \\\ u^{in}(x,t)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\quad t<0.$ (6.50)

The wave equation in (6.50) holds for any function $f(s)$ for all
$t\in\mathbb{R}$ if $|k|=|\omega_{0}|$. The boundary condition in (6.50) can
be satisfied only in the case of nonconvex angle $Q$ of magnitude $\Phi>\pi$
and the wave vector $k$ satisfying the inequalities $k\cdot x\geq 0$ for $x\in
W=\mathbb{R}^{2}\setminus Q$. Then for $\omega_{0}>0$ the boundary condition
holds if

$f(s)=0,\qquad s>0.$ (6.51)

##  7 Limiting amplitude principle

Let us assume that there exists the limit

$f(-\infty):=\lim_{s\to-\infty}f(s),$ (7.52)

and the convergence is sufficiently fast, for example, $f(s)=\theta(-s)$. Then
the incident wave (6.49) admits the long-time asymptotics

$u^{in}(x,t)\sim f(-\infty)e^{ikx}e^{-i\omega_{0}t},\qquad t\to\infty$ (7.53)

which suggests similar asymptotics of solution

$u(x,t)\sim a_{\omega_{0}}(x)e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.54)

Such asymptotics are called as limiting amplitude principle.

Determination of the limiting amplitudes $a_{\omega_{0}}(x)$ for different
diffraction processes is the main goal of the theory of diffraction [7, 65]
(see also [30]). The proof of the asymptotics is the main goal of the
mathematical theory of diffraction. For diffraction by wedges, this
asymptotics has been established for the first time in [26]. Formal
substitution of the asymptotics (7.54) into (6.48) gives a problem of type
(3.1):

$\left\\{\begin{array}[]{rcl}-\omega_{0}^{2}a_{\omega_{0}}(x)&=&\Delta
a_{\omega_{0}}(x),\quad x\in Q\\\ \\\ a_{\omega_{0}}(x)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|.$ (7.55)

However, this boundary problem is  ill-posed since it admits an infinite
number of linearly independent solutions for real $\omega_{0}\in\mathbb{R}$.
Thus, this problem does not allow us to find the limiting amplitude. This fact
is the main peculiarity of the diffraction theory. This can be easily checked
in the case $\Phi=\pi$ when the angle $Q$ is the half-plane, so all solutions
can be calculated by the Fourier transform along the boundary
$\partial\Omega$. For the problems of type (7.55) in convex angles of
magnitude $\Phi<\pi$, this nonuniqueness was discovered in 1973 by one of the
authors [45].

Let us recall how to prove the asymptotics (7.54) and how to calculate the
limiting amplitudes $a_{\omega_{0}}(x)$. First, note that for the incident
wave $u^{in}(x,t)$ the asymptotics of type (7.54) holds by (7.52):

$u^{in}(x,t)\sim f(-\infty)e^{ikx}e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.56)

The reflected wave is defined by geometric optics, and its main properties are
as follows:

$u^{r}(x,t)=-u^{in}(x,t),\quad x\in\partial Q;\qquad u^{r}(x,t)\sim
a^{r}(x)e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.57)

The diffracted wave $u^{d}(x,t)$ is defined by the splitting the total
solution as

$u(x,t)=u^{in}(x,t)+u^{r}(x,t)+u^{d}(x,t).$ (7.58)

Hence, it remains to calculate the corresponding asymptotics for the
diffracted wave

$u^{d}(x,t)\sim a^{d}_{\omega_{0}}(x)e^{-i\omega_{0}t},\qquad t\to\infty,$
(7.59)

Substituting (7.58) into (6.48), using (7.57) and the fact that the wave
equation in (6.50) holds for all $t\in\mathbb{R}$, we get the boundary problem
for the diffracted wave

$\left\\{\begin{array}[]{rcl}\ddot{u}^{d}(x,t)&=&\Delta
u^{d}(x,t)+F(x,t),\quad x\in Q\\\ \\\ u^{d}(x,t)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\qquad
F(x,t):=(\partial_{t}^{2}-\Delta)u^{r}(x,t)\sim
b(x)e^{-i\omega_{0}t},\,\,\,t\to\infty.$ (7.60)

Formal substitution of the asymptotics (7.59) into (7.60), gives the boundary
problem

$\left\\{\begin{array}[]{rcl}-\omega_{0}^{2}a^{d}_{\omega_{0}}(x,t)&=&\Delta
a^{d}_{\omega_{0}}(x)+b(x),\quad x\in Q\\\ \\\ a^{d}(x)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|.$ (7.61)

For $\omega_{0}\in\mathbb{R}$, this system also admits an infinite number of
linearly independent solutions, as well as (7.55). Similar problem of
nonuniqueness arises in every diffraction problem in unbounded regions. The
problem of nonuniqeness was resolved by the discovery of additional features
of the limiting amplitude $a^{d}_{\omega_{0}}(x)$.

The key discovery was the limiting absorption principle (3.4) for the limiting
amplitude of the diffracted wave. In application to problem (7.61), we have

$a^{d}_{\omega_{0}}(x)=\lim_{\varepsilon\to
0+}a^{d}_{\omega_{0}+i\varepsilon}(x),\qquad x\in Q,$ (7.62)

where $a^{d}_{\omega_{0}+i\varepsilon}$ denotes a solution to (7.61) with
$\omega_{0}+i\varepsilon$ instead of $\omega_{0}$.

######  Remark 7.1.

The convergence (7.62) holds for the limiting amplitude
$a^{d}_{\omega_{0}}(x)$ of the diffracted wave $u^{d}_{\omega_{0}}(x,t)$
(formal proof can be found in [30, Section 4.1]). However, it does not hold
for the limiting amplitude $a(x)$ of the total solution $u(x,t)$ although
these amplitudes satisfy quite similar equations (7.61) and (7.55). The
difference is that the initial state of the diffracted wave
$u^{d}_{\omega_{0}}(x,0)$ is of finite energy (in our case zero), while for
the total solution the initial state $(u(x,0),\dot{u}(x,0))$ is the plane wave
(6.49) and its derivative in time at $t=0$.

The limiting absorption principle has been introduced for the first time in
1905 by W. Ignatovsky [19]. Rigorous proofs of this principle for limiting
amplitudes of solutions with finite energy initial states were achieved much
later. The results for the wave and Schrödinger equations in the entire space
and for diffraction problems with smooth boundaries were obtained by Agmon
[1], Eidus [13, 14, 15], Jensen and Kato [20], A.Ya. Povzner [51], B.R.
Vainberg [66] and others.

The convergence (7.62) for stationary diffraction problems has been
established for the first time in 1977 by one of the authors [45]: it was
proven that stationary problem (7.61) and problems (3.2) with
$A=\Delta+\omega^{2}$ and general boundary conditions in convex angles of
magnitude $\Phi<\pi$,
i) for complex $\omega\not\in\mathbb{R}$ admit only a finite number of
linearly independent solutions in appropriate class of functions;
ii) for real $\omega\in\mathbb{R}$ admit an infinite number of linearly
independent solutions,
iii) for real $\omega\in\mathbb{R}$ admit only a finite number of linearly
independent solutions satisfying (7.62).

For the time-dependent diffraction problem (6.48), (6.49), the limiting
absorption principle (7.62) and the limiting amplitude principle (7.54) were
justified in 2006 by the authors [26]. The proofs rely on the analysis of the
Fourier-Laplace transform in time:

$\tilde{u}(x,\omega)=\int_{0}^{\infty}e^{i\omega
t}u(x,t)dt,\qquad\omega\in\mathbb{C}^{+}.$ (7.63)

The function $\tilde{u}(x,\omega)$ satisfies a boundary value problem of type
(7.61) with complex $\omega\not\in\mathbb{R}$. In this case the Helmholtz
operator $A=\Delta+\omega^{2}$ is strongly elliptic. Hence,
$\tilde{u}(x,\omega)$ can be calculated and analysed by the methods described
in previous sections. The limiting amplitude is calculated in [26] using the
limit (7.62).

######  Remark 7.2.

In 1912, A. Sommerfeld discovered the Sommerfeld radiation condition [62] (see
also [52]), which provides the uniqueness of solution to the boundary problem
of type (7.61) in the case when $Q$ is the exterior of a bounded region in
$\mathbb{R}^{3}$. This condition is more practical for numerical calculation
of the limiting amplitudes than (7.62).

##  8 The Sommerfeld diffraction theory and related results

For the angle $\Phi=2\pi$, A. Sommerfeld constructed in 1896 a solution $a(x)$
to stationary diffraction problem of type (7.55) with Dirichlet and Neumannn
boundary conditions. In this case the wedge is the half-plane, which is
represented by the semi-axis $[0,\infty)$ in the corresponding 2D problem. The
main ideas were i) to treat the semi-axis as the cut on an appropriate Riemann
surface, and ii) to extend the known method of reflections to Riemann
surfaces. As a result, A. Sommerfeld constructed a universal integral
representation of a class of branching solutions of the Helmholtz equation on
the Riemann surface in the form of the Sommerfeld integral with a fixed
integral kernel and a with a suitable density function. Further, A. Sommerfeld
chose an appropriate densities to satisfy the boundary conditions.

Sommerfeld’s strategy of costructing the solution remains a mysterious riddle
to this day. This approach is reproduced with some comments in [30, Ch. 5],
see also [48]. However, the Sommerfeld integral representation turned out to
be extremely fruitful, and in particular, was used by G.D. Malujinetz to solve
the problem with the Leontovich boundary condition [34, 35], see also [2].

For any angles $\Phi\in(0,2\pi)$ the stationary diffraction problem (7.55) for
the Dirichlet and Neumannn boundary conditions in the angles of this magnitude
was solved by other methods in 1920 by H.S. Carslaw [8], in 1932–1937 by V.I.
Smirnov and S.L. Sobolev [55, 56, 57, 58, 59], and in 1951 by J.B. Keller and
A. Blank [21].

######  Remark 8.1.

i) In all the works, cited above, the limiting amplitude principle was not
established, and the choice of suitable solution of the  ill-posed problem
(7.55) was not rigorously clarified. Nevertheless, as shown in [28, 29, 46],
all the obtained solutions coincide with the limiting amplitudes calculated in
[26] and admit the Sommerfeld representation.
ii) S.L. Sobolev mentions, in the articles cited above, that the functions of
type (6.49) must be solutions to the wave equation even if the amplitude
$a(s)$ is a discontinuouos function. These remarks later inspired the theory
of weak derivatives of S.L. Sobolev and the theory of distributions of L.
Schwartz.

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As in the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf}, without loss of generality, assume that $\cF_{n,\eps_n}((\nu_n,v_n)) \leq C$ for some $C > 0$. In particular, this implies that $\nu_n = \mu_n$, and by Proposition \ref{prop:Back:TLp}, $\mu_n$ converges weakly to $\nu$. By the uniqueness of weak limits, we must have that $\nu = \mu$.

We have \[
C\geq  \liminf_{n \to \infty} \mathcal{F}_{n,\eps_n}( (v_n,\nu_n) ) \geq \liminf_{n \to \infty} \EnergySnepsn(v_n) \geq \EnergyS(v) = \mathcal{G}((\nu,v))
\]
where the last inequality follows from Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}.
\end{proof}

The $\limsup$ inequality requires two computational Lemmas.

\begin{lemma} \label{lem:Proofs:GConvergence:IllPosed:diracEnergies}
Energy estimates of dirac deltas.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold, $\eps_n$ satisfies~\eqref{eq:Main:Ass:epsLBIllPosed}. Then, there exists $C > 0$ such that, $\bbP$-a.s., for $n$ large enough we have
\begin{equation} \label{eq:Proofs:GConvergence:IllPosed:diracEnergies}
\EnergySnepsn(\delta_{x_i}) \leq \frac{C}{n \eps_n^{2s}}
\end{equation}
for all $x_i \in \Omega_n$.
\end{lemma}

\begin{proof}
With probability one, we can assume that the conclusion of Theorem \ref{thm:Back:TLp:LinftyMapsRate} holds.

By Assumption \ref{eq:Main:Ass:epsLBIllPosed} and Theorem \ref{thm:Back:TLp:LinftyMapsRate}, we can apply \cite[Lemma 22]{Stuart} for $n$ large enough. We recall the variational definition of eigenvalues:
\[
\lambda_{n,n}^s = \sup_{\Vert u \Vert_{\Lp{2}} = 1,\: u \in \mathbb{R}^n} \langle u, \Delta_{n,\eps_n}^s u \rangle_{\Lp{2}(\mu_n)}.
\]
Furthermore, for $x_i \in \Omega_n$,
\[
\Vert \sqrt{n}\delta_{x_i} \Vert_{\Lp{2}} = \sqrt{\frac{1}{n} \sum_{j=1}^n n \delta_{x_i}(x_j)^2} = 1
\]
so that we can estimate:
\begin{align}
\EnergySnepsn(\delta_{x_i}) &= \langle \delta_{x_i}, \Delta_{n,\eps_n}^s \delta_{x_i} \rangle_{\Lp{2}(\mu_n)} \notag\\
&= \frac{1}{n} \langle \sqrt{n} \delta_{x_i} , \Delta_{n,\eps_n}^s \sqrt{n}\delta_{x_i} \rangle_{\Lp{2}(\mu_n)} \notag\\
&\leq \frac{1}{n}\sup_{\Vert u \Vert_{\Lp{2}} = 1,\: u \in \mathbb{R}^n} \langle u, \Delta_{n,\eps_n}^s u \rangle_{\Lp{2}(\mu_n)} \notag\\
&= \frac{\lambda_{n,n}^s}{n} \notag\\
&\leq \frac{C}{n\eps_n^{2s}}\label{eq:Proofs:GConvergence:IllPosed:dirac:eigenvalueBound}
\end{align}
where we used \cite[Lemma 22]{Stuart} for \eqref{eq:Proofs:GConvergence:IllPosed:dirac:eigenvalueBound}.
\end{proof}

\begin{proposition} \label{prop:Proofs:GConvergence:IllPosed:limsup}
$\limsup$-inequality in the ill-posed case.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold, $\eps_n$ satisfies~\eqref{eq:Main:Ass:epsLBIllPosed}.
Assume that $n\eps_n^{2s} \to \infty$ and $\rho \in \Ck{\infty}$. Let $(\nu,v) \in \TLp{2}(\Omega)$. Then, $\bbP$-a.s., there exists $\{(\nu_n,v_n)\} \subseteq \TLp{2}(\Omega)$ such that $(\nu_n,v_n) \to (\nu,v)$ in $\TLp{2}$ and
\begin{equation}\label{eq:Proofs:GConvergence:IllPosed:limsup}
\limsup_{n\to \infty} \cF_{n,\eps_n}((\nu_n,v_n)) \leq \cG((\nu,v)).
\end{equation}
\end{proposition}

\begin{proof}
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line. With probability one, we can assume that the conclusions of Theorem \ref{thm:Back:TLp:LinftyMapsRate} and Lemma \ref{lem:Proofs:GConvergence:IllPosed:diracEnergies} hold.

We note that \eqref{eq:Proofs:GConvergence:IllPosed:limsup} is trivial if $\mathcal{G}((\nu,v)) = \infty$. Hence, we might assume that $\mathcal{G}((\nu,v)) < \infty$ and in particular, this implies that $\nu = \mu$ and $\mathcal{G}((\nu,v)) = \EnergyS(v)$. We therefore need to prove \eqref{eq:Proofs:GConvergence:IllPosed:limsup} on $\{\mu\}\times\mathcal{H}^s(\Omega) \subseteq \TLp{2}$. We start by assuming that $v \in \Ck{\infty}$, which is dense in $\mathcal{H}^s(\Omega)$. Let $\bar{v}_n$ be the restriction of $v$ to $\Omega_n$ and consider the recovery sequence $\{(\mu_n,\bar{v}_n)\}_{i=1}^\infty \subseteq \TLp{2}(\Omega)$. By Theorem \ref{thm:Back:TLp:LinftyMapsRate}, we get transport maps $\{T_n\}_{n=1}^\infty$ from $\mu$ to $\mu_n$. Repeating the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:limsup}, we can show that $(\mu_n,\bar{v}_n) \to (\mu,v)$ in $\TLp{2}$ and that $\limsup_{n \to \infty} \EnergySnepsn(\bar{v}_n) \leq \EnergyS(v)$.

We define the functions \[
v_n(x_i) = \begin{cases}
\bar{v}_n(x_i) &\text{if $i \geq N + 1$,}\\
\ell_i & \text{if $i \leq N$.}
\end{cases}
\]
and estimate as follows:
\[
\int_\Omega \vert v_n \circ T_n - v \vert^2 \,\dd\mu \leq 2 \int_\Omega \vert v_n \circ T_n - v \circ T_n \vert^2 \,\dd\mu + 2 \int_\Omega \vert v \circ T_n - v \vert^2 \,\dd\mu =: 2(A + B).
\]
As in the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf}, we see that $B \to 0$. For the $A$ term, we have
\begin{align}
A &\leq \int_{\{x \spaceBar T_n(x) \neq x_i \text{ for $i \leq N$}\}} \vert v_n \circ T_n - v \circ T_n \vert^2 \, \dd\mu + \sum_{i=1}^N \int_{\{x \spaceBar T_n(x) = x_i\}} \vert v_n \circ T_n - v \circ T_n \vert^2 \, \dd\mu \notag\\
&=\sum_{i=1}^N \vert v(x_i) - \ell_i \vert^2 \mu(\{x \spaceBar T_n(x) = x_i\}) \notag\\
&= \sum_{i=1}^N \vert v(x_i) - \ell_i \vert^2 \mu_n(\{x_i\}) \notag\\
&= \sum_{i=1}^N \frac{\vert v(x_i) - \ell_i \vert^2}{n} \notag
\end{align}
from which we deduce that $(\mu_n,v_n) \to (\mu,v)$ in $\TLp{2}$.

Let $n$ be large enough so that \eqref{eq:Proofs:GConvergence:IllPosed:diracEnergies} holds. We now show that
\[
\limsup_{n \to \infty} \mathcal{F}_{n,\eps_n}(v_n,\mu_n) = \limsup_{n \to \infty} \EnergySnepsn(v_n) \leq \EnergyS(v).
\]
To this purpose, we note that
\[
v_n = \bar{v}_n + \sum_{i = 1}^N \delta_{x_i} (\ell_i - \bar{v}_n(x_i))
\]
and applying Lemma \ref{lem:Proofs:GConvergence:IllPosed:Minkowski}, we obtain:
\begin{align}
\sqrt{\EnergySneps{}(v_n)} &\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + \sum_{i=1}^N \vert \ell_i - \bar{v}_n(x_i) \vert \sqrt{\EnergySnepsn(\delta_{x_i})} \notag\\
&\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + C \sum_{i=1}^N \sqrt{\EnergySnepsn(\delta_{x_i})} \label{eq:Proofs:GConvergence:IllPosed:limsup:linfty}\\
&\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + C \l \frac{1}{n\eps_n^{2s}} \r^{1/2} \label{eq:Proofs:GConvergence:IllPosed:limsup:energy}
\end{align}
where we used the fact that $\Vert v \Vert_{\Lp{\infty}}$ is bounded on $\Omega$ for \eqref{eq:Proofs:GConvergence:IllPosed:limsup:linfty} (and $\bar{v}_n$ is the restriction of $v$ to $\Omega_n$ and so is also bounded in $\Lp{\infty}$) and \eqref{eq:Proofs:GConvergence:IllPosed:diracEnergies} for \eqref{eq:Proofs:GConvergence:IllPosed:limsup:energy}. Taking $n \to \infty$ on the latter right hand side, we have
\[
\limsup_{n \to \infty} \sqrt{\EnergySneps(v_n)} \leq \limsup_{n \to \infty} \sqrt{\EnergySnepsn(\bar{v}_n)} + \limsup_{n \to \infty} C \l \frac{1}{n\eps_n^{2s}} \r^{1/2} \leq \sqrt{\EnergyS(v)}
\]
since by assumption $n\eps_n^{2s} \to \infty$ and $\limsup_{n \to \infty} \EnergySnepsn(\bar{v}_n) \leq \EnergyS(v)$.

Having shown \eqref{eq:Proofs:GConvergence:IllPosed:limsup} on $\{\mu\}\times \Ck{\infty}$, we now use the fact that it is sufficient to establish the existence of a recovery sequence on a dense subset \cite[Remark 2.7]{Trillos3}.
\end{proof}

\subsection{Bounds on Minimizers} \label{subsec:Proofs:Linfty}

\begin{lemma} \label{lem:Proofs:Bounds:UniformBounds}
Uniform bound of energies for minimizers.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold. Assume $\rho \in \Ck{\infty}$. Let $\{(\mu_n,u_n)\}_{n=1}^\infty$ be the minimizers of $\cF_{n,\eps_n}(\cdot)$. Then, there exists $C >0$ such that, $\bbP$-a.s., we have
\[
\sup_{n} \cF_{n,\eps_n}( (\mu_n,u_n) ) \leq C.
\]
\end{lemma}

\begin{proof}
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line.
With probability one, we can assume that the conclusion of Proposition~\ref{prop:Proofs:GConvergence:WellPosed:limsup} holds.

Let $v \in C^\infty(\Omega)$ be a function that interpolates all the points $\{(x_i,\ell_i)\}_{i=1}^N$. Since $\rho \in C^\infty$, we have that $\Delta^s_\rho v \in C^\infty$ implying that $v \Delta^s_\rho v \in C^\infty$.
We have
\[
\EnergyS(v) = \int_\Omega v \Delta^s_\rho v \,\dd\mu < K.
\]
for some $K > 0$.

%Let $v_n$ be the restriction of $v$ on $\Omega_n$.
By Proposition \ref{prop:Proofs:GConvergence:WellPosed:limsup} there exists a sequence $v_n$ converging to $v$ and such that %we obtain
\[
\lim_{n \to \infty} h_n := \lim_{n \to \infty} \sup_{m \geq n } \mathcal{E}^{(s)}_{m,\eps_m}(v_m) =  \limsup_{n \to \infty} \EnergySnepsn(v_n) = \limsup_{n \to \infty} \mathcal{F}_{n,\eps_n}((\mu_n,v_n)) \leq \EnergyS(v) < K.
\]
Let $h := \limsup_{n \to \infty} \EnergySnepsn(v_n)$ and let $\bar{\eps} = K - h > 0$. Then, there exists $n_0$ such that for all $n \geq n_0$, $\vert h_n - h \vert < \bar{\eps}/2$, which is equivalent to $h_n = \sup_{m \geq n} \mathcal{E}^{(s)}_{m,\eps_m}(v_m) < h + \bar{\eps}/2 < K$. Using the latter, we have
\[
\sup_{n} \EnergySnepsn(v_n) = \max\{\mathcal{E}_{1,\eps_1}^{(s)}(v_1),\dots,\mathcal{E}_{n_0,\eps_{n_0}}^{(s)}(v_{n_0}),\sup_{n \geq n_0} \EnergySnepsn(v_n)\}
%\leq \max\{\mathcal{E}_{1,\eps_1}^{(s)}(v_1),\dots,\mathcal{E}_{n_0,\eps_{n_0}}^{(s)}(v_{n_0}),K\}
\leq C.
\]
Since $\{u_n\}_{n=1}^\infty$ are minimizers, we have $\EnergySnepsn(u_n) \leq \EnergySnepsn(v_n)$ which implies
\[
\sup_n \mathcal{F}_{n,\eps_n}((\mu_n,u_n)) =  \sup_n \EnergySnepsn(u_n) \leq \sup_n \EnergySnepsn(v_n) \leq C. \qedhere
\]
\end{proof}

\subsection{Proof of Theorem~\ref{thm:Main:Res:ConsFracLap}}

\begin{proof}[Proof of Theorem \ref{thm:Main:Res:ConsFracLap}]
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line.
%    \item
\paragraph{Well-Posed Case.}
With probability one, we can assume that the conclusions of Lemma \ref{lem:Proofs:Bounds:UniformBounds}, Proposition \ref{prop:Proofs:Bounds:Poincaré}, Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}, Proposition \ref{prop:Proofs:Compactness:L2Compactness} and
Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf} hold.
Using Lemma \ref{lem:Proofs:Bounds:UniformBounds} and Proposition \ref{prop:Proofs:Bounds:Poincaré}, we have
%\[ \Vert u_n \Vert_{\Lp{2}} \leq \Vert u_n - \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\Lp{2}} + \Vert \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\Lp{2}} \leq C \sqrt{\EnergySnepsn(u_n)} + \frac{1}{N} \sum_{i=1}^N \vert\ell_i\vert \leq C. \]
\[ \Vert u_n \Vert_{\addmaths{\Lp{\infty}}{\removemaths{\Lp{2}}}} \leq \Vert u_n - \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\addmaths{\Lp{\infty}}{\removemaths{\Lp{2}}}} + \Vert \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\addmaths{\Lp{\infty}}\removemaths{\Lp{2}}} \leq C \sqrt{\EnergySnepsn(u_n)} + \frac{1}{N} \sum_{i=1}^N \vert\ell_i\vert \leq C. \]
Hence minimisers are bounded in $\TLp{2}$ and $\max\l\sup_{n} \Vert u_{n} \Vert_{\Lp{\infty}},\sup_{n} \mathcal{E}^{(s)}_{n}(u_{n}\r \leq C$.
By Proposition~\ref{prop:Proofs:Compactness:L2Compactness} there exists $u$ and a subsequence converging uniformly and in $\TLp{2}$.
By Propositions~\ref{prop:Proofs:GConvergence:WellPosed:liminf} and \ref{prop:Proofs:GConvergence:WellPosed:limsup} it follows that $(\mu,u)$ is a minimiser of $\cF$.
By uniqueness of the minimiser of $\cF$ it follows that the whole sequence $\{( \mu_n,u_n)\}$ converges in $\TLp{2}$ to $(\mu,u)$.

\paragraph{Ill-Posed Case.}
With probability one, we can assume that the conclusions of Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}, Proposition \ref{prop:Proofs:GConvergence:IllPosed:liminf} and Proposition \ref{prop:Proofs:GConvergence:IllPosed:limsup}  hold.

By Proposition \ref{prop:Proofs:GConvergence:IllPosed:liminf} and Proposition \ref{prop:Proofs:GConvergence:IllPosed:limsup}, we know that $\mathcal{F}_{n,\eps_n}$ $\Gamma$-converges to $\mathcal{G}(\cdot)$.
By Theorem \ref{thm:Proofs:GConvergence:withoutConstraints} $\mathcal{F}_{n,\eps_n}(\cdot)$ satisfies the compactness property.
Hence, by Proposition~\ref{prop:Back:Gamma:minimizers} we can conclude the result.
\end{proof}

\section{Numerical Experiments} \label{sec:NumExp}

\begin{figure}
\centering
\includegraphics[width = \textwidth]{minimizers.png}
\caption{Plots of the discrete and continuum minimizers with the setting described in Section \ref{subsec:criticalBoundary}. The values at the points $(0.1,0.1)$ and $(0.9,0.9)$ are marked with black squares.
\textit{Left}: Discrete minimizer computed with $n = 1733$ points.
\textit{Right}: Continuum minimizer.
\end{figure}

\subsection{Critical boundary between well-posed and ill-posed regimes} \label{subsec:criticalBoundary}

In this section, we will investigate the gap in the upper bound alluded to in Remark \ref{rem:Main:Res:Relationship}. In particular, we will rely on the same methodology as in [53].

To test the gap between the well-posed and ill-posed regime, i.e. $n^{-\frac{2}{s-1}}\lesssim \eps_n\lesssim n^{-\frac{1}{2s}}$ (by Corollary~\ref{cor:Main:Res:ConsFracLap} we know that $\eps_n\lesssim n^{-\frac{2}{s-1}}$ implies asymptotic well-posedness and $\eps_n\gg n^{-\frac{1}{2s}}$ implies asymptotic ill-posedness)
we will consider the following setting: we choose the uniform measure on $[0,1]^2$ with periodic boundary conditions, the kernel function $\eta(t) = 1$ if $t\leq 1$ and $\eta(t) = 0$ else and $s = 16$. This choice of $s$ satisfies the constraint $s > 2d + 9 = 13$ from Remark \ref{rem:Main:Res:LowerBoundS}. The training set consists of the points $(0.1,0.1)$ and $(0.9,0.9)$ labelled $0$ and $1$ respectively.

We will vary the number of points in our graph from $n = 100$ to $n = 5000$. For each $n$, we will also consider a wide range of $\eps_n$ ranging from roughly the connectivity radius to above $n^{-1/2s}$. For each combination of $(n,\eps_n)$, we then compute the discrete minimizer $u_n$ using Lagrange multipliers and compare it to the continuum solution $u$. The continuum solution is computed using a finite-difference scheme on a regular grid of 10000 points on $[0,1]^2$.
The error considered is
\begin{equation} \label{eq:numExp:error}
\text{err}(n,\eps_n,u_n) = \Vert u_n - u \Vert_{\Lp{2}(\mu_n)}
\end{equation}
where, in order to evalutate $u$ on the graph, we use spline interpolation.
Finally, by re-sampling the points for each $n$, we repeat the experiments one hundred times and average the error.

We are interested in finding the value of $\eps_n$ where fractional Laplacian learning switches from the well-posed to the ill-posed regime. In order to compute the latter, we start by smoothing the function $\eps_n \mapsto \text{err}(n,\eps_n,u_n)$ and compute the maximizer of its first derivative and the minimizer of its second derivative which we denote by $\hat{\eps}_n$ and $\eps_n^{*}$ respectively.
We choose $\hat{\eps}_n$ and $\eps_n^*$
greater than the minimizer of $\text{err}(n,\cdot,u_n)$.
Both $\hat{\eps}_n$ and $\eps_n^*$ could be taken as reasonable definitions of the transition point between the well-posed and ill-posed regime and, it is therefore interesting to understand how they scale with $n$.  Using the five largest values of $n$, the best linear fits between  $\log(\hat{\eps}_n)$, $\log(\eps_n^*)$ and $\log(n)$ yield that
\[
\hat{\eps}_n \approx \frac{0.6541}{n^{0.05}} \quad \text{and} \quad \eps_n^* \approx \frac{0.7312}{n^{0.06}}.
\]

For $s = 16$, we have $1/(2s) = 0.03125$ and $2/(s-1) \approx 0.134$. Given the top plots in Figure \ref{fig:error}, we observe that both $\hat{\eps}_n$ and $\eps_n^*$ seem to scale accurately with powers that are different to $1/(2s)$. In fact, we note that $0.06 \approx 0.05 \approx 1/s = 0.0625$. On one hand, this indicates that we should be able to extend the well-posed regime of Theorem \ref{thm:Main:Res:ConsFracLap} to
\[
\l \frac{1}{n} \r^{2/(s-1)} \ll \eps_n \ll \l \frac{1}{n}\r^{1/s}
\]
and, by Remark \ref{rem:Main:Res:LowerBoundS}, we could relax our assumption of $s > 2d + 9$ to $s > d + 4$. If we were able to also tackle the lower bound gap (see Remark \ref{rem:Main:Res:LowerBoundGap}), we could furthermore have $s > d$ in contrast to $s > d/2$ which is conjectured in Remark \ref{rem:Main:Res:Sobolev}.
On the other hand, we are not able to fully confirm the conjecture made in Remark \ref{rem:Main:Res:Relationship} numerically  and it remains an open question to accurately characterize the behaviour of fractional Laplacian semi-supervised learning when
\[
\l \frac{1}{n} \r^{1/s} \ll \eps_n \ll \l \frac{1}{n}\r^{1/(2s)}.
\]

\begin{figure}[t!]
\setlength\figureheight{0.25\textwidth}
\setlength\figurewidth{0.3\textwidth}
\centering
\scriptsize

\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{LinearFitEpsIll2.tikz}
\caption{
The blue line is the fit $\frac{0.6541}{n^{0.05}}$ and the red line is $\hat{\eps}_n$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{LinearFitEpsIll.tikz}
\caption{
The blue line is the fit $\frac{0.7312}{n^{0.06}}$ and the red line is $\eps_n^*$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{Error5000.tikz}
\caption{
Error \eqref{eq:numExp:error} for $n = 5000$. The blue line is the average $\text{err}(n,\eps_n,u_n)$ and the dashed black lines are the 10\% and 90\% quantiles. The red and black vertical lines respectively indicate $\hat{\eps}_n$ and $\eps_n^*$.
\end{subfigure}

\caption{
Plots of the errors between the discrete and continuum minimizers with the setting described in Section \ref{subsec:criticalBoundary}.
\label{fig:error}
\end{figure}

\subsection{Uniform Bounds on Eigenfunctions} \label{subsec:linear}

Proposition~\ref{prop:Proofs:Back:DisReg:EVecAlt} implies that we can upper bound the $\Lp{\infty}$ norm of the discrete eigenvectors $\psi_{n,k}$ by
\begin{equation} \label{eq:numExp:lambda}
\|\psi_{n,k}\|_{\Lp{\infty}}\leq C\lambda_{k}^{d+1}
\end{equation}
when $k\in\{1,\hdots,\ceil{K_n}\}$ where $K_n=\alpha \eps_n^{-d/2} + 1$. We now investigate the following: on one hand, we want to see if the power of $\eps_n$ in the definition of $K_n$, namely $-d/2$, is the lowest we can get; and on the other hand, we are interested in the optimal power of $\lambda_{k}$ in \eqref{eq:numExp:lambda}.

We choose the uniform measure on $[0,1]^2$ with periodic boundary conditions and the kernel function $\eta(t) = 1$ if $t\leq 1$ and $\eta(t) = 0$ else. We proceed as follows: given a number of points ranging from $n = 400$ to $n = 5200$, we seek the smallest $\eps_n$ such that the graph is connected. We then compute $\lb \log\l \lambda_{k} \r, \log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r \rb_{k=1}^{k_n^{(i)}}$ where $k_n^{(1)}  = \alpha_1 \eps_n^{-d/4}$, $k_n^{(2)}  = \alpha_2 \eps_n^{-d/2}$, $k_n^{(3)}  = \alpha_3 \eps_n^{-d}$, $k_n^{(4)}  = n$
and $\alpha_i$ for $1 \leq i \leq 3$ are constants. While we took the arbitrary choice of $\alpha_i = 4$ for $1 \leq i \leq 3$, empirical trials have shown that the overall conclusion does not change significantly. By re-sampling the points for each $n$, we repeat the experiments one hundred times and average the results.

\begin{figure}[t!]
\setlength\figureheight{0.35\textwidth}
\setlength\figurewidth{0.47\textwidth}
\centering
\scriptsize

\begin{subfigure}[t]{0.47\textwidth}
\centering
\scriptsize
\input{eigenPairs1200.tikz}
\caption{
$n = 1200$
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.47\textwidth}
\centering
\scriptsize
\input{eigenPairs5200.tikz}
\caption{
$n = 5200$
\end{subfigure}

\caption{
Plots of $\log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r$ versus $\log\l \lambda_{k} \r$ for $1 \leq k \leq n$ with the setting described in Section \ref{subsec:linear}. The blue line corresponds to the average of $\log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r$, the dashed black lines are the mean plus/minus the standard deviation and the red lines indicate the values of $\log \l \|\lambda_{k_n^{(i),*}}\|_{\Lp{\infty}} \r$ for $1 \leq i \leq 4$ from left to right. \ref{subsec:criticalBoundary}.
\label{fig:eigenvalues}
\end{figure}

\begin{figure}[t!]
\setlength\figureheight{0.25\textwidth}
\setlength\figurewidth{0.3\textwidth}
\centering
\scriptsize

\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k2.tikz}
\caption{
The blue line is the fit $4.05 - 0.63 \log(\Vert \psi_{n,k_n^{(2),*}} \Vert)$ and the red line is $\lambda_{k_n^{(2),*}}$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k3.tikz}
\caption{
The blue line is the fit $0.2 - 0.08 \log(\Vert \psi_{n,k_n^{(3),*}} \Vert)$ and the red line is $\lambda_{k_n^{(3),*}}$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k4.tikz}
\caption{
The blue line is the fit $-0.55 + 0.01 \log(\Vert \psi_{n,k_n^{(4),*}} \Vert)$ and the red line is $\lambda_{k_n^{(4),*}}$.
\end{subfigure}

\caption{
Plots of $\log \l \|\psi_{n,k_n^{(i),*}}\|_{\Lp{\infty}} \r$ versus $\log\l \lambda_{k_n^{(i),*}} \r$ for $2 \leq i \leq 4$ with the setting described in Section \ref{subsec:linear}.
\label{fig:linear}
\end{figure}

From Figure \ref{fig:eigenvalues} there appears to be different regimes of growth for the eigenpairs depending on the value of $1 \leq k \leq n$. In fact, we see that below $\log(\lambda_{k_n^{(3)}})$, $\log(\Vert \psi_{n,k} \Vert_{\Lp{\infty}})$ seems to be increasing monotonically, while it unexpectedly first decreases and then sharply increases from $\log(\lambda_{k_n^{(3)}})$ to $\log(\lambda_{k_n^{(4)}})$. It is the subject of future research to provide explanations for the latter phenomena.

Let us define
\[
k_n^{(i),*} = \argmax_{1 \leq k \leq k_n^{(i)}} \log \l \Vert \psi_{n,k} \Vert_{\Lp{\infty}} \r
\]
and, using the seven largest values of $n$, for $1 \leq i \leq 4$, we will be computing the best linear fits for the points
\[
\lb( \log \l \lambda_{n,k_n^{(i),*}} \r, \log \l \Vert \psi_{k_n^{(i),*}} \Vert_{\Lp{\infty}} \r\rb_{n=400}^{5200}.
\]

For $k_n^{(i)}$ with $1 \leq i \leq 2$, the linear fits in the log-log domain appear to be very accurate as depicted in Figure \ref{fig:linear}, so we are able to confirm the theoretical guarantees of Proposition \ref{prop:Proofs:Back:DisReg:EVecAlt}.
In particular, we obtain
\[
\Vert \psi_{n,k_n^{(2),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(2),*}}^{-0.63}
\]
and $\lambda_{k_n^{(2),*}}^{-0.63} \leq \lambda_{k_n^{(2),*}}^{d+1}$ since $\lambda_k \geq 1$ (in fact, we have $\{\lambda_k\}_{k=1}^\infty = \{ 4 \pi^2 k^2 \spaceBar k \in \bbN \}$ in this particular setting). However, it shows that the bound is not sharp for the flat torus.

The situation for the regime $k_n^{(3)}$ is different and we obtain that \[
\Vert \psi_{n,k_n^{(3),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(3),*}}^{-0.08}.
\]
The much smaller exponent in the above compared to the $k_n^{(2),*}$ setting indicates that we seem to be switching from one regime of growth of $\Vert \psi_{n,k} \Vert_{\Lp{\infty}}$ to another one.

Finally, we note that the linear fit in Figure \ref{fig:linear} for the $k_n^{(4)}$ regime yields
\[
\Vert \psi_{n,k_n^{(4),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(4),*}}^{0.01}
\]
and, making the crude approximation $0 \approx 0.01$, this confirms our intuition that on the flat torus, the graph Laplacian should have uniformly bounded $\Lp{\infty}$-norms of eigenfunctions: indeed, the continuum eigenfunctions are uniformly bounded in $\Lp{\infty}$, so we expect the same behaviour for their discrete counterparts. This suggests that one should be able to pick $\alpha = 0$ in Theorem \ref{thm:Main:Res:ConsFracLap} yielding the (almost) optimal Sobolev bound $s > d/2 + 2$ as discussed in Remark~\ref{rem:Main:Res:Sobolev}. This conclusion also suggests that, on the flat torus at least, one should be able to go above $\eps_n^{-d/2}$ in Proposition \ref{prop:Proofs:Back:DisReg:EVecAlt}.

\section*{Acknowledgements}

The authors would like to thank Nicol\'as Garc\'ia Trillos for his comments and insights on this paper.
MT was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme Grant Agreement No. 777826 (NoMADS).

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\bibliographystyle{plain}

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